diff --git "a/core_physics_books/boltzmann.txt" "b/core_physics_books/boltzmann.txt" new file mode 100644--- /dev/null +++ "b/core_physics_books/boltzmann.txt" @@ -0,0 +1,25839 @@ +LECTURES ON +GAS THEORY + + + +Ludwig Boltzmann + + + + +Pufic. 7^ai/icj*uziuMzi ~Ph.y - A . LC - A . + + + +LECTURES ON GAS THEORY + + +Ludwig Boltzmann + +Translated by Stephen G. Brush + + +DOVER PUBLICATIONS, INC. +NEW YORK + + +~Pu+lc- 7fca^4e«toL2c.a/ P/tyJxci + + + + +~Pu+lc- 7fca^4e«toi2c.a/ P/tyJxci + + + + +LECTURES ON GAS THEORY + + +~Pu+lc- 7fca^4e«toi2c.a/ + + + +Copyright + + +Copyright © 1964 by The Regents of the University of California. +All rights reserve. + + +Bibliographical Note + +This Dover edition, first published in 1995, is an unabridged, unaltered republication of the English +translation originally published by the University of California Press, Berkeley, 1964. The original +work, in German, was published in two parts by J. A. Barth, Leipzig, Germany, 1896 (Part I) and 1898 +(Part II), under the title Vorlesungen iiber Gastheorie. + +The Dover edition is published by special arrangement with the University of California Press, 2120 +Berkeley Way, Berkeley, California 94720. + +This translation was originally “published with the assistance of a grant from the National Science +Foundation.” + + +Library of Congress Cataloging-in-Publication Data + +Boltzmann, Ludwig, 1844-1906. + +[Vorlesungen iiber Gastheorie. English] + +Lectures on gas theory / Ludwig Boltzmann ; translated by Stephen G. Brush. — Dover ed. +p. cm. + +Originally published: Berkeley : University of California Press, 1964. + +Includes bibliographical references and index. + +ISBN 0-486-68455-5 + +1. Kinetic theory of gases. I. Title. + +QC175.B7213 1995 +5 33’.7—dc20 + +94-41221 + +CIP + + +Manufactured in the United States by Courier Corporation +68455503 + +www .doverpublications.com + + +~Pu+lc- 7fca^4e«toi2c.a/ P/tyJxci + + + +CONTENTS + + +Translator’s Introduction x + +PARTI 1 + +Theory of gases with monatomic molecules, whose dimensions are negligible compared to + +THE MEAN FREE PATH. + +Foreword 2 + +Introduction 3 + +1. Mechanical analogy for the behavior of a gas 3 + +2. Calculation of the pressure of a gas 7 + +CHAPTER I 14 + +The molecules are elastic spheres. Extebnal forces and visible mass motion are absent. + +3. Maxwell’s proof of the velocity distribution law; frequency of collisions 14 + +4. Continuation; values of the variables after the collision; collisions of the opposite kind 19 + +5. Proof that Maxwell’s velocity distribution is the only possible one 25 + +6 . Mathematical meaning of the quantity H 29 + +7. The Boyle-Charles-Avogadro law. Expression for the heat supplied 34 + +8 . Specific heat. Physical meaning of the quantity H. 40 + +9. Number of collisions 47 + +10. Mean free path 54 + +11. Basic equation for the transport of any quantity by the molecular motion 57 + +12. Electrical conduction and viscosity of the gas 61 + +13. Heat conduction and diffusion of the gas 67 + +14. Two kinds of approximations; diffusion of two different gases 72 + +CHAPTER II 82 + +The molecules are centers of force. Consideration of external forces and visible motions + +of the gas. + +15. Development of partial differential equations for f and F. 82 + +16. Continuation. Discussion of the effects of collisions 85 + +17. Time-derivatives of sums over all molecules in a region 93 + +18. More general proof of the entropy theorem. Treatment of the equations corresponding to the 101 + +stationary state + +19. Aerostatics. Entropy of a heavy gas whose motion does not violate Equations (147) 110 + +20. General form of the hydrodynamic equations 116 + +CHAPTER III 132 + +The molecules repel each other with a force inversely proportional to the fifth power of + +THEIR DISTANCE. + +21. Integration of the terms resulting from collisions 132 + +22. Relaxation time. Hydrodynamic equations corrected for viscosity. Calculation of B 5 using 142 + +spherical functions + +23. Heat conduction. Second method of approximate calculations 155 + +24. Entropy for the case when Equations (147) are not satisfied. Diffusion 173 + + +^Pufi-C- 7fca^4e«toiZc.a/ + + + +PART II + +Van der Waals' Theory; Gases with compound molecules; Gas dissociation; Concluding + +REMARKS. + + +192 + + +Foreword 192 + +CHAPTER I 194 + +Foundations of van der Waals - theory. + +1. General viewpoint of van der Waals 1 94 + +2. External and internal pressure 195 + +3. Number of collisions against the wall 196 + +4. Relation between molecular extension and collision number 197 + +5. Determination of the impulse imparted to the molecules 199 + +6 . Limits of validity of the approximations made in §4 201 + +7. Determination of internal pressure 202 + +8 . An ideal gas as a thermometric substance 204 + +9. Temperature-pressure coefficient. Determination of the constants of van der Waals’ equation 205 + +10. Absolute temperature. Compression coefficient 206 + +11. Critical temperature, critical pressure, and critical volume 208 + +12. Geometric discussion of the isotherms 211 + +13. Special cases 214 + +CHAPTER II 218 + +Physical discussion of the van der Waals' theory. + +14. Stable and unstable states 218 + +15. Undercooling. Delayed evaporation 219 + +16. Stable coexistence of both phases 221 + +17. Geometric representation of the states in which two phases coexist 223 + +18. Definition of the concepts gas, vapor, and liquid 225 + +19. Arbitrariness of the definitions of the preceding section 226 + +20. Isopycnic changes of state 227 + +21. Calorimetry of a substance following van der Waals’ law 228 + +22. Size of the molecule . 231 + +23. Relations to capillarity 232 + +24. Work of separation of the molecules 235 + +CHAPTER III 239 + +Principles of general mechanics needed for gas theory. + +25. Conception of the molecule as a mechanical system characterized by generalized coordinates 239 + +26. Liouville’s theorem 241 + +27. On the introduction of new variables in a product of differentials 246 + +28. Application to the formulas of §26 250 + +29. Second proof of Liouville’s theorem 253 + +30. Jacobi’s theorem of the last multiplier 257 + +31. Introduction of the energy differential 261 + +32. Ergoden 264 + +33. Concept of the momentoid 267 + +34. Expression for the probability; average values 271 + +35. General relationship to temperature equilibrium 278 + + +We 7fca^4e«toi2ea/ “Pity. + + + +CHAPTER IV + +Gases with compound molecules. + + +283 + + +36. Special treatment of compound molecules 283 + +37. Application of Kirchhoff’s method to gases with compound molecules 284 + +38. On the possibility that the states of a very large number of molecules can actually lie within very 286 + +narrow limits + +39. Treatment of collisions of two molecules 287 + +40. Proof that the distribution of states assumed in §37 will not be changed by collisions 290 + +41. Generalizations 292 + +42. Mean value of the kinetic energy corresponding to a momentoid 294 + +43. The ratio of specific heats, k 296 + +44. Value of k for special cases 298 + +45. Comparison with experiment 299 + +46. Other mean values 300 + +47. Treatment of directly interacting molecules 302 + +CHAPTER V 306 + +Derivation of van der Waals - equation by means of the virial concept. + +48. Specification of the point at which van der Waals’ mode of reasoning requires improvement 306 + +49. More general concept of the virial 306 + +50. Virial of the external pressure acting on a gas 308 + +51. Probability of finding the centers of two molecules at a given distance 310 + +52. Contribution to the virial resulting from the finite extension of the molecules 313 + +53. Virial of the van der Waals cohesion force 316 + +54. Alternatives to van der Waals’ formulas 317 + +55. Virial for any arbitrary law of repulsion of the molecules 319 + +56. The principle of Lorentz’s method 321 + +57. Number of collisions 323 + +58. More exact value of the mean free path. Calculation of W’ according to Lorentz’s method 326 + +59. More exact calculation of the space available for the center of a molecule 327 + +60. Calculation of the pressure of the saturated vapor from the laws of probability 329 + +61. Calculation of the entropy of a gas satisfying van der Waals’ assumptions, using the calculus of 332 + +probabilities + +CHAPTER VI 339 + +Theory of dissociation. + +62. Mechanical picture of the chemical affinity of monovalent similar atoms 339 + +63. Probability of chemical binding of an atom with a similar one 341 + +64. Dependence of the degree of dissociation on pressure 344 + +65. Dependence of the degree of dissociation on temperature 347 + +66 . Numerical calculations 350 + +67. Mechanical picture of the affinity of two dissimilar monovalent atoms 354 + +68 . Dissociation of a molecule into two heterogeneous atoms 356 + +69. Dissociation of hydrogen iodide gas 358 + +70. Dissociation of water vapor 360 + +71. General theory of dissociation 363 + +72. Relation of this theory to that of Gibbs 367 + +73. The sensitive region is uniformly distributed around the entire atom 368 + + +vi. + + +f^uJuc. 7fca^4e«tai2c.a/ + + + +CHAPTER VII 372 + +Supplements to the laws of thermal equilibrium in gases with compound molecules. + +74. Definition of the quantity H, which measures the probabilities of states 372 + +75. Change of the quantity H through intramolecular motion 373 + +76. Characterization of the first special case considered 375 + +77. Form of Liouville’s theorem in the special case considered 377 + +78. Change of the quantity H as a consequence of collisions 379 + +79. Most general characterization of the collision of two molecules 381 + +80. Application of Liouville’s theorem to collisions of the most general kind 382 + +81. Method of calculation with finite differences 385 + +82. Integral expression for the most general change of//by collisions 389 + +83. Detailed specification of the case now to be considered 391 + +84. Solution of the equation valid for each collision 391 + +85. Only the atoms of a single type collide with each other 393 + +86 . Determination of the probability of a particular kind of central motion 394 + +87. Characterization of our assumption about the initial state 399 + +88 . On the return of a system to a former state 400 + +89. Relation to the second law of thermodynamics 401 + +90. Application to the universe 402 + +91. Application of the probability calculus in molecular physics 403 + +92. Derivation of thermal equilibrium by reversal of the time direction 404 + +93. Proof for a cyclic series of a finite number of states 407 + +Bibliography 410 + +Index 433 + + +7fca^4e«toi2c.a/ “Pity. + + + +TRANSLATOR’S INTRODUCTION + + +Gas Theory + +Boltzmann’s Lectures on Gas Theory is an acknowledged masterpiece of theoretical +physics; aside from its historical importance, it still has considerable scientific value today. +It contains a comprehensive exposition of the kinetic theory of gases by a scientist who +devoted a large pail of his own career to it, and brought it very nearly to completion as a +fundamental part of modem physics. The many physicists who are already familiar with +Gastheorie in the original German edition (1896-1898) will not need this Introduction to +remind them of its stature. But perhaps some scientists who scarcely have time to keep up +with the latest publications in their subject may want to know why they should bother to +read a book published more than 60 years ago; and a brief account of the place of the book +in the development of modem physics may be of interest. + +Ludwig Boltzmann (1844-1906) played a leading role in the nineteenth-century +movement toward reducing the phenomena of heat, light, electricity, and magnetism to +“matter and motion” —in other words, to atomic models based on Newtonian mechanics. +His own greatest contribution was to show how that mechanics, which had previously been +regarded as deterministic and reversible in time, could be used to describe irreversible +phenomena in the real world on a statistical basis. His original papers on the statistical +interpretation of thermodynamics, the //-theorem, transport theory, thermal equilibrium, the +equation of state of gases, and similar subjects, occupy about 2,000 pages in the +proceedings of the Vienna Academy and other societies. While some of his discoveries +attracted considerable attention and controversy in his own time, not even the handful of +experts on kinetic theory could claim to have read everything he wrote. Realizing that few +scientists of later generations were likely to study his long original memoirs in detail, +Boltzmann decided to publish his lectures, in which the most important parts of the theory, +including his own contributions, were carefully explained. In addition, he included his +mature reflections and speculations on such questions as the nature of irreversibility and the +justification for using statistical methods in physics. His Vorlesungen Uber Gastheorie was +in fact the standard reference work for advanced researchers, as well as a popular textbook +for students,* for the first quarter of the present century, and is frequently cited even now. +A recent example is the following statement by Mark Kac:f “Boltzmann summarized most +(but not all) of his work in a two volume treatise Vorlesungen Uber Gastheorie. This is one +of the greatest books in the history of exact sciences and the reader is strongly advised to +consult it. It is tough going but the rewards are great.” + +The modem reader will rightly assume that some parts of Boltzmann’s theory must have +been rendered obsolete by later discoveries. In particular, we know that one cannot expect +to develop an adequate theory of atomic phenomena without using quantum mechanics. +However, it turns out that almost all the properties of gases at ordinary temperatures and +densities can be described by the classical theory developed in the nineteenth century, and +for this reason there has been a considerable revival of interest in classical kinetic theory + + +i+ix± 7fca^4e«tai2c.a/ + + + +during the last few years, in connection with rarified gas dynamics, plasma physics, and +neutron transport theory. The reason why the classical theory works is that, while the +internal structure of molecules must be described by quantum mechanics, the interaction +between two molecules can be fairly well described by a classical model which ignores this +structure and simply uses a postulated force law whose parameters can be chosen to fit +experimental data. Aside from phenomena at very high densities or very low temperatures, +the only property that the classical theory fails to account for is the ratio of specific heats. +However, this failure was already well known at the time Boltzmann wrote this book, and +he is therefore quite cautious on this point. He simply concludes that for some unknown +reason all the possible internal motions of a molecule do not have an equal share in the total +energy, and takes this into account as an empirical fact. + +As for various questions of a more fundamental nature, such as the relation between +entropy and the direction of time, quantum theory has thrown new light on these problems +but has not solved them. Boltzmann’s remarks are therefore still of interest. Thus Hans +Reichenbach proposes to define the direction of increasing time as the direction in which +entropy increases, so that in a universe in which entropy fluctuates, + +we cannot speak of a direction of time as a whole; only certain sections of time have directions, and +these directions are not the same. The first to have the courage to draw this conclusion was Ludwig +Boltzmann [Gas Theory, Part II, §90]. His conception of alternating time directions represents one of +the keenest insights into the problem of time. Philosophers have attempted to derive the properties of +time from reason; but none of their conceptions compares with this result that a physicist derived from +reasoning about the implications of mathematical physics. As in so many other respects, the superiority +of a philosophy based on the results of science has become manifest... .* + +The only respects in which the book may be considered out of date are, first, the +emphasis on the inverse fifth-power repulsive force model for calculating transport +properties, and, second, the molecular model used to explain dissociation phenomena. The +reason for choosing the inverse fifth-power force law was that Maxwell (1866)t had +discovered that the calculation of transport properties is particularly simple for this case, +because the collision integrals do not depend on the relative velocity of the colliding +molecules, so that one does not need to know the velocity distribution. Maxwell and +Boltzmann therefore believed that it would be more worthwhile to develop an exact theory +for this model, rather than attempt to work out a more complicated approximate theory for +other force laws. However, it was later shown by Chapman and Enskog (1916-1917) that +the important phenomenon of thermal diffusion does not occur for this particular force law, +and a quantitative treatment of the general case does require a determination of the velocity +distribution in a nonequilibrium state, although it is still not necessary to use the quantum +theory. Boltzmann’s model for dissociation, employing molecules with small “sensitive +regions” on parts of their surfaces, is an interesting historical curiosity, but the need for such +ad hoc assumptions to explain the directional character of chemical bonds has been +el im inated by quantum mechanics. + + +7fca^4e«taiZc.a/ + + +History of the kinetic theory + + + +The development of the kinetic theory of gases has been discussed at length elsewhere* +so only a brief summary will be given here. + +The modem concept of air as a fluid exerting mechanical pressure on surfaces in contact +with it goes back to the early seventeenth century, when Torricelli, Pascal, and Boyle first +established the physical nature of the air. By a combination of experiments and theoretical +reasoning they persuaded other scientists that the earth is surrounded by a “sea” of air that +exerts pressure in much the same way that water does, and that air pressure is responsible +for many of the phenomena previously attributed to “nature’s abhorrence of a vacuum.” +We may view this development of the concept of air pressure as part of the change in +scientific attitudes which led to the mechanico-corpuscular view of nature, associated with +the names of Galileo, Boyle, Newton, and others. Instead of postulating “occult forces” or +teleological principles to explain natural phenomena, scientists started to look for +explanations based simply on matter and motion. + +Robert Boyle is generally credited with the discovery that the pressure exerted by a gas +is inversely proportional to the volume of the space in which it is confined. From Boyle’s +point of view that discovery by itself was relatively insignificant, and though he had +provided the experimental evidence for it he readily admitted that he had not found any +general quantitative relation between pressure and volume before Richard Towneley +suggested the hypothesis that pressure is inversely proportional to volume. Robert Hooke +also provided further experimental confirmation of the hypothesis. It was long known as +“Mariotte’s law” on the Continent, but there are good reasons for believing that Mariotte +was familiar with Boyle’s work even though he does not mention it, so that he does not +even deserve the credit for independent (much less simultaneous) discovery: see any good +book on history of science for details on this point. + +The generalization that pressure is proportional to absolute temperature at constant +volume was stated by Amontons and Charles, but it remained for Gay-Lussac to establish it +firmly by experiments at the beginning of the nineteenth century. + +Boyle’s researches were carried out to illustrate not just a quantitative relation between +pressure and volume, but rather the qualitative fact that air has elasticity (“spring”) and can +exert a mechanical pressure strong enough to support 34 feet of water in a pump, or 30 +inches of mercury in a barometer. His achievement was to introduce a new dimension— +pressure—into physics; he could well afford to be generous about giving others the credit +for perceiving the numerical relations between this dimension and others. He also proposed +a theoretical explanation for the elasticity of air—he l ik ened it to “a heap of little bodies, +lying one upon another” and the elasticity of the whole was simply due to the elasticity of +the parts. The atoms were said to behave like springs which resist compression. To a +modem scientist this explanation does not seem very satisfactory, for it does no more than +attribute to atoms the observable properties of macroscopic objects. It is interesting to note +that Boyle also tried the “crucial experiment” which was to help overthrow his own theory +in favor of the kinetic theory two centuries later, though he did not realize its significance: +he placed a pendulum in an evacuated chamber and discovered, to his surprise, that the +presence or absence of air makes hardly any difference to the period of the swings or the +time needed for the pendulum to come to rest. In 1859, Maxwell deduced from the kinetic + + +x + + +P’sm+uc. 7fca^4e«toi2c.a/ + + + +theory that the viscosity of a gas should be independent of its density—a property which +would be very hard to explain on the basis of Boyle’s theory. + +Newton discusses very briefly in his Pliilosophiae Naturalis Principia Mathematica +(1687) the consequences of various hypotheses about the forces between atoms for the +relation between pressure and volume. One particular hypothesis, a repulsive force +inversely proportional to distance, leads to Boyle’s law. It seems plausible that Newton was +trying to put Boyle’s theory in mathematical language and that he thought of the repulsive +forces as being due to the action of the atomic springs in contact with each other, but there +seems to be no direct evidence for this. Neither Boyle nor Newton asserted that the +hypothesis of repulsive forces between atoms is really responsible for gas pressure; both +were willing to leave the question open. Boyle mentions Descartes’ theory of vortices, for +example, which is somewhat closer in spirit to the kinetic theory since it relies more heavily +on the rapid motion of the parts of the atom as a cause for repulsion. Incidentally, it is +important to realize that there is more to the kinetic theory than just the statement that heat is +atomic motion. That statement was frequently made, especially in the seventeenth century, +but usually by scientists who did not make the important additional assumption that in gases +the atoms move freely most of the time. It was quite possible to accept the “heat is motion” +idea and still reject the kinetic theory of gases—as did Humphry Davy early in the +nineteenth century. + +Despite the tentative way in which it was originally proposed, the Boyle-Newton theory +of gases was apparently accepted by most scientists until about the middle of the nineteenth +century, when the kinetic theory finally managed to overcome Newton’s authority. Thus a +theory which was decidedly a step forward from older ideas nevertheless was able to retard +further progress for a considerable time. + +It is difficult to understand the relative lack of progress in gas physics during the +eighteenth century as compared to the seventeenth. Several brilliant mathematicians refined +and clarified Newton’s principles of mechanics and applied them to the analysis of the +motions of celestial bodies and of continuous solids and fluids, but there was little interest +in the properties of systems of freely moving atoms. The atoms in a gas were still conceived +as being suspended in the ether, although they could vibrate or rotate enough to keep other +atoms from coming too close. This model was rather awkward to formulate mathematically, +as may be seen from an unsuccessful attempt by Leonhard Euler (1727). + +One contribution from this period has been generally recognized as the first kinetic +theory of gases. This is Daniel Bernoulli’s derivation of the gas laws from a “bi ll iard ball” +model—much lik e the one still used in elementary textbooks today—in 1738. His kinetic +theory is only a small part of a treatise on hydrodynamics, a subject to which he made +important contributions. From the viewpoint of the historical development of the kinetic +theory, his formulation and successful applications of the principle of conservation of +mechanical energy were really more important than the fact that he actually proposed a +kinetic theory himself. Bernoulli’s kinetic theory, while in accord with modem ideas, was a +century ahead of its time, for scientists were not yet ready to accept the physical description +of a gas that it employed. Heat was still generally regarded as a substance, even though it +was also recognized that heat might have some relationship to atomic motions; Bernoulli’s + + +xi. + + +^Pu^jc. 7fca^4e«tai2c.a/ + + + +assumption that heat is nothing but atomic motion was unacceptable, especially to scientists +who were interested in the phenomena of radiant heat. The assumption that atoms could +move freely through space until they collided l ik e bi ll iard balls was probably regarded as +too drastic an approximation, since it neglected the drag of the ether and oversimplified the +interaction between atoms. + +Late in the eighteenth century, Lavoisier and Laplace developed a systematic theory of +chemical and thermal phenomena based on the assumption that heat is a substance, called +“caloric.” Laplace and Poisson later worked out a quantitative theory of gases, in which the +repulsion between atoms was attributed to the action of “atmospheres” of caloric +surrounding the atoms; they were able to explain such phenomena as adiabatic compression +and the velocity of sound, as well as the ideal gas laws. + +The caloric theory was being brought to its final stage of perfection by Laplace, +Poisson, Carnot and Clapeyron at about the same time that John Herapath (1790-1868) +proposed his kinetic theory (1820, 1821). Herapath attempted to explain not only the +properties of gases but also gravity, changes of state, and many other phenomena. His +memoir was rejected by the Royal Society, mainly because Humphry Davy (then its +president) thought Herapath’s theory was too speculative, even though Davy himself had +advocated the view that heat is molecular motion. Herapath published several papers on his +theory, but did not succeed in converting other scientists to his views. Another British +scientist, J. J. Waterston (1811-1883), submitted a memoir on the kinetic theory to the +Royal Society in 1845; it was not only rejected, but remained buried in the Society’s +archives until 1891 when it was discovered by Lord Rayleigh. + +During the period 1840-1855, the equivalence of heat, work, and other forms of energy +was established experimentally by Joule, and the general principle of conservation of +energy was formulated by Mayer and others. Joule himself revived Herapath’s kinetic +theory (1848) but did not make any further developments or applications of it aside from +calculating the velocity of a hydrogen molecule. Kronig independently proposed a simple +kinetic theory in 1856, and soon afterwards Rudolf Clausius (1822-1888) and James Clerk +Maxwell (1831-1879) stalled to make substantial progress towards the modem theory. +Clausius introduced the idea of the “mean free path” of a molecule between successive +collisions (1858), and Maxwell, Clausius, O. E. Meyer (1834-1915), and P. G. Tait +(1831-1901) developed the theory of diffusion, viscosity, and heat conduction on this +basis. One of the most surprising results—whose confirmation by experiment helped to +establish the kinetic theory—was Maxwell’s prediction (1859) that the viscosity of a gas +should be independent of density and should increase with temperature. The mean free path +method is still the easiest way to understand transport phenomena, but it does not yield +accurate numerical results and, as mentioned above, it failed to predict thermal diffusion. + +The foundations of the modem theory of transport were laid by Maxwell in his great +memoir of 1866; it is essentially this method which Boltzmann used to make his +discoveries, and which he presents in this book. Maxwell proceeds by writing down +general equations for the rate of change of any quantity (such as molecular velocity) at any +point in space and time resulting from molecular motions and collisions. These +“microscopic” equations are then compared with the corresponding “macroscopic” + + +xi„ + + +7fca^4e«tai2c.a/ + + + +equations (for example, the Navier-Stokes equations of hydrodynamics) and the +coefficients of viscosity, etc., are thus related to certain characteristic functions of molecular +collisions. Boltzmann’s transport equation, derived in 1872, is a special case of Maxwell’s +general equation in which the quantity of interest is the number of molecules having a +certain velocity—in other words, the velocity distribution function. As soon as this function +has been found, all the transport coefficients can also be computed. Much of modem +research in statistical mechanics is based on attempts to solve either the Boltzmann equation +or similar equations for other kinds of distribution functions. In a sense there are two +alternative ways of developing transport theory: one based on solutions of Maxwell’s +equations, and the other on the solution of Boltzmann’s equation, and these two approaches +were followed by Chapman (1916) and Enskog (1917) respectively, the final results being +essentially identical. However, Boltzmann was able to deduce immediately a very +important result from his equation, which was not at all obvious from Maxwell’s original +formulation: a quantity called H, which can be identified with the negative of the entropy, +must always decrease or remain constant, if one assumes that the velocity distributions of +two colliding molecules are uncorrelated. The molecular interpretation of the law of +increasing entropy is thus intimately related to the assumption of molecular chaos and the +relation between entropy and probability. H remains constant only when the gas attains a +special velocity distribution which had previously been deduced by Maxwell (1859) in a +less convincing manner. + +There is apparently a contradiction between the law of increasing entropy and the +principles of Newtonian mechanics since the latter do not recognize any difference between +past and future times. If one sequence of molecular motions corresponds to increasing +entropy, one should obtain a sequence corresponding to decreasing entropy by simply +reversing all the velocities. This is the so-called reversibility paradox ( Umkehreinwand ) +which was advanced as an objection to Boltzmann’s theory by Loschmidt (1876, 1877) +and others. Another difficulty is the fact that a conservative dynamical system in a finite +space will always return infinitely often to a state close to any arbitrary initial state; hence +the entropy cannot continually increase but must go through cyclic variations. This is the +recurrence paradox ( Wiederkehreinwand ) based on a theorem of Poincare (1890) and used +against the kinetic theory by Zermelo (1896). Some scientists held that if there is any +contradiction between macroscopic thermodynamics and the kinetic theory, the latter +should be rejected since it is based on a hypothetical atomic model, whereas +thermodynamics is firmly grounded on empirical observations. Boltzmann’s reply to these +objections is given in this book. + +Another broad field of theoretical research was opened up by J. D. van der Waals +(1837-1923), who published his celebrated investigation of the continuity of the liquid and +gaseous states in 1873. Although van der Waals’ equation of state was supposedly deduced +from the virial theorem of Clausius (1870), and was therefore considered an application of +the kinetic theory, its derivation relied partly on concepts of the older Laplace-Poisson +theory of capillarity, in which molecular motions were ignored. Nevertheless the van der +Waals equation of state gives an excellent qualitative description of condensation and +critical phenomena, and Boltzmann devotes a substantial portion of Part II of these lectures + + +x + + +7fca^4e«toi2c.a/ + + + +to an exposition of van der Waals’ theory. He also describes his calculation of the next +correction term to the equation of state of hard spheres—i.e., what we now call the third +virial coefficient. + +The failure of the kinetic theory to give a satisfactory explanation of the specific heats of +polyatomic gases—and spectroscopic evidence that even monatomic gas molecules must +have internal degrees of freedom which apparendy do not have their proper share of energy +as required by the equipartition theorem —stimulated several theoretical studies of +mechanical models later in the nineteenth century. These studies were directed toward +attempting to prove rigorously the necessary and sufficient conditions for equipartition. It +was generally agreed that a sufficient condition would be that the system passes eventually +through every point on the “energy surface” in phase space before returning to its stalling +point; in other words, no matter what might be the initial velocities and positions of the +particles, they would eventually take on all possible values consistent with their fixed total +energy. This statement is now known as the “ergodic” hypothesis, and it is frequently +asserted that Maxwell and Boltzmann believed that real physical systems have this +property. If the hypothesis were true for a system, then it would be legitimate to replace a +time-average over the trajectory of a single system (which is what one can actually observe) +by a “phase” or “ensemble” average over many replicas of the same system with all +possible sets of positions and velocities consistent with a specified energy. From a +mathematical point of view, the existence of an ergodic system would imply that ann- +dimensional manifold (the energy surface in phase space) can be mapped continuously onto +a one-dimensional manifold (the range of the time variable) so that every point in the +former corresponds to at least one point in the latter. + +While Boltzmann did make certain statements in some of his papers (1871, 1884, 1887) +to the effect that a mechanical system might exhibit such behavior—the Lissajous figure +with incommensurable periods was one alleged example—there is no foundation for the +belief that Boltzmann thought real systems are ergodic. Moreover, Boltzmann did not +clearly distinguish between the “ergodic hypothesis” (which we now know to be false) and +the so-called “quasi-ergodic hypothesis,” that the trajectory of a system may pass arbitrarily +close to every point on the energy surface. The latter hypothesis does indeed provide a +reasonable basis for statistical mechanics, even though it has not yet been proved that it is +valid for real physical systems. To anyone acquainted with the modem mathematical theory +of sets of points and infinite classes, the distinction between the two hypotheses is obvious, +but we must remember that Boltzmann did not accept this theory, although he may have +been aware of it (the fundamental papers of Georg Cantor were published during the same +period that Boltzmann was publishing his papers on gas theory, some of them in the same +journal). On the contrary, Boltzmann often stated that infinitesimal or infinite numbers have +no meaning except as l im its of finite numbers, and for this reason he gave alternative +derivations of some of his theorems using sums of discrete quantities instead of integrals. +To say that a trajectory passes through every point in a region would (to him) be +meaningless unless one meant by this that the trajectory passes arbitrarily close to every +point, and in at least one paper Boltzmann uses the two statements interchangeably.* + +Part of the confusion regarding Boltzmann’s views is due to the fact that the Ehrenfests, + + +x + + +7fca^4e«taiZc.a/ + + + +in their classic encyclopedia article (1911), and later writers following their terminology, +took Boltzmann’s word “ergodic” and used it to refer to a single mechanical system whose +trajectory passes through every point on the energy surface, whereas Boltzmann h im self +had used the word in a completely different sense, to mean an aggregate or ensemble of +systems whose positions and velocities are distributed over all possible sets of values on the +energy surface. The Ehrenfests recognized this distinction, and used the term “ergodic +distribution” for the latter concept, but since then Gibbs’ terminology has come into general +use, and we now use the phrase “microcanonical ensemble” for what Boltzmann called an +Ergode. (Boltzmann’s justification for using Ergoden in gas theory is given in Part n, §35, +and in other remarks scattered through the rest of this book.) + +The statistical mechanics of J. Willard Gibbs (1839-1903) represents the next level of +abstraction and refinement of gas theory, following direcdy on Boltzmann’s introduction of +Ergoden , and relying heavily on the methods of analytical mechanics which Boltzmann +applied to gas theory (see Pail II, Chapter HI). Instead of considering an ensemble of +systems with different initial positions and velocities but the same total energy, Gibbs +proposed to consider the more general case of a “canonical ensemble” of systems having all +possible energies, the number of systems having energy E being proportional to e~P E +(where [i = VkT). Gibbs also introduced the “grand canonical ensemble” in which the +number of particles in each system may vary, the number of systems with N particles being +proportional to e~^ N , where /./ (called the “chemical potential”) is chosen to give the desired +density. These generalized ensembles are easier to deal with mathematically, and enable +one to treat systems in contact with a heat reservoir at constant temperature, or systems in +which chemical reactions may take place. + +There is not space here to describe the modifications of statistical mechanics by quantum +mechanics, associated with the names of Planck, Einstein, Bose, Fermi, and Dirac. It may +however be worth mentioning that Planck’s discovery of the quantum laws of radiation +owed much to Boltzmann’s statistical theory of entropy. At the same time it must be +remembered that during the years preceding the introduction of the quantum theory, the +inadequacies of classical mechanics had become increasingly more evident, as theoretical +physicists tried in vain to construct satisfactory classical models of atoms and of the ether. +There was also a growing reaction against “scientific materialism,” and a movement to +replace atomic theories by purely descriptive theories based only on macroscopic +observables. The most famous leader of this movement was Ernst Mach (1838-1916), +whose criticism of mechanics prepared the way for Einstein’s theory of relativity. In a +series of articles and books, beginning with a critical study of the law of conservation of +energy (1872), Mach attacked the use of atomic models in physics, and asserted that the +basic purpose of science is to achieve “economy of thought” in describing natural +phenomena, not to explain them in terms of hypothetical concepts such as atoms or ether. +Gustav Kirchhoff (1824-1887) expressed similar views in his lectures on mechanics (1874) +but did not develop them as fully, and indeed in his lectures on heat (published +posthumously in 1894) he included a section on kinetic theory. One of the striking features +of Boltzmann’s Gas Theory is the large number of references to Kirchhoff s exposition. + +The school of Energetics included many members of the positivist movement, in + + +xv. + + +^PuJut*. 7fca^4e«tai2c.a/ + + + +particular Wilhelm Ostwald (1853-1932), Georg Helm (1851-1923), Pierre Duhem +(1861-1916) and others. The Energetists proposed to develop thermodynamics, and the +theory of energy transformations in general, on a purely macroscopic basis without any +reference to atomic theories. Their attacks on the kinetic theory were particularly strong +during the period when Boltzmann was writing this book, and this circumstance helps to +explain its polemical flavor. There are many passages justifying various assumptions, +defending the theory against possible criticisms, and attacking rival theories. Boltzmann +knew he could not get away with vague “heuristic” arguments or appeals to authority, since +many of his readers would not be convinced. (It might well be argued that a text written in +such a style is really more effective—provided that it contains no serious errors—than a +modem text expounding a noncontroversial theory which the student is not encouraged to +doubt.) + +According to Lenin* the arguments of Mach and other “idealistic” (in the philosophical +sense) scientists subverted many Marxist philosophers, who accepted them as the latest +results of science. Lenin quotes with approval the views of Boltzmann, who, while “afraid +to call himself a materialist” nevertheless had a theory of knowledge which was “essentially +materialistic.” Consequently Boltzmann now has the dubious distinction of being a hero of +scientific materialism in the eyes of the Marxists.* + +As the leader of the atomist school in the 1890’s, Boltzmann frequently had to engage in +debate with the Energetists. One such debate took place at the meeting of the +Naturforschergesellschaft at Liibeck in 1895. As Arnold Sommerfeld recalls it,f + +The champion for Energetics was Helm; behind him stood Ostwald, and behind both of them the +philosophy of Ernst Mach (who was not present in person). The opponent was Boltzmann, seconded by +Felix Klein. The battle between Boltzmann and Ostwald was much like the duel of a bull and a supple +bullfighter. However, this time the bull defeated the toreador in spite of all his agility. The arguments of +Boltzmann struck through. We young mathematicians were all on Boltzmann’s side; it was at once +obvious to us that it was impossible that from a single energy equation could follow the equations of +motion of even one mass point, to say nothing of those for a system of an arbitrary number of degrees of +freedom. On Ostwald’s behalf, however, I must mention his remark on Boltzmann in his book Grosse +Manner (Leipzig, 1909, p. 405): there he calls Boltzmann “the man who excelled all of us in acumen and +clarity in his science.” + +What did Boltzmann really think about the existence of atoms? One commentator^: has +asserted that “even the most convinced adherents of this [kinetic] theory, such as +Boltzmann, ascribe to it merely the value of a mechanical model which imitates certain +properties of gases and can afford the experimenter certain useful indications, and are very +far from believing bodies to be in reality composed of small particles ...” Indeed, a +curious feature of Boltzmann’s attitude toward the molecular model of gases is his attempt +to justify the use of asymptotic l im its in which the number of molecules in a volume +becomes infinite, while their size becomes in fin itesimal. He is perfectly aware of the fact +that one can deduce finite numerical values for these molecular parameters from experiment +by using gas theory (Pail I, §12; Part II, §22); and he refers to Maxwell’s theory of rarefied +gas phenomena (radiometer effect, etc.) as a triumph of molecular theories over the +phenomenological approach (Part I, §1). Yet his desire to obtain a more intuitively clear +(anschaulich) presentation of his theory impels him to assume that each infinitesimal + + +xv, + + +7fca^4e«tai2c.a/ + + + +volume element in the gas contains a very large number of molecules (Pail I, §6) and, in +attempting to justify this assumption, he goes so far as to say that it is futile to expect to +observe fluctuations from the asymptotic laws resulting from the fact that there may be only +a finite number of atoms in a small space (Part n, §38). This statement was of course +contradicted only a few years later by experiments on Brownian motion and Mil li kan’s oil- +drop experiment. + +We cannot give here a comprehensive analysis of Boltzmann’s views on the philosophy +of science, but must refer to his extensive writings and the studies by Broda and Dugas +cited in the Bibliography. The following quotation is chosen only because it was originally +published in English (in a letter to Nature , February 28, 1895) so that there is no danger of +distorting Boltzmann’s meaning by translation (assuming of course that he is expressing his +own views accurately in English): + +I propose to answer two questions:— + +(1) Is the Theory of Gases a true physical theory as valuable as any other physical theory? + +(2) What can we demand from any physical theory? + +(The first question I answer in the affirmative, but the second belongs not so much to ordinary +physics (let us call it orthophysics) as to what we call in Germany metaphysics. For a long time the +celebrated theory of Boscovich was the ideal of physicists. According to his theory, bodies as well as the +ether are aggregates of material points, acting together with forces, which are simple functions of their +distances. If this theory were to hold good for all phenomena, we should be still a long way off what +Faust ’sfamulus hoped to attain, viz., to know everything. But the difficulty of enumerating all the +material points of the universe, and of determining the law of mutual force for each pair, would be only +a quantitative one; nature would be a difficult problem, but not a mystery for the human mind. + +When Lord Salisbury says that nature is a mystery [Presidential Address to the British Association +meeting at Oxford, 1894], he means, it seems to me, that this simple conception of Boscovich is refuted +almost in every branch of science, the Theory of Gases not excepted. The assumption that the gas- +molecules are aggregates of material points, in the sense of Boscovich, does not agree with the facts. But +what else are they? And what is the ether through which they move? Let us again hear Lord Salisbury. +He says: + +“What the atom of each element is, whether it is a movement, or a thing, or a vortex, or a point +having inertia, all these questions are surrounded by profound darkness. I dare not use any less pedantic +word than entity to designate the ether, for it would be a great exaggeration of our knowledge if I were +to speak of it as a body, or even as a substance.” + +If this be so—and hardly any physicist will contradict this—then neither the Theory of Gases nor any +other physical theory can be quite a congruent account of facts, and I cannot hope with Mr. Burbury, that +Mr. Bryan will be able to deduce all the phenomena of spectroscopy from the electromagnetic theory of +light. Certainly, therefore, Hertz is right when he says: “The rigour of science requires, that we +distinguish well the undraped figure of nature itself from the gay-coloured vesture with which we clothe +it at our pleasure.” [Untersuchungen iiber die Ausbreitung der elektrischen Kraft, p. 31. Leipzig, Barth, +1892], But I think the predilection for nudity would be carried too far if we were to forego every +hypothesis. Only we must not demand too much from hypotheses. + +It is curious to see that in Germany, where till lately the theory of action at a distance was much more +cultivated than in Newton’s native land itself, where Maxwell’s theory of electricity was not accepted, +because it does not start from quite a precise hypothesis, at present every special theory is old-fashioned, +while in England interest in the Theory of Gases is still active; vide, among others, the excellent papers +of Mr. Tait, of whose ingenious results I cannot speak too highly, though I have been forced to oppose +them in certain points. + +Every hypothesis must derive indubitable results from mechanically well-defined assumptions by + + +xi. + + +*Pu*ic- Mo£4cjvi(i£icaZ + + + +mathematically correct methods. If the results agree with a large series of facts, we must be content, even +if the true nature of facts is not revealed in every respect. No one hypothesis has hitherto attained this +last end, the Theory of Gases not excepted. But this theory agrees in so many respects with the facts, that +we can hardly doubt that in gases certain entities, the number and size of which can roughly be +determined, fly about pell-mell. Can it be seriously expected that they will behave exactly as aggregates +of Newtonian centres of force, or as the rigid bodies of our Mechanics? And how awkward is the human +mind in divining the nature of things, when forsaken by the analogy of what we see and touch directly? + +Nevertheless, Boltzmann’s pessimism about the future of the kinetic theory, indicated in his +Foreword to Pail n, deepened in the following years, and led to fits of severe depression, +culminating in his suicide in 1906. This suicide must be ranked as one of the great tragedies +in the history of science, made all the more ironic by the fact that the scientific world made +a complete turnabout in the next few years and accepted the existence of atoms, following +Perrin’s experiments on Brownian motion and an accumulation of other evidence. By 1909 +even Ostwald himself had been converted, as he admits in the Foreword to the fourth +edition of his textbook on chemistry. Atomism had triumphed, though only at the cost of +denying to atoms almost all the properties with which they had originally been endowed. + + +* In his endeavors to satisfy the demands of all his readers, Boltzmann has not forgotten even the +faculty adviser of graduate students in physics. At the end of Part II, §86, he mentions a class of +problems that would be suitable for doctoral dissertations! + +t M. Kac, Probability and Related Topics in Physical Sciences (New York: Interscience, 1959), p. +261. Reprinted by permission. + +* H. Reichenbach, The Direction of Time ( Berkeley and Los Angeles: University of California Press, +1956), p. 128. + +t References for this and other works cited here will be found in the Bibliography at the end of this +volume. + +* See the papers by S. G. Brush and E. Mendoza listed in the Bibliography; also D. ter Haar, +Elements of Statistical Mechanics (New York: Rinehart, 1954), Appendix I (//-theorem and ergodic +theorem), and Rev. Mod. Phys. 27, 289 (1955) (recent developments). A series of selected annotated +reprints of the most important original papers on the kinetic theory is to be published by Pergamon +Press. + +* Boltzmann, Sitzungsber. math.-phys. Classe K. Bay. Akad. Wiss., Miinchen [“Mun. Ber.”] 22, 329 +(1892); the statement referred to is on p. 157 of the English translation in Phil. Mag. [5] 35 (1893). + +* Materialism and Empirio-Criticism Izdanie “Zveno,” Moscow, 1909. + +* See, e.g., the article by Bogolyubov and Sanochkin, Usp. Fiz. Nauk 61, 7 (1957), and Davydov's +introduction to the Russian translation of Gas Theory. + +t A. Sommerfeld, Wiener Chem. Zeitung 47, 25 (1944); see also G. Jaffe, J. Chem. Educ. 29, 230 +(1952). + +$ A. Aliotta, La Reazione idealistica contro la scienza (Palermo : Casa Editrice “Optima,” 1912); +quoted from p. 375 of the English translation by Agnes McCaskill (London, 1914). + + +^Puh.c. 7fca^4e«toiZc.a/ + + + + +PARTI + + +Theory of gases with monatomic molecules, whose dimensions +are negligible compared to the mean free path. + + +NOTE ON LITERATURE CITATIONS + +All the numbered footnotes correspond to Boltzmann’s notes in the original text, but the +numbers start with 1 in each section rather than each page, and the form of citation of some +journals has been changed for the sake of consistency with modem style. Footnotes +indicated by asterisks, daggers, etc., have been added by the translator. More complete +information including titles of articles cited will be found in the Bibliography at the end of +the book. + + +NOTE ON GOTHIC LETTERS + +To forestall possible difficulties in recognition, here are the alphabetical equivalents: + + +ABCDEFGHIJKLMNOPQRSTUVWXYZ + +SRttO¥Qfo62:U80 l 03 + +abcdefghijklmnopqrstuvwxyz + +abcbefgf)iiflmnopqr8tut>tojtn + + +“PufLC- 7^ai/uiMui£ica/ P/tyJxci + + + + + +FOREWORD TO PART I + + +“Alles Vergangliche +1st nur ein Gleichniss!”* + + +I have often before come close to writing a textbook on gas theory. I remember especially +the enthusiastic request of Professor Wroblewski at the Vienna World’s Fair in 1873. When +I showed little inclination to write a textbook, since I did not know how soon my eyes +would fail me, he answered dryly: “All the more reason to hurry up!” At present, when I +no longer have this reason, the time seems less appropriate for such a textbook than it was +then. For, first, gas theory has gone out of fashion in Germany, as it were; second, the +second edition of O. E. Meyer’s well-known text has appeared, and Kirchhoff has devoted +a longer section of his lectures on the theory of heat to gas theory.! Yet Meyer’s book, +though acknowledged to be excellent for chemists and students of physical chemistry, has a +completely different purpose. Kirchhoff s work shows the touch of a master in the selection +and presentation of its topics, but it is only a posthumously published set of lecture notes on +the theory of heat, which treats gas theory as an appendix, not a comprehensive textbook. +Indeed, I freely confess that the interest, on the one hand, which Kirchhoff showed in gas +theory, and on the other hand the many gaps in his presentation because of its brevity, have +encouraged me to publish the present work, which likewise originated from lectures at the +universities of Munich and Vienna. + +In this book I have hied above all to make clearly comprehensible the path-breaking +works of Clausius and Maxwell. The reader may not think badly of me for finding also a +place for my own contributions. These were cited respectfully in Kirchhoff s lectures and +in Poincare’s Thermodynamique* at the end, but were not utilized where they would have +been relevant. From this I concluded that a brief presentation, as easily understood as +possible, of some of the principal results of my efforts might not be superfluous. Of great +influence on the content and presentation was what I have learned at the unforgettable +meeting of the British Association in Oxford and the subsequent letters of numerous +English scientists, some private and some published in Nature .f + +I intend to follow Part I by a second part, where I will treat the van der Waals theory, +gases with polyatomic molecules, and dissociation. An explicit proof of Equation (110a)— +which is only indicated briefly in §16, in order to avoid repetition—will also be included. + +Unfortunately it was often impossible to avoid the use of long formulas to express +complicated trains of thought, and I can well imagine that to many who do not read over +the whole work, the results will perhaps not seem to justify the effort expended. Aside from +many results of pure mathematics which, though likewise apparently fruitless at first, later +become useful in practical science as soon as our mental horizon has been broadened, even + + +~PuJi.c- 7fca^4e«tai2c.a/ + + + +the complicated formulas of Maxwell’s theory of electromagnetism were often considered +useless before Hertz’s experiments. I hope this will not also be the general opinion +concerning gas theory! + +Vienna, September, 1895 + + +Ludwig Boltzmann + + +INTRODUCTION + + +§ 1. Mechanical analogy for the behavior of a gas. + +Clausius has made a sharp distinction between the general theory of heat, based +essentially on the two fundamental principles of thermodynamics, and the special theory of +heat, which starts out by making a definite assumption—that heat is molecular motion— +and then attempts to construct a more precise description of the nature of this motion. + +The general theory of heat also requires hypotheses which go beyond the bare facts of +nature. Nevertheless, it is obviously less dependent on special assumptions than the special +theory, and it is desirable and necessary to be able to separate its exposition from that of the +special theory, and to show that it is independent of the subjective assumptions of the latter. +Clausius has already done this very clearly, and indeed has based the division of his book* +into two parts on just this principle, and it would be useless to repeat his procedure. + +Recently, the mutual relations of these two branches of the theory of heat have changed +somewhat. Through the exploitation of some very interesting analogies and differences, +which the properties of energy exhibit in various phenomena of physics, there has arisen +the so-called Energetics, which is unfavorable to the view that heat is molecular motion. +This view is in fact unnecessary for the general theory of heat and, as is well known, was +not held by Robert Mayer. The further development of Energetics is certainly very +important for science ; however, up to now its concepts are still rather unclear, and its +theorems not very precisely expressed, so that it cannot replace the older theory of heat in +dealing with new special cases where the results are not already known.| + +Likewise in the theory of electricity, the older mechanical explanations of phenomena +by action at a distance, customarily employed especially in Germany, have suffered a +shipwreck. Indeed, Maxwell speaks with the greatest respect of Wilhelm Weber’s theory, +which by determining the conversion factor of electrostatic and electromagnetic units and +by discovering the relation of that factor to the velocity of light has laid the first stone in the +structure of the electromagnetic theory of light; yet one accedes to the contention that +Weber’s mechanical hypothesis about the action of electrical force is harmful to the +progress of science. + +In England, views on the nature of heat and on atomistics have remained relatively +unchanged. But on the continent, where earlier the assumption of central forces, useful in +astronomy, had been generalized to an epistemological demand, and for this reason + + +~Pu+lc- 7fca^4e«taL2c.a/ P/tyJxci + + + +Maxwell’s electrical theory did not receive much notice for a decade and a half (only this +generalization was obnoxious), the provisional character of all hypotheses has now been +generalized, and it has been concluded that the assumption that heat is motion of the +smallest particles of matter will eventually be proved false and discarded. + +It must however be remembered that any similarity between the kinetic theory and the +doctrine of central forces is purely coincidental. Gas theory has in fact a particular kinship +with Maxwell’s theory of electricity; the visible motion of a gas, viscosity, and heat, are +conceived as phenomena that are essentially different in stationary or almost stationary +states, while in certain transitional cases (very rapid sound waves with evolution of heat ; +viscosity or heat conduction in very dilute gases 1 ) a sharp distinction is no longer possible +between visible and thermal motion (cf. §24). Likewise in Maxwell’s theory of electricity, +in borderline cases the distinction between electrostatic and electrodynamic forces, etc., can +no longer be maintained. It is just in this transition region that Maxwell’s electrical theory +has yielded new facts; l ik ewise the gas theory has led to completely new laws in these +transition regions, which appear to reduce to the usual hydrodynamic equations, corrected +for viscosity and heat conduction, as purely approximate formulae (cf. §23). The +completely new laws were indicated for the first time in Maxwell’s 16-year-old paper, “On +stresses in rarefied gases.”* The phenomena which the theories based on older +hydrodynamic experience can never describe are those connected with radiometer action.! +Researches under many different conditions, and quantitative observations, have definitely +proved that the stimulus and explanation for this previously unknown realm of +experimental investigation could only have come from the theory of gases; likewise, the +tremendous fertility of Maxwell’s thcoiy of electricity for experimental investigations for +more than twenty years has seldom been pointed out. + +Although in the following exposition any qualitative difference between heat and +mechanical energy will be excluded, in the treatment of collisions of molecules the old +distinction between potential and kinetic energy will be retained. This does not basically +affect the nature of the subject. The assumptions about the interaction of molecules during a +collision have a provisory character, and will certainly be replaced by others. I have also +studied a gas theory in which, instead of forces acting during collisions, one merely has +conditional equations in the sense of the posthumous mechanics of Hertz,* which are more +general than those of elastic collisions; I have abandoned this theory, however, since I only +had to make more new arbitrary assumptions. + +Experience teaches that one will be led to new discoveries almost exclusively by means +of special mechanical models. Maxwell himself recognized the defect of Weber’s electrical +theory at first glance; on the other hand, he pursued zealously the theory of gases and the +method of mechanical analogies, which goes beyond that which he calls the method of +purely mathematical formulae.! + +As long as a clearer and better representation is not available, we shall still need to go +beyond the general theory of heat, and without impugning its importance, cultivate the old +hypotheses of the special theory of heat. Indeed, since the history of science shows how +often epistemological generalizations have turned out to be false, may it not turn out that the +present “modem” distaste for special representations, as well as the distinction between + + +“PufLC- 7fca^4e«toi2c.a/ P/tyJxci + + + +qualitatively different forms of energy, will have been a retrogression? Who sees the +future? Let us have free scope for all directions of research; away with all dogmatism, +either atomistic or antiatomistic! In describing the theory of gases as a mechanical analogy, +we have already indicated, by the choice of this word, how far removed we are from that +viewpoint which would see in visible matter the hue properties of the smallest particles of +the body. + +We shall first take the modem viewpoint of pure description, and accept the known +differential equations for the internal motions of solid and fluid bodies. From these it +follows in many cases, for example collisions of two solid bodies, motion of fluids in +closed vessels, etc., that as soon as the form of the body deviates the least bit from a simple +geometrical figure, waves must arise, which cross each other ever more randomly, so that +the kinetic energy of the original visible motion must finally be dissolved into invisible +wave motion. This mathematical consequence of the equations describing the phenomena +leads (to a certain extent by itself) to the hypothesis that all vibrations of the smallest +particles, into which the ever diminishing waves must finally be transformed, must be +identical with the heat that we observe to be produced, and that heat generally is a motion +in small—to us, invisible— regions. + +Whence comes the ancient view, that the body does not fill space continuously in the +mathematical sense, but rather it consists of discrete molecules, unobservable because of +their small size. For this view there are philosophical reasons. An actual continuum must +consist of an infinite number of parts; but an infinite number is undefinable. Furthermore, in +assuming a continuum one must take the partial differential equations for the properties +themselves as initially given. However, it is desirable to distinguish the partial differential +equations, which can be subjected to empirical tests, from their mechanical foundations (as +Hertz emphasized in particular for the theory of electricity). Thus the mechanical +foundations of the partial differential equations, when based on the coming and going of +smaller particles, with restricted average values, gain greatly in plausibility; and up to now +no other mechanical explanation of natural phenomena except atomism has been +successful. + +A real discontinuity of bodies is moreover established by numerous, and moreover +quantitatively agreeing, facts. Atomism is especially indispensable for the clarification of +the facts of chemistry and crystallography. The mechanical analogy between the facts of +any science and the symmetry relations of discrete particles pertains to those most essential +features which will outlast all our changing ideas about them, even though the latter may +themselves be regarded as established facts. Thus already today the hypothesis that the stars +are huge bodies millions of miles away is similarly viewed only as a mechanical analogy +for the representation of the action of the sun and the faint visual perceptions arising from +the other heavenly bodies, which could also be criticized on the grounds that it replaces the +world of our sense perceptions by a world of imaginary objects, and that anyone could just +as well replace this imaginary world by another one without changing the observable facts. + +I hope to prove in the following that the mechanical analogy between the facts on which +the second law of thermodynamics is based, and the statistical laws of motion of gas +molecules, is also more than a mere superficial resemblance. + + +~Pu+lc- 7fca^4e«toi2c.a/ P/tyJxci + + + +The question of the utility of atomistic representations is of course completely unaffected +by the fact, emphasized by Kirchhoff,* that our theories have the same relation to nature as +signs to significates, for example as letters to sounds, or notes to tones. It is lik ewise +unaffected by the question of whether it is not more useful to call theories simply +descriptions, in order to remind ourselves of their relation to nature. The question is really +whether bare differential equations or atomistic ideas will eventually be established as +complete descriptions of phenomena. + +Once one concedes that the appearance of a continuum is more clearly understood by +assuming the presence of a large number of adjacent discrete particles, assumed to obey the +laws of mechanics, then he is led to the further assumption that heat is a permanent motion +of molecules. Then these must be held in their relative positions by forces, whose origin +one can imagine if he wishes. But all forces that act on the visible body but not equally on +all the molecules must produce motion of the molecules relative to each other, and because +of the indestructibility of kinetic energy these motions cannot stop but must continue +indefinitely. + +In fact, experience teaches that as soon as the force acts equally on all parts of a body— +as for example in so-called free fall—all the kinetic energy becomes visible. In all other +cases, we have a loss of visible kinetic energy, and hence creation of heat. The view offers +itself that there is a resulting motion of molecules among themselves, which we cannot see +because we do not see individual molecules, but which however is transmitted to our +nerves by contact, and thus creates the sensation of heat. It always moves from bodies +whose molecules move rapidly to those whose molecules move more slowly, and because +of the indestructibility of kinetic energy it behaves l ik e a substance, as long as it is not +transformed into visible kinetic energy or work. + +We do not know the nature of the force that holds the molecules of a solid body in their +relative positions, whether it is action at a distance or is transmitted through a medium, and +we do not know how it is affected by thermal motion. Since it resists compression as much +as it resists dilatation, we can obviously get a rather rough picture by assuming that in a +solid body each molecule has a rest position. If it approaches a neighboring molecule it is +repelled by it, but if it moves farther away there is an attraction. Consequently, thermal +motion first sets a molecule into pendulum-like oscillations in straight or elliptical paths +around its rest position A (in the symbolic Fig. 1, the centers of gravity of the molecules are +indicated). If it moves to A ', the neighboring molecules B and C repel it, while D and E +attract it and hence bring it back to its original rest position. If each molecule vibrates +around a fixed rest position, the body will have a fixed form; it is in the solid state of +aggregation. The only consequence of the thermal motion is that the rest positions of the +molecules will be somewhat pushed apart, and the body will expand somewhat. However, +when the theimal motion becomes more rapid, one gets to the point where a molecule can +squeeze between its two neighbors and move from A to A" (Fig. 1). It will no longer then +be pulled back to its old rest position, but it can instead remain where it is. When this +happens to many molecules, they will crawl among each other like earthworms, and the +body is molten. Although one may find this description rather crude and childish, it may be +modified later and the apparent repulsive force may turn out to be a direct consequence of + + +“PufLC- 7fca^4e«toi2c.a/ P/tyJxci + + + +the motion. In any case, one will allow that when the motions of the molecules increase +beyond a definite limit, individual molecules on the surface of the body can be tom off and +must fly out freely into space; the body evaporates. If it is in an enclosed vessel, then this +will be filled with freely moving molecules, and these can occasionally penetrate into the +body again; as soon as the number of recondensing molecules is, on the average, equal to +the number of evaporating ones, one says that the vessel is saturated with the vapor of the +body in question. + + + + +0 • • B + +• • • + +A A' A' + +E • • C + + +Fig. 1 + +A sufficiently large enclosed space, in which only such freely moving molecules are +found, provides a picture of a gas. If no external forces act on the molecules, these move +most of the time like bullets shot from guns in straight lines with constant velocity. Only +when a molecule passes very near to another one, or to the wall of the vessel, does it +deviate from its rect il inear path. The pressure of the gas is interpreted as the action of these +molecules against the wall of the container. + +§2. Calculation of the pressure of a gas. + +We shall now undertake a more detailed consideration of such a gas. Since we assume +that the molecules obey the general laws of mechanics, then in collisions of the molecules +with each other and with the wall, the principle of conservation of kinetic energy, and of +motion of the center of gravity, must be satisfied. We can make as varying pictures of the +internal properties of the molecules as we like; as long as these two principles are satisfied, +we shall obtain a system which shows a definite mechanical analogy with the actual gas. +The simplest such picture is one in which the molecules are completely elastic, negligibly +deformable spheres, and the wall of the container is a completely smooth and elastic +surface. However, when it is convenient we can assume a different law of force. Such a +law, provided it is in agreement with the general mechanical principles, will be no more +and no less justified than the original assumption of elastic spheres. + +We imagine a container of volume Q of any arbitrary shape, filled with a gas, against +whose walls the gas molecules are reflected exactly l ik e completely elastic spheres. Let a +part of the wall of the container, AB, have surface area cp. We place perpendicular to it, +running from inside to outside, the positive axis of abscissas. The pressure on AB will + + +~Pu+lc- 7fca^4e«toi2c.a/ P/tyJxci + + + + +clearly not be changed if we imagine behind this surface element a right cylinder on the +base AB, in which the surface element AB is lik e a piston which can be displaced parallel +to itself. This piston will then be pushed into the cylinder by the molecular impacts. If a +force P acts from the outside in the negative direction, then its intensity can be chosen so +that the molecular impacts are kept in equilibrium, and the piston makes no visible motion +in either direction. + +During any instant of time dt, there may be several molecules colliding with the piston +AB ; the first exerts a force q\ , the second a force qj, etc., in the positive abscissa direction, +on the piston. Denoting by M the mass of the piston, and by U the velocity in the positive +direction, then one has for the time element dt the equation + +dU + +M — = - p -|- 0i -f qi + • • • +dt + + +If one multiplies by dt and integrates over an arbitrary time /, then : + +M(Ui - U») = - Pt + £ f qdt. + +J 0 + +If P is to be equal to the pressure of the gas, then the piston must not show any noticeable +motion, aside from invisible fluctuations. In the above formula, Uq is the value of its +velocity in the abscissa direction at the initial time, and U\ is its value after an interval of +time t. Both quantities will be very small; indeed, one can easily choose the time t such that +U | = Uq, since the piston must periodically assume the same velocity during its various +small fluctuations. In any case, t/j -Uq cannot continually increase with increasing time, +and therefore with increasing time the quotient (U\ - U 0 )/t must approach the limit zero. +Whence follows: + + +(1) P = 7Zf«* + +t J 0 + +The pressure is therefore the mean value of the sum of all the small pressures that the +individual colliding molecules exert on the piston at different times. We shall now calculate +\qdt for any one collision which the piston experiences during the time t with a molecule. +Let the mass of the molecule be m, and let the component of its velocity in the abscissa +direction be u. The collision begins at time t\ and ends at time t\ + r, before time t\ and after +time f| + r, the molecule exerts essentially no force on the piston. Then + + +7fca^4e«taiZc.a/ + + + + +During the time of the collision, however, the force which the molecule exerts on the piston +is equal and opposite to the force which the piston exerts on the molecule: + + +du + +m — = - q. + +dt + + +We denote by £ the velocity component of the colliding molecule before the collision in +the direction of the positive abscissa axis, and by the same component after the collision, +and we obtain : + + + +Since the same is true for all other colliding molecules, it follows from Equation (1) that: + +(2) P = yIX + +V + + +where the sum is to be extended over all molecules that strike the piston between the +instants 0 and t. Only those which are in collision with the piston at the instants 0 and t are +thereby omitted, which is permissible when the total time interval t is very large compared +to the duration of an individual collision. + +We shall see (§3) that even when only a single gas is present in the container, all the +molecules can be no means have the same velocity. In order to preserve the greatest +generality, we assume that in the container there are different kinds of molecules, which +however are all reflected like elastic spheres from the walls. n\Q molecules will have mass +m | and velocity cq with components £\, rj \, (\ in the coordinate directions. They should be +uniformly distributed, on the average, throughout the interior volume Q of the container, so +that there arc n | in unit volume. Further, there are« 2 fi molecules distributed similarly, +having another velocity C 2 with components £ 2 , // 2 , (2 an d perhaps also a different mass +m 2 . The quantities n 3 , c 3 , T 3 , // 3 , d 3 , m 3 , etc., up to n b into even smaller parallelepipeds. If the +function/is known for one value of t, then the velocity distribution for the m-molecules is +determined at timet. Similarly we can represent the velocity of each mj -molecule by a +velocity point, and denote by + +( 12 ) F(i i, iji, fi, t)di-\driidli\ = F idcoi + +the number of molecules whose velocity components he between any other l im its + +(13) £i and + dfi, tj\ and rji + dij\ } fi and fi + dfi + +and for which therefore the velocity points lie in a similar parallelepiped dco\ . Likewise we +write dojy for df\di]\dC\ and F 1 for F(/|, rji, C,\, t). Moreover, we shall completely exclude +any external forces, and assume that the walls are completely smooth and elastic. Then the +molecules reflected from the wall will move as if they came from a gas which is the mirror +image of our gas, and which is thus completely equivalent to it; the container wall is +thought of as a reflecting surface. According to these assumptions, the same conditions will +prevail everywhere inside the container, and if the number of molecules in a volume +element whose velocity components lie between the li mits (10) is initially the same +everywhere in the gas, then this will also be hue for all subsequent times. If we assume this, +then it follows that the number of m-molecules inside any volume that satisfy the +conditions (10) is proportional to the volume that satisfy the conditions (13) is: + +(14a) $F idcoi. + +From these assumptions it follows that the molecules that leave any space as a result of their +progressive motion will on the average be replaced by an equal number of molecules from +the neighboring space or by reflection at the walls of the container, so that the velocity +distribution is changed only by collisions, and not by the progressive motion of the +molecules. We shall make ourselves independent of these restrictive conditions (made now +to simplify the calculation) in §§15-18, where we shall take account of the effect of gravity +and other external forces. + +We next consider only the collisions of an m-molecule with an m 1 -molecule and indeed +we shall single out, from all those collisions that can occur during the interval dt , only those +for which the following three conditions are satisfied: + +1. The velocity components of the m-molecule lie between the limits (10) before the +collision, hence its velocity point lies in the parallelepiped dco. + + +7fca^4e«toiZc.a/ + + + +2. The velocity components of the m \ -molecule lie between the li mits (13) before the +collision, hence its velocity point lies in the parallelepiped da >\. All m-molecules for which +the first condition is fulfilled will be called “m-molecules of the specified kind,” and +similarly we speak of “m |-molecules of the specified kind.” + +3. We construct a sphere of unit radius, whose center is at the origin of coordinates, and +on it a surface element dk. The line of centers of the colliding molecules drawn from m to +m | must, at the moment of collision, be parallel to a line drawn from the origin to some +point of the surface element dk. The aggregate of these lines constitutes the cone dk. + +(15) Direction mm\ in the cone dk + +All collisions that take place in such a way that these three conditions are fulfilled will +be called “collisions of the specified kind” and we have the problem of determining the +number dv of collisions of the specified kind that take place during a time interval dt in unit +volume. We shall represent these collisions in Figure 2. Let O be the origin of coordinates, +C and C| the velocity points of the two molecules before the collision, so that the lines OC +and OC\ represent these velocities in magnitude and direction, before the collision. The +point C must lie inside the parallelepiped dco, and the point Cj inside the parallelepiped +da>\. (The two parallelepipeds are not shown in the figure.) Let OK be a line of unit length +which has the same direction as the line of centers of the two molecules at the instant of the +collision, drawn fromm to m,\. The points must therefore lie inside the surface element +dk , which is also not shown in the figure. The line C\C = g represents in magnitude and +direction the relative velocity of the m-molecule with respect to the in \-molecule before the +collision, since its projections on the coordinate axes are equal to £ - £\, ij - i ]\, and C - +respectively. The frequency of collisions obviously depends only on the relative velocity. +Hence if we wish to find the number of collisions of the specified kind, we can imagine +that the specified in | -molecule is at rest, while the m-molecule moves with velocity g. We +imagine further that a sphere of radius ff (the sphere a) is rigidly attached to each of the +latter molecules, so that the center of the sphere always coincides with the center of the +molecule, cf should be equal to the sum of the radii of the two molecules. Each time that the +surface of such a sphere touches the center of an m\ -molecule, a collision takes place. We +now draw from the center of each sphere a a cone, similar and similarly situated to the +cone dk. A surface element of area erdk is thereby cut out from the surface of each of +these spheres. Since all the spheres are rigidly attached to the corresponding molecules, all +these surface elements move a distance gdt relative to the specified in | -molecule. A +collision of the specified kind occurs whenever one of these surface elements touches the +center of a specified m \ -molecule, which is of course possible only if the angle & between +the directions of the lines C| C and OK is acute. Each of these surface elements traverses by +its relative motion toward the in | -molecule an oblique cylinder of base o 2 dk and height g +cos ddt. Since there ar efdaim molecules of the specified kind in unit volume, all the +oblique cylinders traversed in this manner by all the surface elements have total volume + + +7fca^4e«toi2c.a/ + + + + +(16) $ = fdu^g cos ddhdt. + +All centers of m 1 -molecules of the specified kind lying inside the volume will touch +during the interval dt one surface element ad/. and hence the number dv of co ll isions of +the specified kind which occur in the volume element during time dt is equal to the number +Z f[) of centers of »q -molecules of the specified kind that are in the volume ® at the +beginning of dt. But according to Equation (14a) this is + +(17) = $F \do)i. + +In this formula there is contained a special assumption, as Burbury 1 has clearly +emphasized. From the standpoint of mechanics, any arrangement of molecules in the +container is possible; in such an arrangement, the variables determining the motion of the +molecules may have different average values in one part of the space filled by the gas than +in another, where for example the density or mean velocity of a molecule may be larger in +one half of the container than in the other, or more generally some finite part of the gas has +different properties than another. Such a distribution will be called molar-ordered [molar- +geordnete ]. Equations (14) and (14a) pertain to the case of a molar-disordered distribution. +If the arrangement of the molecules also exhibits no regularities that vary from one finite +region to another—if it is thus molar-disordered—then nevertheless groups of two or a +small number of molecules can exhibit definite regularities. A distribution that exhibits +regularities of this kind will be called molecular-ordered. We have a molecular-ordered +distribution if—to select only two examples from the infinite manifold of possible cases— +each molecule is moving toward its nearest neighbor, or again if each molecule whose +velocity lies between certain l im its has ten much slower molecules as nearest neighbors. +When these special groupings are not limited to particular places in the container but rather +are found on the average equally often throughout the entire container, then the distribution +would be called molar-disordered. Equations (14) and (14a) would then always be valid for +individual molecules, but Equation (17) would not be valid, since the nearness of the m- + + +7fca^4e«toiZc.a/ + + +molecule would be influenced by the probability that the m 1 -molecule lies in the space . +The presence of the mj -molecule in the space ® cannot therefore be considered in the +probability calculation as an event independent of the nearness of the m-molecule. The +validity of Equation (17) and the two similar equations for collisions of m- or in \ -molecules +with each other can therefore be considered as defining the meaning of the expression: the +distribution of states is molecular-disordered. + +If the mean free path in a gas is large compared to the mean distance of two neighboring +molecules, then in a short time, completely different molecules than before will be nearest +neighbors to each other. A molecular-ordered but molar-disordered distribution will most +probably be transformed into a molecular-disordered one in a short time. Each molecule +flies from one collision to another one so far away that one can consider the occurrence of +another molecule, at the place where it collides the second time, with a definite state of +motion, as being an event completely independent (for statistical calculations) of the place +from which the first molecule came (and similarly for the state of motion of the first +molecule). However, if we choose the initial configuration on the basis of a previous +calculation of the path of each molecule, so as to violate intentionally the laws of +probability, then of course we can construct a persistent regularity or an almost molecular- +disordered distribution which will become molecular-ordered at a particular time. +Kirchhoff 2 also makes the assumption that the state is molecular-disordered in his definition +of the probability concept. + +That it is necessary to the rigor of the proof to specify this assumption in advance was +first noticed in the discussion of my so-called H-theorem or minimum theorem. However, it +would be a great error to believe that this assumption is necessary only for the proof of this +theorem. Because of the impossibility of calculating the positions of all the molecules at +each time, as the astronomer calculates the positions of all the planets, it would be +impossible without this assumption to prove the theorems of gas theory. The assumption is +made in the calculation of the viscosity, heat conductivity, etc. Also, the proof that the +Maxwell velocity distribution law is a possible one—i.e., that once established it persists for +an infinite time—is not possible without this assumption. For one cannot prove that the +distribution always remains molecular-disordered. In fact, when Maxwell’s state has arisen +from some other state, the exact recurrence of that other state will take place after a +sufficiently long time (cf. the second half of §6). Thus one can have a state arbitrarily close +to the Max-wellian state which finally is transformed into a completely different one. It is +not a defect that the minimum theorem is tied to the assumption of disorder, rather it is a +merit that this theorem has clarified our ideas so that one recognizes the necessity of this +assumption. + +We shall now explicitly make the assumption that the motion is molar - - and molecular- +disordered, and also remains so during all subsequent time. Equation (17) is then valid, and +we obtain + +(18) dv = = $Fi(2a>i = /dwFi\ of mj-molecules of the +specified kind, both decrease by one. In order to find the total decrease \dv suffered by fda> +during dt as a result of all collisions of m-molecules with in |-molecules (without restriction +on the magnitude and direction of the line of centers), we must consider f, //, (, dco and dt +as constant in Equation (18) and integrate r/oq and dJ. over all possible values—i.e., we +integrate duty over all space, and dk over all surface elements for which the angle & is +acute. We shall denote the result of this integration by J dv. + +The decrease dn which the number fdco experiences as a result of the corresponding +collisions of m-molecules with each other is obviously given by a completely analogous +formula; we simply denote by , t ]\, C\ the velocity components of another m-molecule +before the collision. All other quantities have the same meaning, except that one replaces +m | by m and the function F by the function/, and a by the diameters of an m-molecule. +When we have instead of dv the expression + + +(19) + + +dn = //i the total decrease of + +fdco resulting from collisions of the m-molecules with each other during dt —one must +obviously consider /, ij, C, dco, and dt as constant and integrate du>\ and dk over all possible +values. The total decrease of fdco during dt is therefore equal to fdv -b fdn . If the state + +is to be stationary, this must be exactly equal to the number of m-molecules whose +velocities at the beginning of dt do not fulfill the conditions (10) but which during this time +interval are changed by collisions in such a way that they now satisfy them—they obtain by +collision a velocity lying between the limits (10). In other words, fdv+fdn must be +equal to the total increase of fdco resulting from co ll isions. + + +§4. Continuation; values of the variables after the collision; collisions of the +opposite kind.* + +To find this increase, we shall next seek the velocities of the two molecules after a +collision of the specified kind. Before the collision one of the colliding molecules, whose +mass is m, has velocity components /, //, (; the other, of mass m |, has components , rjy, +/l. The line of centers drawn from m to m 1 forms at the instant of collision an angle d with +the relative velocity of the molecule m with respect to m |. If the angle e between the plane +of these two lines and any other given plane—for example, that of the two velocities before +the collision—is given, then the collision is completely specified. The velocity components +C, >l', and v'u t'i of the two molecules after the collision can thus be expressed as + + +7fca^4e«toiZc.a/ + + + +explicit functions of the 8 variables i], C, f \, t ]\, C\, ■&, e\ + += ^i({) J l, f, £l> Vh fl>#> € ) +(20) | rj' = fait, tj, f, {i, iji, fi, d r e). + + +We prefer however the geometric construction to the algebraic development of the +functions (20), and we return to Figure 2, p. 39. We divide the segment C,C at the point S +into two parts, such that + +C\S:CS = m'Mi. + +Then the line OS represents the velocity of the common center of mass of the two +molecules; for one sees that its three projections on the coordinate axes have the values: + +mi + mil mi; + mm i + +( 21 ) - 1 - ) - + +m + mi m + mi m + mi + +But these are in fact just the velocity components of the common center of mass. Just as we +have shown that C\C is the relative velocity of the m-molecule with respect to m,, it +follows also that SC and SC | are the relative velocities of the two molecules with respect to +the common center of mass before the collision. The components of these relative velocities +perpendicular to the line of centers OK will not be changed by the collision. The +components in the direction OK are p and p \ before the collision, and p’ and p { after the + +collision. Then according to the principle of conservation of the motion of the center of +mass: + +mp + mipi = mp f + m\p{ = 0, + +and according to the principle of conservation of kinetic energy: + +mp 2 + mipi 2 = mp' 2 + mi pi 2 . + +Whence it follows that + +v' = V, Pi = Vb + +or + +v' = - V, Pi = - VI + + +We 7fca^4e«toi2ea/ lPl±y. i Ze i. + + + + +and one sees at once that, since the molecules must separate from each other after the +collision, only the latter solution is correct, and that hence the two components of relative +velocity with respect to the center of mass, which fall in the direction K\K 2 WOK, will +simply be reversed by the co ll ision. + +From this one obtains the following construction of the lines OC' and OC ( which +represent the velocities of the two molecules after the collision in magnitude and direction. +One draws through S the line K | K 2 \ then one draws in the plane of the lines K\ Kj and +C| C, the two lines SC' and SC \ which are equal in length to the lines SC and SC |, and +are equally inclined on the other side from The two endpoints C' and (J { of the + +latter two lines are at the same time the endpoints of the required lines OC' and OC{ We +can also call them the velocity points of the two molecules after the collision. The +projections of OC' and oci on the three coordinate axes are thus the velocity + +components V'r t'i fl ) Vl f fl of the two molecules after the collision. +These geometrical constructions completely replace the alternative algebraic development +of the functions (20). The points C{, s. and C' obviously fall on a straight line. This +line Q^ represents the relative velocity of the molecule m with respect to the molecule + +m | after the collision, and one sees from the figure that its length is equal to C ] C, whereas +the angle it forms with the line OK is 180° - C. + +Up to now we have considered only one of the specified collisions, and we have +constructed the velocities after the collision for it. We now consider all the specified +collisions, and ask between which limits the values of the variables after the collision lie, +for all these collisions—i.e., for all collisions such that the conditions (10), (13), and (15) +are fulfilled before the collision. Since we assume the time duration of the collision to be +infinitesimal, the direction of the line of centers is the same at the end of the collision as at +the beginning, and it is only a question of finding the limits between which the velocity + +components £'» v', f', ft, Vi t fi are found after the collision. Had we + +calculated the function (20), then we would have considered j?and e as constants and C, //, +C, <(j, }]\, C| as independent variables, and we would have to express +d £' dr,' d{' dU d v { d{i’ in teims of d£ ch / dt d£\ dij\ dC\ by means of the +well-known Jacobian functional determinant. However, we prefer the geometric +construction, and we must therefore answer the question: what volume element would be +described by the points C' and (J { when, without changing the direction of the line OK, +we let the points C and C| describe the volume elements da> and da>\ ? First, let the position +of the point C as well as the direction of the line OK remain fixed, and let C| sweep out the +entire parallelepiped da>\ . From the complete symmetry of the figure, it follows directly that +(J/ describes a congruent parallelepiped which is the mirror image of da>\. Likewise if the +point Cj is fixed and the point C sweeps out the parallelepiped dco, then the point C' +sweeps out a parallelepiped congruent to dto. For all collisions that we earlier called +collisions of the specified kind, the velocity point of the m-molecule lies in the +parallelepiped dco' after the collision, and that of the in | -molecule in c/o.q , and it is always + + +Uuc. 7fca^4e«taiZc.a/ + + + +true that = f/ox/u) i The same result would also be obtained by explicit + +calculation of the functions (20) and construction of the functional determinant 1 + +ar it l + +d£ dri + +We shall now consider another class of collisions of an m-molecule with an in | - +molecule, which will be called the “collisions of the inverse kind.” They are characterized +by the following conditions: + +1. The velocity point of the m-molecule lies in the volume clement do/ before the +collision; the number of m-molecules in the volume element for which this condition is +satisfied is, by analogy with Equation (9), fdoJ, where/ is the value of the function / +obtained by replacing /, ij, C by /', if, C —i.e., it is the quantity/;/', if, C t). + +2. The velocity point of the mj-molecule lies in the volume clement before the + +collision. The number of in \-molecules in unit volume for which this condition is satisfied + +is F i c/co/’ where F\ is an abbreviat i° n for F (£i', Vjl , f / , t)- + +3. The hne of centers of the two molecules at the instant of the collision, drawn from in | +to m, is parallel to some line drawn from the origin of coordinates within the cone d/ 1. (In +those integrals that refer to the collision of identical molecules, there will of course occur in +place of the molecule of mass in | simply an m-molecule whose velocity components are + +Zbdl’ Cl-) + +Figure 3 represents the same collision as Figure 2, the lines having been kept fixed as far +as possible. Figure 4 represents the opposite collision. The arrow directed toward the center +of the molecule always represents its velocity before the collision, while that directed away +from the collision represents its velocity after the collision. In all collisions of the opposite +kind, the relative velocity of the m-molecule with respect to that of the in |-molecule before +the collision is represented by the line ^ Q r in Figure 2. Its magnitude is thus equal + +again to g, and it forms the angle d with the hne of centers drawn from m to ni\ , since we +have likewise reversed the direction of the hne of centers. The angle & must of course be +acute if the cohision is to be possible. The number of inverse cohisions in unit volume +during dt is, by analogy with Equation (18), given by + + + + +7fca^4e«taiZc.a/ + + + +Fig. 3. + + +(22) dv' = f'F i du'dul j \, C \, + + +consider their arguments Yj' } ) f + + +7fca^4e«taiZc.a/ ~Ph-y-&XjcS- + + + +now considers f, //, (, da> and dt constant and integrates over all possible values of doj\ and +dX. Thereby all collisions will be included that can take place between an m-molecule and +an in |-molecule, such that the former has velocity components within the limits (10) before +the collision, but without any other restrictions. The result of this integration, \dv', thus +gives us the increase of fdco resulting from all collisions of m-molecules with in | -molecules +during dt. Similarly, this quantity increases by fdn' as a result of collisions of m- + +molecules with each other, where + +(24) dt i' = /'// dwdwis 2 0 cos #d\dt + +Her £ f { is again used as an abbreviation for F{%\> li > tit 0 + +r, ll, nl, tl are here functions of f, ij, C, f\, ij\. C\, & and e, inasmuch +as the former represent the velocity components after a collision, and are determined by the +initial conditions (10), (13), and (15), in which however both molecules have mass m. + +If we subtract from the total increase of fdco the total decrease, we obtain the net change + +df + +— dadt, +dt + + +which the quantity fda> experiences during time dt. Thus: + + +d[ + +dt + + +dtdw = / dv' - J dv + f dn' - f dn. + + +In the integrals \dv and j dv’ the integration variables are identical, and lik ewise in the +integrals fdn and fdn ' + +If we combine these integrals and divide the whole equation by da) ■ dt, then it follows +from Equations (18), (19), (23), and (24) that: + + +(25) + + +fa/ , + +- = / (fF{ - jF y-g cos dduA + + +dt + ++1 ifH - //i)s s 0 cosi?(WX. + + +The integration is to be extended over all possible da>\ and dX. Likewise, one obtains for +the function F the equation + + +We 7fca^4e«toi2ea/ TRk.y l i .Ze l + + + +( 26 ) + + +IdFi r + +— - } (J'Fl iV 2 0 cosM* >d\ + + +- dt + ++ / (F'Fl - FFi)s[g mtidwdX. + +Here ,V| is the diameter of an m 1 -molecule; in Equation (26), f \, /q. and f) are arbitrary and +should be considered constant in the integral, while f, rj, and C are to be integrated over all +their possible values. In the first integral, f', a, vl, il are the velocity + +components after a collision of the specified kind in the case that one of the colliding +molecules has mass m, and the other has mass mp while in the second integral they refer to +the case that both molecules have mass m \. dFfdt, F, and F’ are abbreviations for dF(f \, +tT\, C|, t)/dt, F(f, >1, C, t ), and F(F rf, t). + +If the state is to be stationary, then the quantities df/dt and dFfdt must vanish for all +values of the variables. This certainly occurs when in all the integrals the integrand +vanishes for all values of the variable of integration—when one therefore has for all +possible collisions of the m-molecules with each other, the in | -molecules with each other, +and of m-molecules with in | -molecules, the three equations + +( 27 ) //,=/'/,', FFi-FFi,JFi-rFi. + +Since the probability of the originally specified collision is given by Equation (18), and that +of the opposite one by Equation (23), the general validity of the third of Equations (27) is +equivalent to the statement that, however cho, daj\, and d/. | may be chosen, the originally +specified (or briefly, “direct”) collision is as probable as the opposite one. In other words, it +is equally probable that two molecules should separate from each other in a certain way as +that they should collide in the opposite way. The same follows from the two other parts of +the system of Equations (27) for the collisions of m-molecules with each other and of m | - +molecules with each other. However, one sees at once that a distribution of states must +remain stationary when it is equally probable that two molecules separate from each other +after a collision and that they collide with each other in the opposite way. + +§5. Proof that Maxwell’s velocity distribution is the only possible one. + +We shall deal later with the solution of Equations (27), which presents no particular +difficulty. It leads necessarily to the well-known Maxwell velocity distribution law. For this +case the two quantities df/dt and dF/dt vanish, since the integrands of all the integrals +vanish identically. It still remains to be shown that the Maxwell velocity distribution, once +established, is not altered by further collisions. It has still not been proved that Equations +(25) and (26) cannot be satisfied by other functions that do not make the integrands vanish +for all values of the variables of integration. One may give such possibilities as little weight +as he wishes; I find myself inclined to dispose of them by a special proof. Since this proof + + +7fca^4e«taiZc.a/ + + + +has, as will appear, a not uninteresting connection with the entropy principle, I will +reproduce it in the form given by H. A. Lorentz.* + +We consider the same gas mixture as before, and retain the earlier notation. +Furthermore, we denote by If and IF the natural logarithms of the functions/ and F. The +result obtained on substituting in If for /, C the velocity components of a particular gas +molecule of mass m at time t, we call the value of the logarithmic function corresponding to +the molecule considered at the time considered. Similarly, we obtain the value of the +logarithm function corresponding to any in \ -molecule at any time when we substitute in +IF j the velocity components f \, rft, C\ of that molecule at that time. We shall now calculate +the sum H of all values of the logarithm functions corresponding at a particular time to all +m-molecules and in | -molecules contained in a volume element. At time t, there are in the +volume element fdco m-molecules of the specified kind, i.e., m-molecules whose velocity +components lie between the li mits (10). These clearly contribute a term/- If - dco to the sum +H. If we construct the analogous expression for the in | -molecules and integrate over all +possible values of the variables, then it follows that: + +(28) H = f J-lf-du + / Fi-lFi-dui. + +We now seek the change experienced by H during a very small time dt. This will be due to +two causes: 1 + +1. Each m-molecule of the specified kind provides at time / the term If in the expression +(28). After time dt, the function/experiences the increment + + + +dt + + +Hence /experiences the increment: + + +1 + +7 + + + +and each m-molecule of the specified kind provides in the expression (28) the term + + + +All m-molecules of the specified kind provide together the contribution + + +7fca^4e«toiZc.a/ + + + + +in (28). Applying the same arguments to all other m-molecules and m 1 -molecules, one +finds for the total increment of H, resulting from the variation of the quantities If and IF +under the integral signs in Equation (28), the value : + + + +df r dF i +— dtdoi -f* I — dtdwi. +dt Jdt + + +But this is nothing more than the variation of the total number of molecules in unit volume, +which must be equal to zero, since neither the size of the container nor the uniform +distribution of molecules should undergo any change. + +2. The collisions change not only If and IF but also fdco and F\daj\ —i.e., the number of +molecules of the specified kind changes a little. The variation clH of H due to this second +cause will, according to the above, be equal to the total variation of FI during dt. In order to +find it, we again denote by dv the number of collisions of the specified kind in unit volume +during dt. Each such collision decreases both fdto and F\du>\ by one. + +Since each m -molecule contributes the addend If to (28), and each m 1 -molecule +contributes IF \, the total decrease of H resulting from these collisions will be + +(If + Wi)dv + + +Each of these collisions increases fdco' by one, and consequently H is increased by Ifdv. +Finally, each of these collisions increases F{d\ by one. Consequently the inverse colhsions will +increase Ft by + + +(IJ + lFi-lf-lFlW += (If + IFi - lf f - IFDf'Fl dudui (cells), so that the presence of the velocity point of +a molecule in each such volume element is to be considered an event equiprobable with its +presence in each other volume element—just as previously we considered the drawing of a +black or white or blue sphere to be equally probable. Instead of a, the number of drawings +of white spheres, we now have the number tiy(o of molecules whose velocity points lie in +the first of our volume elements; instead of b, we have the number n 2 to of molecules whose +velocity points he in the second volume element oj, and so forth. Instead of Equation (34) +we have + + +n\ + +( 35 ) Z =- + +(n 1 w)!(n 2 w)!(7i3w)! • • • + +for the relative probability that the velocity points of n \ oj molecules lie in the first volume +element, and so forth. Then n = (ni+n 2 +ny I- ■ ■ ) oj is the total number of ah molecules in +the gas. For example, the event that all molecules have equal and equally directed velocities +corresponds to having ah velocity points lying in the same cell. Here we would have Z = +n \/n ! = 1; no other permutations are possible. It is already very much more probable that + + +7fca^4e«ta£Zc.a/ + + + +half the molecules have one particular velocity and direction, and the other half have +another velocity and direction, than that all have the same velocity and direction. Then half +the velocity points would be in one cell and half in another, so that + +n\ + +Z =-etc. + + + +Since now the number of molecules is extraordinarily large, ti \ (j) , n 2w , etc., can be +treated as very large numbers. + +We shall use the approximation formula + + +p! = V2^)’ + +where e is the base of natural logarithms and p is an arbitrarily large number. 1 +Denoting by l the natural logarithm, we see that: + +/[(ftiw)!] = -)- 2)^1 "1" - 1) -f- !(/o> 4" Z2 tt). + +Omitting - 1 , compared to the very large number n | <0 and forming the similar expressions +for (n 2 &>)!, (« 3 &>)!, etc., one obtains : + +IZ = - o)(niln\ + 7i 2 Zn 2 • • •) + C, + +where + +C = I(»0 - n(h - 1) - - (Jh + 12j) + +td + +has the same value for all velocity distributions and is therefore to be considered a constant. +Then we ask for the relative probability of the distribution of different velocity points of our +molecules in our cells, where of course the cell divisions, the size of a cell co, the number of +cells C and the total number of molecules n and their total kinetic energy are to be +considered constant. The most probable distribution of velocity points of the molecules in +our cells will be the one for which /Z is a maximum; hence the expression + +w[ni/ni + W2ZW2 + • • • + + +'nfLc. 7fca^4e«toi2c.a/ lPl±y. i .Zc. 1 + + + +will be a maximum. If we write dfdijdt for to and/f is chosen very small, the velocity points of many molecules +would still always lie in it. The order of magnitude of the volume chosen as volume +element is completely independent of the order of magnitude of the volume elements u> and +dfdijdf. + +Even more dubious is the assumption we shall make later, that not only the number of +molecules in the volume element whose velocity points he in a differential volume, but also +the number of molecules whose centers are in such a volume element, is infinitely large. +The latter assumption is no longer justified as soon as one has to deal with phenomena in +which finite differences in the properties of the gas are encountered in distances that are not +large compared to the mean free path (shock waves of 1/100 mm thickness, radiometer +phenomena, gas viscosity in a Sprengel vacuum, etc.). All other phenomena take place in +such large spaces that one can construct a volume element for which the visible motion of +the gas can be taken as a differential, yet which still contains a large number of molecules. +This neglect of small terms whose order of magnitude is completely independent of the +order of magnitude of the terms occurring in the final result must be carefully distinguished +from the omission of terms that are of the same order of magnitude as those from which the +final result is derived (cf. beginning of § 14). While the latter omission causes an error in the +result, the former is simply a necessary consequence of the atomistic conception, which +characterizes the meaning of the result obtained, and is the more permissible, the smaller the +dimension of the molecule compared to that of the visible bodies. In fact from the +standpoint of atomistics, the differential equations of the doctrines of elasticity and +hydrodynamics are not exactly valid, but rather they are themselves approximation +formulae which become more nearly exact as the space in which the visible motions occur +becomes large compared to the dimensions of molecules. Likewise, the distribution law for +molecular velocities is not precisely correct as long as the number of molecules is not +mathematically infinite. The disadvantage of giving up the supposedly exact validity of the +hydrodynamic differential equations is however compensated by the advantage of greater +perspicuousness. + +§7. The Boyle-Charles-Avogadro law. Expression for the heat supplied. + +We now proceed to the solution of Equations (27). These are only a special case of +Equations (147) which we shall treat in §18. From these equations it follows, as we shall +prove explicitly there, that the functions/and F must be independent of the direction of the +velocity, and can only depend on its magnitude. We could already give this proof in the +same way here for a special case. In order not to repeat ourselves, we shall assume without +proof that neither the form of the container nor any special circumstance influences the +distribution of states. Since then all directions in space are equivalent, the functions/and F +must be independent of direction, and can be functions only of the magnitudes of the + + +34 _ + + +ru*i.C- 7fca^4e«tai2c.a/ PA.ylic.1 + + + +relevant velocities c and q. If we set/ = e = 3 RT. + +If the normal gas is at another density, then the temperature T is the same when C 2 has +the same value. Hence R is also independent of density and the formula P = p'C 2 /' 3 + +goes over to P = Rp'T. The constant R can be chosen so that the difference between the +temperature of the gas when in contact with melting ice and when in contact with boiling +water is equal to 100. The absolute value of the temperature of melting ice is thereby +determined. Then this must be in the same ratio to the temperature difference (100) between +boiling water and melting ice as the pressure of hydrogen at the latter temperature is to the +pressure difference between the two temperatures (all pressures being taken at the same +density). This proportion gives the temperature of melting ice as 273. + + +^Pi mJuc. 7fca^4e«taiZc.a/ + + + +For another gas, for which lower-case letters are used, one obtains in the same way +p = pc 2 /3’ and since at equal temperatures yjigL = ■ it follows from Equation + +(51) that + +_ MC 2 M 3 R + +(51a) c ! = — = Z — RT = — T = ZrT, + +mm it + +where /,/ = m/M is the so-called molecular weight, i.e. the ratio of the mass or weight of a +molecule (a freely moving particle) of the gas in question to the mass of a molecule of the +normal gas. If we put this value of ^.2 into the equation p=pc 2 /Z , then we obtain for + +any other gas: + + +R + +( 52 ) p = — P T = r pT , + +M + +where r is the gas constant of the gas considered, but R is a constant which is the same for +all gases. Equation (52) is the well-known expression for the combined Boyle-Charles- +Avogadro law. + +§8. Specific heat. Physical meaning of the quantity H. + +We now imagine a simple gas in an arbitrary volume Q. We introduce the amount of +heat dQ (measured in mechanical units), which raises the temperature by clT and increases +the volume by dQ.. We set dQ = dQ \ + r/(f 4 , where dQ \ represents the heat used in +increasing the molecular energy, while clQ^ represents that used in doing external work. If +the gas molecules are perfectly smooth spheres, then in collisions no forces act to make +them rotate. We assume that such forces in general do not exist. Then, when the molecule +happens to have some rotational motion already, it will not be changed by the addition of +the amount of heat qQ. Hence the total amount of heat dQ | will be consumed in increasing +the kinetic energy with which the molecules move among each other, which we call the +kinetic energy of progressive motion. We have considered only this case up to now; +however, in order not to have to repeat the same calculations later, we shall now perform +the following calculations for the more general case in which the molecules have another +form, or consist of several particles moving with respect to each other (atoms). Then in +addition to progressive motion, intramolecular motion will also be present, and work can be +performed against the forces holding the atoms together (intramolecular work). In this case +we set dQi = dQ 2 + dQ 2 and denote by dQ 2 the heat consumed in increasing the kinetic +energy of progressive motion, and by dQ 2 the heat consumed in increasing the kinetic +energy of intramolecular motion and in performing intramolecular work. By kinetic energy +of progressive motion of a molecule, we mean the kinetic energy of the total mass of the + + +^PiUuc. 7fca^4e«taiZc.a/ + + + +molecule, considered to be concentrated at its center of mass. + +We have shown that when the volume of a gas is increased at constant temperature, the +kinetic energy of progressive motion and also the law of distribution of the different +progressive velocities among the molecules both remain unchanged. The molecules just +move farther apart—i.e., there is a greater interval between two collisions. Although we +have not investigated the internal motion, we could take it as probable that in a simple +expansion at constant temperature, on the average neither the internal motion during the +collision, nor that during the motion from one collision to the next, would be changed by a +mere decrease in the collision frequency. The duration of a collision would still be +vanishingly small compared to the time between two successive collisions. Like the kinetic +energy of progressive motion, the intramolecular motion and the intramolecular potential +energy can depend only on temperature. The increase of each of these energies is therefore +equal to the temperature increment clT multiplied by a function of temperature, and if we set +dQ-x, = then f J > can depend only on temperature. We can return at any time to the + +previous case, absolutely smooth spherical molecules, simply by setting J3= 0. The number +of molecules in the volume Q of our gas is nQ, and since the mean kinetic energy of +progressive motion of a molecule is racV2’ the total kinetic energy of progressive +motion of all molecules is + + + +or, when one denotes the total mass of the gas by k, equal to + + +k + + +2 + + + +since clearly k = pQ. = nniQ. + +Since, further, the total mass k of the gas is not changed by the addition of heat, the +increase of kinetic energy of progressive motion is + +k _ + +— dc 2 , + +2 + + +If we measure the heat in mechanical units, then this is also equal to dQ 2 - But according +to Equation (51a) + + + +mJuc. 7fca^4e«taiZc.a/ + + + +and hence + + +m + +dQi = — dT, + +2/i + +3(1 + P)kR + +dq\ = dqt + dq^ =- dT. + +2/u + + +The external work done by the gas is p ■ dQ ; this is therefore also equal to the quantity +r/<2 4 measured in mechanical units. Now the total mass of the gas remains constant, hence + + +dti = kd + + + +and according to Equation (52), + + +hence + + +1 _ R T +t> up + + + + +If one substitutes all these values he obtains, for the total heat added, the value: + + +(53) + + +dQ — dQi + dQi = + + +Rk + +H + +Rk +H L + + +3(1 + (?) [T + +- - -dT + pi[- + +2 \p + +■3(1 + 0) /I + + +If the volume is constant, then dQ/k = d( 1 Ip) = 0, and the added heat will be + + +7fca^4e«taiZc.a/ + + + +m + +dQ, = — (1 + fldT. + +2/i + +On the other hand, if the pressure is constant, then d(T/p) = (dT)/p and the added heat +will be + + +Rk , + +dQ p = — [3(1 + j?) + 2 }dT + +2/i + + +If one divides dQ by the total mass k , then he obtains the heat added per unit mass. If +one divides by dT , then he obtains the amount of heat that must be added to raise the +temperature by one degree, the so-called specific heat. Therefore the specific heat per unit +mass of the gas at constant volume is: + + + + + +On the other hand, the specific heat per unit mass at constant pressure is: + + + +[ 3(1 + 0 ) + 2 ]. + +2/i + + +In both expressions all quantities except /3 are constant. The latter can be a function of +temperature. Since R refers only to the noimal gas and hence has the same value for all +gases, the product j p ■ p and also the product j v ■ p will have the same values for all gases +for which [3 has the same value—e.g., in particular for all gases for which [} is equal to zero. +The difference of specific heats, j p - j v , is for all gases equal to the gas constant itself: + +R + +(55a) ip - % = r = — • + + +The product of this difference and the molecular weight p is for all gases a constant equal to +R. The ratio of specific heats is + + + +7fca^4e«taiZc.a/ + + + +Conversely, + + +2 + + + +In the case that the molecules are perfect spheres, which we have assumed previously, /3 += 0 and hence ^ = 1 ^ • This value was in fact observed by Kundt and Warburg for + +mercury gas,* and more recently by Ramsay for argon and helium;f for all other gases +investigated up to now, k is smaller, so that there must be intramolecular motion. We shall +come back to this subject in Part II. + +The general expression (53) for clQ is not a complete differential of the variables T and +p; however if one divides it by T, then since (3 is a function only of T, he obtains a complete +differential. If f3 is constant, then one has + + + +dQRk + +— = —l[7»B/«a-M) p -i] + const . + +T m + + +This is therefore the so-called entropy of the gas. + +If several gases are present in separate containers, then naturally the total added heat is +equal to the sum of the amounts of heat added to the individual gases, so that whether or +not they have the same temperature, their total entropy is equal to the sum of the entropies +of each gas. If several gases whose masses are k\, k 2 , ■ • ■ , whose partial pressures arep 1? +P 2 , ■ ■ ■ , and whose partial densities are p \, p 2 , • • • , are mixed in a container of volume f2, +then the total molecular - energy is always equal to the sum of the molecular energies of the +components. The total work is (pi+P 2 +- • -)dCl, where + +R R + +ft = Jci/pi = /C2/P2 • ' ‘ Pi = — P\T, P2 = — P2T • • • + +Pi M 2 + + +Hence it follows that the differential of the heat added to the mixture has the value: + + + +flE- + + +3(1 + 0) +2 + + +T + pTd + + + +From this it follows that the total entropy of several gases, when /ihas the same constant +value for each one, is + + +We 7fca^4e«toi2ea/ lPl±y. i .Ze l + + + ++ const. + + +„ k r + +(58) RY l -l[T wnnB) p-' + +M + +where some are in different containers, while others may be mixed; except that in the latter +case p is the partial density, and naturally all the mixed gases must have the same +temperature. Experience teaches that the constant is not changed on mixing, as long as p +and p do not change. + +Since by now we have learned the physical meaning of all the other quantities, we shall +deal with the physical meaning of the quantity denoted by H in §5; for the present we shall +have to li mit ourselves to the case considered in §5, where the molecules are perfect +spheres, and hence the ratio of specific heats is K = 1 ^ • + +We obtain, according to Equation (28), for unit volume of a single gas, H = \flfdo)\ for +the stationary state we have + + +f = ae~ + + +hmc + + +hence + + +H = lajfdu - hmj c 2 fdw. + +Now however j fdoj is equal to the total number n of molecules, and furthermore + + + +3n + +2 hm + + +hence + + +H = n(la - f). + +Furthermore, according to Equations (44) and (51a), + +3 __ 3 RM + +-= c 2 =- T, + +2 hm m + + +hence + + +1 + +h = -—, +2RMT + + +7fca^4e«toiZc.a/ + + + +and according to Equation (40) + + + +Hence, aside from a constant, + +H = nl{pT~ 31 *). + +We saw that -H represents, apart from a constant, the logarithm of the probability of the +state of the gas considered. + +The probability of simultaneous occurrence of several events is the product of the +probabilities of the events; the logarithm of the former probability is therefore the sum of +the logarithms of the probabilities of the individual events. Hence the logarithm of the +probability of a state of a gas of doubled volume is -2 H; of tripled volume, -3 H\ and of +volume Q, -QH. The logarithm of the probability SB of the arrangement of the molecules +and distribution of states among them in several gases is + +Qnl(pT~ l «), + +where the sum is to be extended over all gases present. This additive property of the +logarithm of the probability is already expressed by Equation (28) for a gas mixture. + +If we multiply by RM, which is constant for all gases (M is the mass of a hydrogen +molecule), we obtain + +k + +RMM = - £KAfM(pT-»») = fl£- Kp-'T'i'). + +In nature, the tendency of transformations is always to go from less probable to more +probable states. Thus if SB is smaller for one state than for a second, then to facilitate the +transformation from the first state to the second, the action of another body may be +necessary, but this transformation will still be possible without permanent changes in any +other bodies. On the other hand, if SB is smaller for the second state, the transformation +can occur only if another body takes on a more probable state. Since the quantity +RMm which differs only by a constant factor and an addend from -H, increases and +decreases with we can assert the same things about it as about SB- But the quantity +RMm is in our case, where the ratio of specific heats is equal to 11, in fact the total +entropy of all the gases. + +One sees this at once if he sets /3= 0 in the empirically correct expression (58). The fact +that in nature the entropy tends to a maximum shows that for all interactions (diffusion, heat + + +7fca^4e«toiZc.a/ + + + +conduction, etc.) of actual gases the individual molecules behave according to the laws of +probability in their interactions, or at least that the actual gas behaves l ik e the molecular- +disordered gas which we have in mind. + +The second law is thus found to be a probability law. We have of course proved this +only in a special case, in order not to make it too difficult to understand because of too +much generality. Moreover, the proof that for a gas of arbitrary volume Q the quantity +QH —and for several gases the quantity Y. ElH —can only decrease through collisions, and +thus is to be considered as a measure of the probability of states, was only hinted at. This +proof can easily be given explicitly, and will be given at the end of §19. We still need to +generalize and deepen our conclusions. + +Even if one concedes validity to the gas theory only as a mechanical model, I still +believe that this conception of the entropy principle, to which it has led, strikes at the heart +of the subject in the correct way. In one respect we have even generalized the entropy +principle here, in that we have been able to define the entropy in a gas that is not in a +stationary state. + +§9. Number of collisions. + +We shall now consider again the same mixture of two gases as in §3, and adopt all the +notation used there. We proceed from the number of collisions, given by Equation (18), +which occur between an m -molecule (a molecule of the first kind of gas of mass m ) and an +m 1 -molecule (a molecule of the second kind of gas of mass m \) in unit volume during time +dt, that satisfy the three conditions (10), (13), and (15). + +We consider now only the state of thermal equilibrium, for which we found in §7 the +equations (41) and (42). + +We ask first, how many collisions in all, without any restriction, take place between an +m-molecule and an in |-molecule in unit volume during time dt. We obtain this by dropping +the three restrictive conditions to which the collisions have previously been subjected—i.e., +when we integrate over the differentials. In order to find the limits of integration, we +represent the velocities c and cq of the two molecules before collision by the lines OC and +OC\ in Figure 5. The line OG should be parallel to the relative velocity C | C of the m- +molecule with respect to the in | -molecule before the collision, and the sphere with centre O +and radius 1 (the sphere E ) intersects the point G. The line OK should have the same +direction as the line of centres drawn from m to in |, and intersects the sphere E in the point +K. Hence KOG is the angle denoted by d. We allow the position of the line OK to vary in +such a way that & increases by dd as the angle e of the two planes KOG and COCy +increases by d e. The circle shown in Figure 5 should be the intersection of the sphere E +with the latter plane, which we can choose as the plane of the diagram, when we imagine +the coordinate axes (of which we are now completely independent) to be lying in some +oblique way. When d and e take all values between & and 3+d&. and e and e+de, then +the point K describes on the sphere E a surface element of surface area sin & ■ dd ■ de. As +indicated in §4, we can choose this surface element as the surface element dA, so that we + + +7fca^4e«taiZc.a/ + + + +obtain, according to Equation (18), + + + +Fig. 5. + +dv = fduF'iduigc 2 cos d sin ddddedt. + +We now leave fixed the two volume elements dco and da>\ within which the points C +and C\ lie, but integrate dv with respect to & and e over all possible values, i.e., we +integrate d from 0 to rd 2 and e from 0 to 2rc (see the conditions mentioned in §3). The +result of the integration will be denoted by dv \, and thus we obtain 1 + + + +This is therefore the total number of collisions that occur between an m-molecule and an +m | -molecule in unit volume during time dt, such that before the co ll ision: + +1. the velocity point of the m-molecule lies in the volume element da>, + +2. the velocity point of the in \-molecule lies in the volume element da>\. + +On the other hand, condition (15) has been dropped, so that the direction of the line of +centers is not subject to any restrictive condition. We shall now denote the angle COC \ in +Figure 5 by cp , the point C being held fixed while C| varies so that the line OC\ takes all +values between cq and c \ +dc \, and the angle cp takes all values between cp and cp+dcp. We +thereby obtain the surface element, denoted by dcf in Figure 5, of area c \ dc | dcp at a +distance C\A = c\ sin cp from the line OC. If we allow this surface element to rotate around +the line OC as axis, it will sweep out a ring of volume 27TC 2 { sin cpdc | dcp. The two + +integrations over cp and cq can always be carried out in the same way each time, so that the +velocity point Cj of the in | -molecule always lies inside the ring/?. We find the total + + +We 7fca^4e«toi2ea/ lPl±y. i Ze i. + + + +number dv of collisions in unit volume in time dt between an m-molecule and an m | - +molecule such that the velocity point of m lies in dco while that of m \ lies in the ring R by +integrating the expression dv j with respect to clop over all volume elements of the ring R. +In other words, we simply set + + +rfcoi = 2 ttCi sin ipdcid

c. Hence one has + + + +x + +g sin c. + +3ci + +r\ + +One therefore must split the integration in Equation (62) into two parts:" + + +Wtc, 7fca^4e«tai2c.a/ “Pity. i .Zc. 1 + + + + +dvi = |tt V/dcrtfl £ + +/. + + +2 cl + 3c 2 +F ,c,-At + +o C + +" 2 o 2 + 3c! ■ + +4 I fiCi-cfci + +Cl + + +The above quantity + +SO + +_1 + + +If we replace all the quantities referring to the second kind of molecule by those +referring to the first—i.e., replace ti\, in |, and <7 by n, m, and s —then the above quantity v c +becomes + + +/ v + +, 2 mhc 2 + 1 r cVhm . ' + +(71) n c = ns 2 i /— + +e~ hmc H-=- I e~ x dx + +} km + +CD + +> + +so + +_1 + + +The number H c is the number of times that the m-molecule moving with constant +velocity c through the mixture collides on the average with another m-molecule in unit +time. + +The quantity cln c given in Equation (43) specifies how many of the n m-molecules have +on the average a velocity between c and c+dc, dn c ln is therefore the probability that the +velocity of an m-molecule lies between these limits, and when one follows an m-molecule + + +We 7fca^4e«toi2ea/ “Pity. i .Ze l + + + +through a sufficiently long time interval T, then the fraction of the time T during which its +velocity lies between c and c+clc will be equal to Tdnjn. During this time Tdn c /n, the m- +molecule collides vj'dnjn times with a in | -molecule, and l\ c Tdtl c / U times with +another m-molecule. Hence each m-molecule colhdes a total of ( T/n)\v c dn c times with an +m 1 -molecule and (T/n)fn c dn c times with another m-molecule. Therefore in unit + +time each m-molecule will collide on the averager = (l/n)\v c dn c times with an in \ - +molecule and ]j — ( | ft f\\ c ({fl - times with an m-molecule, so that there will be +(,+n) collisions in all. + +Integration of Equation (69) yields : + + +where + + +V + + +16 _ + +— n^hWnVlJ i -f J 2 ), +3 + + + + +-hmct 2 + +e cac + + + +2 2 + +2 -hmicl C + +C\e - + +Cl + + + + +3 (m + 2 mO j ?r + +8 ml v h\m + mi) 5 + + + + +2 -h mjc] Cl “(" 3C + +C\e - + +c + + + +In the latter integral, c has all values from zero to infinity, but cq can only run through +those values smaller than the given value of c. If one interchanges the order of integrations, +then C| has all values from zero to infinity whereas c assumes all values greater than the +given (q Hence : + + +Wte, 7fca^4e«tai2c.a/ “Pity. i .Zc. 1 + + + +QO + + +Jl + + +r •-**!*! 2 r + += I c Ciaci I + + +2 o 2 + +2 -/iroc 1 Ci + OC + +C C -flCi. + + +q + + +Since one can label the variables at will in a definite integral, we can exchange the +labels c and c j. Hence we obtain for / 2 an expression that differs from the one given for / 1 +only by having the symbols m and ni\ permuted. One thus obtains 7 2 by permuting m and +mi in J|: + + +J o = + + +3 (mi + 3m) + + +7r + + +8 m 3 ^ /i 7 (m + mi ) 5 + + +hence + + +(72) + + +V = + + +j lm + mi) /m + mi + +4/ —:-= irahii 4/- c + + +hmrrii + + +m i + + += TS^iVfc’) 2 + (ci) 2 = 2 y— a^iVc 2 + c 2 . + +• 3 + + +If one writes n, m, and .s' in place of n \, m\, and a, then it follows that: + + +(73) + + +n = 2 ns 2 + + + +mh + + += ms 2 c\/2. + + +Since in unit volume there are n m-molecules, each of which collides v times in unit time +with an mi-molecule, there are in all + + +(74) + + +vn — 2(r 2 nn\y/j + + +'m + mi + + +hmrrii + + +collisions between an m-molecule and an in |-molecule.* But since in a colhsion of two m- +molecules, there are always two molecules of the same kind, the number of co ll isions +between m-molecules is : + + +7fca^4e«taiZc.a/ + + + + + +A similar formula holds for the collisions of m 1 -molecules with each other. + + +10. Mean free path. + + +Let there be n m-molecules in unit volume; let the first have velocity C |, the second c 2 , +etc. Then ^ = |^ . . • )/h is the mean velocity. We shall call it the + +number-average. Since the state is stationary, q does not change with time. If we multiply +the last equation by clt and integrate over a very long time T, then: + + + + +T + +hdt • • • + + +Since during a very long time all molecules behave in the same way, all the summands are +equal and it follows that ^ = where + +- 1 C T + +’• ’ TJ, + + +is the time-average of the velocity of any single molecule. + + + +T + +cdt = Tct + + +is the sum of all distances that it traverses during time T. But since it collides T(*- t-n) +times with another molecule during this time, the mean distance traversed between two +successive collisions (the arithmetic mean of all distances between two successive +collisions) is: + + +(76) X = + + +1 + + +v + n / /m + mi + +7r ( + +37t$ 2 r = knc\m — = kpc\ — • +dz dz dz dz + +If we denote the total mass of gas between the bottom and the layer z by M, and the + +velocity of its center of mass in the abscissa direction by J, then + +- ) + +where Yjn^ is the sum of the momenta of all particles in the abscissa direction. As a result +of the molecular motion of the gas, more momentum T will be transported downwards than +upwards in unit time through unit surface. Hence during the time clt the quantity Yjn^ +experiences the increment + + + +We 7fca^4e«toi2ea/ lPl±y. i .Ze l + + + +Yadt + +while M remains unchanged. Here ie 7fca^4e«tai2c.a/ + + + +Hence we may well assume that the volume of a liquid is not more than ten times as large, +and generally not smaller, than it would be if two neighboring molecules were at that +distance which is in the gas their minimum distance in a collision, so that e is between 1 +and 10. The density of liquid nitrogen was found by Wroblewsky to be not much different +from that of water.* Also, from the atomic volume it follows that the difference between +the two densities cannot be so large that it need be taken into account in this approximate +calculation. If we set the two equal, then we find for nitrogen at 15° and atmospheric +pressure: (v„/vy) = 813; and we obtain, when we set e = 1, s = 0.0000001 cm = one +mil li onth of a mi ll imeter. Hence we may take it as probable that the mean distance of the +centers of gravity of two neighboring molecules in liquid nitrogen, as well as the smallest +distance to which two colliding molecules in gaseous nitrogen can approach on the +average, lies between this value and the tenth pail of it. + +For the number n = (\/ \Z2tt s 2 X) of molecules in 1 cc of nitrogen at 25 °C at + +atmospheric pressure, one obtains a number which in any case falls between 91 and 250 +tril li ons. + +Substitution of this value into Equation (90) gives: T = (23 • 10 9 /sec). This would be the +absolute conductivity in electrostatic units. The electromagnetic specific resistance would +therefore be: + + +(9* 10 20 cm 2 /F sec 2 ) = (4-10 10 cm 2 /sec). + +A cube of nitrogen, 1 cm on each side, has resistance (4 • 10 10 cm/sec) = (40 ohm) while +an equal cube of mercury has resistance (1/10600) ohm. Since nitrogen actually is a much +poorer conductor than mercury, it follows that the hypothesis that the molecule is a +conducting sphere is incorrect. + +The order of magnitude of the diameter of a molecule was later calculated by Lothar +Meyer, 6 Stoney, 7 Lord Kelvin, 8 Maxwell, 9 and van der Waals, 10 and they found, by +several completely different methods, values in agreement with the one above. + +In order to find the dependence of the viscosity coefficient on the nature and state of the +gas in question, we replace p by nm, and A by its value according to Equation (77). One +obtains: + + +kmc + +y/2 7rS 2 + +and by virtue of Equations (46) and (51a): + +2k fRMTm + + +TftnthdfuntlcLnl + + + +Thus the viscosity coefficient is independent of the density of the gas, and proportional +to the square root of its temperature. The independence of density, which is of course valid +only as long as the condition of our calculation is satisfied—namely, that the mean free path +is small compared to the distance between top and bottom—was verified by experiment, +especially by Kundt and Warburg.* As far as the dependence on temperature is concerned, +Maxwell’s experiment gave a viscosity coefficient proportional to the first power of the +temperature ( loc. cit.), which is correct only for easily coercible gases, especially carbon +dioxide. For less coercible gases, several later observers found close agreement with the +formula developed here for the temperature coefficient of the viscosity though most values +lie between the value of this calculation and the one found experimentally by Maxwell. 11 + +The first remark to be made is that a more rapid increase of the viscosity with +temperature than the square root of the absolute temperature cannot be attributed to the +inaccuracy of our calculation, since one perceives immediately the following: when one +raises the temperature without changing the density, then on the assumption of elastic, +negligibly deformable molecules, the molecular motion is on the average completely +unchanged except that its velocity increases in proportion to the square root of the absolute +temperature. It is as though time were shortened in this ratio, and hence it follows that the +amount of momentum transported in unit time must increase by the same amount. On the +other hand, according to Stefan, 12 s could decrease with increasing temperature. This +would have the following meaning. The molecules are not absolutely rigid, rather they are +somewhat flattened by the collision, so that their diameter appears to decrease, and indeed it +decreases more the higher the temperature of the gas. Maxwell* assumes that the molecules +are centers of force, which exert on each other a force that is negligible at large distances, +but as they approach becomes a very rapidly increasing repulsive force, which is to be +chosen as an appropriate function of the distance. In order to explain the temperature +coefficient of the viscosity which he found, he sets this function equal to the inverse fifth +power of the distance. I once remarked that one could obtain all the essential properties of a +gas by substituting for this repulsive force a purely attractive force which is an appropriate +function of the distance, and one could thereby explain dissociation phenomena and the +well-known Joule-Thomson experiment.! Because of our ignorance as to the nature of +molecules, all these models must of course be considered simply as mechanical analogies, +which one must regard as all being on an equal footing, as long as experiment has not +decided between them. In any case, however, it is probable that the diameter of a molecule +is not a precisely defined quantity. Nevertheless, the neighboring molecules in the liquid +state must be at such a distance that they interact strongly with each other, and the +interaction of more than two of them is no longer an exceptional case. Hence they are at +distances about the same as that at which gas molecules already experience a significant +deviation from rectilinear paths. The quantities denoted above by s and cr represent nothing +more than the order of magnitude of this distance. In order that the calculation may remain +meaningful, we shall return to the assumption that the molecules are almost undeformable +elastic spheres. Then it follows from the last formula for the viscosity coefficient that it is +proportional to the square root of the mass of a molecule, for different gases at the same +temperature, and inversely proportional to the square root of the molecular diameter. + + +7fca^4e«toiZc.a/ + + + +§13. Heat conduction and diffusion of the gas. + +In order to calculate the heat conductivity from Equation (88), we have to assume that +the top and bottom planes are maintained at two different temperatures. G is then the +average heat contained by a molecule. The mean kinetic energy of progressive motion of a +molecule is + + +m + +2 + + + +The total average energy of internal motion of a molecule will be set equal to + + +m + + + + +hence the total average molecular motion is + +1 + 0 — + +- me 2 . + +2 + +or, by virtue of Equation (57), + +1 _ + +- me 2 . + +3 (« - 1 ) + +Since according to our hypothesis heat is nothing but the total energy of molecular +motion, the amount of heat G pertaining to a molecule will be measured in mechanical +units. If we assume that the ratio of specific heats, n, is constant, which is probably true at +least for the permanent gases, then + +dG 1 dc 2 + +— =- m - + +dz 3 (k - 1 ) dz + +Now according to Equation (51a), + + +__ m + +c 2 =- + + +Wtc, 7fca^4e«tai2c.a/ lPl±y. i .Zc. 1 + + + +where, as before, /./ = ( m/M ) is the molecular weight of the gas. We thereby obtain + + +dG Rm dT +dz (k - l)/n dz + +hence according to Equation (88), + +kRpc\ ST + +r =-- + +(k - l)/i dz + +The coefficient of dT/dz is what one calls the heat conductivity ^ of the gas. It therefore +follows that + + + +8 * + + + + +2k + + +(k — l)n (k — 1)$ 2 + + +mPT + +t 3 m + + +The dependence of the heat conductivity on density and temperature is therefore, as +long as k is constant, the same as that of the viscosity. In particular, since k depends hardly +at all on density for permanent gases at constant temperature, the heat conductivity is +independent of density, which was confirmed by the experiments of Stefan* and of Kundt +and Warburg, t + +For different gases for which k has nearly the same value, the heat conductivity +coefficient is proportional to the quotient of the viscosity coefficient divided by the +molecular weight—or, as the last expression in Equation (92) shows, inversely proportional +to the square of the diameter and to the square root of the molecular weight. Hence it is +significantly larger for the smaller and lighter molecules than for the larger ones. This has +been confirmed by experiment. + +If we denote by j p and j v the specific heats of the gas referred to constant mass, at +constant pressure and constant volume, respectively, where heat is again to be measured in +mechanical units, then we have (Eq. 55a) + +R 7p + += 7 ? “ 7 » = 7 t >(* — 1 ) “ (k — 1 ), + +M k + +hence + +1 + +( 93 ) 8 = 7 .» = - + +K + + +7fca^4e«taiZc.a/ + + + +In the last formula the unit of heat is arbitrary. If one puts, for air at 0°C and atmospheric +pressure, + + +g-Calor. + +k = 1.4, 7 P = 0.2376- 5 - + +(gram mass) X (1° C.) + +and for its value given above, then it follows that: + +g-Calor. + +2 = 0.000032—-- + +cm/sec. 1° C. + +For the heat conductivity of air, different observers have found values lying between +0.000048 and 0.000058 in the above units. 1 Taking account of the fact that our calculation +is only an approximate one, we consider this agreement sufficiently good. + +In order to calculate the diffusion of two gases, we shall again return to the gas cylinder +considered in §11. Let the gas be a mixture of two simple gases. A molecule of the first +kind of gas has mass m, and diameter s; a molecule of the second kind has mass ni\ and +diameter In the layer z there are (in unit volume) n molecules of the first and n 1 of the +second kind of gas, where n and n\ are to be functions of z. Also, the number dn c of +molecules of the first kind for which the magnitude of the velocity lies between c and c+dc +will be a function of z. One then finds by considerations similar to those in § 11 that in unit +time + + +dn c + +dyicj = —c sin# cos +2 + +molecules of the first kind move through unit surface with velocities between c and c+dc, +the angle between their direction of motion and the negative "-axis lying between d and S ++ dd. These molecules come on the average from a layer whose z-coordinate has the value +z+X c cos d, for which therefore we can write in place of dn c : + +ddn e + +dn c -\-\ c md - + +dz + +If one integrates over & from 0 to rd 2, then the number of molecules of the first kind of +gas that pass in unit time through unit surface at any angle but with velocity between c and +c+dc has the value: + + +We 7fca^4e«toi2ea/ “Pity. i .Ze l + + + +cdn e c\ c ddn c + +-_L-. + +4 6 dz + + +l ik ewise the number of molecules that pass from below to above has the value: + + +cdn c c\ c ddn c + +4 6 dz + +Hence there will be a net flow of + + + +c\ c ddn c + +d% = - + +3 dz + + +molecules of the first kind of gas from above to below. On the simplifying assumption that +the velocities of all molecules are the same, one would have in place of < m e simply the +total number of molecules of the first kind that pass in unit time through unit surface +from above to below, in excess of those that pass from below to above, and instead of dn c +one would have simply the total number n of molecules of the first kind in unit volume in +the layer z. One would then have: + + +cX dn + +9J =- + +3 + +The occurrence of different velocities for molecules of the same kind will be taken +account of only in the simplest case, where both kinds of gas have molecules of the same +mass and diameter. In this case, which Maxwell calls self-diffusion, we assume that during +the diffusion Maxwell’s velocity distribution holds for the molecules of each kind of gas in +each layer, so that Equation (43), + + + + +/hV + +dn c = 4 n M — c 2 e~ hnci de + +' 7T + + +remains unchanged, except that n is a function of z, whence one obtains: + + +ddn c 4 dn /hV + +-= — \ -c 2 e~ hmc dc. + +dz dz * 7T + + +k^ie. 7fca^4e«tai2c.a/ lPl±y. i .Zc. 1 + + + +Moreover, /. c has the same value as it would in a simple gas in which there are n+n | +molecules in unit volume. A c is then given by Equation (78) in which v c = 0 but Jt c is +given by Equation (71). In the latter equation, n+ni is to be substituted for n, and s means +the diameter of a molecule, which is the same for both kinds of gas. Substitution of all these +values in Equation (94) and integration over c from 0 to oo yields for the total number of +molecules of the first kind that pass through unit surface from above to below, in excess of + +dn r* 4x 3 , + +— I - e~ x dx, + +dz J o i{x) + +a formula which one could have obtained directly from Equation (87) by replacing T and G +b y SR and n/(n+n \). Thus the probability that a molecule belongs to the first kind of gas +may be treated exactly lik e the quantity Q, introduced in §11, pertaining to a molecule, and +T then means the number of molecules of the first kind that pass through unit volume in +unit time from above to below, in excess of those that pass in the opposite direction. Self¬ +diffusion thus takes place, according to our approximate formulas, just as we imagined in +§12 that electrical conduction might; one merely replaces the electrical charge of the +molecule by the property of belonging to one or the other kind of gas. But there is an +essential difference when one assumes that the electrical charges of two colliding molecules +are equalized in a collision. However, since our formulas are constructed in such a way that +after the collision any direction in space is equally probable for each molecule, it must +follow from this that electrical conduction is just as rapid when the molecules behave lik e +perfect insulators in collisions with each other, and like perfect conductors in collisions with +the top and the bottom. Then electrical conduction would be completely analogous to +diffusion. + +If one introduces in Equation (96) the quantity k defined in Equation (89), then he +obtains: + +dn SR dn + +SR = k\c — -- + +dz p dz + +If one multiplies on both sides by the constant m, then it follows that: + +d(nm) SR d(nm) + +SRm = k\c — = -- * + +dz p dz + +S Jim is the net mass of the first gas that passes through the surface from above to +below, while nm is the mass of the first gas in unit volume in the layer z, hence c)(nm)/dz is +its gradient in the z direction. The factor multiplying this expression in the last equation is + + +those that pass from below to above, the value: + +(96) SR =- = - + +3irs : V km [n + nj + + +7fca^4e«toiZc.a/ + + + +therefore what one calls the diffusion coefficient. This gives for air at 15°C and +atmospheric pressure, based on the above value for 0^, the value 0.155 cm 2 /sec; while +Loschmidt has found values between 0.142 and 0.180 for different combinations of gas +that behave l ik e air. If one considers the dependence of the quantity p on temperature and +pressure, then he finds that the diffusion coefficient is directly proportional to the ^ power + +of the absolute temperature, and inversely proportional to the total pressure of the two +gases. At equal temperature and equal total pressure, the diffusion constant for self¬ +diffusion is, like the heat conductivity, inversely proportional to the quantity as + +one finds from Equation (96), since h and n+n \ are constant. + +In this simplest case of diffusion, where the mass and diameter of a molecule are the +same for both gases, the aggregate of the two gases behaves lik e a single stationary gas. If +we denote by dN c & dn c & and dtl 1 „ the total number of molecules of both gases, the + +number of molecules of the first kind of gas, and of the second kind, respectively, for +which the velocity lies between c and c+dc and its direction forms an angle between & and +d+dd with the positive z axis; then according to Equation (38): + + + +hhn 3 + + +(n + ni)ck~ hmc2 dcmdd$. + + +One might think that consequently at least in this simple case our calculations would be +exactly correct. However, we shall see that when the molecules are elastic spheres, the +faster molecules diffuse more rapidly and the slower ones less rapidly. 3 Where n is small, +that is at a place where the molecules of the other kind of gas predominate over the +diffusing molecules, then for large values of c the quantity dn c $ will be larger than + +n + +■ dN cj, + +n + n i + + +whereas for smaller values of c it will be smaller than this. At the same place the reverse +must be true for the other gas. Hence the exactness of the equation we obtained, + +n + +dn e ,i =- dN'j + +n + n i + + +is in doubt. Likewise it is doubtful whether among the molecules that collide in a layer +(according to Clausius, the ones sent out from a layer) all directions of the velocity are +equally probable. + + +14. Two kinds of approximations; diffusion of two different gases. + + +We 7fca^4e«toi2ea/ lPl±y. i .Ze l + + + +One might think from what has been said up to now that Equation (87), and Equation +(88), which was derived from it with the coefficient (89), are strictly correct; this would be +an error. In fact, in deriving them we made the assumption that the velocity distribution +would not be altered by the quantity Q associated with the molecules. In many cases, for +example viscosity, when the visible motion is small compared to the mean velocity of a +molecule, the velocity distribution will be altered only slightly; yet the value of the quantity +cln c in Equation (83) will always be different from its value rfyi' in Equation (84). Hence +there is added to the expression (85) a term of the form + +C + +— G(z){dn c - dnl) + +4 + +which is of the same order of magnitude as the expression (85) itself. Also, the assumption +that all directions of motion of a molecule are equally probable is doubtful. + +We assumed finally that each molecule transports through the surface AB that amount +G(z.+a' cos &) of the quantity Q which a molecule would possess on the average in the +layer in which it experienced its last collision. This assumption is also arbitrary. This +amount can differ for molecules that leave the layer in different directions and with different +velocities; it can therefore be a function O of c and d, so that dG/dz cannot be taken out in +front of the integral sign in the following integrations over & and c. The amount of Q +transported by a molecule through AB would then depend not only on the layer where it +last collided, but also on the place where it collided the next to last time, and perhaps also +on the place of the collision before that. + +This is related to a situation already discussed in comparing diffusion and electrical +conduction. It can happen that in a collision each of the colliding molecules retains the +amount of Q that it had before the collision; however, an equalization can also take place. If +we call Q electricity, then the former corresponds to the case when the molecules are +conducting but are covered with an insulating layer that is penetrated in collisions with the +top and bottom but not in collisions between two molecules; the latter corresponds to the +case when the molecules are composed of a conducting substance right up to their surface. + +In these two cases the function ® can be different, so that the transport of Q would be +unequal even though the mean value of G in a layer z would be the same in both cases, +namely + + +(Ox- Go )(2 - 2 .) + +Go i-• + +Zl “ ZO + +In fact it is more probable that a molecule will continue in nearly the same rather than the +opposite direction after a collision. One sees this in the formulas (201) and (203) derived +later on. Hence the transport of Q is more obstructed and hence slower when it is equalized +between two colliding molecules than when it is not. + + +7fca^4e«taiZc.a/ + + + +Numerous researches have been carried out to take account of the terms neglected +because of all these assumptions, especially by Clausius, O. E. Meyer, and Tait.* +Nevertheless, in the case of elastic spheres the perturbation of the velocity distribution by +viscosity, diffusion, and heat conduction has not yet been calculated exactly, so that in all +the relevant formulas terms have been neglected which are of the same order of magnitude +as those which determine the result, so that they are not essentially better than the ones +obtained here in a simpler way. + +Such omissions, which make the results mathematically incorrect in the sense that they +are not logical consequences of the assumptions made, are to be distinguished (as we +explained at the end of §6) from assumptions that are physically only approximately +correct, for example the assumption that the duration of a collision is small compared to the +time between two collisions. As a consequence of the latter assumptions the results will +also be physically inexact, i.e., their validity can only be determined by experiment. +However, these physical approximations do not prevent the results from being +mathematically correct, since they provide the limiting case to which the law must approach +ever more closely as the physical assumptions are realized more exactly. + +We shall now calculate the diffusion of two gases when the mass and diameter of a +molecule are different in the two gases, but only on the simplifying assumption that the +velocities of all molecules of the first kind of gas are equal to c, while those of all molecules +of the second kind are equal to c\. + +Equation (95) then holds for the first kind of gas. The mean free path will then of course +be calculated from Equation (68). However, since the whole calculation is only an +approximate one, we shall not take account of the occurrence of different velocities here, +since this simplifies the calculation, and use Equation (76). Thus we obtain for the number +of molecules of the first kind of gas that pass in unit time through unit surface from above +to below, in excess of those going the other way, + + +dn + +M = ® i-> + +dz + +whence + + + +C + + +3tt + + +a /A , S + S A 2 t /m+mi +shi\/2 + -) n x + + +m + + +Similarly one finds for the number 0^ j of molecules of the second kind of gas that pass +downwards through the surface in excess of the number passing upwards the value: + + +We 7fca^4e«toi2ea/ “Pity. i .Ze l + + + +bU\ dn + +911 = “ ®2 - = + ®2 — > + +dz dz + + +since (n+ny) is constant in the gas as a whole. Here we have: + + + +There now arises the difficulty that the diffusion constant does not come out the same +for both gases—i.e., according to the formulas, more molecules pass through each cross +section in one direction than in the other. This actually happens in diffusion through a very +narrow passage or a porous wall. However in our case where we assume that the mixture is +at rest and the effect of the side walls is negligible, the pressure must always be equalized, +and therefore according to Avogadro’s law an equal number of molecules must move in +each direction. + +Our formula gives a false result. Similarly, Maxwell’s first formula for heat conduction* +gave a visible mass motion of the heat-conducting gas. Clausiust and O. E. Meyer:;: have +obtained other formulas for heat conduction, which avoid this visible mass motion, but for +which the pressure is different at different places in the gas. Although this actually occurs in +very dilute gases, as calculation and experiment both agree for the radiometer,** such great +pressure differences as would follow from the formulas are inadmissible. 1 This is therefore +a clear proof of the inaccuracy of all these calculations. + +In the case of diffusion, with which we now concern ourselves, O. E. Meyer* has +removed the contradiction by superposing on the molecular motion calculated here—in +which more molecules pass through unit surface in unit time downwards than + +upwards—an equal but oppositely directed flow of the mixture. Since the mixture consists +o f n+n | molecules, n of the first kind and of the second, the flow of the mixture is +imagined to be such that n(N\ - N)/(n+n |) of the first kind and //1 W| -N)/(n+n{) of the +second kind pass downwards in excess of those going upwards. Hence according to this +superposition, + + + +n(9?i - 91) +n + ftj + + +tti9! + n% + +n + n\ + + +n®i + Wi® 2 bn +n + iii dz + + +molecules of the first kind pass downwards in excess of those going upwards, and equally +many molecules of the second kind pass upwards in excess of those going downwards. +The diffusion coefficient is therefore now + + +We 7fca^4e«toi2ea/ “Pity. i .Ze l + + + +rti$i + n®! + +- ) + +n + ni + +where 5)i and have the values just found. According to this formula the diffusion +coefficient would depend on the mixing ratio, and therefore would not have the same value +in different layers of the gas mixture, so that for the stationary state n and n | would not be +linear functions of z. Stefan 2 has developed another approximate theory of diffusion on +other principles, and he finds that the diffusion coefficient should not depend on the mixing +ratio. Experimentally the question is still open. Yet such a strong variation of the diffusion +coefficient as that given by the above formula seems to be excluded. + +As for the various complicated revisions undergone by these various theories of +viscosity, diffusion, and heat conduction, the comparison with experimental results for +different kinds of gases, and the conclusions which may be drawn about the molecular +properties of various gases, I cannot go into them here. They may be found, rather +exhaustively collected, in O. E. Meyer’s Kinetische Theorie der Case. Of the works +published later, those of Tait 3 may be mentioned.* + + + +One sees at once that the first two terms represent the increase in H by what is called the first cause in the +text, and that for the reasons given in the text it vanishes. The other two terms represent the increase in H +due to the second cause and give, after substitution of the values of dF/dt from Equations (25) and (26): + + +(29) + + +dH + +dt + + +•jVtfFl -JFtidp+jWH +J IF i (f'F i -JF j)dp + + ++ / lFi(F'Fl -FFi)dn, + + +where dp=o 2 g cos ddiodw id A, dr = s 2 g cost9dwdw\dA and f/ 7 * j = $~g cosddaidui\dA. All the +integrations are extended over all possible values of the variables. + +One sees at once that the sum Jf'lf'dv>' + [F'i IFld a collision in which they are |j', f', $1 , 1 ){ , fj + + +before and //, d, d|, //j, d] afterwards. Then by symmetry one gets + + +— = / (/Fi — fF\ ) c. + + +If moreover the mi-molecules are identical with the ///-molecules, there being in all n of them in unit +volume, and furthermore c =c\, and s is the diameter of the molecules, then one obtains for the number +of collisions that one molecule experiences with identical ones of the same velocity moving in all +directions, in unit time, the value + + + + +4 + +-xnA, + + +The mean path (from one collision to the next) will be + + + + +3 + + +3 + +4 + + +X + + +r* + + +This is the value calculated by Clausius for the mean free path; it differs numerically somewhat from +that derived in the text, which was calculated by Maxwell. [R. Clausius, Ann. Physik [2] 105, 239 +(1859); J. C. Maxwell, Phil. Mag. [4] 19, 19 (1860); Clausius defended his original value in a note in +Phil. Mag. [4] 19, 434 (1860). Maxwell did not bother to give a detailed proof of his own result, but this +was done by W. D. Niven in Maxwell’s Scientific Papers , Vol. I, p. 387. See S. G. Brash, Amer. J. Phys. +30, 271 (1962); Bernstein, Isis 54, 206, (1963).—T R .] + +If one has in unit volume n molecules with diameter s and n\ with diameter sj =2o - s, and if all n of +the former move with the same velocity c, while all n\ of the latter move with a different velocity c \, +distributed uniformly in all directions in space, then one of the n molecules experiences v'+v" collisions, +and its mean path is: + + +( 68 ) + + +X' = + + +3c 2 + + +v' + v" 4t n$ 2 c 2 + 7r c. + + +1 Burbury, Nature 51, 78 (1894). See also Boltzmann, Weitere Bermerkungen liber Wurmetheorie, +Wien. Ber. 78 (June 1878), third-to-last and next-to-last pages. + +2 Kirchhoff, Vorlesungen iiber Warmetheorie, 14th lecture, §2, p. 145, line 5. + +* See Part II, §77, for corrections to this section. For further discussion of opposite collisions, see R. +C. To 1 man, The Principles of Statistical Mechanics (New York: Oxford University Press, 1938), +Chapter 5. + +3 Cf. Wien. Ber. 94, 625 (1886); Stankevitsch, Ann. Physik [3] 29, 153 (1886). The fact that the + + +We 7fca^4e«toi2ea/ “Pity. i .Ze l + + + + +angles z? and e also depend on the positions of c and c | does not weaken the force of the arguments in the +text. One can first introduce in place of z?and e two angles which determine the absolute position of OK +in space, then transform /, //,-■■ C\ into v, • • • e' and finally again introduce z? and e. + +* H. A. Lorentz, Wien. Ber. 95,115 (1887). + +4 One can present the proof given in the text in the following more analytical form. We shall +certainly include all required values when we integrate the integrals in Equation (28) over all variables +from - oo to + oo. Velocities that do not occur in the gas will not contribute to the integral, since for these +velocities either/ or F must vanish. Hence the limits are constant and one obtains dffldt by +differentiating with respect to t under the integral sign, which gives: + +This proof is somewhat shorter, but it appears to depend on certain mathematical conditions— +permissibility of differentiation under the sign of integration, etc.—which affect only the ease of +demonstrating the theorem and not its validity, since it is really a question of very large but not infinite +numbers. The theorem was proved without introducing definite integrals in my paper in Wien. Ber. 66, +Oct. 1872, Section II. + +5 See Schlomilch, Comp, derhoh. Analysis, Vol. 1, p. 437,3d ed. + +6 Boltzmann, Wien. Ber. 76 (Oct. 1877). + +* J. Loschmidt, Wien. Ber 73, 128, 366 (1876), 75, 287,76, 209 (1877). See also E. P. Culverwell, +Phil. Mag. [5] 30, 95 (1890); Nature 51, 581 (1895). G. H. Bryan, B. A. Rep. 61, 85 (1891); Nature 52, +29 (1895). G. J. Stoney, Phil. Mag. [5] 23, 544 (1887). J. Larmor, Nature 51,152 (1894). H. W. Watson, +Nature 51,105 (1894). + +7 Cf. Nature 51,413 (Feb. 1895). + +* M. Planck, Mun. Ber. 24,391 (1895). + +8 It would be necessary to prove that the following cases are not possible: 1. Besides Maxwell’s law +there is another molecular-disordered stationary distribution of states in which each velocity does not +have the same probability as the opposite one; and a third distribution that transforms to the second one +on reversing the velocities. 2. Besides the Maxwell (most probable) distribution—which does not in +general transform to a molecular-ordered state on reversing velocities, since a molecular-ordered state is +as probable as a disordered state—there exists a rare molecular-disordered stationary one, which on +reversal transforms into a molecular-ordered one. 3. There are also stationary, molecular-ordered state +distributions. 2 and 3 are also related to the case of the presence of external forces. The impossibility of +case 3 cannot be proved from the minimum theorem, and probably cannot be proved in general without +special conditions. Clearly the concept “molecular-disordered” is only a limiting case, which an +originally molecular-ordered motion theoretically approaches only after an infinitely long time, though +actually very quickly. + +* The subscript w stands for wahrscheinlichste, “most probable.” + +* Kundt and Warburg, Ann. Physik [2] 157,353 (1876). + +t Ramsay, Proc. R. S. London 58,81 (1895); Rayleigh and Ramsay, Phil. Trans. 186,187 (1896). + +9 If one substitutes in this formula the value 1 for fdu>, n for F\da>\, c for g, and 1 for dt, then he gets + +* An interesting application of Eq. (74) was made by Fiichtbauer and Hoffmann [Ann. Phys. [4] 43, +96 (1914)] who tested Lorentz’s “collision broadening” theory of spectral lines for cesium atoms in a +nitrogen discharge. They found that the Lorentz theory could not explain the observed line width except +by assuming a collision rate 32 times greater than that calculated from Boltzmann's kinetic theory +formula. This left the field open for Holtsmark’s theory based on the Stark effect [Holtsmark, Ann. +Physik [4] 58,577(1919)]. + +* Boltzmann here ignores the “ergodic problem” concerning the validity of equating these averages, +not because he is unaware of it but presumably because he does not wish to complicate the discussion of +the elementary mean free path concept. It will be recalled that the basic postulate in this book is not the +earlier so-called ergodic hypothesis—that the system will eventually pass through every set of values of + + +7fca^4e«toiZc.a/ + + + +positions and momenta consistent with its energy—but rather the assumption that the gas is always +molecular-disordered (§3). + +11 On substituting the values for v c and 1^, one easily sees that the quantity A c approaches the limit A,. +(Eq. 60) with increasing c. In fact, when the molecule considered moves with a very large velocity, the +others must behave as if they were at rest. The mean free path is of course unchanged when all the +velocities are increased or decreased by the same amount; A therefore does not vary with temperature at +constant density, provided the molecules can still be considered as negligibly deformable elastic bodies. + +* R. Clausius, Ann. Physik [2] 115, 1 (1862); see in particular p. 432 of the English translation in +Phil. Mag. [4] 23(1862). + +12 Boltzmann, Wien. Ber. 96,905 (Oct. 1887). + +13 Tait, Trans. R. S. Edinburgh 33,74 (1886). + +14 Boltzmann, Wien. Ber. 84,45 (1881). + +15 Tait, Trans. R. S. Edinburgh 33,260 (1887). + +16 Maxwell, Phil. Trans. 156, 249 (1866); Scientific Papers 2,24. + +17 O. E. Meyer, Ann. Physik [2] 148, 226 (1873). + +18 Kundt and Warburg, Ann. Physik [2] 155, 539 (1875). + +19 Loschmidt, Wien. Ber. 52,395 (1865). + +* The subscript/stands for fliissig, “liquid” or “fluid.” + +2(1 Boltzmann, Wien. Ber. 66,218 (July 1872). + +* Wroblewsky, C. R. Paris 102,1010 (1886). + +21 L. Meyer, Ann. Chem. Pharm. 5 (Suppl.) 129 (1867). + +22 Stoney, Phil. Mag. [4] 36,132 (1868). + +23 Kelvin, Nature 1,551 (March 1870); Amer. J. Sci. 50, 38 (1870). + +24 Maxwell, Phil. Mag. [4] 46, 463 (1873); Scientific Papers 2,372. + +25 Van der Waals, Die Continuitat des Gasfdrmigen und Fliissigen Zustandes (Leipzig, 1881), Chap. + +10 . + +* Kundt and Warburg, Ann. Physik [2] 155, 337,525, 156, 177 (1875). + +26 Cf. O. E. Meyer, Die kinetische Theorie der Gase (Breslau: Maruschke & Berendt, 1877), p. 157 ff. +[See also Hirschfelder, Curtiss, and Bird , Molecular Theory of Gases and Liquids (New York: Wiley, +1954), Chap. 8 .—Tr.] + +27 Stefan, Wien. Ber. 65 (2) 339 (1872). + +* Maxwell, Phil. Trans. 157,57 (1867); Scientific Papers 2,36. +t Boltzmann, Wien. Ber. 89,714 (1884). + +* Stefan, Wien. Ber. 65,45 (1872); 72,69 (1876). + +t Kundt and Warburg, Ann. Physik [2] 155, 337,525, 156, 177 (1875). + +28 O. E. Meyer, Die Kinetische Theorie der Gase, p. 194. From Winkelmann’s experiments Kutta +found, according to an improved approximate formula, the value 0.000058 (Miinchn. Dissert. 1894; +Ann. Physik [3] 54, 104 [1895]). + +29 Loschmidt, Wien. Ber. 61, 367 (1870); 62, 468 (1870). + +30 This follows from the manner in which g occurs in J* gbdb COS* 0 ( t4 ' §§18 and 21). + +* R. Clausius, Ann. Physik [2] 115, 1 (1862); Ann. Physik [3] 10, 92 (1880); Die Kinetische Theorie +der Gase (Braunschweig: F. Vieweg, 1891), chap. ii. O. E. Meyer, Die Kinetische Theorie der Gase +(Breslau, 1877). P. G. Tait, Proc. R. S. Edinburgh 33, 65, 251 (1886-1887); Phil. Mag. [5] 25, 172 +(1888); Trans. R. S. Edinburgh 35, 1029 (1890); Proc. R. S. Edinburgh 15, 225 (1889). See also J. H. +Jeans, Phil. Mag. [6] 8, 700 (1904); The Dynamical Theory of Gases (London: Cambridge University +Press, 1904). G. Jiiger, Wien. Ber. 102, 253 (1893), 105, 97 (1896), 108, 447 (1899), 109, 74 (1900), + + +7fca^4e«toiZc.a/ + + + +127, 849 (1918). M. v. Smoluchowski, Rozprawy Wydzialu mat.-przyr. Ak. Um. (Krakowie) A46, 129 +(1906). + +* Maxwell, Phil. Mag. [4] 19, 19, 20, 21,33 (1860). + +t Clausius, Ann. Physik [2] 115, 1 (1862); Die Kinetische Theorie derGase, chap. iv. + +$ Meyer, Die Kinetische Theorie der Case (Breslau, 1877). + +** G. J. Stoney, Phil. Mag. [5] 1,177,305 (1876); 6,401 (1878); G. F. Fitzgerald, Sci. Trans. Dublin +1, 57 (1878); O. Reynolds, Proc. R. S. London 28, 304 (1879); Phil. Trans. 170, 727 (1880); Maxwell, +Phil. Trans. 170, 231 (1880); W. Sutherland, Phil. Mag. [5] 42, 373, 476 (1896), 44, 52 (1897). For +summaries of later work see M. Knudsen, Kinetic Theory of Gases (London: Methuen, 1934); L. B. +Loeb, The Kinetic Theory of Gases, 2d ed., (New York, 1934); E. H. Kennard, Kinetic Theory of Gases +(New York, 1938). + +31 Kirchhoff, Voriesungen iiberdie Theorie der Wanne, ed. by Max Planck (Leipzig: B. G. Teubner, +1894, p. 210). + +* Meyer, Die Kinetische Theorie der Case. + +32 Stefan, Wien. Ber. 65,323 (1872). + +33 P. G. Tait, Trans. R. S. Edinburgh 33,65,251 (1887); 36,257 (1889-1891). + +* As Boltzmann remarked, a satisfactory solution of these difficulties (regarding especially the +dependence of the diffusion coefficient on concentration) could not be found until the perturbation of +the velocity distribution was calculated. This was first done for general types of force laws by S. +Chapman, Phil. Trans. A216, 279 (1916), A217, 115 (1917) and D. Enskog, Kinetische Theorie der +Vorgduge in mdssig verdunnten Gasen (Uppsala: Almqvist and Wiksells, 1917). It was found that for +elastic spheres and most other types of force law, there is a slight dependence on concentration, but not +as great as that predicted by Meyer’s theory. More precise values of the numerical coefficients in the +expressions for viscosity and thermal conduction were also obtained, and the relation between the +temperature-dependence of these coefficients and the force law was clarified (see Rayleigh, Proc. R. S. +London A66, 68 [1900] for a deduction of the temperature dependence by dimensional arguments). But +the most significant new result of the Chapman-Enskog investigations was the prediction of thermal +diffusion, which was then confirmed experimentally by Chapman and Dootson, Phil. Mag. [6] 33, 248 +(1917). For details of the calculations, comparisons with experiment, and a survey of the contributions +of various scientists to the theory, see the monograph by S. Chapman and T. G. Cowling, The +Mathematical Theory of Non-uniform Gases (London: Cambridge University Press, 2d ed., 1953). + + +mJuc. 7fca^4e«ta£Zc.a/ + + + +CHAPTER II + + +The molecules are centers of force. Consideration of external forces + +and visible motions of the gas. + +§15. Development of partial differential equations for f and F. + +We now pass to the consideration of the case when external forces act, and any arbitrary +interaction may take place during collisions. In order to avoid the necessity of generalizing +the formulas later on, we consider again a mixture of two gases, whose molecules have +masses m and in \ respectively. We call them again, for brevity, m-molecules and in | - +molecules respectively. Each molecule will be almost completely unaffected by the others +during the greatest pail of its motion; only when two molecules come unusually close to +each other will there be a significant change in the magnitude and direction of their +velocities. Simultaneous interactions of three molecules occur so rarely that they can be +disregarded. In order to achieve a precise exposition, we think of the molecules as material +points. As long as the distance r of an m-molecule from an m \-molecule is greater than a +specified very small distance a, no interaction takes place; however, as soon as r is smaller +than a, the two molecules may exert on each other any force, whose intensity ifir) is a +function of their distance of separation, and which is sufficient to produce a significant +deviation from their rectilinear paths. As soon as r becomes equal to a we say that a +collision begins. For the sake of simplicity we exclude those force laws which can cause +the molecules to stick together, even though these are especially interesting since they may +lead to a clarification of dissociation phenomena;* then after a short time, r will again be +equal to a, and at this instant, which we call the end of the collision, the interaction ceases. +For collisions of m-molecules or of mj -molecules with each other, we replace a and dir) +by 5 and T(r), sq and v P 1 (r), respectively. The case of elastic spheres is now a special case +obtained on assuming that the functions if), 4k and Tj represent repulsive forces whose +intensities increase without li mit as soon as r becomes the least bit smaller than a, s, or sq, +respectively. Everything presented up to now is therefore obtained as a special case of the +equations to be developed here. In addition to these molecular forces, we now include +those forces acting on the molecules that arise from external causes, called briefly external +forces. We draw any fixed coordinate system in the gas. The components mX, mY, mZ of +the external forces acting on any m-molecule should be independent of time and of the +velocity components, and for all m-molecules they must be the same functions of the +coordinates x, y, z of the molecule in question. X, Y, and Z are thus the so-called +accelerating forces. The corresponding quantities for a molecule of the second kind will +have the subscript 1. The external force can indeed vary from one place to another in the +gas, but it should not vary markedly as long as the coordinates do not change by an amount +large compared to the sphere of action (characterized by the lengths a, s, jq). Finally, we +also do not exclude the case that the gas is in visible motion. A priori one can assume + + +7fca^4e«taiZc.a/ + + + +neither that all directions of velocity are equally probable, nor that the velocity distribution +or the number of molecules in unit volume are the same in all parts of the gas, nor that they +are independent of time. + +We fix our attention on the parallelepiped representing all space points whose +coordinates he between the l im its + +(97) x and x + dx, y and y + dy , z and z + dz. + +We set do = dxdydz, and we always call this parallelepiped the parallelepiped do. + +We assume, following the principles mentioned earlier, that this parallelepiped can be +infinitesimal yet still contain many molecules. The velocity of each m-molecule that finds +itself at time t in this parallelepiped shah be represented by a line starting at the origin, and +the other end point C of this line will again be called the velocity point of the molecule. Its +rectangular - coordinates are equal to the components f , rj, C of the velocity of the molecule +in those coordinate directions. + +We now construct a second rectangular parallelepiped, which includes ah points whose +coordinates he between the l im its + +(98) £ and £ + d£, v and 17 + drj, f and f + df. + +We set its volume equal to + + +d£dr/df = du + +and we cah it the parallelepiped dco. The m-molecules that are in do at time t and whose +velocity points he in d completely unchanged. The number of m-molecules that satisfy the +conditions (97) and (98) at time t+dt is, according to Equation (99), + +dn' = }{x, y, 2 , i|, f, t + dt)dodu + +and the total increase experienced by dn during time dt is + +d} + +(101) dn' - dn = —dodudt, + +dt + +The number dn experiences an increase as a result of four different causes. + +1. All m-molecules whose velocity points he in dco move in the x-direction with velocity +F, in the y-direction with velocity //, and in the z-direction with velocity C. + +Hence through the left of the side of the parallelepiped do facing the negative abscissa +direction there will enter during time dt as many molecules satisfying the condition (98) as +may be found, at the beginning of dt, in a parallelepiped of base dydz and height fdt, viz. + +t = $•/(*, y, 1, % f, t)dydzdodt + +molecules (cf. p. 33 and 88). Then, since the latter parallelepiped is infinitesimal and is +infinitely near to do, the numbers £ and fdodco contained in the two parallelepipeds are +proportional to the volumes fdydz.dt and do of the parallelepipeds. Likewise one finds, for +the number of m-molecules that satisfy the condition (98) and go out through the opposite +face of do during time dt, the value : + +£ f{x + dx , yz, £, ?j, f, t)dxdzdudt. + +By similar arguments for the four other sides of the parallelepiped, one finds that during +time dt, + + +/ d/ df df\ + +— (£ —b*? —bf — ) do- do)dt +\ dx dy dz) + +more molecules satisfying (98) enter do than leave it. This is therefore the increase +which dn experiences as a result of the motion of the molecules during time dt. + +2. As a result of the action of external forces, the velocity components of all the +molecules change with time, and hence the velocity points of the molecules in do will + + +84 _ + + +ru*i.C- 7fca^4e«tai2c.a/ PA.ylic.1 + + + +move. Some velocity points will leave dco, others will come in, and since we always +include in the number chi only those molecules whose velocity points he in clco, cln will +l ik ewise be changed for this reason. + +£, ij, and ( are the rectangular coordinates of the velocity point. Although this is only an +imaginary point, still it moves like the molecule itself in space. Since X, Y, Z are the +components of the accelerating force, we have: + + + +Thus ah the velocity points move with velocity X in the direction of the x-axis, with +velocity Y in the direction of the y-axis, and with velocity Z along the "-axis, and one can +employ completely similar arguments with respect to the motion of the velocity points +through dco as for the motion of the molecules themselves through do. One finds that, out of +the velocity points belonging to m-molecules in do, there will enter from the left of the +surface of the parallelepiped clco parallel to the yz plane, in time dt, + +X’/(z, y, 2,1, v, {,t)dodiidtdt + +of them, while + +X-f(x, y,z,l + 4 if, 1, l)dodi\d$dt + +of them will go out through the opposite surface. If one employs similar considerations for +the other four surfaces of the parallelepiped dco, he finds that in all + +/ df 8} 8f\ + +V,= -\X-+Y- + Z-)doiudt + +\ d£ 8y 8z) + +more velocity points of m-molecules (in do) enter dco than leave it. + +Since, as noted, a molecule is included in cln when it not only lies in do but has its + +velocity point in dco, this represents the increase of cln resulting from the motion of the + +velocity points. But those molecules that enter clo during the time dt, while during the same +time dt their velocity points enter clco, are not taken account of, nor are those that enter clo +while their velocity points leave dco during df, on the other hand, those that leave clo during +dt while their velocity points enter or leave dco are counted in V 1 as well as in V 2 and are +therefore counted twice. Yet this leads to no error, since the number of all these molecules +is an in fin itesimal of order (dt) 2 . + +§16. Continuation. Discussion of the effects of collisions. + + +We 7fca^4e«toi2ea/ “Pity. i Ze i. + + + +3. Those of our dn molecules that undergo a collision during the time dt will clearly +have in general different velocity components after the collision. Their velocity points will +therefore be expelled, as it were, from the parallelepiped by the collision, and thrown into a +completely different parallelepiped. The number cln will thereby be decreased. On the other +hand, the velocity points of m-molecules in other parallelepipeds will be thrown into dco by +collisions, and cln will thereby increase. It is now a question of finding this total increase V 3 +experienced by cln during time dt as a result of the collisions taking place between any m- +molecules and any nq -molecules. + +For this purpose we shall fix our attention on a very small fraction of the total number v\ +of collisions undergone by our dn molecules during time dl with in | -molecules. We +construct a third parallelepiped which includes all points whose coordinates lie between the +l im its + +(102) {1 and fi + $ 1 , t/i and rj i -f dif\ } fi an d fi + dfi* + +Its volume is (Iqj 1 = ' l constitutes the parallelepiped do)\ . By analogy with + +Equation (100), the number of in \ -molecules in clo whose velocity points lie in da>\ at time +t is: + +(103) dN 1 — F \dodu\. + +F\ is an abbreviation for F(x, y, z, F \, i/\, (\). + +We now ask for the number v 2 of collisions that take place during dt between one of our +dn m-molecules and an m | -molecule, such that before the collision the velocity point C | of +the latter molecule lies in dco\. We again denote by C and Cj the velocity points of the two +molecules before the collision, so that the lines OC and OCy drawn from the origin to C +and Cj represent in magnitude and direction the velocities of the two molecules before the +collision. The line C | C = g gives also the relative velocity of the m-molecule with respect +to m \, in magnitude and direction; the number of collisions clearly depends only on the +relative motion. Furthermore, we assume that a collision always occurs between an m- +molecule and an in | -molecule as soon as they approach to a distance smaller than a. The +problem of finding v 2 is thus reduced to the following purely geometrical question. In a +parallelepiped do there arc dN\ = F\ cloda>\ points. We call them again the in | -points. +Moreover, fclocko points (the m-points) move therein with velocity g in the direction C| C, +which we call the direction g for short. v 2 is just equal to the number of times that an m- +point comes so close to an m 1 -point that their distance is less than a. Naturally we assume a +molecular-disordered, i.e., completely random, distribution of the m-points and the in | - +points. In order not to have to consider those pairs of molecules that are interacting at the +beginning or the end of the time dt, we assume that while clt is indeed very small, it is still +large compared to the duration of a collision, just as clo is very small but still contains many +molecules. + +In order to solve this purely geometrical problem, one can completely ignore the +interaction of the molecules. The motion of the molecules during and after the collision will + + +7fca^4e«taiZc.a/ + + + +of course depend on the law of this interaction. However, the collision frequency can be +affected by this interaction only insofar as a molecule that has already collided once during +dt may collide again, with its altered velocity, during the same interval dt, yet such effects +would certainly be in finitesimals of order (dt) . + +We define a passage of an m-point by an in j -point as that instant of time when the +distance between the points has its smallest value; thus m would pass through the plane +through m 1 perpendicular to the direction g, if no interaction took place between the two +molecules. Hence V 2 is equal to the number of passages of an m-point by an ni\ -point that +occur during time dt, such that the smallest distance between the two molecules is less than +a. In order to find this number, we draw through each in \ -point a plane E moving with m \ +perpendicular to the direction g, and a line G parallel to this direction. As soon as an m- +point crosses E, a passage takes place between it and the m\ -point. We draw through each +m |-point a line m\X parallel to the positive abscissa direction and similarly directed. The +half-plane bounded by G, which contains the latter line, cuts £ in the line m | H, which of +course again contains each -point. Furthermore, we draw from each in \ -point in each of +the planes E a line of length b, which forms an angle e with the line ni\H. All points of the +plane E for which b and e he between the li mits + +(104) b and b + db, t and t + dt + +form a rectangle of surface area R = bdbd e. In Figure 6 the intersections of ah these lines +with a sphere circumscribed about m \ are shown. The large circle (shown as an ellipse) lies +in the plane E: the circular arc GXH lies in the half-plane defined above. In each of the +planes E, an equal and identically situated rectangle will be found. We consider for the +moment only those passages of a n m-point by an ni\ -point in which the first point +penetrates one of the rectangles!?. 1 In their motion relative to mj, each of the m-points +travels the distance gdt during time dt, in a direction perpendicular to the plane of all these +rectangles. Therefore during time dt all those m-points that were initially in any of the +parallelepipeds whose base is one of these rectangles and whose height is equal to gdt will +go through the surface of one of these rectangles. (Cf. pp. 33, 38, and 113. The state should +again be molecular-disordered.) The volume of each of these parallelepipeds is therefore + +II = bdbdtgdt , + +and since the number of m\ -points, and consequently the number of parallelepipeds, is +equal to F \dodn>\, then the total volume of all the parallelepipeds is : + +£n = Fidoduigbdbdtdt. + +Since these volumes are infinitesimal, and lie infinitely close to the point with +coordinates x, y, z, then by analogy with Equation (99) the number of m-points (i.e., m- +molecules whose velocity points lie in dco) that are initially in the volumes LI I is equal to : + + +mJuc. 7fca^4e«taiZc.a/ + + + +( 105 ) + + +vi = n = JFidodudwigbdbddt. + + + +Fig. 6 + + +This is at the same time the number of m-points that pass an in | -point during time dt at a +distance between b and b+db, in such a way that the angle e lies between e and e + de. + +By x >2 we mean the number of m-points that pass an m 1 -point at any distance less than a +during dt. We find Vo by integrating the differential expression v z over e from 0 to 2n, and +over b from 0 to a. Although the integration can easily be carried out, it is better for our +purposes merely to indicate it. Hence we write: + +nh + +\. The number (earlier denoted by +v |) of all collisions of our dn molecules during dt with in | -molecules is therefore found by +integrating over the three variables . tj\. C\ whose differentials occur in d(u\. from - oo to ++ oo ; we indicate this by a single integral sign, so that we obtain : + +/» r* 9 n 2r + +(107) v\ = do-dwdt I I I fFigbdwidbde. + +j J 0 d 0 + +In each of these collisions, provided it is not merely a glancing one, the velocity point of +the m-molecule involved is expelled from the parallelepiped dot, and hence dn decreases by +one. + +In order to find out for how many m-molecules the velocity points lie in da) after a +collision with an mj -molecule, we simply need to ask how many collisions occur in a + + +We 7fca^4e«toi2ea/ lPl±y. i Ze i. + + + +manner just the opposite of that considered above. + +We shall consider again those collisions between m-molecules and m 1 -molecules, +whose number was denoted by V 3 and is given by Equation (105). These are the co ll isions +that occur in unit time in the volume element do in such a way that the following conditions +are satisfied: + +1. The velocity components of the m-molecules and the m \ molecules lie between the +li mits (98) and (102), respectively, before the interaction begins. + +2. We denote by b the closest distance of approach that would be attained if the +molecules did not interact but retained the velocities they had before the collision. +We denote by P and P\ the points at which they would be found at the instant when +this smallest distance is realized, and by g the relative velocity before the interaction. +Then b, and the angle between the two planes through g parallel to P\ P and to the +abscissa axis, respectively, lie between the li mits (104) (cf. footnote 1). + +We call all these collisions, for short, direct collisions of the kind considered. For them +the velocity components of the two molecules after the collision he between the l im its + + + +(' and £' + ij' and ij' + dif, and £' + +ii and {/ + #1,1)1 and 1 j{ + rfiji/ f 1 and + df {. + + +We denote by P^P the smallest distance to which the two molecules would approach if +they always had the same velocities as those with which they separate from each other after +the collision; and by g' the relative velocity after the collision. Thus for ah direct collisions +the length of the segment P\P' and the angle of the planes through g' parallel to P\P' and +to the abscissa axis, respectively, he between the l im its + +(109) V and h' + db\ t' and e' + dt + + +All collisions that occur during time dt in the volume element do such that the values of +the variables before the collision lie between the li mits (108) and (109) will be referred to as +inverse collisions. For them the direction of g' is to be reversed. They clearly follow just the +opposite course as the direct collisions, and after the collision the values of the variables he +between the limits (98), (102), and (104). + +Since we assume that the force law acting in the cohisions is given, the values +f', tf, vi, Si, V and e' of ah the variables after the colhsion can be + +calculated as functions of the values of the same variables P, rj, C, P \, tj \, C \, b, and e before +the colhsion. Just as for the number of direct cohisions we obtained Equation (105), so one +obtains for the number of inverse cohisions the value : + + + +7fca^4e«toiZc.a/ + + + +Here we have written: do/ for di'dr,'dt '; do>{ fr d{i dr,l df{ ; and /' and + +F{ for fix, y, z, *7\ C» 0 and F(«, 2 /, 2 , f i, n 1 , f 1,0 In order to be able to +perform the integration, we must express all the variables as functions of /, //, C, f \ , r/\ , T|, +b, and e. + +Later on we shall study explicitly the motion during the interaction (§21). Here we only +note the following. The motion of m relative to m.\ (i.e., relative to three coordinate axes +passing through m 1 and always parallel to the fixed coordinate axes, on which the +quantities g, g',b,b', e and e' depend) we call the relative central motion. It is just the same +central motion that one would obtain for the same force law by holding m \ fixed and letting +m originally have the relative velocity g in a line that has the perpendicular distance b from +m |. The latter material point must then have mass mm \ /(m+m \ ) instead of its actual mass, +g' is nothing but the velocity of m at the end of the relative central motion ; b' is the +perpendicular distance of the line described by m as it recedes from m \ at the end of the +relative central motion. From the complete symmetry of any central motion it follows at +once that g' = g, b' = b (cf. Figure 7, §21). The symmetry axis of the path of m in the +relative central motion, which we call the line of apses, is the line connecting m \ with that +point where m has, in its entire relative central motion, the smallest separation from m \. It +plays the same role for central motion as does the line of centers for elastic collisions. The +plane of relative central motion will be called the orbital plane. It contains the four lines g, +g\ b, and b'. One sees that de = de' if he introduces for de the angle of rotation dd of the +line of apses, so that /, C, /\. rft, C\ transform into V, f', U, *n', ti and + +then again introduces de' for dd; then the expression for de in terms of dd and the values +of the variables before the collision must be exactly equal to that for de' in terms of dd and +the values of the variables after the collision. + +The proof that f/co c/co r T d(j) i = (/id / has already been given for elastic +spheres. Since we used there only the theorem of kinetic energy conservation and the +theorem of conservation of motion of the center of mass in constructing the proof, and since +these theorems are still valid here, the proof can be constructed in the same way ; in place +of the line of centers one has of course to introduce the line of apses. Taking account of all +these equations, one can also write : + +( 1 10) i 3 = fF[ dodudw\dtgbdbdt + +In Pail II we shall prove a general theorem, of which the law that here + +(110a) du'dulg'b'db'dt' = duduigbdbdt + +is only a special case. It is only to avoid the necessity of repeating here for a special case, as +a useless digression, results that will be developed in general later on, that we have merely +indicated briefly the proof of this theorem which is undoubtedly correct. + +As a result of each “inverse” collision, the number dn of m-molecules in the + + +7fca^4e«taiZc.a/ + + + +parallelepiped do whose velocity points lie in the parallelepiped da> increases by one. The +total increment q experienced by dn as a result of collisions of m-molecules with ni\- +molecules is found by integrating over e from 0 to 2 rr, over b from 0 to a, and over , //|, +C\ from - oo to + oo. We shall write the result of this integration in the form: + + both before and after the collision. These glancing +collisions are counted in (105) and hence also in v 1; and are subtracted from V 3 , although +in these collisions the velocity point of the m-molecule is not expelled from da> but is +merely pushed from one place to another inside this parallelepiped. Yet this entails no error. +It is precisely because the velocity point of the m-molecule also lies in dco after the co ll ision +that these collisions are also counted in the expression ( 110 ) for i 3 and hence also in i\, and +they are therefore added back to V 3 . + +These collisions are simply to be understood as collisions in which the velocity point of +the m-molecule is actually expelled from dto at the beginning of the collision, but at the end +of the collision it is thrown back into the same parallelepiped. Indeed, in Equation (112) we +can extend the integration over b to values greater than a. We would thereby include in the +number of passages in v l5 and exclude from those in V 3 , some passages in which no +change of the velocities and directions actually occurred. But the number of these collisions +will therefore be counted in the expression for q, and added back again to V 3 . Obviously +the li mits of integration cannot be chosen differently in the two expressions, (107) for y | +and (111) for/j. In Equation (112), where q -vj is united into a single integral, on the +other hand, the integration over b can be extended as far as one wishes, since as soon as b +is larger than a, the quantities {', v', S', fi', vl, Si will be identical to T, //, C, , +7l , Ct; hence }'F{ =/F, and the integrand vanishes. This remark is important in all +cases where the interaction of the molecules falls off gradually with increasing distance, so +that no sharp boundary of the sphere of action can be specified. In such cases one can + + +Uuc. 7fca^4e«taiZc.a/ + + + +integrate in Equation (112) over b from 0 to oo, and since these l im its are permissible in all +other cases, we shall retain them in the future. When no distance can be specified at which +the interaction of two molecules drops off precisely to zero, we shall still assume of course +that the interaction decreases so rapidly with increasing distance that the case where more +than two molecules interact simultaneously can be neglected. + +The number of molecules that collide during dt but move in such a way that even +without a collision they would have left do or their velocity points would have left dco is of +course an infinitesimal quantity of order (dt) . + +4. The increment V 4 experienced by dn as a result of collisions of m-molecules with +each other is found from Equation (112) by a simple permutation. One now uses <0, rft, C\ +and £i, vi, ti for the velocity components of the other m-molecule before and after +the collision, respectively, and one writes f\ and f { for + +/(*, y, *, h, in, h, 0 + +and + +1 {x,y,z, + + +Then: + + + +Vi = dodwdt + + + + +Since now E 1 +V 2 +V 3 +V 4 is equal to the increment dn'-dn of dn during time dt, and +this according to Equation (101) must be equal to (df/dt)dodoxlt, one obtains on substituting +all the appropriate values and dividing by dodoxlt the following partial differential equation +for the function /: + + +(114) + + +a/ df df df df a/ a/ + +-+i i +»-+r-+x-+f-+z i + +dt dx dy dz dx dy dz + += [[* f (f'F! -JFJgbduM + +J J n J n + + +0 0 +oo /»it + + ++ + + +f f * f ’(/'// -JfOgbduM. + +J J 0 * 0 + + +7fca^4e«taiZc.a/ + + + +Similarly one obtains for F the partial differential equation : + + +dF i dF i dF i dF i dFi dFi dFi + +—+ ti —+ m —+ fi — + Xi—+ Fi—+ ft — + +M dx dy dz dx dy dz + + + +/* /» 00 /* It + + + +JFi)gbdudbdt + + + +- FF\)gbdu\(Mt + + +Here we have used the abbreviation F' for F(x, y, z, F, if, C, t). + + +§17. Time-derivatives of sums over all molecules in a region. + +Before we go further, we shall develop some general formulas useful in gas theory. Let +cp be an arbitrary function of x, y, z, F, rj, C, t. The value obtained by substituting therein the +actual coordinates and velocity components of a particular’ molecule at time t will be called +the value of cp corresponding to that molecule at time t. The sum of all values of cp +corresponding to all the m-molecules that he in the parallelepiped do and whose velocity +points he in the parallelepiped da> at time t is obtained by multiplying cp by the number +fdodto of those moleclules. We denote it by + +(116) = vjdodw. + +Similarly we choose for the second kind of gas any other arbitrary function of.i', y, z, +F, ip t, I and denote by + +(117) = $\F\dodui + +the sum of the values of corresponding to all the in | -molecules lying in do whose +velocity points he in dap. T | is the abbreviation for T(.i', y, z, F\ , rj\, C,\, t). + +If we keep do constant in these expressions and integrate with respect to dco and dap, +over all possible values, then we obtain the expressions: + +(US) YcM = doj , respectively, that + + +7fca^4e«taiZc.a/ + + + +correspond to all molecules in do at time t, without any restriction on the velocities. + +If we also integrate do over all volume elements of our gas, then we obtain the +expressions: + +(119) L„,» f =JJ vfdodu and = // Vidatoi + + +for the sums of all values of cp and corresponding to all gas molecules of the first and +second kinds, respectively. + +We shall now calculate the increment (d 'Ld, J uio ( pldt)dl experienced by the sum Y. c i 0K do ( P +during an infin itesimal time dt, if there is no change in the two volume elements do and dco +in size, shape, or position. According to the latter condition, which is expressed by the +symbol d/dt, one should differentiate only with respect to the time. Since during the time +interval dt, cp changes by (dcpldt)dt and / by (df/dt)dt, we obtain from Equation (116): + + +d + +dt + + + +f d

0). + +This sum experiences an increase tp' - tp because of each of these collisions, and since +according to Equation (105) the number of these direct collisions is given by v 3 , one +obtains the total increase B^(tp)dodt experienced by the sum 'L (j)do tp as a result of co ll isions +of an m-molecule with an m | -molecule when he integrates the product (tp' - (pYp, keeping +do and dt constant, over all values of all the other differentials. One thus obtains: + + + +dip dip\ + +Y- + Z-)dt. + +drj df ) + + +7fca^4e«taiZc.a/ + + + + + + +2f + +W " ')/'// gbduduidbdt. + + +One may still consider the two colliding molecules to play the same role, so that one can +interchange, in the last two formulae, the letters with subscript 1 and those with no +subscript, without changing the value of B 5 (cp). Taking the arithmetic mean of the values of +B 5 (cp) obtained by this peimutation, one gets from (135): + + +(137) BM + + + +-

, doj \, db, and de, then we obtain the increase of Y. IjIC i 0 ( P resulting +from collisions of m-molecules with each other, i.e., the quantity B 5 ((p)dodt. However, we +must divide by 2, since we have counted each collision twice, and therefore we obtain +Equation (137). Had we considered only the inverse collisions, then we would have +l ik ewise obtained Equation (138). + +The special cases of Equation (136) obtained by assuming that cp is independent of the +time and of the coordinates x, y, z will be treated in §20. + +We shall now set: + +(140) — = C\{(p) + C 2 W + C 3 W + Ci{ + +and at time t+dt a contribution larger by the amount + +/ d

-?' - Miffl - //>) + + +0 ^ 0 + +•gbdodwdwidbdt + + +We have not considered here any effects due to motion of the walls of the container, +since the molecules that would collide in volume elements entering the space considered +because of such motion would provide terms only of order (dt) . + +Since the expression for the time derivatives of the quantities denoted by E^ W| t / 0 T [, +'L,, n j 0 < b i and are constructed in just the same way, we shall not write them out. + +Since A, B , and C are only the increments of definite quantities resulting from specified +causes, most authors express them as derivatives of those quantities. Maxwell* writes (3/30 +E ow / r /p, Kirchhoff t ( D/Dt ) E w . f / r /p for B$( = IF, where l signifies the +natural logarithm. Then + +Em ^ — If ~ J f fl/dodoif + += ^ i = / / f llFidodui, + +and we shall set + +( 144 ) H = 2 ... IJ + IF , = J fflfdodu + / / FMn ■ + + +One has according to equation (141): + + +(145) + + +cm + cm+cm = ff *. i. i, f, 0 , *>' = W, y, z, v', f', t). + +so that the last of Equations (147) becomes + +(148) ,' + +— + 2m, A({ + rthB + + + + +With a suitable choice of the four factors, the coefficients of all twelve differentials will +vanish; hence + + +1 1 d$ i + +—— 4* — U + B = —■—b + # = 0 + +m mi a£i + + +IQ + + +7fca^4e«tai2c.a/ + + + +or + + +1 dip 1 d$i + +- 7 r~ 7 r =2A((i - () - + +m mi dfi + +Likewise it follows that + +1 ^ 1 d$i + +— 7-— = 2i%-q). + +m orj m i orji + +El im ination of A—which, as an undetermined factor, cannot in any case be set +identically equal to zero—yields: + + + + +1 Mi + +Mi dqj + + +(fi - I). + + +This equation contains, aside from the variables x, y, z, t, which we always consider +constant, only the six completely independent variables f, tj, C, £\. >]], C \. If one takes the +pai'tial derivative with respect to C, it follows that: + + +3V + + +(ii - q) + + +dfy + +drjdf + + +tti - {)• + + +Further pai’tial differentiation of this equation with respect to rj\ gives: + +3V + +— = 0 . + +m + +Differentiation with respect to gives however: + +av + +— = o, + +a,ar + + +and by cyclic permutation it follows that: + + +IQ + + +^ufuc. 7fca^4e«tai2c.a/ + + + +dfy + +— = 0 . + +dijdt] + + +These three equations express the fact that cp must break into three summands, of which the +first contains only T, the second only //, and the third only C. + +Similarly for the function we obtain : + +Mi M, Mi + +(152) -=-=-= 0. + +dfadrn d(idfi dr/tf, + +Differentiation of Equation (151) with respect to £ gives: + +i av i df l a$>, + +(153) ( 11 - 1 ) = + +m d £ 2 m dij mi drj i + +since + +av + +— = o. + +d^drj + +Further differentiation of Equation (153) with respect to ?/ 1 gives however : + +l av l Mi + +■ —— ^ ■' ■■ ■ ■ ■ ■ • + +m d ( 2 mi di}\ + +Since the two expressions on the left and right hand sides contain completely different +variables, they can be equal only when both are independent of all variables and therefore +are equal to a quantity independent of £ rj, (, fJi, (i- + +Since the y and z axes enter the equations to be solved in exactly the same way, one can +l ik ewise prove the equation + + +i av i m, + +m d( 2 m 1 dfj + +and also, that the last expression must be equal to + + +IQ + + +aV^lc. 7fca^4e«tai2c.a/ + + + +1 dfy + +— ■ — — i • + +m drj 2 + +Hence all these second derivatives are equal to one and the same quantity - 2 h, + +independent of f, rj, C, f \. ij\. and . From all these equations one easily draws the + +o 'y + +conclusion that cp must be equal to -hm(£ +t] +( ) plus a linear function of £, //, C. The +coefficients of the latter can be written in such a form that one can write, without loss of +generality, + + +)’] + Ik + + +where u, v, w and / 0 are new constants, which however l ik e h can still be functions of x, y, +z, and t. Whence it follows that: + + + + + +and one obtains similarly + + + + + +The functions f and F must in any case have this form when the three equations (147) +are satisfied for all values of the variables. One sees easily that, conversely, when f and F +have this form, Equations (147) are in fact satisfied, provided only that u\ = u, rq = v, vrq = +w. The quantities/o, Fq, u, v, w, h can be any functions of x, y, z, t. + +These functions are to be determined in such a way that the two equations + + + +and + + +df d} df dj dj + +f + {- + 7- + l- + X- + + +dt dx dy dz + + + + + +dF dF dF dF dF + +—!{ — + >)—If—h Tfj —• + +dt dx dy dz d% + + +dF 3F 1 + +Y l - + Z l - = 0 + +dy df + + +are satisfied; Equations (114) and (115) reduced to these, since their right hand sides vanish +identically. + +The number of molecules at time 1 in do whose velocity points are in do> is: + +fdodo, = + + +10 . + + +7fca^4e«toi2c.a/ + + + +If one sets + +(158) { = f + «, i; = 5 + v, f = j + w, + + +then he obtains just Equation (36), except that t), 3 re P' acc C *7> C + +From this one sees immediately that all the considerations attached to Equation (36) +remain valid, except that all the gas molecules have a common progressive motion through +space, whose velocity components are u, v, w. When u = U\, v = v 1; w = vtq, these are the +components of the visible velocity with which the entire gas in do moves. If u, v, w differ +fromz^, v|, vtq, respectively, then u, v, w would be the components of the velocity with +which the gas of the first kind moves through the gas of the second kind in do. + +One can also see this in the following way. The number of m-molecules that lie in do at +time t is : + + +dn = do J fda = rfo/ 0 III e -hm[ ()*+ (f-«)*+ (f _w ). + + +On substituting (158) it follows that: + ++00 + + +(159) dn = do/o + + +Ilf + + +e -hm ( ib\ = doja + + +h 3 m 3 + + +If one multiplies this by m and divides by do, then he obtains the partial density of the +first kind of gas : + + + +The mean value £ of the velocity component along the abscissa direction of all m- +molecules in do is : + + + +This is clearly also the v-component of the velocity of the center of mass of the first kind +of gas in do. If a surface element parallel to the yz plane moved with this velocity in the +abscissa direction, an equal number of molecules of each kind would go through it, as + + +IQ + + +7fca^4e«toi2c.a/ + + + +follows directly from the concept of mean velocity. One can therefore call C the velocity +with which the first kind of gas moves in the abscissa direction in do. + +On substituting (158), the numerator of Equation (161) is transformed to: + + +dn = do + + + + + + +One sees at once that the first teim vanishes, but that the second reduces to udn. Hence + +( 162 ) u = l + +Since J is the relative velocity of a gas molecule with respect to a surface element moving +with velocity u, and f is an even function of JT, one sees at once that through each surface +element _L to the v-axis, on the average as many molecules of the first kind go in as go out. + + +19. Aerostatics. Entropy of a heavy gas whose motion does not violate +Equations (147). + + +On substituting the values (154), Equation (156) yields many solutions each of which +describes a state of the gas when it is under the influence of specified external forces in a +container at rest. The walls of the container satisfy the conditions employed in the +derivation of Equation (146). Thus after the cessation of all heat conduction and diffusion +phenomena, there is no persistent flow of heat into or out of the gas. We shall now look for +these solutions. It is clear that none of the relevant quantities here can depend on time. + +Moreover, the equations u = v = w = u j = vy = viq = 0 must hold. Hence according to +Equations (154) and (155): + + + + + +where / 0 , Fq and h can still be functions of the coordinates. If we substitute this into +Equation (156), it follows that: + + +11 . + + +IPu+uc- 7fca^4e«tai2c.a/ + + + +/ dh dh dh \ + +- m({ 2 + 1} 2 + f 2 ) ({— + v — + f — ) + +\ dx dy dz/ + ++ - 2hmf a x'j + - 2hmfjj + ++ f(~ ~ 2/wn/ozj = 0. + +Since this equation must be valid for all values of f, ij, and C, we see that: + +dh dh dh +- = - = - = 0 . +dx dy dz + +Therefore h must be constant everywhere in space. + +Furthermore, the coefficients of T, //, C in the following terms must vanish separately. +This can happen only when X, Y, and Z are the partial derivatives of one and the same +function - x of the coordinates. If this condition is not satisfied, then the gas cannot in +general be at rest. If it is satisfied, then: + +(164) /o = + +where a is an absolute constant, since in each volume element do the quantity/ 0 is constant, +Equation (163) must have the same form as Equation (36). The velocity distribution in each +volume element is therefore exactly the same as it would be if only one kind of gas were +present, and the same external forces acted on an equal partial density of it. In other words, +in spite of the action of the external forces, every direction in space is equally probable for +the direction of motion of a molecule. Since the problem to which the equations treated at +the beginning of §7 refer is only a special case of the problem treated here, we see that the +assumption made there without proof, viz., that all directions of motion of a molecule are +equally probable, has now been proved. Because of the identity of the forms of the +equations, the equations developed in §7 and the conclusions drawn there from are now +applicable without change to each volume element. Hence, again corresponding to +Equation (44), the mean square velocity of an m-molecule is: + +_ 3 + +c 2 =-, + +2 hm + + +u + + +' if^ufLC- 7fca^4e«toi2c.a/ + + + + +i.e., under the action of external forces the mean kinetic energy of each molecule also +remains the same; lik ewise for the second kind of gas, + + + +3 + +2/wti + + +and the constant h must have the same value for both kinds of gas. Let p be the partial +density of the first kind of gas in do, and let p be the partial pressure which this kind of gas +would exert on the wall if it were present alone in do', then according to Equations (160) +and (164), + + + +Since furthermore dn/do is the number of molecules in unit volume, then according to +Equation (6): + +me 2 ' dn pc 2 p + +3 do 3 + +Hence pip has the same value everywhere in the gas. Since now the gas in each volume +element has exactly the same properties as it would have at the same partial density and +equal energy if no external forces acted, then just as in the latter case we have pip = rT. +The gas constant r is, as earlier (§8), equal to \hmT, Since furthermore pip has the +same value everywhere and is equal to rT, then the temperature is also the same +everywhere in spite of the action of the external forces. + +For the second kind of gas one finds, completely independently of the presence of the +first, + + + + +F o = Ae- nm '*\ + + +where + + +Xi — — / {Xidx + Y4y + Zidz). + +The two gases therefore do not disturb each other in the equilibrium state. Thus in air +completely at rest and in complete thermal equilibrium, each constituent of the air would +form an atmosphere according to the laws given above, just as if the others were not +present ; except that for each gas, h and therefore temperature must have the same value. +According to Equation (165): + + +11 . + + +7fca^4e«tai2c.a/ + + + +(1G7) P = poe-w-tx-w' = poc <*r*>' rJ \ + +and likewise according to (166), + +( 168 ) p = poe (x r x),,T . + +Here p, p, and x are the values of these quantities at any position with coordinates x, y, +z, and p 0 , Pq, y {) are the values of the same quantities at any other place with coordinates +x 0 , >’(), zq. These are the well known formulas of aerostatics (barometric height +measurements). + +We shall now, following Bryan,* treat the following case which, although it does not +occur in nature, is of theoretical interest. The two gases in the container will be separated +by an arbitrary surface 6 j into two parts, a left part T | and a right part. To the right of the +surface S 1 is a second surface S 2 , which lies everywhere very close ter S|. The space +between .S'] and S 9 will be called t, and the space to the right of S 2 will be called T 2 . Let % +be zero throughout T |, and let it have positive values between and S 2 , becoming infinite +as one approaches S 2 . Thus an m-molecule experiences no force in T\ , but in t a force +begins to act, pushing it away from S 2 toward Sj and becoming in fin ite if the molecule +approaches S 2 . Conversely in T 2 no force acts on the in | -molecules; however, when these +enter x, a force pushes them away from toward S 2 , and becomes in fin ite near S |. Thus +■/1 is zero in T 2 , positive in t, and becomes infinite near S- + +If there are initially no m-molecules in T 2 , then none can ever get into T 2 \ if there are +initially some m-molecules in T 2 , where no force acts on them, then each molecule that +reaches the surface S 2 is expelled into T 1 and can never return. Hence we can assume in +any case that the space T 2 contains no m-molecules, and likewise the space T\ contains no +m-molecules. This illustrates the formula; for one has + +f = ^ f = ^g-AmiCef+^xi). + +In the space T \, ^ = 00 , hence F = 0; in T 2 , x = °°, hence /= 0. Also, Equation (167) +gives the value zero for the partial density when y is infinite. This in both T\ and T 2 only +pure gases are present; only in x where both y and y j are present does one find a mixture. +Our formula gives theimal equilibrium only when h has the same value for both gases, +which according to Equation (44) implies that the mean kinetic energy of a molecule must +have the same value for both gases. This is exactly the same condition that we also found +for the thermal equilibrium of two mixed gases. The mechanical conditions just discussed +are of course somewhat different from those for two gases separated by a solid heat- +conducting wall; but they have a certain similarity. We can imagine a third surface S 3 to the +right of S 2 and everywhere near it. By an appropriate choice of y for three different kinds +of gas, it can be arranged that molecules of the first kind are present only to the left of S 2 , +those of the second kind only to the right of S 2 , and those of the third kind only between S| +and S3. This third gas thus facilitates the heat exchange between the first and the second + + +11 . + + +7fca^4e«toi2c.a/ + + + +kinds; equality of mean kinetic energy for each kind of gas is then the condition of thermal +equilibrium. Since experience shows that the condition of thermal equilibrium of two +bodies is independent of the nature of the body that facilitates the heat exchange, then the +assumption made in §7, that equality of mean kinetic energy is the condition of thermal +equilibrium when the heat exchange takes place in some other way—e.g., through a solid +wall separating the gases—is seen to be very probable. + +The solution of Equations (156) and (157) found in this section is the only one possible, +provided that + + +U = v = W = Ui = Vi = Wi = 0 + +and that everything is independent of time. However, if one assumes that these quantities +are different from zero, then each of these equations has many solutions, representing +motions in which H does not decrease and hence the total entropy does not increase. One +can have a gas mixture moving with constant velocity in some fixed direction in space. +There are also many other solutions. One sees immediately that, if the wall of the container +is an absolutely smooth surface of revolution, at which the molecules are reflected like +elastic spheres, then + + +d + +dt + + + +reduces to C^(lf) + C$(lf). Then entropy neither flows in nor out of the gas. For the +stationary state we must have clH/dt = 0, hence Equations (147) and also (156) and (157) +must hold. Such a possible stationary state consists however in a uniform rotation of the +entire gas mixture around the axis of rotation of the container, as if the gas were a solid +body. This state must certainly be represented by Equations (154) and (155). If the z-axis is +the axis of rotation, then in this case + +u = U\ - - by, v = v\ = + bx. w = w\ = 0. + +One can then satisfy Equations (156) and (157); / 0 andF 0 will be functions of +Vx 2 +y 2 : and will thus show the density variations created in the gas by centrifugal + +force. Other solutions of these equations, in which t can also occur explicitly, can also be +found. 4 For example, there is one remarkable solution in which the gas flies outward from +a center equally in all directions in such a manner that, first, there is no viscosity; and +second, while the temperature of course drops as a result of the expansion, it drops equally +everywhere in space so that no heat conduction occurs. We shall no longer concern +ourselves with these matters, however; we shall only ask what value the quantity H has in +all these cases. + +If we denote by H' the contribution to H for the first kind of gas, in Equation (144), + + +11J_ + + +Kuiuc- Ma£4cjvi(i£icaZ + + + +then: + + +H' =J J doduflf. + +In all cases where Equation (147) is not violated, f is given by Equation (154). If one +sets, in accordance with Equation (160), + + +then: + + + +The integration over dco = d£di]dC can be carried out at once ; it is to be extended over +all values of £, //, C from - oo to + oo and one obtains, on doing this: + + + +Now mdn = dm is the total mass of the first gas in the volume element. If we multiply +Equation (169) by the mass M of a molecule of the standard gas (hydrogen), further by the +gas constants of this gas, and finally by - 1, and if we again denote by // = m/M the +molecular weight of our gas relative to the standard gas, then : + + +C Rdm + +r / + +/hhn\ + +3 " + +-MKH' = - — + +i(pi + +/— + +1 7. + +J n + +. \ i + +/ + + +Since according to Equations (44) and (51a), + + + + +11 . + + +mJuc. 7fca^4e«tai2c.a/ + + + +then: + + +_ 3 3 R + +c 2 = — = — T +2 hm p + + +) = lipT-w) +1 + + + +miw + + +) + + + +the last logarithm is however a constant. Further, + + + +Rm + +M + + +which is a constant in any case. If one combines all the constants, then it follows that: + + + +-MRH + + +"/ + + +Rdm + + +Hp-'P' 2 ) + Const. + + +According to Equation (58), however, this is just the sum of the entropies of all the +masses in all the volume elements, and hence it is the total entropy of the first kind of gas, +and one sees from Equation (144) that one simply adds the entropies of the two parts of the +gas mixture. Neither a progressive motion of the gas nor the action of external forces has +any effect on the entropy, as long as Equations (147) hold, and therefore the velocity +distribution in each volume element is given by Equations (154) and (155). Thus we have +completed the proof—which was given only incompletely in §8—that H is identical to the +entropy, aside from a factor -RM which is constant for all gases, and an additive constant. + + +§20. General form of the hydrodynamic equations. + +Before we pass to the consideration of further special cases, we shall develop some +general formulas. Since u, v, w are the components of the velocity with which the first gas +moves as a whole, one easily sees that during time dt there will flow through the two faces +perpendicular to the abscissa axis of the elementary parallelepiped dxdydz the amounts +pudydzdt and + + +d{pu) + +pu +- dx + +dx + + +dydzdt + + +ii. + + +jMcifte, 7fca^4e«toi2c.a/ T^h.y.A.Lc.A. + + + +of gas, respectively. + +The sum of these quantities, added to the total amount of gas that flows through the +other four faces, is the total increase + + +dp + +—dxdydzdt + +dt + + +of the amount of the first kind of gas in the parallelepiped; whence it follows that: + + + +dp d(pu) d(pv) d(pw) + + +—h + +dt dx + + +H-h + +dy + + +dz + + + +This is the so-called continuity equation. If one imagines an equal parallelepiped +doadxdydz moving in space at a velocity whose components are u, v, w, then during time +dt the coordinates of the molecules contained therein will increase on the average by udt, +vdt, and wdt. The average acceleration is therefore : + +du du du du + +— + U— + V — + W— • + +dt dx dy dz + +if + +YjM = pdxdydz + +is the total mass of these molecules, then their total momentum measured in the abscissa +direction increases by + + +/du du du du\ + +(172) — -f w — + t>— \-w— • pdxdydz, + +\dt dx dy dz] + +This increase in momentum will be in pail created by the external forces acting on the +total gas mass Yjn , whose components are + +If only one kind of gas is present, the total momentum will not change, because of the +conservation of the motion of the center of mass in collisions; however, it will be changed +by entrances and exits of molecules in do. If we denote by tj, C the velocity components + + +11 . + + +^PuJuc. 7fca^4e«tai2c.a/ + + + +of some molecule, and set (see Eq. [158]) £ == n_|_£, 7] — f = + +then t), J arc the components of motion of the molecule relative to the volume + +element do. If in unit volume there ar efdu) molecules whose velocity points lie within dco, +then there will enter through the left of the side surface, facing the negative abscissa +direction, of the parallelepiped do, during time dt, + +ifdudtdydz + + +molecules, whose velocity points he within da>; these transport momentum + + +mi(u + i)fdudtdydz + + +into the parallelepiped. Since f“f+t + + + +so that the total momentum transported through the left side of the parallelepiped do is + + +mdydzdtf ffdu = P, + +where the integration is to be extended over all volume elements doj. +\fdoj is the total number of molecules in unit volume, hence + +mjfdos = p + +is the density of the gas. The quantity + +J ffdu +Jfdu + +we call the mean value ^2 of ah £ 2 . Hence + +P = pfdydzdt. + + +ii. + + +7fca^4e«tai2c.a/ + + +Through the opposite face, momentum + + + +dydzdt + + + +will be transported. Similarly one finds that through the sides of do perpendicular - to the y- +axis the amounts + + +pfldxdzdt + +and + + +- , )) , +pW + —— dy +9y . + + +dxdzdt + + +respectively of momentum in the abscissa direction will be transported. If one applies the +same considerations to the last two sides, and finally sets the total increase (172) of +momentum in the abscissa direction equal to the sum of the total momentum transported +and the increase caused by the action of external forces, then it follows that: + +du du du\ + ++ w—b v — f- w — +dx dy dz) + +d(Pt 8 ) d(pfl) d(ffl) + +dx dy dz + +with two similar equations for the y and z axes. These equations, as well as Equation (171), +are only special cases of the general equation (126) and were derived from it by Maxwell +and (following him) by Kirchhoff.* One can see this in the following way: + +Let ip be an arbitrary function of x, y, z„ 6, >h C 6 which may be equal to the function +earlier denoted by

M d(m) dirt?) + +dx dy dz + +Since X, Y, Z are not functions of £, //, £ it follows from Equation (130) that: + + +mB,{ip) + + +dip + += p X- + +L a? + + +dip dip dip’ + +-+Y-+Z- . + +d% dt\ df. + + +If one collects all these terms, then Equation (126) reduces in this special case to: + + + +aW + a(pfr) a(w) a(pM + +dt dx dy dz + +dp dp dp 1 r . + +- P I-+f- + Z- = m[5,W + + +. a$ + + +From this equation Maxwell calculated the viscosity, diffusion, and heat conduction, +and Kirchhoff therefore calls it the basic equation of the theory.* If one sets cp = 1, he +obtains at once the continuity equation (171); for it follows from Equations (134) and (137) +that 5 4 (1) = B 5 ( 1) = 0. Subtraction of the continuity equation, multiplied by cp, from (177) +gives (using the substitution [158]) ; 5 + +dp dp dp dp d(plp) + +p- + pu~ + p v~ + pw- + - + +dt dx dy dz dx + +( 178 ) ■ + ++ + +,= m[/W) + + + +aW) aw) + +dy dz + + +dip dw dip + +-P X-+Y-+Z- +- d£ dij df + + +We 7fca^4e«toi2ea/ “Pity. i Ze i. + + + + +If one has only one kind of gas, then B^icp) always vanishes. If one substitutes in the +above equation : + + + + +then + + +

any function ofx, y, z„ and 1. ( d4>/dt)dt is the increase experienced +by the value of this function at a point A fixed in space during time dt. Now let the point A +move with a velocity (w, v, w) equal to the total velocity of the first kind of gas in the +volume element do. During the time dt, A becomes A'. If one now substitutes for the value +of the function ® at time t+dt at A' its value at time t at A, and divides the difference by the +elapsed time, then he obtains + +d4> d d + +-b u - b v -b w — > + +dt dx dy dz + +which value we shall denote by d

)) a(pfj) + +H-1- + +dz + + + + +dj\ dx + + +dy + + +(dv df dv df dv df' + ++ P ( —I- + —+ — + +\ dx dt) dy dt) dz dt) + +_ df/ d{pl t)) d(p^) d(pt)g) + +dt) \ dx dy dz + + +(dw df dw df dw df’ + ++ p (— r—+ —+ —* — + +\dx dl dy dl dz dl + +_ df / d(p») d(pt)$) d(p$ 2 ) v + +dj\ dx d?/ dz + + +If one sets + + +here, then, since £ = Q, + + +k^ie. 7fca^4e«tai2c.a/ !Pl±.y. i .Zc. 1 + + + +(189) + + +„ , n d ? , , + +mB S (f ! )=p —+ — + —— + + +at dx dy + + +dz + + +f _ du -du _ dv> + ++ 2p(f ! - + fl)- + {j- + +dx dy dz/ + + +If one sets f = then + + +(190) + + +d(«) % J t)) a(pf? J ) a(pfi)j) + +«fli(n) = p—+—+—+ + + +dt + + +dx + + +dy + + +dz + + +du _du —du —dv —dv _ dv + ++ p]r>) - + + >18- + ?- + a- + nr}> + +dx d?/ dz dx 5?/ dz + + +which is exactly correct. + +If we now make the assumption that the state distribution corresponds approximately to +Maxwell’s law, then Equations (181) are approximately correct. In addition, + +fi + +always rapidly approaches the Maxwellian distribution. Any mean value that vanishes for +the latter distribution will therefore be quite small, as we shall see in the next section when +we consider the effect of collisions explicitly. In this approximation, Equation (189) +transforms (taking account of Eq. [186]) to + + + +. . . = Q, for as a result of collisions the distribution of states + + +*(’) + +\p / du + +(191) mBi(f) = p —-— + 2 p - • + +dt dx + + +We now construct similar equations for the y and z axes, add all three equations +together, and recall that + ++ AjO) 2 ) + Bbil 2 ) - +1) 2 + 1 2 ) = 0 + +since the total kinetic energy of two molecules is not changed by a collision. Thereby we +obtain: + + +We 7fca^4e«toi2ea/ lPl±y. i .Ze l + + + + +0 , + + +it) + +\ p / /du dv dw\ + +3 p — -h 2p ( —I-1-) + +dt \dx by dz) + +or, using the continuity equation (184), + +4 -) + +\ p / 2 p dp dp 5 p dp + +3p-- - = 3--- - = 0. + +dt p dt dt p dt + +The integration gives, when one follows the gas mass in a volume element along its +trajectory, pp~ 5/3 = constant, the well-known Poisson relation between pressure and +density.* Conduction of heat is neglected here. In general we do not know anything at all +about heat radiation. The ratio of specific heats is, in the case considered here, 5/3. Since +the internal state of the gas is nearly the same as that of one in thermal equilibrium, moving +with unifoim velocity components u, v, w, the Boyle-Charles law is valid. Thus p = r p T, +hence Tp _5/3 = constant. Any compression is connected with an adiabatic temperature +increase, and any expansion with a temperature decrease. + + +* See Part II, chap. vi. + +1 b is the smallest possible distance of the two colliding molecules that could be attained if they +moved without interaction in straight lines with the velocities they had before the collision. In other +words, b is the line P\P, where P\ and P are the two points at which m\ and m would be found at the +moment of their closest approach if there were no interaction. Thus e is the angle between the two +planes through the direction of relative motion, one parallel to P\P and the other to the abscissa axis. + +* Maxwell, Phil. Trans. 157,49 (1867). + +t Kirchhoff, Vorlesungen ilber die Theorie der Wcinne. + +$ See also Benndorff, Wien. Ber. 105,646 (1896) for calculations of Bftp) + +2 One sees at once that on this assumption a completely smooth flat wall would experience no +resistance to its motion in a direction lying in its own plane. + +3 We understand as usual by S an arbitrary surface completely enclosing the gas, which can +everywhere be taken as close to the walls as one wishes, and the integral over do extends only over all +volume elements within this surface, while that over dS extends over all surface elements. We denote by +K'dt the number of molecules that in time dt go out through the surface S in excess of those that come in; +by L'dt, the decrease in the number of molecules lying within the surface S, so that we always have K' + +L' =0. However, K' and L' are not identical with the quantities called K and L in the text, since in +calculating (d/dt) X (JK q If we have followed each molecule along its path during time dt. The sum has +therefore included the same molecules at the beginning and end of dt, and the difference of these two +sums was divided by dt. We thus assume that the volume element do moves along with the molecules +being considered, and that these same molecules always remain within the surface S. This is not the case +as soon as S does not move with the molecules. We shall always sum over the same space element at the + + +We 7fca^4e«toi2ea/ TRk.y l i .Ze l + + + + +beginning and end of dt, so that (d/dt) E (jJ 0 If is simply the expression given by Eq. (120), integrated +over do and du>, in which of course If is to be substituted for cp. Then one obtains, on substituting the +values (121-125): + + +(145a) + + + + + + +*1 + +dri + + +df' + + +C4 (If) + C5 ( If) are the same quantities as before. The first term of the double integral is likewise the +same as in Eq. (145), and therefore it is equal to K. Also, the last three terms can, by integrating by parts +with respect to f, ij, (, be put in the same form as the corresponding terms inEq. (145). By direct +integration of the fifth term with respect to f, the sixth with respect to //, and the seventh with respect to + +^ QB + +f, these can also be reduced to zero, since///' vanishes for infinite f, //, or ( (since must be + +finite). The sum of the second, third, and fourth terms of the double integral (145a) gives (since d(flf-f) += lfdf) on integrating over x, y, and z, the two surface integrals + +/ / dodSJN - j f dodSNfl} + + +both to be extended over the surface S (which is now considered fixed). The first is the quantity +previously denoted by K; the second represents, when it is multiplied by dt , the amount of the quantity H +(defined by Eq. 144) that is carried into the surface S by the motion of the m-molecules, in excess of that +carried out. There cannot be any creation of //inside the gas, any more than in the case considered in the +text. The total amount of H contained within the surface S can only increase by an amount less than, or at +most equal to, the amount of this quantity carried into the surface from without. + +The quantity -H, which is proportional to the entropy, will never be changed when visible motion is +produced, or its direction is changed by external forces, or any other change occurs, as long as no +molecular motion arises thereby through collisions. Even if initially one gas is in one half and another +gas in the other half of the container, the entropy is not changed by their progressive motion. The +mixing of course leads to a more probable state, but the velocity distribution is less probable, since each +gas has an average motion in a particular direction. As soon as this average motion is annihilated by +collisions (transformed into disordered molecular motion) then H decreases and the entropy increases. + +* Bryan and Boltzmann, Wien. Ber. 103,1125 (1894). + +4 Boltzmann, Wien. Ber. 74,531 (1876). + +* Maxwell, Phil. Trans. 157, 49 (1867); Kirchhoff, Vorlesungen iiber die Theorie der Wdnne +(Leipzig, 1894). + +5 For a better understanding of §3 of the 15th lecture of Kirchhoff’s Vorlesungen we make the +following remark: + +Since cp is a function only of A //, (, it transforms by the substitution (158) into a function of + +f -t-U, t) ~\~V, J + W, and therefore + +dtp dip dip + +d £ dll dt + +when in the last two derivatives

(w+jr, v-l-1), w +1) and taking partial derivatives with respect to u, holding +constant the mean values of the coefficients containing £, t), these coefficients do not need to be +considered as functions of u, v, w, or their derivatives with respect to the coordinates. Nor are u, v, w to +be regarded as functions of x, y, z- Then + + +hence also + + +dp + +du + + +dp + +du + + + + +dp + +di + + +dp + +du + + +The same holds of course for the other two coordinates. + +6 As Poincare (C. R. Paris 116,1017 [1893]) remarks, the derivatives of

y, ^ j . In the following equations, however,^ is a function ofy^ j and + +is the expression obtained when one substitutes, in the expression (137) for j). Di> }i) ancl 50 for,h Since r'. >>'• + +i'. Jl , l)i , }l are given functions of fl, J, f,, (),, + +integrations over the latter 8 variables directly. + +* S. D. Poisson, Ann, chim. phys. 23,337 (1823). + + +J 1 , b and e, one can carry out the + + + +13 , + + +mJuc. 7fca^4e«tai2c.a/ + + + +CHAPTER III + + +The molecules repel each other with a force inversely proportional to + +the fifth power of their distance. + + +21. Integration of the terms resulting from collisions. + + +We now consider cases where Equations (147) are not satisfied; in order to be able to +calculate the values f, rf, C of the variables after the collision as functions of their values +before the collision, we must now consider the collision process in more detail. + +We imagine a molecule of mass m (the m-molecule) to enter into collision, i.e., into +interaction, with another molecule of mass in | (the ni\ -molecule). At any time t, let x, y, z, +be the coordinates of the first and x\, y\, z\ those of the second molecule. The force that the +two molecules exert on each other will be a repulsion in the direction of their line of centers +r, whose intensity ifir) is a function of r. The equations of motion are: + + +(191a) + + + + + +ii - x + +- } + +r + + + +x - X\ +r + + +with four similar equations for the other coordinate axes. + +In order to find the relative motion of the two molecules, we construct through m\ a +coordinate system whose axes remain parallel to the fixed coordinate axes, but move +parallel to themselves in such a manner that they always go through the in |-molecule, +which is therefore at any time the origin of the second coordinate system. The coordinates +of the m-molecule with respect to this second coordinate system, and thus its coordinates +relative to the m 1 -molecule, are: + +a = x - xi f b = y - y h c = z - z\. + +If one substitutes + +mmi 1 1 1 + +SO? =- f hence — = —\— > + +m + mi 2R m mi + +then he finds easily from Equations (191a) + + + +13 , + + +“PiLh-d. 7fca^4e«tai2c.a/ + + + +with two similar equations for the other two coordinate axes. Since also we have +r 2 = a 2+ b 2 + c 2, these equations represent just the central motion that the m- +molecule would perform if its mass were W and it were repelled by the fixed in \ -molecule +with the same force dir). Therefore we need to discuss only this latter central motion, +which we call the relative central motion, or the central motion Z. It always takes place in +the plane that includes ni\ and the initial velocity of m, which we have already called in +§16 the orbital plane. The initial velocity of the m-molecule is considered to be that velocity +which it has at a great distance from m\, before the collision, and which we already +denoted by g in the same section. The line g drawn from the fixed in | -molecule in Figure 7 +will represent it in magnitude and direction. Its extension in the opposite direction is to be +called ot 0. We specify the position of m at any time t by its distance r from in | and by the +angle ft which r forms with m0. The work performed from the beginning of the co ll ision +up to time t by the force dir) is: + + + +Fig. 7 + + + +The integration can begin at r = oo since dir) = 0 for distances greater than the sphere of +action. For the present we consider only the central motion Z of the m-molecule around the +mass since we know that the actual motion of m relative to in \ is exactly equivalent. +For this central motion Z, the kinetic energy before the collision is j 2’ but at time t it +is: + + + + +fdr\ l + +l\7i) + + + + +The equation of energy conservation for the central motion Z therefore reads: + + +13 , + + +7fca^4e«tai2c.a/ + + + +(192) + + +ski r / and p are increasing, one must choose the positive sign of +the square root until it vanishes. We now specialize the function xp in order to carry out the +integration, by substituting: + + +(194) i(r) = -4 • + +pn+1 + +This is the repulsion between an m-molecule and an m 1 -molecule at distance r. At the +same distance the repulsion between two m-molecules will be equal to frf|/r" +1 , while that +between two /U|-molecules will be 1 • + +Then: + +K 2 R 2 K(m + m\)p n + +R = —> —=—-— • + +nr n 3 )lg 2 nmmig 2 b n + + +n + + +7fca^4e«tai2c.a/ + + +If we substitute: + + + + +b = a + + +K(m + mi) + +L mmi0 2 + + +l/n + + +then: + + + +In order to avoid all discussions about the value which the quantity under the square +root sign may have, we assume that the force is always repulsive, so that dir) is always +positive, and hence R and 2pP/na n are also positive. Since, according to Equation (193), /3 +always increases with time, and since the square root cannot change its sign without going +through zero, we see that p must always increase up to the point when + + + +The smallest positive root of this equation we denote by p(a). For given n, it can be a +function only of a. When n is positive, as we assume, then moreover p- + 2ff/na" can be +equal to unity for only one positive value of p; hence Equation (196) has no other positive +roots. For p = p(a), the moving body reaches that point A (the perihelion) of its orbit at +which it is closest to m \, and at which its velocity is perpendicular to r. Since the quantity +under the square root sign would become negative if p increased any further, and constant +p corresponds to a circular orbit (which is impossible for a repulsive force), then p must +again decrease; hence the root must change its sign. Because of the complete symmetry of +the problem, a congruent branch of the curve will then be described, which is the mirror +image of the pail described up to that point (with respect to a plane through ni\A +perpendicular to the orbital plane). The angle between the radius vector p( a) = in \ A and +the two asymptotic directions of the orbital curve is: + + + +It can also be computed as a function of a, as soon as n is given. 2& is the angle + + +13 + + +7fca^4e«tai2c.a/ + + + +between the two asymptotes of the orbital curve, and therefore between the lines along +which (in its motion relative to m\) the m-molecule approaches the m \-molecule before the +collision and recedes from it after the collision. (The foimer line is opposite to the direction +of motion of the molecule before the collision; the latter is in the same direction as its +motion after the collision.) + +The angle between the two lines g and g', representing the directions of the relative +velocities before and after the collision, is n -2 &. (These are the line DC and the extension +of BD beyond D , in Figure 7.) + +When each of the two colliding molecules is an elastic sphere, only one modification is +required in Figure 7. The sum of the two radii is m\D = a. The m-molecule moves relative +to m | not in the curve BAC but rather in the broken line BDC\ for b < a, we have: + +b + +(198) d = arc sin — + +<7 + + +but for large values of b, &= tt/1. + +We now imagine, in Figure 8, a spherical surface of center in | and radius 1; it is +intersected at G and G' by two lines drawn from m | parallel to g and g', and at X by a line +through m 1 parallel to the fixed abscissa axis. Then the greatest circular arc GG' on this +sphere is equal to rr -2d. + + + +The angle e has been defined in §16 in the following way. We draw through m \ a plane + + +13 , + + +Pi£^ie 7fca^4e«tai2c.a/ + + + +E perpendicular tog. Further, we draw through m | G two half-planes, one of which +contains the line b, and the other the positive abscissa axis. The first we called the orbital +plane, e was then the angle of the two lines in which these two half planes intersected the +plane E\ therefore also the angle of the two half-planes themselves, or also the angle of the +two great circle arcs GX and GG’ on our sphere, where one is always supposed to +understand by “great circle arc” the largest circular arc that is smaller than re. + +For the spherical triangle we have: + +(199) cos (G'X) = cos (GX) cos (GG 1 ) + sin (GX) sin (GG 1 ) cost. + +Now however: + +4 .GG'= * - 2 .J, g' cos (G'X) = +g cos (GX) = {- h, g sin (GX) = vV - ft - h) 2 , + +where the positive sign of the root is to be taken, since GX < re. + +If we multiply Equation (199) by the magnitude g = g' of the relative velocity before or +after the collision, then: + + +- {/ = ft - ii) cos (ir - 29) + Vff 2 - ft - £i) 2 sin 29 cos c. + +If we multiply this equation by rri\ and add it to the equation + ++ Mi{i = + Wifi = (m + + mifi - m{, + + +then: + +mi _ + +(200) ?*{+- [2fti- {) cos 2 1 ?+V 0 2 - ft - h) 2 sin 29 cos t ]. + +m+mi + +If only one kind of gas is present, then we have to put m\ = m, K = K\. Then: + +(201) (' = { + ({!- {) cos 2 # + \/g 2 - ((- (i) 2 sin d cos $ cos«. + +If we again denote £ - u, £ - u, rj - v ... by . . . then we obtain for + +£, t), l an equivalent equation: + +(202) j' = | + (u - j) cos 2 d + y/g 2 - (f - Ji) 2 sin t? cos cos t. + + +13 , + + +7fca^4e«tai2c.a/ + + + +In order to find , we have to integrate the quantity + +(f ' 2 + fi 2 - f 2 - + +over e from zero to 2jt. The entire orbital curve thereby remains unaltered. Then we have +to integrate over b, keeping t, 9 J.ti, Pi. Ji constant. Then follows the integration +over these quantities. Since the equations (201) and (202) are equivalent, we can obtain the +expression for B 5 (£ 2 ) by simply writing f, tj, C for £ in Bdl 2 ) + +Leaving out the terms containing the first power of cos e, we get + +j' 2 - f = 2(jif - f) cos 2 d + (|i - j) 2 cos 4 d + ++ i [(7 2 “ (f - hY] sin 2 2d cos 2 e + +2 2 2 2 2 2 += (f i - j) cos d - p sin d cos d + ++ (g 2 - p 2 ) sin 2 d cos 2 d cos 2 e. + +The components of the relative velocity with respect to the coordinate directions will be +denoted by p, (J, T, so that + + + +P = (- h = f - fi + +q = n - 1)1 = t) - h + +t = f - fi = i ~ Ji. + + +If one forms the expression +permuting J and T l, it follows that: + + + +/2 2 2 2 2 2 2 ++ fi - f = 2 r(g - 3p ) sin d cos d. + + +Since at the moment we are considering only one kind of gas, we must set m \ = m and +K = K]_, and one has according to Equation (195): + + + + +2k, V' + + +g 2/n a. + + +\ m / + + +13 , + + +7fca^4e«tai2c.a/ + + + +Since t, t>, it ?i, >h, Ji and hence also g are considered constant in the integration +over b and e, which now concerns us, it follows from this that: + + + + + +1 /» + +g~ 2ln da. + + +b)bdbdt + +12Ki\ 2ln r * + += 2ir(g 2 - 3p 2 ) (-) (?) , then he obtains under the integral sign a + +factor g 1_ ( 4/ 'd; therefore since n must in any case be positive, there will in general be a +negative or fractional power of g, which makes the integration very difficult. Only for n = 4 +does g drop out completely, and the integration becomes relatively easy. Since we have set +the repulsion between two molecules equal to K/i Jl+] , this implies that two molecules repel +each other with a force inversely as the fifth power of the distance. One then obtains, as we +shall see, a law of dependence of the viscosity, diffusion, and theimal conduction +coefficients on temperature which for compound gases (water vapor, carbon dioxide) +appears to be in good agreement with experiment, though not for the most common gases +(oxygen, hydrogen, nitrogen). Other phenomena from which one could infer such a force +law are not known. We certainly do not wish to assert that gas molecules actually act lik e +point masses between which there is a repulsive force inversely proportional to the fifth +power of the distance. Since it is here only a question of a mechanical model, we adopt this +force law first introduced by Maxwell,* for which the calculation is easiest. 1 Moreover, +with this law the repulsion increases so rapidly with decreasing distance that the motion of +the molecules differs little from the motion of elastic spheres (aside from glancing +collisions, which are of little importance). In order to show this, Maxwell" has published a +very instructive figure* in which the paths of the centers of a number of molecules are +shown, projected towards a fixed molecule with the same velocity, and repelled by various +force laws. In order to compare these paths with the path that would be followed by elastic +spheres, we can do the following: we imagine in Maxwell’s figure a marked circle whose +center is S, and whose radius is Maxwell’s dotted line, so that its radius is the smallest +distance to which the centers of the two molecules approach for each force law. Now if the +molecules are elastic spheres, whose diameter is this smallest distance, and if we hold one +of them fixed and throw the others at it (not all at the same time, of course, but one after + + +Hence + + +(205a) + + +[TV+h + +J n J n + + +” l “ + + +13 , + + +7fca^4e«toi2c.a/ + + + +another so they won’t interfere with each other) then Maxwell’s figure undergoes the +following modification. The center of the fixed molecule is still S. The centers of the +moving ones will come from the same directions as in Maxwell’s figure, but will be +reflected like very small elastic spheres from the marked circle. + +One sees that the resulting paths for elastic spheres are indeed quantitatively different +but not essentially qualitatively different from those following from the new Maxwellian +law. + +In the following we set n = 4 with Maxwell. Then it follows from Equation (205a) that: + +f 2 ' ,2 ,2 2 2 / Kl A 2 + +(206) (?' + tl - * - h)9bdbd* = A/-—(9* - 3p 2 ), + +Jo Y 2m g + +where + +/» 80 + +(207) At = 4tt I sin 2 1 ? cos 2 a* da + +J 0 + +is a pure number. 3 + +According to Equation (197): + + + +The upper limit is the only positive value for which the quantity under the square root +sign vanishes. & is therefore expressible in terms of a complete elliptic integral and a +function of a. The integral (207) was evaluated by Maxwell by mechanical quadrature. He +obtained the result:* + + +(209) A* = 1-3682 • • • + +We now have, according to Equation (137), + +(210) Bi(| ! ) =jJJf j (f ! + h - l ~ hjfftfbdostkudbdt. + + +14 + + +IRuJix*. 7fca^4e«tai2c.a/ “Ph-ySALcS. + + +Substitution of (206) gives: + + + +(211) Bt(f ') -V|i.//V- 3p‘)//i<^i. + +We have: + +g — 3t> =i) + i)i + f + fi - 2{ - 2{i - 2t(i)i - 2!"fi + 4{{i += >) 2 + >)i + l + Ji - 2|* - 2fJ - 2l)p, - 2jji + 4f|i. + + +In the integration over d we can take 7), //|, C] or ri. Pi, 8i outside the +integral sign. According to Equation (175): + + + +Since, however, the two colliding molecules are equivalent—or, if one prefers, since +one can give any labels to variables over which one integrates in a definite integral—we +also have: + + + +and since y = l) = ^ = ()• wc have: + + +(213) + + +Btf) = + + +2m 5 + + +dipV + f 2 - 2{ 2 - rv - M + 2H) + + +j 2 -2f 2 ) = + +2m* + +K _ _ + +—AiP 2 (c 2 - 3f 2 ). + +2m* + + +c = V? 2 +t) 2 +i 2 is the total velocity of a molecule relative to the average motion of + + +14 + + +7fca^4e«tai2c.a/ + + + +all molecules in the volume element. + +The quantity b.(») is calculated by Maxwell by a coordinate transformation. We +imagine new x and y axes placed in such a way that the old x and y axes are rotated by an +angle X in the xy plane. We denote the corresponding quantities in the new coordinate +system by capital letters: + +j = I cos X — $ sin X, 1) = 9 cos X + I sin X, +p = $ cos X - 0 sin X etc. + +If we substitute these values in Equation (206), we obtain the same terms with factors of +cos X, cos X sin X, and sin X. If we set X = 0, we see that the former must separately be +equal; if we set X = nil, we see that the latter must l ik ewise separately be equal. Hence the +terms multiplied by sin X cos X on the right-and left-hand sides of the equality sign must +separately be equal. Setting them equal to each other, we obtain: + + + +Since the new coordinate axes are just as good as the old ones, one can again use lower¬ +case letters instead of capitals. If one carries out the further integrations exactly as for +Equation (206), it follows that: + +f 1 C C C* C u + += - J j J J (?v + ii (i - n - fib) + +(214) ■gbff l du)daidbde + +§22. Relaxation time. Hydrodynamic equations corrected for viscosity. +Calculation of B 5 using spherical functions. + +We now have to substitute this value into the general equation (187). We first consider a +special, completely ideal case: a single kind of gas fills an infinite space. There are no +external forces. The number of molecules in any one volume element do, whose velocity +components lie between the limits / and / + d£, rj and rj + dr], C and C + d(, will at time t = +0 be equal to /(T, rj, C, Q)dodfdi]dC, where the function/ is the same for all volume + + +7fca^4e«taiZc.a/ ~Ph-y-&XjcS- + + + +elements. For any later time t, let this number be equal to/(T, //. C, t)dodfdi}dC. Since all +volume elements are subjected to the same conditions, rj, C, t) also has the same value +for all volume elements. If + + + + +where a, h, u, v, w are constants, then we would have a gas in which Maxwell’s +distribution holds, but which moves through space with constant velocity components u, v, +w. Then we would have + + +(S - «) 2 = (v - v)* = (f - v>y, ({- «) (»I - v) = (( - u) (f - w) + + += (7)-v)(f-u>)=0. and the distribution of states, seen from the viewpoint of an + +observer moving with the gas, would not change with time. >j, C, 0) is some other + +function of T, C, then at the initial time a velocity distribution differing from Maxwell’s, +but still the same in each volume element, holds. This distribution changes with time, but +the components of the visible motion of the gas + + +- /{/<*<■> - Jnfdu - Itfd + +« = £ = T7T’ » = V = T—’ W = f = + + +w + + +ffd* + + +J }du + + +Jfd « + + +naturally do not change with time, because of the conservation of the motion of the center +of mass. If we again set £ — U = 7) — V = — W = fa then in general now + + +f 2 - t) 2 , f 2 - i 2 , >) 2 - J 2 , P), n “d t)j + + +differ from zero, and we ask ourselves how these quantities change with time. First, since +nothing depends on x, y, or z, it follows that, from (188): + + +(215) + + +3f + += f). + +dt + + +If one now substitutes j OF then it follows with the help of Equations + +(213) and (214) that: + + +df /Ki _ _ dp} /Ki _ + + +Similarly, to the first of these equations, we have: + + +14 + + +‘Pu+lc- 7fca^4e«toi2c.a/ + + + + +and consequently + + +/K, _ + + +Since everything is independent of x, y, z, the differential quotients with respect to t are +to be taken in the usual sense. Since furthermore all volume elements are equivalent, +exactly as many molecules will flow in through each lateral surface as flow out through the +opposite surface. The density p must therefore remain constant. Hence integration of these +equations, using the index 0 to characterize the values at time zero, gives the results: + +f - t ) 2 = (f$ - + +Multiplication by p yields (recalling the notations [179]): + +X,-Y t = (Xl - + +Similar equations follow of course for the other coordinate axes. In the simple special +case now considered, the difference in the normal pressure in two different directions (e.g., +X x - Yy) and lik ewise the tangential force (e.g., X y ) simply decreases with increasing time in +geometric progression. The time after which it becomes e times smaller is the same for all +these quantities, and is equal to + + +d{f - t) 2 ) +dt + + + + +1 /2m> + +u#VT, =T ' + +Maxwell* calls this the relaxation time. We shall see that it is very short. + += Ptf + +but these quantities are still approximately equal. We shall therefore calculate their +deviations from a quantity nearly equal to one of them. For this purpose we choose their +arithmetic mean. Since, according to the assumptions necessary for the validity of +Equations (181), this is equal to the quantity denoted by p, we shall again denote it by p, +and therefore put + + +We now return to the general case. In general we no longer have pj-2 + + + + +14 + + +7fca^4e«toi2c.a/ + + +(217) + + +V = 7 (f - 1) 2 + f) = 7 C*. + +O U + + +If we denote the right-hand side of Equation (189) by T and substitute the value (213) +on the left-hand side for B*(t) , then it follows that: + + +(218) + + +_ - 1 /2m* + +c 2 - 3| ! = — \ —t. + +Ki + + +We look for the small difference between the two quantities ^2 _ ^2 +»> 2 +i 2 and +3r 2 This, and hence the right-hand side of the above Equation (218), are small first-order + +infinitesimal quantities; hence we need to retain on the right-hand side only the terms of the +largest order of magnitude. The terms of smaller magnitude are also smaller than ^2 — 3t 2 +. In the expression for £ we can therefore set + + +f>? = pt) ! = Pi 2 = p, ri = fj = uj = p = ti) 2 = fj 2 = 0. + + +We saw that then (see Eq. [191]) + + +r = p + + +,p/ du + +— + 2p-. +dt dx + + +We wish to find j2 and thence X x and its dependence on the instantaneous state; we must + +therefore eliminate the term that contains a time-derivative. This is easy, since we find to +the same degree of accuracy + + +i V - + + +2p / du dv dw \ +3 \dx dy dz) + + +P«.7ie 7fca^4e«taiZc.a/ + + +Hence in first approximation + + + + +2p / du dv dw\ + +r = —(2 -. + +3 \ dx dy dz / + + +The following terms in the expression for X provide terms of smaller magnitude in +c*-3r’. and therefore we can ignore them. Hence, according to Equation (218), + + + +therefore, since we set + + +pc 2 = 3 P’ + + + + +dv + + + +) + + +_ 2p /2m 3 / du dv + +X, = pf = p - A —(2- + +9iip V KA dx dy + + + +We now wish to substitute the value (214) for B.(n) into Equation (190). On the +right-hand side of this equation we can set p^2 __ p^2 — p^2 _ p for the same reason +as before, and set the terms containing odd powers of£ ? ty, orj under the bar symbol +equal to zero. We thereby obtain: + + +(218a) + + +p /2m 3 / dv du\ + +TiWiy) + + +If one substitutes the abbreviation + + + +v /2m 3 + +-{/— = pr = SR, + +U iP y K x + + +14 + + +aV^lc. 7fca^4e«tai2c.a/ + + +then he obtains the following values: + + + +29?/ du dv dw\ + +^ 3 \ dx dy dz) ) + + +29?/ dv du dw + +P-— 2- + +3 \ dy dx dz + + +( 220 ) + + +29?/ dw du dv\ + +^ 3 \ dz dx dy/' + + +— I dv du\ + + +_ (dw du\ + + +__ (dv dw\ + +Y v = Z u = = - 9?( —+ —). + +\ dz dz I + + +These equations are not completely exact, of course; however, they are one degree more +exact than the equations X x = Y y = Z-=p, X y = Y x = X z = Z x - Y- = Z v = 0. Substitution of +these values into the equations of motion (185) yields + + +du dy +P dt dx + + +1 d (du dv dw + +9? Au +-( —H-h — + +3 dx\dx dy dz + + +- pX = 0 + + +dv dy + +(221) \p~ + 7 " + +dt dy + + +1 d (du dv dw' + +9 ? hv -)- 1 —|- 1 - + +3 dy\dx dy dz , + + +- pF = 0 + + +dw dy +P dt dz + + +1 d (dll dv dw' + +9? A w + -— + — H - + +3 dz \ dx dy dy , + + +- pZ = 0. + + +Here ^ is considered constant, which is also not strictly correct, since ^ is a function + + +7fca^4e«taiZc.a/ + + + +of temperature, and temperature changes during compression or rarefaction. However, +since the actual temperature-dependence of ^ is still in doubt, and since the gas moves, in +the case of less violent motions, almost like an incompressible fluid, so that there is no +significant compression or rarefaction, this error is not important. Equations (221) are the +well-known hydrodynamic equations corrected for viscosity.* These equations are satisfied +and one obtains therefore a possible motion, when he sets p equal to a constant and X = Y = +Z = 0, v = w = 0, and u = ay. Each layer of gas parallel to the xy plane moves with velocity +ay in the x-direction. a is the velocity difference between two such layers at unit distance +from each other. One of these layers must clearly be artificially held fixed, and another +must be artificially maintained in a state of constant motion. The tangential force on unit +surface of these layers has the value according to Equations (220), and ^ is therefore +the quantity that we already called the viscosity coefficient in §12. From Equation (219) it +follows that it is proportional to p/p and therefore to the absolute temperature, but at a given +temperature it is independent of pressure and density. The latter statement is also true when +the molecules are elastic spheres, but then ^ is proportional to the square root of the +absolute temperature. Of course the mean free path cannot now be calculated from the +numerical value of since the beginning and end of a collision are not sharply defined; +this value can only provide a relation between the mass m of the molecule and the constant +K | in the force law. It also permits the calculation of the relaxation time 7 - = /p. From + +the value of ^ for nitrogen used in § 12 , we find a relaxation time of about x = 2 • 10 _ 10 +sec at atmospheric pressure and 15°C. + +We now proceed to the calculation of B 6 ( |p 2 ) , etc. There is no difficulty + +in raising the expression ( 201 ) to the third power and then performing the integration, as we +have done for the calculation of Btf) . The same coordinate transformation as before + +gives the values of «.( n*) and W) and the other 5 5 ’s, which contain as +arguments of the function terms of third order in l), and J. B 6 (m) must be found by +a spatial (3-dimensional) coordinate transformation. We shall adopt here another method +which Maxwell, in his paper “On stresses in rarified gases ,” 4 indicated in bracketed notes +added in the last months of his life. + +Any function p of the n’th degree in x, y, z, which satisfies the equation + +d 2 V d 2 fl d 2 p + +— + — + — = 0 + +dx 2 dy 2 dz 2 + +we call a (solid) spherical function of the n’th degree. If we substitute x = cos X, y = sin X +cos v, c = sin X cos v, then it is transformed to a spherical surface function of the/z’th +degree: p (n \X, v ). Further, we denote the coefficient of x” in the power series that arises +from the expansion of + + +14 + + +7fca^4e«tai2c.a/ + + + +(222) (1 - 2ia + xV' 2 + +by P in Hp) (zonal spherical function or spherical function of one argument). Now let G and +G' be any two points on a spherical surface with polar coordinates X, v and X', v' and let G t +be the symbol representing n + 1 arbitrary other points on the same spherical surface. Let +the polar coordinates of G t be /.,■ and v- r Then 5 + +»-2n+l + +(223) p w (X'/) = I cfcm + +i-1 + +where is the cosine of the spherical angle G'G r The c,- are constant coefficients which +can be determined. Now hold G and G, constant, while G' describes a circle such that the +spherical angle GG' always remains constant. Its cosine is called /./. Finally, denote by e the +angle between the great circle GG' and a fixed circle drawn through G. Then: + + +| /» 2t t*2n+l q. /» 2t + +- p«(X', v')di = £ ~ P w (si)dt. + +2?r J o t-i 2tcJ o + + +Furthermore: 6 + + +/, + + +2r + + +PM(si)d( = 2rP <%)•?<»>(*<), + + +where ,v ; - is the cosine of the spherical angle GG,. One has therefore: + +2t i*2n+l + +pM(\',y , )d( = 2rP<*>(n)- E c,P<"»(«.)• + + +f + +J o + + +t -1 + + +As in Equation (223), the latter sum has the value p (n HX, v). One therefore obtains the +formula: 7 + + + + +2r + +p<">(X', v')d« = 2tP M (n)-p M (\, v). + + +We wish to show the application of this theorem to the calculation of B 5 in a special +case, and in particular to calculate fis(jp) + +As before let{ f v , f, ( t , f|, fj, 1)', f', fri {{ be the + + +14 + + +7fca^4e«tai2c.a/ + + + +velocity components of the two molecules before and after the collision; let +X, h, U, hi, X', ti, lit %it I / bet h e same velocities relative to the +average motion of all the m-molecules in the volume element, so that therefore +£ — £ = 7 1 — X) = V * * * etc. where u, v, w are the components of the mean + +velocity of all m-molecules in the volume element. Further, let + +P = i I - lx, q = v - n= P - Pi, r = 1 - fi= J - )i, + +p'= r- ('.=i'- f'., q'=o'- »'.=pi', f = r- r.=f- if. + +be the components before and after the collision, respectively, of the velocities g and g f +respectively of the molecule that has before the collision velocity components <^, //. ( +relative to the other one that has velocity components rjC\ ■ The latter will again be +called the m 1 -molecule, even though it actually has mass m. Finally, we denote by + +u=j+h=l'+?i, p=P+Pi=P'+Pi, >»=j+li=s'+ji + +twice the velocity components of the center of gravity of the system formed by the two +colliding molecules, in its motion relative to the mean motion of all m-molecules in the +volume element. These are equal before and after the collision. Then + +4f p = p q + uq + Dp + utt + +4fipi = p q - uq — Dp + u» + +4fp' = p'q' + uq' + Pp' + utt + +4{ipi = p'q' - uq' - Pp' + utt + + +hence + +(225) 2(f'p' + {'ip/ - fp - fipi) = p'q' - pq. + +We now construct around m\ a sphere of radius 1. The lines through in \ parallel to the +abscissa axis, representing the relative velocities g and g\ should intersect this sphere in the +points X, G, and G' respectively (Fig. 8). X, v and a', v' are the polar coordinates of the +points G and G' (i.e., X and X ' are the angles Xm | G and Xm | G', while v andv' are the +angles made by the planes GmX and G'mX with the xy plane). Since p, Q, T and p / ^ +q'^ ■£* are the projections of g and g' on the coordinate directions, we have + + +15 + + +ifi-c. 7fca^4e«toi2c.a/ + + + +hence + + +p = <7 cos X, q = g sin X cos v t r = g sin X sin v , +p' = 0 cos X', q'= 0 sin X'cos /, v'= g sin X'sin v, + + +pq = pV s) 0S »)> PV = ff 2 p (s) (X', /), + +where // 2 \A, v) is the spherical function cos X sin X cos v. Earlier we denoted by e the +spherical-triangle angle XGG', and by tc - 2d the angle Gm\G'. Then, according to the +theorem cited on spherical functions, + + + + += 2irp<»M •?%), + + +where /j = cos(tt - 2d). On expanding Equation (222) one finds: + +P (2) {n) = in 2 - 2 = 2 cos 2 (2$) - \ = 1 - 6 sin 2 cos 2 + + +Hence + + + ++ = - irgY 2) {\ r) *6 sin 2 i> cos 2 t> + += - 6 x|)qsin 2 0 cos 2 tf. + + +Whence follows, by comparison with Equation (208), + + + +- fp - fipi)<2« = - U, + + + + +- |p - (ith)gbffiduduidbd( + + + +15, + + +7fca^4e«tai2c.a/ + + + +whence finally, according to Equation (212), we obtain: + + + +Since Equation (226) is valid for any spherical function of the second degree, it follows +in general that: + + +Btif'Kh »)] = - 34*« p ( »(f, 4 ) + +V 2m 6 + +e.g. B t {f - If) = - 3 V J (d -if). + +V 2m 6 + +When/is not a function ofv, y, z, andX = F = Z = 0 (and the effect of the wall +vanishes) it follows from Equation (188) that + +(227) p-- = + +at + +Hence if f is any spherical function of the second degree, it follows in general that + +(228) | = f oe -jA„v5^r , )i i + +Therefore + +, , 1 /K + +,229) + +is the reciprocal of the relaxation time for all spherical functions of the second degree in +J, tj, and £—i.e., the time in which, by the action of collisions, the mean value of that +spherical function decreases to 1/e of its original value. This con fir ms our earlier result. + +We now pass to spherical functions of the third degree, e.g., r*-3nf By analogy +with Equation (225) we find + + +15, + + +“Pul+lc- 7fca^4e«toi2c.a/ + + + +C+ |C- 1 -6-8tfr + I^-»-!*)] + += 3u(p' ! - q' 2 - p 2 + q 2 ) - 6#(p'q' - pq). + +If we denote the expression in brackets by , then according to the theorem on spherical +functions. + + +/ >2 . , ,2 + + +f +J 0 + + +2r 3ir 3 + += —(up 2 - uq 2 - 2bpq) — (,u 2 - 1). + + +2 + + +2 + + +Now recall that /u 2 - 1 = - 4 sin 2 & cos 2 d. If one substitutes + +u=t+fi, »=P+Pi, P“I _ lii apphes Equation (212) and + +assumes that f=p = j=0. then it follows, on taking account of Equation (208), that + + +(230) + + +l r C C* C + +Bi(f - 3ft) 2 ) = — J J J J tyfigbdaduidbdi + +9 /Ki - — 3pp _ — + + +The equality is vahd for any spherical function of the third degree. In general, + + +(231) + + +3pp + + +P. l)] = ~ — + +2 myi + + +The reciprocal of the relaxation time of a spherical function of the third degree is +therefore + + +3 _i +2 9} ’ + +Any function of the third degree in J, ty, £ can be represented as a sum of spherical +functions of the third degree and of the three functions + +*(f 2 +t} 2 +g 2 ), t)(* 2 +l? 2 -H 2 ), a(f 2 +t) 2 +i 2 ) each multi P lied by some + +constant. The latter three functions are the products of the spherical functions of the first +degree by the expression (t 2 +9 2 +i 2 ) .The relaxation time of these latter products still + + +15 + + +~*Puii.c- 7fca^4e«tai2c.a/ + + + +has to be found. +We have + + +2ti(i\)' + i) + n(n+h+it)-i(i+J + i) + + +- f.(t! + + i!)] = «(p' 2 - p ! ) + #(pV - pq) + t»(pV - pt). + + +If we denote the expression in brackets by T then we have + +r u ru d id + +*dt = + | t ( 2 P 2 - q 2 - d) + t pq + r P r + +J n + + +6 + + +2 + + +2 J + + +3ir(fi ! - 1). + + +(231a) + + +hence + + +(232) + + +/* « nit + +I gbdb I = — [u(2p 2 - q 2 - r 2 ) +J o d o 2 + ++ 3bpq + 3totjr]4 2 \/— + +V m + + +#6fc(? 2 + P 2 + J 1 )] = — j j J J Vffigbduduidbde += - 2.v J ^ (F+tT ! +fT J )=~ (7+fT 2 +fT 2 ) • + +oWlji + + +Therefore: + + +2 vp + + +(233) B.[(f 2 +P 2 +S 2 )p (1) (f, p, (f 2 +D ! +lV l) (l, 1, l)- + +3 mvt + +The reciprocal relaxation time of the product of (t’+P’+i’) by a spherical +function of the first degree is + + +15 + + +7fca^4e«toi2c.a/ + + + +2 p + +3 9J + + +23. Heat conduction. Second method of approximate calculations. + + +We now wish to set ^ in Equation (188) and retain only terms of the largest order +of magnitude, thus neglecting the deviation of the state distribution from that which holds +for a gas moving with constant velocity, so that r , =t)r J =p>)=o Thereby we +obtain from Equation (188): + + + +Since the present approximate calculation is based on calculating the terms as if the +Maxwell distribution were valid, if one writes X), J for //, C then he can apply +Equation (49) provided he writes therein X}, $ in place of T, //, C. Hence + + +Therefore + + +p? = MfY = 3 — i pi* - p. + +P + + + + +If one substitutes f=?» 2 then it follows using the same approximations that + + + + + +15 + + +7fca^4e«tai2c.a/ + + +Since now + + + +then + + +Likewise + + +hence + + +—- f + +?y = f‘-i) , = -> + + +V + +d\- + + +mBfatf) = P + + +dx + + + +mBii? - 3fl) ! ) = 0 + + + +and according to Equations (230) and (232): + +f - 3jt) 2 = f - 3h 2 = 0 + + +(234) + + +p(f + ft) 2 + f j 2 ) = - + + + +15 + + +7fca^4e«tai2c.a/ + + +Whence it follows that + + + +9 » + +f =- + + +p + +d\- + + +2 p dx +similarly it follows that + + +?! — +— > M 2 + + +3 # + + +V +31 - + + += H - + + +2 + + +fa + + + + +j 2 i) = l)j 2 = + + +3 9 } \p) +2 p dy + + + + + +dz + + +These values can be used to carry the approximate solution of Equations (189) and +(190) one step further than has been done up to now. + +We next add to Equation (189) the analogous equations for they and z axes. Now we +have B b (% 2 ) -\~B b (Xf 2 ) 4 -B b (l 2 ) =0 If one takes account of Equation (234) and +the two equations obtained from it by cyclic permutation, as well as the continuity equation +(184), and if one substitutes finally for pr2 = X Xf p Jt) = A ^ etc - the values given in + +Equation (220), it follows that: + + +15 + + +Pu.^.e. 7fca^4e«toi2c.a/ + + + +i - +3 p \p + +2 A + + +v dp 15 + ++_ + +p dt 4 + + +a + + + + + + +2 /aw a^ au>\ 2 /ay aiy\ 2 /aw dwA 2 + +3 \ax ay aj \a« ay/ ax/ + +/aw dv\ r + ++ -+- . + +\dy ax/J + + +H ere Zp/ P = f +t) 2 +j 2 is the mean square velocity of thermal motion of a + +molecule in the volume element do. By thermal motion, we mean the motion of the +molecule relative to the visible motion of the gas in do, the latter having velocity +components u, v, w. pdo is the mass of all molecules in do. + + + +dt + + +is therefore the increment of heat measured in mechanical units, i.e., the increase of the +kinetic energy of theimal motion of all molecules contained in do during time dt. However, +in this case the volume element do does not remain fixed in space; rather, during time dt it +must experience that deformation and progressive motion which is required in order that +each point in it may move with velocity components u, v, w. The same molecules therefore +remain in do, aside from the exchanges effected by molecular motion. The amount of heat +supplied by the latter will then be included in the calculation as that conducted and created +by viscosity. + +From §8 we find that the amount of compressional work added to a gas during time dt is +- pdkl = - pkdi 1/p). In our case, k = pdo and d( 1/p) = -(1 /pr)(dp/dt)dt. Hence the term + + +15 + + +7fca^4e«tai2c.a/ + + + +V dp + +--dido +p it + + +in Equation (236) represents the work done by the external pressure p during dt on do, and +hence the heat of compression produced by the pressure p. If one applies the same +considerations by which the work of deformation of an elastic body is calculated, then he +finds that the last teim with the factor ^ outside the differential sign in Equation (236), +when it is multiplied by dodt, expresses the total work done by additional forces that must +be added to the pressure in order to obtain the forces X x , X y ... given by Equations (220). 8 +This term therefore corresponds to the heat developed by viscosity. The next to last term, +multiplied by the factor 15/4, must therefore represent (if one multiplies it by dodt) the heat +introduced by heat conduction into the volume element. If we imagine the volume element +to be a parallelepiped with edges dx, dy, dz, and draw the x-axis from left to right, the y-axis +from back to front, and the z-axis from below to above, and denote by T the temperature, +by ^ the coefficient of heat conductivity—then, according to the old Fourier theory of heat +conduction (which is established by experiment, at least approximately) + +dT dT dT + +8 — dydzdt. 8 — dxdzdt and 8 — dxdydt +dx dy dz + + +are the quantities of heat that leave the parallelepiped on the left, in back, and below, +respectively; + + +and + + +dT + +d . + +f dT\ + +i + + +—+- + +8 — + +]dx + +dydzdt , + +dx + +dx + +( dx / + +i + +dT + +d , + +{ dT \ + +{ 1 + + +A A + +8 - + +)iy\ + +dxdzdt + + +L dy dy\ dy! + + +r dT d / dT +8 — + - 8 — +_ dz dz\ dz + + +dz + + +dxdydt + + +are the quantities of heat that come in on the opposite sides. The total increment of heat +caused by heat conduction into the parallelepiped do during time dt is therefore + + +159 + + +7fca^4e«tai2c.a/ + + + + +The term multiplied by 15/4 in Equation (236) is small. We can therefore neglect higher +powers of this quantity, and treat the gas as if u, v, w were constant, while ty, 3 arc +given by the Maxwell velocity distribution. Its internal state will then be determined only +by Xj } J- and we can apply the formulae of §7 and §8 as if it were a gas at rest. If r is +the gas constant of our gas, and R that of the normal gas, while m//i is the mass of a +molecule of the latter, then according to Equation (52) we have + + + +R + +-T. + + +Hence the term with the factor 15/4 in Equation (236), multiplied by dodt, has the +following form: + + +15 R[ B( BT\ B / BT\ B / BT + +4 it LcteV Bx / 3y\ dy ) Bz\ Bz j J + +This agrees completely with the empirical expression (237), if one sets + + +dodt. + + +(238) + + +_ 15 fiiR + +4 7 + + +In order to make this independent of the thermal units used, we introduce instead of R +the specific heat. Since we assume there is no intramolecular motion, the quantity /? in +Equation (54) is zero; this equation then gives: + + +hence 9 + + + +7 * + + + + +j + + + +5 + +r x + + +16 + + +7fca^4e«tai2c.a/ + + + +This value is 5/2 times as large as that given by Equation (93) and is about as much +larger than the experimental values as the latter value is smaller than them. One should not +expect quantitative agreement in cases where the assumptions made (e.g., // = 0) arc clearly +not satisfied. Since R, ju, and hence also y v are constants, ft depends on temperature and +pressure in the same way that ^ does. + +We have thus obtained all the formulae accepted by the so-called descriptive theory, +except that a coefficient in the terms representing viscosity, which remains arbitrary in the +descriptive theory, has here a particular’ value. In the descriptive theory + +(p-X,)-(3/29t) is equal to + + +while here it is equal to + + +du / du dv dw\ + +3-€(— + -+—) + +dx \dx dy dz) + + +du (du dv dw + +3-i —|-1- + +dx \ dx dy dz + + + +i + + +Therefore in the descriptive theory, in the expression for X x -p, the compression-dependent +expression + + +du dv dw + +—I-1- + +dx dy dz + +is multiplied by a coefficient that is independent of the coefficient of du/dx, while in our +theory the latter coefficient is exactly three times as large as the former. The same holds for +Y y and Z-. The latter coefficient must indeed here, as also in the descriptive theory, be twice +the coefficient of + + +dw dv + +- 1 - + +dy dz + +in the expression for K_, and thus it is twice the experimentally measurable viscosity +coefficient.* + +In the light of our theory, all these formulas are approximate ones. There is no difficulty +in carrying the approximation further. The equations thus extended to a higher degree of +approximation would not necessarily agree with experiment everywhere, since many of our +hypotheses are still arbitrary, but they will probably be useful guidelines when experiments + + +16 + + +7fca^4e«tai2c.a/ + + + +are initiated. It will be difficult, but not completely hopeless, to test them by experiment, +and it is to be expected that they will teach us new facts going beyond the old +hydrodynamic equations. In order to indicate briefly how the approximation is to be carried +further, we shall substitute in Equations (189) and (190) the values just found. From the +latter, it follows by Equations (214, 235, 220, 52, 238) that + + +(239a) + + +9? + +Xy = Pl t) = - - + + +drt) du du du +P~-\-X y - b ^l/“ + Yt — + + +p L dt + + +dx + +dv + + +dz + + +dz + + +dv dv dv ++ X x —b X y —b X, — +dx dy dz + +2 a /. dT\ 2 d / an a( P m) + +5 dx\ by) 5 by\ bx) bz J + + +If one makes the substitution f=tw in Equation (188), he obtains only terms that +vanish to the present degree of accuracy. One can therefore set + + +mBi{ fi)}) = - + + +3 P — + +n' m + + +equal to zero, and therefore also + + +b(m) + + +dz + + + +ForX r , X y ... one is to substitute the values on the right side of Equation (220). Further, +according to (218a), + + +dp) + +dt + + +d + +dt + + +%(bv +Lp Vto + + + +> + + +and since one uses here only the terms of largest magnitude, + + +lti + + +7fca^4e«tai2c.a/ + + + +K d +H- + +p dx + + +d Iffl +dt + + +SR / dv du + + +— ( ~+ ~ + +p\dx dy + + +du dv dw + +—|-1- + +dx dy dz + + +/I dp \ SR d /I dp \ 1 /dv bu\d SR +\P dy ) p by\p dx / p\dx by) dt + + +Similarly X x , Y z ... can be calculated. One would thus obtain some very complicated +expressions, which would appear to Continental physicists as strange as did at first those in +Maxwell’s electrical theory. Yet who knows whether many terms of these equations may +not some day play an important role? Here we shall indicate only a special case already +considered by Maxwell. 1. Let there be neither mass motion nor external forces acting on +the gas; hence u=v = w= X = Y = Z = 0. 2. Suppose that some heat flow is taking place. +Then the derivatives with respect to t will vanish, hence, according to Equation (239a), + + +X,= Y, + + +£91 +5 p + + +a + +, + +Lai + + +dy / dy\ dx I. + + +In this special case, Equation (189) gives: + + +391 + +Y. + Z,-2X, = — + + +. 9{pfrj) . a(pf’j) + + +p l ai + +Therefore, taking account of Equations (235), + +2X, -Y,-Z, + + ++ + + +dy + + ++ + + +dz J + + +691 + + +a / an a / ar\ a / ary +5p L ail ai/ + ap\ ay/ + a^ \ dz ).I + + +Id + + +"‘Puh.c- 7fca^4e«tai2c.a/ P4^.i.Zc.i. + + +hence since X x + Y y + Z_ = 3 p. + + + +x,= + + += v + + + +29 ? + +r d i + +f dT + + +3— + +? — + +bp + +L a/ + +v fa + +49 ? + +d / + +dT\ + +— + +- ? + + +5 + +fa\ + +fa / + + +\ + LUE\±i l >T + + ++-(«— + +dz\ dz + + +since for stationary heat flow + +a / dT\ a /ar\ a/ ar + +- — + - ?— +fa \ dx) dy \dy / dz\ dz + + += 0 . + + +In this case we also have + + +dX x dY x dZ s + +- 1 - 1 -= 0 ; + +dx dy dz + + +therefore the volume elements in the interior of the gas are in equilibrium. But the +customary view (cf. the last page of Kirchhoff s lectures on heat theory, already cited, +where what is said otherwise about the old heat conduction theory is quite hue)—that in +stationary heat flow the pressure can be everywhere equal—is shown to be false. The +pressure varies from place to place, and at a given point it may be different directions, and +not exactly noimal to the surface. + +Hence, if a solid body is completely surrounded by a heat-conducting gas, it will in +general staid to move, since the pressure is not equal everywhere. Maxwell was quite +correct in attributing the cause of radiometer phenomena to this effect.* Moreover, a gas in +contact with a solid wall cannot remain at rest if the wall cannot exert a finite tangential +force on the gas. These motions, created by pressure-differences inside the gas, are not to +be confused with those which arise through the action of gravity, in consequence of the +different density of warmer and colder gases. The latter motions canot play any role in the +radiometer, since the axis of rotation is vertical. Also, our formulas do not apply to the latter +motions, since we set X = Y = Z = 0. + + +Up to now we have followed the ingenious methods devised by Maxwell, and applied +by Kirchhoff and others. These methods permit one to avoid calculating the velocity +distribution function/lx, y, z, f, //, C, t). There is another method which proceeds on the +contrary from a calculation of this function. Although the latter method has not been used, I +will say a few words about it here, since to calculate the entropy we need to know/. + +The stalling point is the general equation (114) in which, since we consider only one +kind of gas, the next to last term vanishes. If we write, instead of the previously used + + +16 + + +7fca^4e«tai2c.a/ + + + +constants a, h, u, v, w, + + +k + +e a , —) uo, Vo, Wo, +m + + +then we know that the equation will be satisfied if we put + + + +j = ({-no) l + (r-'o) *+ ) *1 + + +as long as a, k, u 0 , v 0 , vv 0 are constants. Then « () , v 0 , vv () are the velocity components of the +gas as a whole. + +Now let k, a, u 0 , v 0 , vv 0 be functions of v, y, z, t; their variations (i.e., their derivatives +with respect to these variables) are assumed to be so small that only small correction terms +need to be added to the expression (240) in order to satisfy Equation (114). We shall +represent these corrections in the form of a power series. Since a, k, uq, vq, w’q are arbitrary, +we can choose their values in such a way that the terms multiplied by rj, and C in the +power series will vanish. This can therefore be done without any loss of generality. +Moreover, we can choose the coefficients of , if, and C to be such that their sum is +equal to zero. We introduce the variables + +(241) £o = { “ Mo, t)o = rj - t'o, Jo = J* “ Wo + + +and put + + + +where + + +(243) / (0) = e°“*W>+}o) + +and + +(244) b n + 622 *f ^33 = 0. + +The left side of Equation (114) is now transformed to + + +Id + + +7fca^4e«toi2c.a/ + + + +a/ a/ a/ a/ + +l = —b (fo + Mo) —b (t)o + fo) —b (jo + Wo) — + +dt 3x dy dz + +df df df + ++ 1 —+ Y— + Z-- + +3fo a>)o 5jo + +Since all the differential quotients are small, we can replace/by /°) in them. If we write +Co f ° r ro+t>o+$o and d(/dt for d/dt + u 0 d/dx + v 0 d/dy + w () d/dz, then we find: + + +f 1 d^a 2 d 0 k + +— I = —— Co — + fo + +/<°> it it + + ++1) 0 + + +(245) + + +da ( +- 2k \ + + + + +L dx \ dt + + +- X + + +1 + +Oj + +© + +< doi'o Y + + +aa + +(down \"l + +—b + +— -1 + ++ Jo + +~2k \ + +-— z) + +Lay ' + +^ dt )\ + +L dz + +{ dt ). + + ++ 2 k + + +du + + +2 C7U0 2 W O 2 + + +dv i + + +dw 0 + + +/dv o + + +Xo ——h t)o ——h go ——h t)o^ol — + — + +dx dlj dz \ dz dy + + +/dwo duA /duo dv 0 \ + +2 / dk dk dk + +— Co fo—Mo —b Jo - +\ dx dy dz , + + +If one considers the coefficients b to be small, so that their products and squares may be +ignored, then the right side of Equation (114) becomes + +r = f f f 7YW«e[6, 1 (f' 2 + rl) + +J J n J n + + +, , /2 . /2 2 2 1 ++ &22O) + t)l ”1) “ t)i) • * * J. + + +Id + + +7fca^4e«tai2c.a/ + + + +In order to avoid an accumulation of too many indices, we shall drop the index zero +from the quantities 3 unt ^ we S cl to Equation (246)—i.e., it will not be explicitly + +indicated that they arise by subtracting the quantities uq, vq, h’q rather than u, v, w from the +corresponding quantities £, rj, £ Since and /1 0) it is Uq, v 0 , Wq and not u, v, w that + +are subtracted from £, ij, C, we find just as before that: + + + +Hence: + + + +The same result holds for the products £ } and t)$. Since now + + + +it follows that &ll^ 2 "h^22t) 2 + &333 2 can be represented as a sum of spherical +functions of the second degree, and we have + + +f f " f + bt® + M) + +J J 0 * 0 + +3 / 2 Ki + +2 r m 2 + + +where the abbreviation ^ has been used for j '2 +r,' 2 + +similar meanings. + +If one sets + + +-i’-i? 9 and ^ have + + +16 + + + + + +v / /2 / /2 i S + +3£i = fc + fiCi -fc -fiCi + +S 0 l = l)C +>hCi - t)C ->)iCi + +~ M 2 . / / 2 2 2 + +3 l = J C + J1C1 - JC - J1C1, + +then he finds likewise, according to the principles of the previous section (cf. Eq. [231a]): + +J gbdbj d(h = - A t ^ ~ [2(f* - {!)({- {,) + +- (F + fi)(>> ~ ^i) 1 - (f + Ii)(i ~ Ji)* + ++ 3(i> ! -i)!)(t-r,) + 3(} , -}!)(f+ri)]. + + +f f f /i ( 0 , rfui(?MMf(cifi + C $1 + Cj3i) = + +J J o J 0 + +- 2 A 2 p J ^ [(ciji + c 2 t } 2 + Cl]) c 2 + +r 2m 3 + + +" 77 (ClJ + Cfl + C 3 J)], + +2k + + +hence finally + + +Id + + +7fca^4e«tai2c.a/ + + + +(246) + + +f T /K\ (2 2 2 + +r— = ~ 3A2P \/ — \ ^lljo + Mo + &33$0 + +/< 0) V 2 m* { + +2 2 + ++ Mojo + Mojo + Motyo + ~ Co(ciJo + C2P0 + Cajo) + +o + + +5 + + +- — (C 1 X 0 + c 2 1)2 + C 3 Jo)^ • + +3fc + + +Equation (114) must be satisfied identically. Hence expressions (245) and (246) must be +equal for all values of xg, y 0 , and zq. The terms independent of xg, y 0 , zq must be equal; +hence: + + + +Since by 1 + b 2 2 + ^33 + 0 , the terms of second order in Jo gi yc: + + +16 + + +7fca^4e«toi2c.a/ + + + +(248) + + +djc 2k/du Q dvo dwo\ + +——h ~(-1-1-) = 0, + +dt 3 \ dx dy dz ) + + +In* + + +bn = + + +bn = + + +2k + +9A2P + +2k + +9A2P + +2k + + +2m 3 / dvo dw 0 duo\ +Ki\dy ^ dz dx / + + +2m 3 / duo dwt dv 0 + +— (- 1 - 2 — + +Ki \ dx dz dy + + +9A 2 p + + +2m z /duo dv 0 dt^o\ +\ dx cty 32 / + + +623 = “ + + +fcl3 - ~ + + +fcl2=- + + +2fc + +34 2 p + +2k + +3 A 2 P + +2fc + + +2m l /dvo dwo +Ki \ dz dy + + +2m z /dwt dua\ +K\ \ dx ^ dz / + + +3 A 2 P + + +2m z /duo 3v 0 \ +Ki \ dy dx /’ + + +Cl = + + +1 + +/2m 3 dfc + +2^4 2 P + +V Y { 7x + +1 + +/2m 8 dfc + +2 A 2 P + + +1 + +/2m 3 3fc + + +T + + +7fca^4e«taiZc.a/ P4y.i.Lal + + + +Setting equal the terms containing first powers of Jo, Jo gives finally (taking +account of the values found for cy, c 2 and C3): + +do^o 1 da 5 dk + +- X+ - + +dt 2 k dx 4 k 2 dx + +doVo 1 da 5 dk + +=- y + - + +dt 2 k dy 4 k 2 dy + +doWo 1 da 5 dk + += - Z + -= 0. + +dt 2 k dz 4fc 2 dz + +Since by j + /> 2 2 + ^33 = 0, and each term containing an odd power of l}o or Jo +vanishes on integration, it follows that (if one writes dat and cIojq for d£dijd( and + +dXodVfodio)- + +///>-///> + +Hence + + + + +if we do not carry the approximation far enough to produce any correction to the density of +the gas. Likewise: + + +2, .(0) + + +/ ($0 + tyo + Jo)/dw = / (f0 + t)o + Jo)/ dw0. + + +Hence the mean square velocity of motion of the molecule relative to a point moving + + +with velocity u () , v 0 , wq is equal to 3/2 k. + +On the other hand, uq, v 0 , h’ () are only approximately equal to the components of visible +velocity of the gas in the volume element do. These components are actually defined as + + +i, v, f . Now we have £ — Uq “I" £()' an< ^ f urt h ermore + + +7fca^4e«toiZc.a/ + + + +_ _ /jo/du _ _ f ilctf dwo _ 5ci + +U ~ JJda ~ //<»><*«o “ ¥ ' + +If we denote the exact components V, l of the visible motion of the gas by u, v, w, +and those of the motion of a molecule relative to the visible motion by t), j, then we +obtain in this approximation + +5ci 5c 2 5 c 3 + +u = Wo H— t v = To H-» w = ii>o H- + +2fc 2fc 2fc + + +5ci 5c 2 5 c 3 + +'-'•‘i" + +Furthermore, + +2 2 2 + +P a a a P / a a a 25 Ci “j” C 2 ~i“ C 3 ^ +p = -(f +>) +3)=-^o + l)o + Jo-- — + +/I 25 Ci + c 2 -f c 3 + +_P \2fc _ 12 + +One has therefore as a first approximation + +d 0 d pl + +U = Uq, V = Vq. W = Wq, — = —) k = — = — > + +A dt 2p 2 rT + + +P /fc 3 \ /p 6 / 2 p -3/2 + +a=i[«4/-]=im= + +\ wi\/8t 3 + + +H + + +pT + + +'-3/2 + + +r i + +Hence, according to Equation (247), + +pp" 6/2 = const, or pT _3/2 = const. + + +my/Wr* + + +17 + + +“PiLh-d. 7fca^4e«tai2c.a/ + + + +which is Poisson’s law. Furthermore, + + +1 p da 5 dp 3 dp 1 dk 1 dp 1 dp + +2k p dx 2p dx 2 p dx k dx p dx p dx + +hence + +1 /da 5 dk\ 1 dp +2 k\dx 2k dx) p dx + +Therefore Equations (249) yield: + +du 1 dp dv 1 dp dw dp + +-X + —- =-F + —- =-Z + — = 0. + +dt p dx dt p dy dt dz + + +If we wish to carry the approximation one step further, we can make the above +substitutions in the terms that are small, and thereby find: + + + += - P + + +/ lotyojdwo + +Jf w du> + + + +pb n + +4 k 2 + + +pbn + +2k + + +V + +3i4 2 p + + +'2m 3 /dv dw\ /dv dw\ + +Ki Vae ^ dy) \d* + dy) + + +Likewise the rest of Equations (220) follow, and there is again no difficulty in extending +the degree of approximation. + + +§24. Entropy for the case when Equations (147) are not satisfied. Diffusion. + +Up to now we have calculated H only under the restrictive assumption that Equations +(147) are satisfied. We now wish to calculate it under the general assumption that/is given +by Equation (242), so that viscosity and heat conduction are present. We assume a simple +gas. Then + + +H = J J JlJdodu. + + +173 _ + + +7fca^4e«tai2c.a/ + + + + + +Since/is given by Equation (242), it will be approximated by + + +2 2 2 + +If = a - k(i o + tyo + Jo) + 4 - — + +L + + +where the expression in parenthesis in Equation (242) has been denoted by 1 +A. + +We now wish to construct the expression H for the gas contained in the volume element +do. The value thus found will be multiplied by -RM and divided by do. Let this quantity be + +J = - RMjJlfdoi. + +Jdo is then the entropy of the gas contained in do. + +If we now substitute the above values for/ and If we obtain first a teim independent of +the coefficients b and c This is the entropy (divided by do) that the gas in do would have if +it had the same energy (heat) content and the same progressive motion in space, and +obeyed the Maxwell velocity distribution law. It can be calculated just as in §19, and has, +as shown there, the value + + +Rp + +— 1 ( T* V 1 ) + +M + +apart from a constant. Second, we obtain terms linear in the coefficients b and c. These all +vanish. Since + + +/ toWo exp (- fc(f l + l)o + Jo*) I d «0 = o + +if one of the numbers a, b, c is an odd integer, the coefficients of Zq 2 , ^ 13 , 7>23> c l, c 2 ar| d +C 3 all vanish. However, if all three numbers a, b, c are even integers, then the integral does +not change its value under cyclic permutations of t)j and Jo- Hence bn, b 2 2 and Z > 33 +have the same coefficients, and the sum of the terms in question vanishes in any case, since + +fen + f>22 + feaa = 0. + +Since we are omitting higher order terms, there still remain in the expression for J only +terms of second order in the coefficients b and c. Their sum is + + +17 ; + + +7fca^4e«tai2c.a/ + + + +3 A ,3 4 + + +a a + + +Rp J4 . _ . . . + +J 1 = ” ~~(bnh + ^22^0 + ^3jJo + 2 &li 622 |ot)o + +2m + + +a a + + +a a . . a a a . ,a aa + + ++ 2611633 ^ 0^0 + 2622633 ^ 0^0 + & 12 jot)o + 613 ^ 0^0 + +a a a 1 44 a 94 334 + ++ 623^0^0 t CifoCo + C2^oCo + C3J0C0). + + +The next terms to be added in Equation (242) (which we have not calculated) will of +course be of the same order of magnitude as these, but it is not improbable that they would +also vanish on integration +We now find: + + +4 4 4 + +fo = >)o = Jo = + + +3 + +4fc* + + + +and one finds easily: + + +34 3 4 34 ■* 6 + +foCo = l)oCo = J0C0 = — Co = + + +35 + +8 fc»’ + + +Since + +1 RT + +2k m + +we have therefore + + +J\ = - + + +R'T'p + +~V~ + + +(222 + +1 3 ( 6 n + 622 + 633) + 2(6 h 6 o 2 + 611633 + 622633) + + +2 2 2 ++ 612 + 613 + 623 + + +5-7-9 5 »v r/ary /st y nry + +16 RtfV A dx ) \ dy / \ bz /. + + +n. + + +7fca^4e«tai2c.a/ + + + +On substituting the value of b, one finds, writing 9 for + + +bu bv bw + +—I-1- + +bx by bz + + +the following value for the total entropy of the gas contained in the volume element do: + + +Rpdo MWPpdo ( /bu 1 Y + +Jd0 = ; ( r»V)-— 2 ---a + +2p pV { \bx 3 / + + +/bv 1 \ 2 / bw 1 \ 2 /dv bw\ 2 + ++ 2 - 0 +2 - 0 + - + — + +\by 3 / \ bz 3 / \bz by/ + + +/ bw bu \ 2 / bv bu\ 2 5*7*9 u ~/bT\ 2 +\ dx ds / \ dx dp/ 64 RP A bx j + + +(250) + + +/bT\ 2 lb r r \ 2 + ++ (— ) + (— ) +\ by / \bz / + + +Rpdo + + + + +49? 2 R 2 Ppdo j ~/buV“ / bvY /bw\ 2 +pY \ \bx) \by) \bz), + + +2 /bu bv bw \ 2 /bv bw\ 2 + +-( —I-1-) + ( —I-) + +3 \bx by bz) \ bz by ) + + +/bw bu\ 2 / bu bi A 2 + ++ - + - + - + - +\dl bz) \by bx) + + +5-7*9 p r/ary /ary /wy + +A 64 Rp\bxj \ 5?/ / [bz/ + + +The sum of all the terms containing derivatives of u, v, w with respect to x, y, z. is what + + +7fca^4e«taiZc.a/ + + + +Lord Rayleigh calls the dissipation function of viscosity.* The sum of the last three terms +has been called by Ladislaus Natanson the dissipation function of heat conduction, t + +The energeticist holds that the different forms of energy are qualitatively different; to +him, an energy halfway between kinetic energy and heat is very strange. Hence the oft- +emphasized principle of the superposition of properties of different energies contained in a +body. The principle is valid for the static state, and for completely stationary visible motion, +where to a certain extent the forms of energy can be separated. On the other hand, if the +above equation is correct, then in the presence of viscosity and heat conduction the entropy +of the gas is not the same as it would be at the same temperature and velocity, when there is +no dissipation. Thus we have to deal with a kinetic energy that is, so to speak, half visible +kinetic energy and half transformed into thermal motion, so that in the expression for the +entropy it appears in a form that would not have been foreseen from the laws of static +phenomena. If we deform a completely elastic body by means of an external force, then we +get back all the energy we put into it, in the form of work, when it returns to its initial state. +If we produce viscosity in a gas by means of an external force, then the work done is +transformed into heat energy. After the removal of external forces, this transformation +becomes complete after a time considerably greater than the relaxation time has elapsed. +While the external forces are acting, our equations predict that the entropy at each instant is +somewhat smaller than it would be if the energy lost from visible motion were completely +transformed into heat. Instead, this energy is in a state intermediate between ordinary heat +and visible energy, and part of it can still be transformed back into work, since the Maxwell +velocity distribution does not yet hold exactly. This description of the dissipation of energy, +based on a purely mechanical model, seems to me especially remarkable. + +Now suppose that two kinds of gas are present. Let m be the mass of a molecule of the +first kind, and m 1 the mass of one of the second kind. The mean value u of the velocity +components £ of all molecules of the first kind found in a volume element will be called the +x-component of the total velocity of the first kind of gas in this volume element. It need not +be equal to the mean value u\ of the velocity components £\ of all molecules of the other +kind of gas in the same volume element, rq will be called the x-component of the total +motion of the second kind of gas in the volume element do. v, w, v\ and w\ have similar +meanings. Let p and pj be the partial densities of the two kinds of gas—i.e., p is the total +mass of all molecules of the first kind contained in do, divided by do, and similarly for pj. +Let p and p\ be the partial pressures—i.e., the pressure that each kind of gas would exert +on unit surface if the other were not present. Let P =p + be the total pressure. Finally, +let J and£i, f) j, be the excess of the velocity components over the total + +velocity components of the corresponding kind of gas: + +i = u + f, >i = v + ij, I = w + a + +{l = u l + flj 'll = 1*1 + ?1> fl = Ml + }i. + +Then the continuity equation is valid for each kind of gas, as we already proved before we + + +17 + + +Maf4cjvi(i££caZ + + + +made the assumption that only one kind of gas is present. Hence: + + + +dp + +d(im) + +a ((w) + +d/)W) + +-b + +-+ + + ++ -= 0 + +dt + +dx + +dy + +dz + +dpi + +d(pitti) + +a(piPi) + +9{m) _ + +-b + +-+ + + ++-= 0. + +\dt + +dx + + +dz + + +We shall now imagine that the volume element do moves during time dl with the +velocity components u, v, w of the first kind of gas in this volume element. The difference +between the values of any quantity at time l + dl in the volume element in its new +position, and at time t in the volume element in its old position, divided by dt, we denote by +d$/dt, so that: + + +— =-b u -by-bw —. + +dt dt dx dy dz + +A similar meaning is given to: + +d$ d$ d$ d$ + +— =- + U\ -+ V\ -+ 101 - + +dt dt dx dy dz + + +In constructing the latter quantity, one imagines that the volume element is moving with +velocity components u\, iq, nq. Then the two continuity equations can also be written: + + + +f dp / du dv dw\ + +—+p —' = 0 + +dt \ dx dy dz + +d\pi (du\ dvi dW\ . + +— + pi — + — + — l = o. + +dt \ dx dy dz + + +We ignore the deviations from the Maxwell velocity distribution law. Then: + + +17 + + +7fca^4e«tai2c.a/ + + + +p = p f 2 = p y = p j 2 , n = n = t)« = o. + +pi = Piii = pi>)I = Piii, Mi = fiji = wi = o. + + +The mean kinetic energy of a molecule cannot in any case be very much different for +the two kinds of gas. One has approximately: + + +m _____ _ mi ______ + +7 ft*+ *’ + *•) = -«! + ’/? + !!)• + +L L + + +Since to the present degree of approximation we can ignore the squares of the small +velocity components u, v, w with which the gases diffuse through each other, compared to +£ 2 , t] 2 ,... we also have: + +m{? +1) 1 + S ! ) = TOi(t| +1)| + jJ). + +We set these quantities again (cf. Eq. [51a]) equal to 3 RMT, and we call T the +temperature in do. Here M is the mass of a molecule of some third gas (the normal gas) and +R is a constant corresponding to the temperature scale to be chosen (the gas constant of the +normal gas). Since each of the original two gases behaves l ik e a gas at rest, + +, v R R + +(253) p = rpT = - pT, pi = r { piT = - piT), + +p Pi + +where r and /q are the gas constants of the original two gases, and // = m/M, /./| = m\!M. + +We shall now set tp = £ = U -}- £ in Equation (187). Then: + +_ _ _ _ dip dp dp + +

-{)p + +J n + + +4tC 0S 2 lM). + + +We also set (Eq. [195]): + + +6 = + +“K(m + mi) + +mmi + + +db = + +" K(m + mi) + +mmi + + +4. Therefore: + +1 + +» oo /» 2 t + + +11/n + + +g Iln.Q + + +”11/n + + +g~ 2,n da + + +o ^ o + + += Midi - I) + + + + +/» 00 + +J n + + +mmi(m + mi) J 0 + +Maxwell calls this definite integral A | and finds* + + +4ttcos 2 i )ada. + + +18 . + + +7fca^4e«tai2c.a/ + + + +A x = 2-6595. + + +(255) + +We set: + +(256) + +and obtain + + +^3 - A\. + + +K + + +mm\[m + wii) + + +n it + +((' “ Qgbdbde = - {). + +j + +Whence it follows, furthermore, that: + +m# 4 ({) = At[mf fdu-niij hFidwi - mf tfdwmij F\du\\. + +Now according to Equation (175): + +mj fdu = p, mf tfdw = p£ = pu + +and since clearly the same result holds for the second kind of gas, + +inifFidui = pi, wii/fiFidwi = p\Ui f + + +we have + + +mB t (() = A 3 ppi(ui - u) + +and Equation (254) reduces to + +du dp + +(257) p -7 H - pX + A 3 ppi(w - U\) = 0. + +dt dx + +Likewise one obtains for the second kind of gas, + +du\ dpi + +(257a) Pi ——I- piXi + Aipp\(u\ - u) = 0. + +dt dx + + +18 . + + +^Pi l * lc - 7fca^4e«toi2c.a/ “Ph-ySALcS. + + + +These are the familiar hydrodynamic equations. According to our present assumptions, +viscosity and heat conduction cannot be important. Only the last term describes the +interaction of the two kinds of gas. On these assumptions, therefore, this interaction has +exactly the same effect as if one added to the force X pclo, which acts from outside on the +gas of the first kind in do, the contribution - A^ppfu - u\ )do. We can arrange things so +that this gas is unaffected by the other forces that act on it, and encounters only this +resistance to its motion through the second kind of gas. Since the same holds for the y and z +axes, this resistance is equal to the product of the partial densities of the two gases, their +relative velocity V (u — u i ) 2 + (t; — v i) 2 + ( w— Wi ) 2 , the volume do of the + +volume element, and the constant At. It has the direction of this relative motion, and acts on +each kind of gas against the relative motion. If we set

+ +This equation has the same form as the one established by Fourier for heat conduction. +Both natural processes therefore follow the same law. In our special case, the diffusion +takes place just as if instead of a cylindrical mass of gas one had a homogeneous metal +cylinder, whose upper half is maintained initially at a temperature of 100°C, and whose +lower half initially has zero temperature; through the entire surface of the metal cylinder, no +heat may enter or leave, either by conduction or radiation. + +$ is the diffusion constant. It is directly proportional to the square of the absolute +temperature T, and inversely proportional to the total pressure P. It is independent of the +mixing ratio, so that it is constant at all times during the diffusion process for all layers of +the container. If the molecules behaved like elastic spheres, $ would be proportional to +the ?J power of T, and would depend on the mixing ratio. The dependence on P remains +the same in both cases. + +A simple definition of the diffusion constant $ can be obtained in the following way. +We multiply Equation (263) by — nT)/RT and obtain: + +R'T 1 dp dp + +pu =--= -$-• + +AytpiP dx dx + +pu is clearly the total amount of gas that goes through unit cross section in unit time. It is +proportional to the gradient dp/dx of the partial density of the gas in the direction of the axis +of the container. The proportionality constant is just the diffusion constant. + +If we retain the assumption of inverse fifth power forces, then we cannot draw any +conclusion about K from the force constants K\ and Kj. Thus from the properties of two +gases by themselves, we can draw no conclusion about their interaction with each other. +However, we may be able to draw such conclusions if we imagine that the repulsive force +is transmitted by means of compressible ether-shells. We can then ascribe the value s to the +diameter of the ether-shell of an m-molecule, and the diameter ,V| to the ether-shell of the +m 1 -molecule. In a collision, the centers of two m-molecules will approach up to a min im um +distance s, on the average. If we imagine one such molecule held fixed and the other one +moving toward it with mean kinetic energy [, then the latter will have zero velocity at the + +distance s. Then: + + +18 . + + +7fca^4e«tai2c.a/ + + + +(264) + + + + +A n m | -molecule will, however, approach an m-molecule up to a minimum distance +equal to the sum of the radii (5 + .v | )/2 on the average. If we again fix one molecule and let +the other approach it with the mean kinetic energy of all the molecules, then its velocity will +be annihilated at the distance (5 + ,v j )/2, which gives: + +4 K + +1 = - * + +(s + s,)< + +From these equations it follows that: + +2n y/2pt _ RTM* 1 ' V2m, + +-> \/Kt =- + +3d, R 3A, R, + +difir /y n y m y + +6 \Z 2 /l 2 \/ mmi(m + Mi) w® V®i/ + + +_ 6vwr /iTTm i + +d.fi y ,1 ' + +V® + vrT/ + + +This equation pemtits one to calculate the diffusion constant of two gases from their +molecular weights and viscosity coefficients. It agrees approximately with experiment. +However, one should not suppose that it is exactly correct. Nevertheless, it seems to have a +more rational basis than any other formula yet proposed for this puipose. + +If in Equation (264) we set: + + +then + + + +K\ = 2 mV, + + +hence + + +K = + + +pm + + +%AipS 2 y/ c 5 + + +18 . + + +7fca^4e«tai2c.a/ + + +Now we have + + + +hence + + +V + + +p + + + + +R =--0.0812- + +94 2 S 2 s 2 + +According to Equation (91), + +9 } = knmc\, + +1 + +X =-= • + +imsV 2 + + +Further, according to Equation (89): + +k = 0.350271, + + + +Therefore: + + +9} = 0.350271 —= +T\/3ir + + + + +s l + + += 0.0726 + + + + +One sees that the numerical coefficient is only insignificantly different. + +The concepts of mean free path and number of collisions are not suited to the theory of +repulsive forces proportional to the inverse fifth power of the distance. In order to define +them, one must make a new arbitrary assumption. One must, for example, decide that an +encounter of two molecules is to be considered a collision if the relative velocity is +deflected through an angle greater than 1°. + +It would be of the greatest interest to carry the approximation further in the calculation + + +18. + + +IPm+lc- 7fca^4e«tai2c.a/ + + + +of diffusion, as well as to calculate the entropy of the two diffusing gases. In the first case +there would probably be fluctuations of temperature during the diffusion, whose calculation +according to the principles established would not be difficult; likewise, it would be easy to +calculate a new dissipation function, that of diffusion, by determining the entropy of the +two diffusing gases. However, we shall not pursue this matter any further.* + + +* Maxwell, Phil. Trans. 157 , 49 (1867). + +1 Likewise the assumption of an attraction proportional to the inverse fifth power of the distance +produces a similar simplification (cf. Wien. Ber. 89 , 714 [1884]). However, one must then assume that +for distances small compared to the distance at which the interaction becomes strong, the force follows a +different law, according to which the attraction remains finite or is transformed to a repulsion, since +otherwise the molecules would not separate in a finite time after collisions. In the text we always assume +a repulsion inversely as the fifth power. + +2 Maxwell, Phil. Mag. [4] 35 , 145 (1868); Scientific Papers 2 , p. 42. + +* Here is Maxwell’s figure: + + + +3 Likewise one finds easily: + + +(208) 2?r gbdb sin* 0 cos* 0 = A + +Jo + +* Later calculations ofA 2 were made by K. Aichi and T. Tanukadate, reported by H. Nagaoka, +Nature 69 , 79 (1903), and by S. Chapman, Mem. Proc. Manchester Lit. Phil. Soc. 66 , No. 1 (1922). The +values found were 1.3704 and 1.3700, respectively. + +* Maxwell, Phil. Trans. 157 , 49 (1867), esp. Eq. (130). Maxwell actually called it the “modulus of +the time of relaxation.” + +* Now usually called the Navier-Stokes equations; however Boltzmann, following Maxwell (op. +cit.) has suppressed the second viscosity coefficient (bulk or dilatational viscosity). This point is +discussed in §23. + +4 Maxwell, Phil. Trans. 170 , 231 (1879); Scientific Papers!, p. 681. + +5 Heine, Handbuch der Kugelfunctionen (2d ed.), p. 322. + +6 Heine, op. cit., p. 313. + +1 1 thank Prof. Gegenbauer for this proof of Maxwell’s theorem. + +8 Cf. Kirchhoff, Vorlesungen iiberdie Theorie der Wdnne (Teubner, 1894), p. 118. + +9 As a result of an error in calculation, Maxwell (Phil. Mag. [4] 35, 216 (1868), Scientific Papers 2, + + + +18 . + + +7fca^4e«toi2c.a/ + + + + +77, Eq. [149]) found for ^ only ^ of the above value, as I noted in Wien. Ber. 66, 332 (1872). Poincare +made the same remark: C. R. Paris 116 , 1020 (1893). + +* The result of Maxwell and Boltzmann, that a gas does not have two independent viscosity +coefficients, is certainly not true in general for real gases, nor is it a strict consequence of the kinetic +theory except perhaps for certain idealized models. For a list of recent papers on "bulk viscosity” see S. +G. Brush, Chem. Revs. 62 , 513 (1962). + +* Maxwell, Phil. Trans. 170 , 231 (1879). + +* Rayleigh, Proc. London Math. Soc. 4,357 (1873); Phil. Mag. [5] 36,354 (1893). + +t Natanson, Rozprawy Krakow 7 , 273, 9 , 171 (1895); Phil. Mag. [5] 39 , 455,501 (1895). + +* Maxwell, Phil. Trans. 157,49 (1867). According to the calculations of Aichi and Tanukadate, +reported by Nagaoka, Nature 69 , 79 (1903), Aj = 2.6512; according to Chapman, Mem. Proc. +Manchester Lit. Phil. Soc. 66, No. 1 (1922), A\ = 2.6514. + +* Loschmidt, Wien. Ber. 61 , 367, 62 , 468 (1870). + +* Modern research on the properties of gases is reviewed by J. S. Rowlinson, J. E. Mayer, H. Grad, L. +Waldmann and others in Handbuch der Physik, Vol. XII (Berlin: Springer, 1958). Older work is +surveyed by J. R. Partington, Advanced Treatise on Physical Chemistry, Vol. I (London: Longmans, +Green, 1949). + + +19 , + + +^Pi mJuc. 7fca^4e«ta£2c.a/ + + + +PART II + + +Van der Waals’ theory; Gases with compound molecules; Gas +dissociation; Concluding remarks. + + +19 . + + + + + + + +FOREWORD TO PART II + + +���The impossibility of an incompensated decrease of entropy seems to be reduced to an +improbability.” 1 + +As the first pail of Gas Theory was being printed, I had already almost completed the +present second and last part, in which the more difficult parts of the subject were not to +have been treated. It was just at this time that attacks on the theory of gases began to +increase.* I am convinced that these attacks are merely based on a misunderstanding, and +that the role of gas theory in science has not yet been played out. The abundance of results +agreeing with experiment, which van der Waals has derived from it purely deductively, I +have tried to make clear in this book. More recently, gas theory has also provided +suggestions that one could not obtain in any other way. From the theory of the ratio of +specific heats, Ramsay inferred the atomic weight of argon and thereby its place in the +system of chemical elements—which he subsequently proved, by the discovery of neon, +was in fact correct.* Likewise, Smoluchowski deduced from the kinetic theory of heat +conduction the existence and magnitude of a temperature discontinuity in the case of heat +conduction in a very dilute gas.f + +In my opinion it would be a great tragedy for science if the theory of gases were +temporarily thrown into oblivion because of a momentary hostile attitude toward it, as was +for example the wave theory because of Newton’s authority. + +I am conscious of being only an individual struggling weakly against the stream of time. +But it still remains in my power to contribute in such a way that, when the theory of gases +is again revived, not too much will have to be rediscovered. Thus in this book [this Part] I +will now include the parts that are the most difficult and most subject to misunderstanding, +and give (at least in outline) the most easily understood exposition of them. When +consequently parts of the argument become somewhat complicated, I must of course plead +that a precise presentation of these theories is not possible without a corresponding formal +apparatus. + +I thank especially Dr. Hans Benndorf for collecting numerous literature citations during +my absence from Vienna. + +Volosca, Villa Irenea, August, 1898 + + +Ludwig Boltzmann + + +1 Gibbs, Trans. Conn. Acad. 3,229 (1875); p. 198 in Ostwald’s German edition. + +* In addition to the works cited in the footnote on p. 24, see: R. Mayer, Mechanik der Wdnne + + +19 . + + +7fca^4e«tai2c.a/ + + + + +(Stuttgart, 1867), p. 9. E. Mach, “Die okonomische Natur der physikalischen Forschung, “ Wien. +Almanach 293 (1882); Uberdie Erhaltung der Arbeit (Prague, 1872); Monist 1, 48, 393 (1891), 5, 167 +(1894); Die Prinzipien der Wdrmelehre (Leipzig: J. A. Barth, 1896, 2nd ed., 1900), pp. 362-364, 429- +431. K. Pearson, The Grammar of Science (London: Scott, 1895), pp. 200, 214, 311. J. B. Stallo, The +Concepts and Theories of Modem Physics (New York: Appleton, 1882), chap, viii. P. Du hem. La +Theorie Physique, son Objet et sa Structure (Paris: Chevalier et Riviera, 1906). J. Ward, Naturalism and +Agnosticism (New York: Macmillan, 1899). A. Aliotta, La reazione idealistica contro la scienza +(Palermo, 1912). G. Hirn,Mem. Acad. Sci. Bruxelles 43 (1881), 46 (1886); C. R. Paris 107,166 (1888). +W. Ostwald, Lecture at the 64th meeting of the Deutscher Naturforscher und Arzte, Halle, 1891, in his +Abhandlungen und Vortrage, p. 34. H. Poincare, Nature 45, 485 (1892); Thermodynamique (Paris: +Gauthier-Villars, 1892), p. xviii. F. Wald, Die Energie und ihre Entwerthung (Leipzig: Engelmann, +1889). V. Lenin, Materialism andEmpirio-Criticism (Moscow, 1908), chap. 5. + +* Ramsay and Travers, Proc. R. S. London 62 , 316, 63,437 (1898). Ramsay, Mem. Proc. Manchester +Lit. Phil. Soc. 43 , No. 4 (1900); The Gases of the Atmosphere (London: Macmillan, 1896), pp. 207-232. + +t Smoluchowski, Wien. Ber. 107 , 304 (1898), 108 , 5 (1899); Ann. Phys. [3] 64 , 101 (1898); Prace +Mat.-Fiz. 10,33(1898). + + +193 _ + + +7fca^4e«tai2c.a/ + + + +CHAPTER I + + +Foundations of van der Waals’ theory. + +§ 1. General viewpoint of van der Waals. + +When the distance at which two gas molecules interact with each other noticeably is +vanishingly small relative to the average distance between a molecule and its nearest +neighbor—or, as one can also say, when the space occupied by the molecules (or their +spheres of action) is negligible compared to the space filled by the gas—then the fraction of +the path of each molecule during which it is affected by its interaction with other molecules +is vanishingly small compared to the fraction that is rectilinear, or simply determined by +external forces. Then the Boyle-Charles law holds for the gas in question, whether the +molecules are simply material points or solid bodies, or when they are compound +aggregates. The gas is “ideal” in all these cases. + +Gases found in nature only partly satisfy these conditions of the ideal gas state, and +hence a theory that takes account of the finite extension of the spheres of action of the +molecules is very much to be desired. + +Such a theory was given by van der Waals,* who considered the molecules to be +negligibly deformable elastic spheres, as we did at the beginning of Pail I. He generalized +the theory in two ways: + +1 . he does not assume that the space actually occupied by the elastic spheres +representing the molecules is vanishingly small compared to the total volume of the gas; + +2 . he assumes that, in addition to the instantaneous elastic forces that act during +collisions, there is also an attractive force between the molecules, which acts in the line of +their centers, and whose intensity is a function of the distance of centers. We call this +attractive force the van der Waals cohesion force. + +The necessity of assuming an attractive force between molecules follows directly from +the possibility—now demonstrated for all gases—of liquefaction, since the simultaneous +existence of a liquid and a gaseous phase of the same substance at the same temperature +and pressure in the same container is understandable only if there is a force between +molecules that causes them to rebound at collisions, and also an attractive force. + +This attractive force can be demonstrated directly by the following experiment. One +suddenly brings a container filled with a compressed gas into communication with another +container filled with the same gas in a dilute state. As it flows out, the gas in the first +container does work against the pressure and cools itself; in the latter container there is first +a visible streaming, which turns into heat as a result of viscosity. If there were only a +repulsive force between the molecules, then the heat finally created must be completely +equivalent to the cooling in the first container. If there is also an attractive force, then this +equivalence is not complete; rather there is a net loss of heat, since the average distance of +the molecules will be larger, and hence a definite amount of heat must be used to overcome + + +19 . + + +7fca^4e«tai2c.a/ + + + +the attraction. + +The experiments conducted by this method by Gay-Lussac 1 and later by Joule and Lord +Kelvin 2 did not give a definite answer to the question of the presence of this attractive +force, but the latter two scientists have proved experimentally, by a more indirect method, +the existence of this attractive force, by expansion experiments with gases. 2 They showed +that a gas which (without any addition of heat from outside) is pushed by pressure through +a porous stopper experiences thereby a small cooling, while calculation shows that a +completely ideal gas would not change its temperature. + +The simultaneous existence of an attractive force and an elastic core for a molecule has +of course a certain improbability. In particular, it appears to be diametrically opposed to the +assumption made in Part I, Chapter El, that two molecules repel each other with a force +inversely proportional to the 5th power of their distance. Nevertheless, both assumptions +can provide a certain approximation to the truth, if the molecules actually exert a weak +attraction at great distance and repel each other at very small distances with an inverse fifth +power force. The attraction must then turn into repulsion as the distance decreases, so that +the former does not come into consideration in collisions, because of the greatly +predominating repulsion at very small distances. + +We shall have to leave a more precise formulation of the possible assumptions to the +future, and in the following we shall not be concerned with the exact relation between the +hypotheses discussed in Part I and the assumptions of van der Waals. From the viewpoint +of our theory, the latter assumptions are considered to form a picture that is correct in many +but not all respects. Indeed, up to now, realizing our ignorance about the actual properties +of molecules, we have not pretended that our assumptions are precisely realized in nature. +On the other hand, we have laid the greatest weight on the requirement that the calculations +must be exactly correct—i.e., that the results must be logical consequences of the +assumptions. The resulting development of mathematical methods was our principal +purpose. If we know the consequences of various kinds of assumptions, it should be easier +to find experiments that will test them, and at the same time it should be made possible that, +as our knowledge progresses, mathematical methods for the investigation of newly +discovered laws should be readily available. + +Unfortunately, van der Waals had to abandon mathematical rigor at a certain point in +order to carry out his calculations. Nevertheless, his theory has proved to be of great +practical value, for the resulting formula gives in general a sufficiently good description of +the behavior of a gas up to its point of liquefaction, even if it does not achieve complete +quantitative agreement with experiment. From this one is justified in concluding that its +foundations could hardly be replaced by completely different ones. + +In this chapter I will derive the equation of van der Waals in the simplest and shortest +possible way, leaving further refinements to Chapter V. + +§2. External and internal pressure. + +A container of volume V contains n identical molecules, which are completely elastic, +negligibly deformable spheres of diameter a. The space occupied by the spheres + + +19 . + + +7fca^4e«tai2c.a/ + + + +themselves will be fairly small but not completely negbgible compared to the total volume +of the container. It will be shown that the formulas obtained are also approximately +applicable to the state of the substance when it is no longer a gas but rather a liquid. +Therefore in the following we shall call it simply a substance, not a gas, although we shall +still always have in mind such cases when its state approximates that of a gas. + +Between the centers of two molecules there acts an attractive force (the van der Waals +cohesion force), which vanishes at macroscopic distances but decreases so slowly with +increasing distance that it may be considered constant within distances large compared to +the average separation of two neighboring molecules.* Consequently the van der Waals +cohesion forces exerted on each molecule in the interior of the container by the surrounding +molecules are very nearly equal in all directions in space, and they balance each other in +such a way that the motion of the individual molecule is like that of the usual gas molecule, +and is not noticeably modified by the cohesion forces. Hence even though we did not treat +such forces in Pail I, we can still calculate the molecular motions by the same principles +established there. + +The van der Waals cohesion force has an appreciable effect only on the molecules that +are very near to the surface of the substance. These molecules will thus be acted on by two +forces. The first is the counter-pressure of the wall on the gas; the second is the cohesion +force. The intensity with which the first force acts on the molecules lying on unit surface +we call p, while that of the latter we call so that the total force on these molecules is + +(1) Pi = P + Pi. + +Now suppose that a part DE of the wall of the container has surface area Q. The total +force + + +% = fl(p + + +which acts on the molecules at the surface DE and which they exert back on the wall (in +the equilibrium state) is, according to §1 of Part I, equal to the total momentum (in the +direction of the noimal N to the surface DE) that the molecules would cany-through this +surface in unit time, if it were placed in the interior of the gas, plus the momentum +corresponding to the velocities with which these molecules are reflected from the surface +back into the interior of the gas. + +§3. Number of collisions against the wall. + +We first pick out of all the molecules only those for which the magnitude of their +velocity, c, lies between c and c+dc, and the angle & formed by the direction of the +velocity with the outwardly directed normal N at the surface DE lies between d and 3+d■&; +furthermore, the angle e between a plane noimal to DE containing the velocity direction +and a fixed plane normal to DE must he between e and e+c/e. We call the set of these + + +19 . + + +Pi£h;e 7fca^4e«tai2c.a/ + + + +conditions: + + +“the conditions (2).” + + +All molecules that satisfy (2) will be called molecules of the specified kind, and we ask +first: how many molecules of the specified kind collide with the surface DE during a very +short time dt? + +Each molecule is to be considered a sphere of diameter a, so that it collides with DE +when this sphere touches it. During the time interval dt, the centers of all the specified +molecules travel nearly the same distance, cdt, in nearly the same direction. We shall find +the number of collisions between molecules of the specified kind and the plane DE during +dt as follows: + +We allow DE to touch, at each of its points, a sphere whose diameter is equal to the +molecular diameter a. The centers of all these spheres lie in a second plane of surface area +Q. Through each point of this second plane, we draw a line that is equal in length and +direction to the path travelled by the specified molecules during dt. All these lines fill up an +oblique cylinder 7 of base Q and height + +(3) dh = cdt cos + +and hence of volume Qdh. One sees easily that the molecules that collide with DE during +dt are just the ones whose centers lie in 7 at the beginning of the time interval dt. + +§4. Relation between molecular extension and collision number. + +In order to find the number dz of these latter molecules, we first determine quite +generally the probability that, for a given configuration of the other molecules, the center of +a particular molecule lies within the cylinder j. This molecule cannot be less than a +distance a from the center of any other molecule. We find the volume available to the +center of this molecule, when the positions of the other are fixed, as follows: we construct a +sphere of radius cf, which we call the covering sphere, around the center of each of the n - +1 other molecules. Its volume is 8 times the volume of the molecule, if it were considered as +an elastic sphere. The total volume, 4n(n - 1 ) + +r,-r, + +from which one may also determine the value of the constant a. If one calculates this value +for several values of v, he can find out how well the term added to p on the left side of van +der Waals’ equation (22) agrees with experience. Thus one can test the validity of van der +Waals’ assumption that the cohesion force extends to distances large compared to the +average distance of two neighboring molecules. + +§10. Absolute temperature. Compression coefficient. + +We can never actually determine the temperature by means of an ideal gas, since no + + + +2ft + + +% J ti3uc- 7fca^4e«tai2c.a/ + + + +known gas, not even hydrogen, possesses exactly the properties that we ascribe to an ideal +gas. The most rational definition of temperature is of course that based on Lord Kelvin’s +temperature scale. As is well known, this scale is derived from the maximum work that can +be gained by transforming heat from one temperature to a lower one. However, since the +direct experimental determination of this work is always performed very inexactly, one is +forced to calculate it from the equation of state of some body. Now the deviations of +hydrogen from the ideal gas state are rather small, so that if one treats these deviations on +the basis of van der Waals’ assumptions, one ought to be able to construct the absolute +Kelvin temperature scale with an accuracy that can hardly be surpassed at the present +time. 9 One can then use the equations developed above for the determination of the +absolute temperature, except that one cannot assume that T |, T 2 , and 7) can be determined +by comparison with another more ideal gas. One can first express the temperature +differences in terms of numbers, by means of the proportion (25), if he merely chooses any +arbitrary temperature unit, for example as indicated above. As a control, one can determine +the temperature at several different densities. If one finds pressures pi, p 2 , p 3 , at +temperatures T j, T 2 , T 3 for specific volume v, and pressures j){ f p' p ■[ at the same +temperatures for specific volume v', then the pressure must satisfy the relation + +Pi ~ Pi-Pi - Pi = pi ~ pi-pi ~ pi + + +if the gas is to satisfy van der Waals’ equation with sufficient accuracy. + +If V 1 and Vi are the pressures corresponding to temperatures T | and T 2 at a specific +volume v', then one can write Equation (26) as follows: + + + + + + +Ti-Tt + + + + + + +If one defines T | to be the temperature of melting ice, and T 2 to be the temperature of +boiling water, and sets T 2 - 7) = 100, then all the other quantities in the last two +expressions in this equation are accessible to observation, and one can therefore calculate +Ti . Moreover, one can determine the value of the constant a for hydrogen. + +Since one now knows the absolute temperature, he can immediately determine the +values of r and b for hydrogen, according to the method given earlier. Here the following +remarks must be made: when we consider the van der Waals equation (22) as simply a +given fact of experience, then we must write on its right-hand side, instead of T, some +functional) of Kelvin’s absolute temperature. The absolute temperature itself would not +then be determinable without some empirical information about specific heats or Joule- + + +2ft + + +7fca^4e«tai2c.a/ + + + +Thomson cooling, etc. 10 + +In order to test the relation between p and v at constant temperature T —i.e., the pressure +coefficient of density—we write the van der Waals equation in the foim + +rT a a - rbT + +pv =-= rT -- • + +b v v + +1 -- + +v + +As long as v is large compared to b, and also to a/rT, Boyle’s law is nearly valid; pv is +almost constant at constant temperature. The gas is far from the region of liquefaction. As +long as a>rbT, the correction to Boyle’s law resulting from the van der Waals cohesion +force will dominate that due to the finite extension of the molecular cores, and pv = p/p will +increase with increasing volume. The pressure coefficient of density, clp/clp, decreases with +decreasing pressure. For any gas at very high temperatures, aT k , dp/dv cannot vanish, +and hence cannot become positive; the isotheim falls off with increasing v. When T < T/., +dp/dv goes through zero to a negative value, and then again goes through zero to a positive +value. The ordinate of the isotherm has a min im um and a maximum. For T = Tj., dp/dv is +always negative; it is only once equal to zero, when v = 3b. Hence p decreases with +increasing v, but at this point only by an amount that is an in fin itesimal of higher order than +the in finitesimal increase in volume. This point is called the critical point. We give the +values of v, p, and T corresponding to this point the index k and call them the critical +values,* thus: + +(28) v k = 36, T t = 8o/27rf». + +For the corresponding value of p, the critical pressure, one finds from Equation (22): + +(29) Vk = a/276*. + +v k , Tf. and p k are therefore three real positive values. The former is larger than the min im um +volume b of the substance. From Equation (27) one finds (with T constant): + + + +9 9 + +and one easily sees that for the critical values, d p/dv vanishes, as was to be expected, +since we have seen that for the critical values the isotherm has a completely regular +max im um-min im um [point of inflection]. + +In addition I shall describe an algebraic property of the critical quantities. If we bring all +the quantities to the same side of the equality sign in Equation (22), eliminate the fractions, +and collect powers of v, then this equation becomes + +(30) py 3 - (bp + rT)v 2 + av - ab = 0. + + +2ft + + +7fca^4e«toi2c.a/ + + + +For given values of p and T, this is a third-degree equation for v. We shall denote its left- +hand side by/(y). If not only /(y) but also/(v) and f'(v) vanish for a particular set of values +of p, T, and v, then this third-degree equation has three equal roots for v; here/(v) is the +first derivative, and f'(v) the second derivative of/(y) at constant p and T. + +If one keeps only T constant, then it follows from Equation (30) that + +dp + +(v 3 -bv t )—= -f(v), +dv + + + +d 2 p dp + +- bv 2 ) -b (3^ 2 - 2 bv) — = + +dv 2 dv + + + +The quantities dp/dv and cl~p/dv 2 are the same ones mentioned above, which we have +proved vanish for the critical values of p, v, and T. For these values, therefore, not only /(y) +but also/(v) and/'(v) vanish—i.e., Equation (30) has three equal roots forv, if one +substitutes the critical values for p and T. Now the coefficient of v , taken negative and +divided by p, is equal to the sum of the roots. The coefficient of the term that does not +contain v, likewise taken negative and divided by p, is equal to the product of the roots. +Finally, the coefficient of v, taken positive and divided by p, is equal to the sum of the +products of each pair of roots.* Hence one finds for the values p k and Tj. (for which Eq. +[30] has three equal roots, whose common value we denote by v^,) the three equations + + +rT k 2 a + +3y* = b H- ) dVk = — + +Pk Pk + + +3 ab + +v k = —> + +Pk + + +from which follow the values of v k , p k , and T k already found. For those values of the +temperature for which the ordinate of the isotherm has no minimum, there corresponds to +each value of p only one value of v, so that Equation (30) has only one real root, which is +greater than b\ however, for any temperature for which the ordinate of the isotherm has a +minimum p\ and a max im um p 2 , Equation (30) has three real roots for v, greater than b, +when p lies between p | and p 2 , as one can perceive at once from the form of the isotherms. + +Up to now we have made no particular stipulation about the units of pressure and +volume. The formula will become especially simple if we choose the critical volume and +the critical pressure pj. of each substance as the units of volume and pressure, in discussing +the properties of that substance. We shall also ignore the empirical unit of temperature +derived from the freezing and boiling points of water, and choose for each gas its absolute +critical temperature T k as the unit of absolute temperature. We therefore set: + + +21 . + + +7fca^4e«tai2c.a/ + + + + +v = Vk'jo = 3 boo, + + +V = PW + + +a + +276 2 + + +T = T k r = + + +8a + +27r6 7 + + +Thus we measure the volume by w (the ratio of the volume to the critical volume) and +likewise the pressure and temperature by rt and r. + +These three quantities a>, re, and r we call the reduced volume, reduced pressure, and +reduced temperature, or—when there is no question of comparison with another system of +units—simply the volume, pressure, and temperature of the substance. + +We have of course introduced different units for each gas—which we shall call the van +der Waals units—but this disadvantage is outweighed by the advantage that the equations +become much simpler. Since we can calculate a , b, and r, and hence also v^, p^, and for +each gas from its empirical properties, we can transform from van der Waals’ units to any +other units whenever we wish. If in Equation (22) we replace p, v, and T by n, co, and r, +then we obtain (after dividing through by a factor that can never be zero) + +8 r 3 + +(32) t -- + +3co — 1 w 2 + + +All the constants characterizing the gas have dropped out of this equation. If one bases +measurements on the van der Waals units, then he obtains the same equation of state for all +gases. Van der Waals believes that this equation is valid up to the liquefaction of the gas, +and indeed even into the liquid region. Only the values of the critical volume, pressure, and +temperature depend on the nature of the particular substance; the numbers that express the +actual volume, pressure, and temperature as multiples of the critical values satisfy the same +equation for all substances. In other words, the same equation relates the reduced volume, +reduced pressure, and reduced temperature for all substances. + +Obviously such a broad general relation is unlikely to be exactly correct; nevertheless, +the fact that one can obtain from it an essentially correct description of actual phenomena is +very remarkable. + + +§12. Geometric discussion of the isotherms. + +In order to gain some insight into the relation represented by Equation (32), we shall +draw on the positive abscissa axis OQ from the origin of coordinates O the reduced volume +(o as the abscissa OM. Over the point M we erect the ordinate MP, representing the reduced +pressure rr, parallel to the ordinate axis OIL Then each state of the gas, characterized by its +pressure and volume, is represented by a point P in the plane. The corresponding reduced +temperature is the value of r that would be obtained from Equation (32) for the assumed + + +21 + + +7fca^4e«toi2c.a/ + + + +values of to and tt. If one assumes than van der Waals’ equation is correct, the reduced +pressure would become infinite at ^ = A for any positive r. As we pointed out earlier, +one can compress the substance to the volume ^ = A only by exerting an infinite + +pressure, and since the pressure must increase as the volume decreases, the smaller volumes +for which the formula gives negative pressures are impossible. + +We must therefore limit our considerations to abscissas >i + +By an isotherm we mean the locus of all points representing those states of our +substance for which the temperature has a fixed value. The equation of an isotherm is any +relation between tt and to that follows from Equation (32) if we substitute any arbitrary +constant value for r. We obtain the set of all possible isotherms by letting r take all possible +values from a very small positive value to + oo. From Equation (32) it follows that for each +r, tt has a very large positive value when to is slightly larger than A. On the other hand, tt + +has a very small positive value when to is very large. Moreover, at constant r it follows that + + + +Since this expression is finite for A < to < oo, all isotherms between w = A and to = oo +must be continuous curves. As to approaches the limit they approach asymptotically the +line AB, which is parallel to the ordinate axis at a distance A from it, on the side of positive +ordinates. As to becomes very large, the isotherms lik ewise approach the abscissa axis on +the side of positive ordinates. In the foimer case tt has a very large positive value, while +clrr/dto has a very large negative value; in the latter case, tt is small and positive, while +dTr/dto is small and negative. All isotherms have two branches, going to infinity on the +positive side of the abscissa axis. On the other hand, it is possible for tt to be negative +between ^ — A and to = oo; the curve representing Equation (32) for constant r can dip +down below the abscissa axis. + +In order to form a picture of this behavior, we note first that, as a glance at Equation (32) +shows, smaller values of r will always correspond to smaller values of tt, when to is the +same. Therefore each isotherm must he below the isotherms corresponding to higher +temperatures, so that for each abscissa to the isotherm at the higher temperature has a larger +ordinate than the isotherm at the lower temperature; two isotherms can never intersect. + +We now discuss the expression for dTr/dto, Equation (33). It is a continuous function of +to for to between -A and + co. For very large values of to, and also when to is slightly larger + +than the second term dominates, so that dTr/dto is negative, as we mentioned before, +dTr/dto cannot become positive within this interval without going through zero. The latter +event can happen only for + + +7fca^4e«taiZc.a/ + + + +(34) + + +(3u - 1)’ +T ” 4 ( 0 " + + +according to Equation (33). Not only when to is slightly larger than A but also when to is +very large, the right hand side of this equation has a very small positive value. Its value +changes continuously with to in this interval; as one can find by well-known methods, it has +a single maximum value, 1, for to = 1. Thus there are three cases to be distinguished: + +1. For r > 1, Equation (34) cannot be satisfied, and clrr/cloj cannot vanish but must be +negative in the entire region considered, so that the isotherm (marked 0 in Fig. 1) sinks +continuously toward the abscissa axis as to increases. + +2. Let r = 1, so that the isotherm corresponds exactly to the critical temperature. Then, +according to what we have said about the right-hand side of Equation (34), clrc/clto vanishes +only for to = 1. According to Equation (32), then rr = 1 also. The substance therefore has its +critical temperature, volume, and pressure. This state (the critical state) is represented by the +point A' of Figure 1, whose abscissa and ordinate are both equal to 1. On taking the +derivatives at constant r, it follows from Equation (33) that: + + + +Fig. 1. + + +21 . + + +f^&L+LC- 7fca^4e«toi2c.a/ + + + +9 9 + +For the critical state we therefore have also d, rc/dur = 0 (as was to be expected since we +know that cl 2 p/dv 2 vanishes for the critical state). However, dr’p/dc? is negative. The +isotherm therefore has an inflection point. Its tangent is parallel to the abscissa axis, but the +ordinate decreases with increasing co on both sides. The same isotherm has a second point +of inflection at co = 1.87; it therefore turns its concave side downwards between this value +and the abscissa 1, but for other abscissas it turns its convex side downwards. Curve 1 in +Figure 1 represents the critical isotherm. + +The two inflection points first appear for isotherms corresponding to a reduced +temperature r = 3 7 • 2 -11 = 1.06787, where they both occur at ^ 11 For smaller r + +they separate from each other. For larger r, the isotherms fall to the positive abscissa axis +without inflection points, and drc/dco decreases steadily. + +3. Suppose the temperature is below the critical temperature: 0 < r < 1. Then, as one +sees from Equation (33), drc/dco is positive for co = 1, while for large values of co and for +values near i it is negative. Hence drc/dco must vanish for a value of oj greater than 1, and +also for a value between 1 and i. For none of these values can d 2 rc/dco 2 also vanish, since + +3 9 9 + +from Equation (35) it follows that drc/dco and d rc/dcor can simultaneously vanish only +when co = 1. This follows also from the fact that Equation (30), which differs from the +present equation only in the choice of units, must then have three equal roots for v, and as +we saw this is possible only at the critical pressure, volume, and temperature. For the value +of co between A and 1, at which drc/dco changes from a negative to a positive value as co + +3 9 9 + +increases, and d rc/dcor is positive according to Equation (35), n therefore has a minimum, +while for the other value of co it has a maximum, drc/dco cannot vanish for a third value of +co, since Equation (34), which gives the condition for this, can be written in the form + +(36) W - (3(o - 1) ! = 0. + +The polynomial is negative for co = 0 and positive for ^ = A, so that its third root must he + +somewhere between these two values of co, and hence in the interval that we do not +consider. For ah isotherms corresponding to temperatures below the critical temperature, +the ordinate rc has a minimum for one abscissa co between A and 1, and a maximum for one +abscissa greater than 1. Curve 3, Figure 1, shows its general form. + +§13. Special cases. + +We now consider two special cases of the third case. + +3a. Let r be slightly smaller than 1, say 1 - e. The two ordinates for which rc has an +extremal value are then close to 1, and we can write them in the form 1 + q. Substitution of +r = 1 - e and co = 1 + £ into Equation (36) yields (retaining only terms of the first order of +magnitude) { = i \/4 3 the abscissa for which it cuts the abscissa axis again, then: + + +8r _ 3 + +3 wj “1 W3 + + +3 9 + +—j 4— = 8 r. + +to a w 3 + + +The transformation to the latter equation is permissible since we are not interested in values +of u >3 lying near A; indeed we already know that there is such a solution of the equation, + +corresponding to the intersection of the descending branch 5a with the abscissa axis. +Instead we now want the solution for which is large, and we see that in this case will +be approximately 9/8 r. Thus for very low temperatures, the ordinates of the curve 5b do +not become positive until the volume is very large. At co = 3 7 • 2 -11 , therefore, d~~r/dor +cannot vanish. + + +21 . + + +7fca^4e«toi2c.a/ + + + +CHAPTER II + + +Physical discussion of the van der Waals’ theory. + +§ 14. Stable and unstable states. + +Let us now consider the physical meaning of the diagrams in Figure 1. Each point P of +the quadrant bounded by the two infinite lines AQ, and AB represents a certain volume and +a certain pressure, and thus a certain state of the substance, since Equation (32) provides the +corresponding pressure. We call this state simply the state P. Each curve PQ lying in this +quadrant (which, like the point P, is not shown in the figure) therefore represents a +variation of state, in particular the sequence of different states corresponding to different +points on the curve. We say that the substance experiences the state variation PQ as it +passes through all the states represented by the different points on this curve. + +The isotherm 0 in Figure 1 represents, for example, a compression of the substance at a +constant temperature Tq > 1, stalling from a very large volume. The pressure increases +continuously as the volume decreases, and for large volumes it is nearly inversely +proportional to the volume, since then the quantities b and a/v~ in Equation (22) are +relatively very small. The substance then behaves almost like an ideal gas. On the other +hand, if co is slightly larger than 1/3, then the volume is nearly equal to 1/3 of the critical +volume; then the isotherm rises very rapidly and approaches the line AB asymptotically. In +this case, a very small decrease in volume results in a very large increase in pressure; the +substance is almost incompressible, and behaves like a liquid. The transformation from the +gaseous to the liquid state takes place very gradually; no break in the continuity of the +transition is ever noticeable. This is also hue for the isotherm 1 corresponding to the critical +temperature, in Figure 1, except that at the critical point the tangent of the isotherm is +parallel to the abscissa axis, so that an in fin itesimal isothermal change in volume +corresponds then to a higher order infinitesimal change in pressure. + + + +21 . + + +7fca^4e«tai2c.a/ + + + +Now suppose that we compress the substance at a temperature below the critical +temperature. Then we follow an isotherm for which r < 1, for example isotherm 3 in Figure +1, which corresponds to the temperature r 3 . We draw this isotherm again in Figure 2, and +denote by C and l) the points whose ordinates CC\ and 1)1) \ have the minimum and +max im um values, respectively. The lines drawn from C and D parallel to the abscissa axis +intersect the isotherm again at E and F, respectively. Let the projections of the latter two +points on the abscissa axis be £j and F \, respectively. As long as the pressure is less than +EE\, we find ourselves on the branch LE of the isotherm. At a given temperature, only one +state of the substance is possible for each pressure, and Boyle’s law is obeyed with greater +and greater accuracy as the pressure decreases. On the other hand, as soon as the pressure +becomes equal to EE | two completely different states (phases of the substance) are possible +at the same temperature and pressure, represented by the two points E and C on the +isotherm, which have equal ordinates. The phase represented by the point E (or simply, the +phase E) corresponds to a huger specific volume, hence to a smaller density, while the +phase C corresponds to a larger density. The former phase is the vapor, the latter the liquid. +If the isotherm corresponds to a temperature that is only slightly below the critical +temperature, then the two points C and E will be very close together, and the substance will +have only slightly different properties in these two states. Far below the critical temperature, +however, the liquid and vapor phases are completely different. + +As soon as the pressure becomes equal to DD |, there are again two points D and F on +the isotherm, each of which represents a possible phase at that temperature and pressure. +However, if the pressure is between EEy and DD | say GG |, then we have three points on +the isotherm: G, H , and J, to which this pressure corresponds. One easity convinces h im self +that the state corresponding to the middle point H is unstable. + +Suppose the substance is in a cylindrical container closed with an easily displaceable +piston, and it is initially in the state H , so that at equilibrium the pressure HH\ is exerted in +the piston. Now if we push the piston in a little bit, without changing the external pressure, +then the volume will decrease somewhat. We assume that the substance is surrounded by a +good heat conductor, so that its temperature remains constantly equal to that of its +surroundings. Then, as the nature of the isotherm near// shows, the pressure of the +substance on the piston will decrease; the piston will therefore be pushed in by the external +pressure until the volume becomes equal to OC\. Since the same thing would happen for +an infinitesimal isotheimal expansion, the least motion of the piston will cause the volume +to change by a finite amount from its original value. Hence for any pressure between EE | +a n d DD | only two stable phases are possible at the temperature r 3 corresponding to +isotherm 3. + +§15. Undercooling. Delayed evaporation. + +What would happen if the critical volume of unit mass of the substance were between +DC | and OD j at temperature r 3 ? Such a volume does not correspond to any stable state of +the substance at temperature r 3 , yet it must be a possible volume since there must be some +kind of transition between the smaller volume of the liquid phase and the larger volume of + + +21 . + + +7fca^4e«toi2c.a/ + + + +the vapor. This paradox is resolved by the possibility that one pail of the substance may be +in the liquid phase, while simultaneously another part is in the vapor phase, so that the +points representing the two coexisting phases must he on the same isotherm if there is to be +thermal and mechanical equilibrium, and must have the same distance from the abscissa +axis, since the temperature and pressure must be equal for the two phases. Under the +influence of gravity, the heavier liquid phase will of course collect at the bottom of the +container and the vapor will stay on top. + +Coexistence of the liquid and vapor phases can also occur if the volume is between OF \ +and OC\, or between OD \ and OE\. If the substance first undergoes the state variation LE +and then its volume decreases still further isotheimahy, then there are two possibilities, +according to our present considerations. The further state variation can be represented by +the curve ED, so that all of the substance is in the same state at every instant. However, at +any moment it can happen that, while one pail of the substance continues along the curve +DE, another part can go from a state G over to a state J having the same temperature and +pressure. Further decrease in the volume can then cause a larger amount of the substance to +pass from G to J, instead of continuing along the curve GD. In fact, if there is no dust or +other substance present that can initiate condensation, a vapor can be compressed without +condensation, even though under other circumstances—especially in the presence of a +small amount of the same substance in the liquid state—it would already have stalled to +condense. Since this state is more often reached by cooling than by compression, it is called +the undercooled vapor. When condensation finally occurs, a larger amount of it suddenly +liquefies irreversibly (i.e., the liquid-vapor mixture thereby produced cannot be transformed +back into undercooled vapor in such a way that it goes through the same sequence of states +in reverse). + +The same thing happens of course with evaporation. The substance first passes through +the states on the curve MF, so that it is initially a highly compressed liquid. When the +volume has increased isothermally to the state F, it can continue along the curve FJC; or, at +any point, the coexistence of two phases can begin, so that from then on a pail of the +substance is transformed into the vapor state. On further expansion, one pail of the +substance will remain at the state J on the branch FC, while the other will go over to the +state G, which has the same height, on the curve DE. Evaporation may be delayed when +the liquid and the wall of the container are free of air. However, on further expansion or +heating, a large amount will suddenly evaporate (evaporation- or boiling-delay). The latter +process is lik ewise not reversible. + +If we find ourselves on an isotherm which, l ik e isotherm 4 in Figure 1, dips below the +abscissa axis, then the pressure may even become negative. Mercury extracted from a +barometer tube provides a good example of this. If one gradually draws the mercury out of +a barometer tube (which is sealed at the top) then the pressure at the upper end continually +decreases; the mercury itself goes through the sequence of states MFJC in Figure 2. The +mercury column still does not break apart, even after it has become higher than the +barometer position, which shows that the point C, where the ordinate of the isotherm has its +minimum, lies below the abscissa axis, lik e isotherm 4 in Figure 1. Finally the column +breaks and the mercury quickly evaporates; of course this is barely noticeable because of + + +7fca^4e«tai2c.a/ + + + +the small tension of mercury vapor. However, if distilled water is placed in the barometer +tube above the mercury, then the copious evolution of vapor in the water is visible to the +eye. + +A negative pressure can also occur in distilled water at room temperature. Hence for +water the isotherm corresponding to room temperature also dips below the abscissa axis, +l ik e isotherm 4 in Figure 1. On the other hand, for ether the isotherms do not go below the +abscissa axis for easily observable temperatures. If one had put ether above the mercury in +the experiment described above, then he could make the mercury column so long that the +pressure in the ether would be smaller than the saturation pressure of ether vapor, though +not so low that it would be negative. + +In processes that one usually calls boiling-delay (superheating), the substance is in +contact with its own vapor. The state is not then an equilibrium state, but on the contrary +violent evaporation takes place on the upper surface of the liquid; the temperature is equal +to the boiling temperature corresponding to the pressure of the vapor standing over it. It is +hotter in the interior, where the same state as in evaporation-delay is found, and this state +can be maintained only by continuous evaporation at the surface and heat conduction +through the interior. + +§16. Stable coexistence of both phases. + +It appears that our description of the nature of state variations is not uniquely +determined. However, it is to be expected that a unique determination can be achieved if +we exclude irreversible transitions, such as undercooling and delayed evaporation. + +For this puipose we consider a certain sample q of our substance, whose mass shall be +equal to 1. It is initially in the state L (Fig. 2), and it will be compressed isothermally. As +soon as it reaches E, it will from time to time be brought into contact with liquid at the same +temperature and pressure. Initially, when q is still near the state E, this liquid will be in the +state of delayed evaporation. It will explosively evaporate into q. Nevertheless we shall +restore the previous state of q, compress it again a little bit, and again bring it into contact +with the liquid at the same temperature and pressure. We repeat this process until eventually +we come to a state in which, when q is brought into contact with the liquid at the same +temperature and pressure, none of the liquid evaporates into q, and none of q condenses; +thus liquid and vapor are in equilibrium. At this temperature the liquid is in the state J in +Figure 2, and the vapor is in the state G. + +This equilibrium state cannot depend on the mixing ratio of the two phases, but only on +the state at the surface of contact, for the molecules at the surface of contact are the only +ones that are in equilibrium with the other phase. However, the size of the surface of +contact cannot be important, since each part is subjected to similar conditions. 1 Hence any +arbitrary quantity of the substance in phase J can be in equilibrium with any arbitrary +quantity of the same substance in phase G, as soon as equilibrium between the two phases +is possible at all. + +The state G then forms the boundary between the normal and the undercooled vapor, +while J forms the boundary between the noimal and superheated liquid. + + +7fca^4e«toi2c.a/ + + + +If one compresses q, excluding bodies that can effect condensation, then it can pass +through states along the curve GD. In each such state, however, if it is brought into contact +with the liquid at the same temperature and pressure, it will suddenly condense in an +irreversible way. But if it is compressed while in contact with the liquid, starting from the +state G, then it will condense more and more until it is completely transformed into the +liquid state, and this process is reversible, since the substance can be evaporated again in +the same way, if one increases the volume at constant temperature. + +The positions of the points G and J, which form the boundary between the normal and +the superheated or undercooled states for a given isotherm, were found by Maxwell in the + +'•y + +following way, by introducing a hypothesis. As is well known, for any reversible cyclic +process one has \clQ/T = 0, where dQ is the added heat and T the absolute temperature. The +heat is to be measured in mechanical units. Maxwell assumes that this equation still remains +valid even when unstable states intervene between the initial and final states, for example +those states represented by the branch CHD in Figure 2. + +If the cyclic process takes place at constant temperature, then the factor 1 IT can be taken +outside the integral sign, and one is left with \dQ = 0. dQ is equal to the excess dJ of the +internal energy of the substance over the externally performed work. The latter is equal to +pdv when, as assumed here, the external force is just the normal pressure force whose +intensity on unit surface is equal to p for all surface elements. + +Since )dJ vanishes for any cyclic process, one has therefore \pdv = 0, for which one can +also write \mioj = 0, since the choice of units is completely arbitrary. We now consider unit +mass of the substance. It undergoes the following cyclic process at constant temperature. +Initially all its parts are in the phase J, then it is transformed bit by bit into phase G. The +pressure remains constant and equal to JJ\ , but the volume increases from OJ \ to GGj. +The external work thereby performed, \rcdoj, is equal to the product of the pressure and the +volume increase, hence it is equal to the area of the rectangle JJ^GiG = R. Now we +imagine that we return to the original state by following the curve GDHCJ. The volume +will decrease, so that external work is done on the substance. This work is equal to the +negative of the integral J ndu) taken along the entire state variation. Since co is the abscissa +and 7i the ordinate of the curve, the integral is equal to the area J \JCHDGG \J \ = O, which +is bounded above by the curve JCHDG, below by the abscissa axis, and on the right and +left by the two ordinates JJ\ and GGj. At the end of the process the substance returns to its +original state K; \ndco extended over the entire state variation is therefore equal to the +difference R - O, which is equal to the difference JCH - HDG of the two shaded areas in +Figure 2. If one makes the Maxwell hypothesis that the second law must be valid even +though pari of the imaginary path goes through unstable states, then he finds the following +result: the line GHJ, which connects the two phases G and J, which are in contact at +equilibrium (the two-phase line) must be drawn in such a way that the two shaded areas in +Figure 2 turn out to be equal. If this condition is not satisfied, then the two phases G and J +cannot be in equilibrium, even though they lie on the same isotherm and are at the same +distance from the abscissa axis. (Concerning the equation that expresses this condition, cf. +§60.) + + +222 _ + + +7fca^4e«toiZc.a/ + + + +17. Geometric representation of the states in which two phases coexist. + + +If in the future we always mean by GHJ a line parallel to the abscissa axis, for which +the two shaded areas are equal, then the results concerning the behavior of a substance +under isothermal compression at a temperature r 3 can be expressed as follows. As long as +the volume is greater than OE | it is in the vapor state. If the volume is between OE\ and +OG |. the liquid phase still cannot coexist with the vapor phase. Condensation can occur +only when a salt, or rather a body whose particles attract the particles of the substance more +strongly than they attract each other, is present. The liquid thereby formed will dissolve the +salt or cover the body, and, if an infinite amount is not present, the vapor pressure drops as +this process continues (premature condensation). If no such body is present, the substance +will remain gaseous until its volume becomes equal to OGy. Here, if it is brought into +contact with the least amount of the same substance in the liquid state, further isothermal +compression causes condensation, and the vapor pressure does not increase until all of the +substance has liquefied, since vapor of a higher pressure cannot exist over the liquid phase +(normal condensation). If no body is present to facilitate noimal condensation, then the +substance can be still further compressed without condensation, so that its states are +represented by the curved line GD (under-cooled vapor). However, if condensation sets in +—which must happen in any case if the volume becomes less than OD |—then a finite +amount of the substance suddenly liquefies, and the pressure drops to the value GG \ if the +temperature is held constant. The substance behaves in a similar way if it is initially liquid +and is gradually expanded; instead of bringing a small amount of liquid in contact with the +vapor, one now creates in the liquid an empty or vapor-filled cavity. + +We must omit from each isotherm the section CHD , since it corresponds to states that +cannot be physically realized. Moreover, we shall consider neither delayed evaporation +(superheating) nor undercooled or prematurely condensed vapor, but only normal +condensation, which represents the directly reversible transition from the liquid to the vapor +state. We then have to retain only the parts MJ and GL for each isotherm, in Figure 2. In +the intermediate region one part of the substance will be liquid, in a state represented by the +point/, while the other will be gaseous, in the state (?, both parts being at the same +temperature and pressure. Each such intermediate state can be represented by the two +points G and / simultaneously, each point having a weight corresponding to the fraction of +the substance in that state. It is preferable to represent these states by different points on the +line JG (the two-phase line). The ordinate NN\ (Fig. 2) of an arbitrary point N on this line +represents the pressure, which is the same for both coexisting phases. The abscissa ONy +should be chosen so that it is equal to the total volume of the substance, i.e., the sum of the +liquid and gaseous parts. The larger the liquid fraction, the closer to / will lie the point +representing the state, and conversely. If we denote by x the mass of the liquid, and by 1 - +x the mass of the vapor, in the state N, then x has the following property: since OJ\ is the +specific volume of the liquid, and OG \ that of the vapor, then x • OJ\ is the volume of the +liquid and (1 - x) • OG | that of the gaseous part of the state represented by N. Since the +sum of the volumes is equal to the abscissa ONy, one has the equation + + +7fca^4e«toi2c.a/ + + + + +x-OJ i + (1 - x)'OG\ = ONi, + + +from which it follows that: + +NiGi _ NG JJfi _ JN + +17Fi~Jg' ~ x ~ Jth ~ JG’ + +x NiGi NG + +If one imagines that the mass x of the hquid is concentrated at the point J, and that of the +vapor, 1 - x, at the point G, then N is the center of gravity of the system formed from the +two masses. The rule that the reciprocal of the abscissa always represents the density does +not of course hold for points on the two-phase line. On the contrary, if p| is the density of +the liquid, and p 2 that of the gas, then the abscissa is + +x 1 - x + +ON i = — +- + +Pl P2 + +If we represent the states where hquid and gas coexist in this way, and ignore the +undercooled vapor as well as the superheated liquid, then the isotherms will have the form +shown in Figure 3 instead of that in Figure 1. The isotherm labeled 3 in Figures 1 and 2 is +likewise marked as 3 in Figure 3. The part JG is a straight line, and a points on this line +represents a state in which a part x of the substance is liquid, while the remainder 1 - x is +gaseous, and the two parts are both at a pressure AW|, while the sum of their volumes is +ONi ■ x and 1 - x are then given by Equations (37). Also, for those isotherms for which the +pail representing delayed evaporation sinks below the abscissa axis, there is a line JG, +always lying above the abscissa axis, for which the two shaded areas in Figure 2 are equal; +for the area between the abscissa axis and the pail of the isotheim that goes below the axis +is always finite, while the area between the pail of the isotherm corresponding to larger +abscissas and the abscissa axis becomes logarithmically infinite as the abscissa goes to +infinity. Hence equating the two shaded areas in Figure 2 always leads to a two-phase line +lying above the abscissa axis. + + + +7fca^4e«toi2c.a/ + + + + +Fig. 3. + + +§ 18. Definition of the concepts gas, vapor, and liquid. + +We may denote the region lying above the critical isotherm 1, shaded horizontally in +Figure 3, as the gas region. The points in this region that have large abscissas will in fact +represent states close to the ideal gas state. The smaller abscissas, corresponding to points +lying near the lineAB, represent of course states in which the substance behaves like a +liquid, but since these states can be transformed isotheimally without discontinuity into +states that are undoubtedly gaseous, we shall count them as belonging to the gas region. An +example is provided by compressed air at ordinary temperatures. + +The region filled by two-phase lines will be called the two-phase region; it is shaded +vertically in Figure 3. Since the dotted curve of Figure 1, which is the geometrical locus of +all maxima and minima of the isotherms, lies entirely within the two phase region in any +case, the curve bounding this region has a smaller curvature than the dotted curve at the +critical point, and in any case it has no cusp. + +Beneath the gas region and to the right of the two-phase region lies the vapor region, in +which the substance behaves l ik e a gas; on the left of the two-phase region is the liquid +region (bounded by the lineAB) in which we call the substance a “condensable fluid.” +These two last-named regions are shaded with slanted lines in Figure 3. They are +characterized by the property that no state of one can be transformed isotheimally into a +state of the other without condensation. + +A typical isotheim going through these regions is curve 3 of Figure 3. If one starts from +the low-density region, he can make the following statements about the properties of the +substance under isothermal compression : in the vapor region, as long as the volume is +large, Boyle’s law is obeyed approximately. If one decreases the volume, deviations from +this law become more and more appreciable. As soon as one reaches the two-phase region, +the pressure remains constant on further compression, and an increasing proportion of the +substance becomes liquid. After all the substance has been liquefied, the pressure increases +rapidly on further compression. + +Two extreme cases are illustrated by isotherms 2 and 5. The first lies near the critical +isotherm. Here the liquid phase differs but little from the vapor. Condensation lasts only a + + +Wte, 7fca^4e«tai2c.a/ “Pity. i .Zc. 1 + + + + + + + + + + + + + + + + + + + + + + + + + + +short time, so that it has only the character of a temporary irregularity in the compressibility. +Isotherm 5, on the other hand, corresponds to a temperature far below the critical; in +general the vapor exerts only a vanishingly small pressure, and indeed it is hardly +noticeable at all. As soon as the pressure reaches an appreciable value, and no other +substance is mixed with the one considered, then it can exist only as a liquid, which has a +definite volume that can be altered only slightly by pressure. The compressibility of the +liquid at temperatures far below the critical temperature is therefore very small. Of course, +according to our diagram, the vapor pressure can only approach asymptotically to zero. A +trace of vapor must always be present even at the lowest temperatures. + +§19. Arbitrariness of the definitions of the preceding section. + +We have based our definitions of the concepts gas, vapor, and liquid on isothermal state +variations. This is of course an arbitrary procedure, which can at best be justified by saying +that in practice one usually tries to keep the temperature equal to that of the surroundings, +and thus as nearly constant as possible. We could also consider state variations such that the +substance is in a cylindrical container closed by an airtight, easily movable piston, subject +to a constant pressure. Suppose the temperature is initially very high. As heat is lost, the +volume decreases. We call such a state variation an isobaric one; it is represented by a line +parallel to the abscissa axis (the isobar). If the constant pressure acting on the substance is +greater than the critical pressure, then it goes over from an approximately gaseous state to +an approximately liquid state, without discontinuity. If, on the other hand, the pressure is +smaller than the critical pressure, then the temperature drops with increasing volume only +until the two-phase region is reached. Since the isobar then coincides with the isotherm, the +temperature remains constant until all of the substance is liquefied. If one wanted to base +the definition of the concepts “vapor” and “gas” on isobaric compression, then the +separating line between the two states would be a line parallel to the abscissa axis passing +through the critical point, since above this line, isobaric transformation of any state into any +other state never involves any condensation, whereas below it, isobaric compression +always goes through the two-phase region. + +One would obtain yet another distinction between vapor and gas if he used adiabatic +variations, i.e., variations not involving addition or loss of heat. In order to compute these, +one must set the differential of the added heat, dQ, equal to zero (cf. §21). We shall not +pursue this further, and we remark only that if one does not adopt some definite criterion— +isothermal, isobaric, or adiabatic—for state variations, then a distinction between states that +can or cannot be continuously transformed into other states is not in general possible. For +one can go from any state into any other state without passing through the two-phase +region, and thus without condensation or evaporation. One can see this in the following +way: let points in the two-phase region have the significance that their ordinates are equal +to the pressure, and their abscissas are equal to the total volume of the liquid and vapor +states combined, so that any point of the quadrant BAD. in Figure 3 represents a possible +state of the substance. Let two arbitrary states of the substance be given. In each of these +states, let the entire substance be in the same phase, so that the two points, called P and Q, + + +7fca^4e«tai2c.a/ + + + +are both outside the two-phase region. We can always connect them with a curve that goes +above K from P to Q without going through the two-phase region. This curve represents a +continuous sequence of states by which the substance can be transformed from the state P +to the state Q without any part of the substance ever being in a phase different from the rest. +However, we can also draw a curve from P toward the left boundary line AJK of the two- +phase region, then across this region from left to right, and finally above it to Q . This curve +would represent a state variation in which the substance starts out in the liquid phase, +gradually evaporates, and then when it has completely vaporized, passes continuously to +the final state Q. Conversely, a curve might go from P, above the two-phase region to a +point lying on the right-hand boundary curve KGQ of the two-phase region, then across +this region to a point on the curve AJK, and then above the two-phase region to Q. Such a +curve would represent a state variation in which the substance is transformed from the state +P continuously to a vapor state, then condenses, and finally ends up in the state Q. + +A substance can even be transformed from a liquid state to a vapor state, not by +evaporation, but rather by condensation. One merely needs to stall with a small volume, +heat the substance above the critical temperature, then expand it to a large volume, cool it +below the critical temperature, then condense it, then again heat the liquid above the critical +temperature, and then transform it to the desired final state. Likewise one can transform a +vapor to a liquid by evaporation. + +Obviously one would call a substance a liquid, vapor, or gas, when its state is +represented by a point near the lower pail of the line AB, or near the curve KGQ, or above +the critical isotherm far from AB, respectively. In the intermediate region, however, these +states can gradually transform into each other, so that if one desired a sharp boundary he +would have to establish it by some arbitrary definition. + +§20. Isopycnic changes of state. + +If we enclose a fixed amount of a substance (say unit mass) in a tube closed on both +sides, and gradually warm it, then we produce very nearly a change of state at constant +volume (isopycnic change of state). If the volume is exactly equal to one-third of the critical +volume (in general it cannot be any smaller than that) then the pressure is always infinite, +except at absolute zero temperature. At any other volume, we would always find ourselves +in the two-phase region at sufficiently low temperatures. Hence some of the substance will +be found as a liquid in the lower pail of the tube. Above it stands the vapor of the same +substance, whose pressure is nearly zero at very small temperatures, and increases with +temperature. The boundary between liquid and vapor is called the meniscus. Since we +assume that the volume is constant, the change of state must be represented by a line +parallel to the ordinate axis, passing through the two-phase region—for example, the line +N\N in Figure 3. In the state represented by the point A, the mass of the liquid part of the +substance is, according to Equation (37), + + +^Pi mJuc. 7fca^4e«tai2c.a/ + + + +and that of the vapor part is + + +x = + + +NG + +Jg + + +1 + + + + +We now have to distinguish three cases: 1. The line N\N representing our state variation +lies to the right of the line KK |, which is parallel to the ordinate axis and goes through the +critical point. The chosen constant volume is then larger than the critical volume. Then, as +the temperature increases, NG becomes smaller and smaller relative to JG. The amount of +liquid in the tube decreases with increasing temperature, and the meniscus drops. Finally, +when the line NyN has reached the boundary of the two-phase region, the entire substance +becomes vapor. 2. The line NN\ lies to the left of KK |. Then JN will instead decrease +relative to NG, the meniscus will rise, and the entire substance becomes liquid at the +moment when the boundary of the two-phase region is reached. 3. When the substance has +precisely the critical volume OK the ratio of TV to NG always remains finite as the +substance is heated at constant volume, until the critical point is reached. The meniscus +always stays between the upper and lower ends of the tube, until finally it vanishes at the +critical temperature, since the liquid and vapor both have the same properties then. + +We saw that the boundary of the two-phase region becomes nearly horizontal in the +neighborhood of the critical point. Therefore the meniscus almost vanishes if NN\ lies in +the neighborhood of KK\ . Theoretically it must remain in the interior of the tube until the +temperature is almost equal to the critical temperature, and then it moves very quickly to the +upper or lower end of the tube.* This cannot be observed, since it has previously already +become so indistinct that one can no longer see it. Moreover, small impurities in the +substance cause significant perturbations at the critical point. + +§21. Calorimetry of a substance following van der Waals’ law. + +Since we have adopted a definite mechanical model, there is no difficulty in determining +an expression for the differential of the added heat, dQ. As in Equation (19) and in Part I, +§8, let q2 be the mean square velocity of the center of mass of a molecule (progressive + +motion), so that i^2 is the mean kinetic energy of the motion of the center of mass of the + +molecules found in unit mass of the substance. If the temperature of unit mass is raised by +dT, then the part of the added heat used to increase this kinetic energy is (in mechanical +units) + +dQi = ]d{c 2 ) = %rdT. + + +7fca^4e«tai2c.a/ + + + +The latter relation follows from Equation (21). + +Although in the derivation of van der Waals’ law we assumed that the molecules behave +almost lik e elastic spheres in collisions, in general we shall not exclude intramolecular +motion; we set (as in Pail I, §8) the work done against intermolecular forces by the applied +heat equal to + +(37a) dQz = pdQi. + + +Since the van der Waals cohesion force acts on each molecule almost equally in all +directions, it will not influence the internal motion. The same results that we shall derive in +§§42-44 for an ideal gas of compound molecules will therefore be valid for this internal +motion. It does not depend on how often the molecules collide with each other, but only on +the temperature, so that /3 can only be a function of temperature. If the molecules are rigid + +2 ; but if they are rigid solids of some other +shape, then /? is equal to 1. If intramolecular motions are present, and if/is the number of +degrees of freedom of a molecule, then the corresponding fraction of the average kinetic +energy will be f — ]. One may also add a contribution for the work done by + +intramolecular forces, which can be a function of temperature. + +At present we shall not go into these details, but simply consider /? to be some function +of temperature in Equation (37a). + +The specific heat of unit mass at constant volume is therefore: + + +solids of revolution, then, as we shall see, ft + + + +dQi 4 dQ) + +It + + +3 + +-r( 1 + 0. + + +If the volume simultaneously increases by dv, then the work done to overcome the +external pressure is pdv, while that done to overcome the internal molecular pressure is +adv/v . Hence the total heat added to unit mass is + + +(38) dQ = + + +3r(l+fl + +2 + + +dT + + + + +rT + +- dv. + +v-b + + +Since van der Waals’ kinetic hypothesis not only determines the equation of state but also +permits one to make statements about the specific heat, it is clear how one can determine +the specific heat by using a substance that satisfies the conditions of the hypothesis, but not +for a substance that merely satisfies empirically the van der Waals equation of state. The +entropy is: + + + +^PuJi.c- 7fca^4e«tai2c.a/ + + + +which, if /? is constant, reduces to : + + +rt [(v - 6)r*a+«rt] + const. + +where l means the natural logarithm. Setting this quantity constant yields the equation for +adiabatic changes of state. The specific heat of unit mass at constant pressure is + +_ 3r(l + j8) r _3r(l + (5) r + +2 ^ 2 a(v - b)' 1 2 ^ 2a{v - b) + +1 - 1 - + +rW vipv 2 + a) + + +and the ratio of specific heats is + + +K=l + + + +3(1 + 0) + + +1 - + + +2 a{v - b) +rTv z + + +n + + +If the gas suddenly expands into a vacuum (as in Gay-Lussac’s experiment) so that the +specific volume and density have initially the values v and p, and afterwards the values v' +and p', then the work done against molecular attraction, per unit mass of the gas, + + + +a + +v + + + +a(p - p'), + + +is therefore independent of temperature. + +However, if the gas expands adiabatically in a reversible manner, then it follows from +Equation (38) that + + +(dT\ 2T T + +W/s " ~ 3(1 + 0)(r - b) ~~~ + +Let unit mass of the substance be originally in the liquid state, and then evaporate at +constant temperature T, at the saturation pressure p of the vapor, corresponding to this +temperature. Then one has to set clT = 0 in Equation (38). The total heat of vaporization is + + + +23 + + +lIic- 7fca^4e«toi2c.a/ + + + +therefore : + + + + ++ p(v' -»). + + +where v and p are the specific volume and density of the liquid, and v' and p are the same +quantities for the vapor at the same temperature. + +The last term of this equation represents the work needed to overcome the external +pressure on the vapor. If one neglects the density of the vapor compared to that of the +liquid, then: + + +I = ap + +is the work of separation of the liquid particles. + +One can now calculate the constant a from the deviations of the dilute heated vapor +from the Boyle-Charles law. From this, one obtains for the liquid state of the same +substance the co-called internal or molecular pressure (i.e., the difference between the +pressure in the interior of the liquid near the surface, and that on the outside of the surface): +ap 2 . The heat of vaporization of the liquid, measured in mechanical units (more precisely, +the heat of separation of its particles) is ap, a quantity that can be compared with +experiment. + +The latter result is independent of the assumption from which we have obtained van der +Waals’ equation, and depends only on the form of that equation, so that it would remain +correct if one one were simply given this equation as an empirical law. + +§22. Size of the molecule. + +By calculating the constant b from the deviations of a gas from the Boyle-Charles law, +one can improve Loschmidt’s determination of the size of a molecule (see §12 of Part I). +We can now calculate the amount of space actually occupied by the molecules in unit mass, +since this is equal to 1/4 b, whereas in Part I we had to use the assumption that the volume +of the molecules in the liquid state cannot be smaller than the total volume of these +molecules themselves, and should not be more than ten times that volume. + +From Equations (77) and (91) of Part I, one obtains for the viscosity coefficient of a gas +the value 9? = kpc/ 7rn<7 2 \/2’ where, according to Equation (89) of Part I, k can +differ only slightly from ^ , if one understands by c the mean velocity of progressive motion +of the gas molecule.* a is the diameter of a molecule, p is the mass in unit volume, n is the +number of molecules in unit volume, so that p/n = m is the mass of a molecule. One can +therefore write also + + +23 , + + +7fca^4e«tai2c.a/ + + + +kmc + + + + +further, we have: + + +2™ 2 + +b = - + +3m + + +hence + + + +<7 = —=— ; + +y/2 kc + +where the mean velocity c can be calculated with sufficient accuracy from Equations (7) +and (46) of Part I. + +I will abstain from giving numerical values, since the choice of the most reliable +numbers would not be possible without an analysis of the experimental data, and this +would be completely out of place in the present book. + +§23. Relations to capillarity. + +Van der Waals obtained the term a/v 2 in his formula by a route somewhat longer than +the one we used in §7; he employed the arguments by which Laplace and Poisson had +derived the basic equations of capillarity.* Since this connection with capillarity is of some +importance, I will summarize here the original derivation of van der Waals. + +The following considerations are applicable to both liquids and gases, but more +especially to the former, wherefore we shall call the substance under consideration “the +fluid” for short. We assume that the attraction of two mass particles m and m' acts in the +direction of their line of centers, and is a function of their distance/, which we call mm'F(f). +We set: + + + +mm= m + + +so that mm'xif) is the work required to bring the two particles m and m' from the distance/ +to a much larger distance. In any case F(f) will have a value different from zero at +molecular distances. We assume that it decreases more rapidly than the inverse third power +of/as/increases, so that not only F(f) but also x(f) and 1 'if) always vanish when/does not +have a very small value. From our hypothesis it follows that the force acting between m + + +23 + + +“Pul+lc- 7fca^4e«toi2c.a/ + + + +and m' is equal to - mm 'dx(f) / df. + +We now construct a sphere K of radius b in the fluid, 3 denote by do a surface element of +K. and further construct the right cylinder Z, situated outside this sphere on the surface +element do, which has a very great length B - b. The center O of the sphere is chosen as +origin of coordinates, and we place the positive abscissa axis along the axis of the cylinder. + +We now pose the problem of finding the attraction dA exerted by the fluid in K on the +fluid contained in the cylinder Z. + +For this purpose, we construct in Z the volume element dZ, which lies between the cross +sections with abscissas x and x+dx. Its volume is dodx, hence it contains mass pdodx of +fluid (the density p is assumed to be constant everywhere). + +We then cut out from the sphere K a concentric spherical shell S, which lies between +spherical surfaces of radii u and u+du, and from this shell we cut out a ring R, for which +the line connecting it with the origin of coordinates makes an angle between & and &+dQ +with the positive abscissa axis. The volume of R is 2tcu sin ddudd. When multiplied by p, +this is the mass contained in the ring. + +Any fluid particle of mass in' that lies in R and has abscissa x' exerts on any other fluid +particle m that lies in the cylinder Z and has abscissa x the attraction + +, d X (!) + +-mm - + +df + + +whose component in the direction of the negative abscissa axis is + + +d x (f) x-x 1 +- mm - + +df f + + +- mm' + + +dxif) + +dx + + +The total fluid mass contained in the ring R therefore exerts an attraction + + +- 27 Tprdodxu 1 sin OdudO + + +dm + +dx + + +on the mass in dZ. The total attraction of the fluid in the sphere K on the fluid in the +cylinder Z can be found by choosing the following order of integration, which is the most +convenient one for this calculation: + + +r b r D d r v + +dA = - 2irp 2 do I udu I dx —I x(/)wsi +d o d u dxd o + + +sin Odd. + + +In carrying out the integration over & we consider just the spherical shell S, so that we + + +23 , + + +7fca^4e«tai2c.a/ + + + +can consider u constant. Substituting/ for u as the variable of integration, we therefore +obtain + + +ux sin = fdf + + +and the li mits for/will be x - u and x+u. Hence: + + + +l + +u sin M = — + + +x + + + +-m)-iK* + u)], + + +where ip is the function defined at the beginning of this section. The integration over x will +now be performed by setting x = B in this expression and subtracting from this the value +obtained by setting x = b in the same expression. + +It is to be recalled that the function ip always vanishes if the argument of the function is +not very small. B and b are very large compared to molecular dimensions, and therefore +also larger than all values of u that come into consideration. Hence ip(B+u), i//(B-u) and +ifib+u) vanish. Only tfib - u) can take a value different from zero, and one obtains: + + +hence + + + + +r 1 + +x(f)u sin ddd - - ij/ib- + + + + +If one introduces the variable z = b - u in the definite integral, then it is transformed to + + + +6 + +2 ^/{z)dz. + + +Since t/Xz) vanishes for large values of z, one can write oo instead of b for the upper limit of +the integral. If one sets + + + +23 , + + +7fca^4e«ta£2c.a/ + + +then + + + +dA 2 ap 2 + +o r + +- = OLD 1 -- + +do b + +If the cylinder Z is surrounded by fluid on all sides, then clearly all the forces acting on it +are neutralized. Hence the fluid surrounding the sphere K must exert a force on the fluid in +the cylinder exactly equal and opposite to that which the fluid in K exerts. Now if the fluid +inK is taken away, then there remains only a fluid mass whose outer surface is the +spherical surface K. dA is then the fraction exerted by this fluid mass on the cylinder Z +placed over one of its surface elements do, in a direction toward the inside, which was +denoted in §2 by pjdo. + +If the surface is plane, or if its radius of curvature is so large that \lb can be neglected, +we obtain the value already found in §7: p { = ap 2 . However, according to Equation (39) +the constant a can be expressed in terms of the law of attraction of the molecules. + +The term -2 apr/b shows that this expression requires a small correction if the surface is +curved. This correction gives rise to capillary phenomena, as is well known; when the +surface is not spherical, it takes the form + + +(40) - ap 2 ( —|- V + +\R, %/ + +where 91 j and 9 ?9 are the two principal radii of curvature of the surface. + +§24. Work of separation of the molecules. + +We shall now express the heat of vaporization in terms of the function x. We construct +around a fluid particle of mass m a spherical shell S, which is bounded by two spherical +surfaces of radii/ and f+df, in which therefore the amount of fluid mass is Arrpfdf. Since +mm'xif) is the work that must be done in order to bring the fluid particles m and m' to +infinite separation, if they are initially at a distance/, then the work required to bring m +from the midpoint of the spherical shell S to a great distance is: + +4 * mPxWf. + +The total work needed in order to bring m from the interior of the fluid to a point far away +from all other fluid particles is therefore: + + +B = 47rpm + + + +29 + + +7fca^4e«tai2c.a/ + + + +If unit mass of the fluid contains n particles, we have mn = 1. If one assumes that in the +vapor each particle is already far away from the sphere of action of all the others, then the +work done in evaporating unit mass of the fluid, to overcome the cohesive force, is + + + +To this we must of course add the workjpdv done against the external pressure in +evaporation. + +^ is therefore only half of nB, since in the expression for nB the work of separation of +each particle from each other one is counted twice. In §21 we found the value ap for the +work of separation where a is given by the first of Equations (39). Partial integration of +the right hand side of this equation gives in fact: + + + +The value of ^ obtained previously therefore agrees with the one found here. + +One would obtain the work of separation directly by integration of the first of Equations +(39), if he first calculated the work of separation of a particle at a distance h from another +one in a plane layer of thickness dh in the fluid. This would be: + + + + +When multiplied by n and integrated over h from zero to infinity, this gives the total work +of separation + +By means of a similar formula we can solve the following problem. Let there be given a +cylinder of cross-section 1; we draw through it anywhere a cross-section AB and ask for the +work necessary to separate the fluid on one side from the fluid on the other side. + +We first calculate the work of separation of a layer of thickness dx at a distance x from +the bottom of the container, lying below AB, from another layer of thickness dh above AB. +We set m = pclx in Equation (41). Then the work of separation is equal to + +WiMx f /x(M. + +J h + +Here h is the distance between the two layers. We keep this fixed for the moment, and +integrate over all allowed values of x. If c is the distance between AB and the bottom, then + + +22 + + +Pi£^ie 7fca^4e«tai2c.a/ + + + +when h is constant we have to integrate from x = c -h to x = c, which gives: + + +/i oo /»e /»oo + +2 rfdh\ fx(f)df dx = 2rpM I / X (M + +4 k 4 e-k J h + +If one integrates h over all possible values from zero to oo, he obtains for the total work of +separation of the fluid above AB from that below AB the value : + +/» 00 /» 00 + +2irp 1 2 I W/i I / X (M = 2ap 2 4 . + +Jo J h + +Since in this process of separation the surface area of the fluid increases by 2 units, the +work required to increase the surface area of the fluid by 1 is just half of this, and is +therefore equal to ap . However, this quantity is at the same time the coefficient of + +1 1 + +in the basic equation of capillarity, (40). In fact it is well known that this coefficient +represents the work required to increase the fluid surface by 1 unit. + +These equations would be considerably more complicated if one introduced the +improvement that Poisson made in the old Laplace theory of capillarity, by taking account +of the variation of density during the transition from the interior to the surface of the fluid. +Yet the form of the equations obtained remains the same; only the expression for the +constant in terms of a definite integral is different. Since capillarity theory is only of +incidental interest here, I will not go into this further, and merely refer to Stefan’s +treatment. 5 + + +1 The curvature of the surface of contact will have an effect, if this is very small. Over a concave +upper surface (as at the meniscus in a capillary tube) the vapor pressure is less than the hydrostatic +pressure of the vapor column that lies between the level of the meniscus and that of the plane liquid- +surface outside the capillary tube. Over a surface that is convex to the same degree, it would be larger by +the same amount. (Cf. end of §23.) + +2 Maxwell, Nature 11, 357, 374 (1875); Scientific Papers 2, 424. See also Clausius, Die Kinetische +Theorie der Case (Braunschweig, 1889-1891), p. 201. + +* See van der Waals’ remark on this point, mentioned at the end of §61. + +* According to the Chapman-Enskog theory, the correct value of kin this formula is 0.499. + +* Laplace, Traite de mecanique celeste (Paris: J. B. M. Duprat, 1798-1825), Supplement au X e +Livre. Poisson, Nouvelle Theorie de Taction capillaire (Paris: Bachelier, 1831). + +3 This is not to be confused with the constant earlier denoted by b. + +4 a is the quantity given by the second of Eqs. (39). + + +23 , + + +7fca^4e«tai2c.a/ + + + + +5 Stefan, Wien. Ber. 94,4 (1886); Ann. Phys. [3] 29,655 (1886). + + +23 , + + +7fca^4e«ta£2c.a/ + + + +CHAPTER III + + +Principles of general mechanics needed for gas theory. + +§25. Conception of the molecule as a mechanical system characterized by +generalized coordinates. + +In all calculations up to now (except for the specific heat) we have treated the gas +molecules as completely elastic spheres or as centers of force, without any internal +structure. Many circumstances show that this assumption cannot exactly conform to reality. + +All gases can be made luminous, and their light then sometimes provides a wonderfully +complicated spectrum. This would be impossible for simple material points; moreover, the +vibrations of elastic spheres could scarcely produce the observed spectral phenomena, even +if one took account in the calculations of the internal motions of the elastic substance, +which we have so far ignored. + +Furthermore, the facts of chemistry compel the assumption that, in chemically +compound gases, the molecules consist of many heterogeneous parts. One can show that +even the molecules of the chemically most simple gases must consist of at least two +separate parts. For example, if Cl and H are bound into C1H, then the C1H gas occupies at +equal temperature and pressure exactly the same space that the Cl and H gases did together. +Since, according to Avogadro’s law (Pail I, §7), there are the same number of molecules in +all gases at equal temperatures and equal pressures, a molecule of chlorine and a molecule +of hydrogen must combine to form two molecules of C1H; hence both the chlorine and the +hydrogen molecules must have been composed of two parts. One half of the chlorine +molecule combines with one half of the hydrogen molecule to form a C1H molecule; the +other two halves of the two original molecules combine to form another C1H molecule. + +In order to take into account this undubitable composite structure of the gas molecule, +one will have to consider it as an aggregate of a definite number of material points, held +together by central forces. One does not obtain very good agreement with experiment in +this way; on the contrary, for many gases the thermal phenomena, at least, are better +interpreted by assuming that the molecules are rigid nonspherical bodies. Thus it appears +that the connection of the parts of these molecules is so intimate that they behave like rigid +solids with respect to thermal phenomena, even though in other cases the constituents +appeal’ to vibrate against each other. + +In view of this circumstance it would be best to make our assumptions about the +properties of molecules so general that all these possibilities can be included as special +cases. We shall therefore obtain a mechanical model that will have the greatest possible +capacity to explain new experimental results. + +We shall consider the molecule as a system of whose nature we know no more than that +its changes of configuration are determined by the general mechanical equations of +Lagrange and Hamilton. It is then a question of studying those properties of a mechanical + + +23 , + + +7fca^4e«toi2c.a/ + + + +system that will later be needed in the most general way. + +Let the state of an arbitrary mechanical system by given. The positions of all its parts +will be uniquely determined by p independently variable quantities p \, p 2 ,.... p fl , which we +call the generalized coordinates. Since the geometric nature of the system, and the masses +of all its parts, are given, we also know the kinetic energy L of the system as a function of +the velocity-changes of its coordinates. This is a homogeneous quadratic function of the +derivatives p { f p . 2 f • • • } pj of the coordinates with respect to the time, whose + +coefficients can be any functions of the coordinates. The partial derivatives of the function +L with respect to p' arc the momenta q, so that one therefore has for each value of i: + + +dL(p, p') + + + +The q are therefore linear functions of the p and the coefficients in these functions can +again be functions of the p. Conversely, one can express the p' as functions of the q. If one +substitutes the appropriate values into Lip, p'), then he obtains L as a function of the p and +q. This function Lip, q) is therefore also determined by the geometric nature of the system.* + +The forces acting on the various parts of the system should likewise be exactly +specified. They should be derived from a potential function V, which is a function only of +the p, and whose negative partial derivatives with respect to the coordinates give the force, +so that for each arbitrary displacement of the system, the increase dV of this function +represents the work done by the system. If the kinetic energy of the system increases by dL +at the same time, then according to the conservation of energy, we have dV + dL = 0. + +Not only the geometric nature of the system in question but also the forces acting on it +are given. The equations of motion of the system are thereby determined. If one wishes to +calculate the actual values of all the coordinates and momenta at some time t, then the initial +state of the system must also be given. One might be given the values of the coordinates +and their time-derivatives at the initial time (time zero). However, one might equally well +be given the values of the coordinates and momenta at time zero, since the momenta are +given as functions of the p'. We shall denote the values of the coordinates and momenta at +time zero by PpP 2 , - , Pjj, Qp C?2> - . Q > The values ppp 2 , - , /> q\, •••> of the +coordinates and momenta at time t are to be considered as given functions of these values +and of the elapsed time 1 . + +Since L and V are given as functions of the p and q, we can calculate L and V for each +instant of time as functions of P, Q, and I. When we put these into the integral + + + +then it is also a function of the initial values P , Q , and of the elapsed time t, since the entire +motion is determined by P and Q, and the integral can be calculated as soon as we are + + +24 + + +IPuJix*. 7fca^4e«toi2c.a/ + + + +given the l im its of integration. + +We saw just now that the 2p quantities p, q are given as functions of P, Q, and 1 —i.e., +there are 2 p equations between 4//+1 quantities p, q, P, Q, and 1 . From these 2 p equations +we have to determine the 2 p quantities p, q as functions of the others. However, we can +also imagine that we have solved these equations for the 2 p quantities q, Q, so that the q +and Q are expressed as functions of the 2/r+l other quantities p, P, t. For the moment we +shall indicate a function of these latter variables by putting a bar above it. Thus q , means +that the quantity q, is to be considered expressed as a function of p, P, and t. Since we can +find W as a function of P, Q, and t, we can also express therein the Q as functions of p, P, +and t, so that W itself (now denoted by J J) becomes a function of p, P, and t. It is well +known 1 that + + +Whence it follows at once that + + +(42) + + +dW . + +dW + +c?* + +II + +1 10 + +II + +1 + +1 «■© + +l + +dQi + + +dp,- + + + +; + + +where the bar means that in the differentiation with respect to any one p, all the other p, all +the P and the time are to be considered constant; similarly in the differentiation with respect +to any one of the P. i and j can take any integer values from 1 to p, independently of each +other. + + +§26. Liouville’s Theorem.* + +When one wishes to discuss any curve whose equation contains an arbitrary parameter, +it is customary to consider simultaneously all the curves obtained by giving this parameter +all its possible values. We are now dealing with a mechanical system (characterized by +given equations of motion) whose motion depends on the values of the 2 p parameters P, Q. +Just as one can represent a curve infinitely many times, each time with a different value of a +parameter, so we can represent our mechanical system infinitely often, so that we obtain +infinitely many mechanical systems, all of the same nature and subject to the same +equations of motion, but with different initial conditions. Among this infinite number, or +family, of mechanical systems, we are given certain ones for which the initial values of the +coordinates and momenta are specified within infinitesimally close l im its, e.g., + + +24 + + +^Pi mJuc. 7fca^4e«ta£2c.a/ + + + + +(43) + + +P i and P [ + dP\ } P 2 and P 2 + dP 2 • • • and + dP^ +Qi and Q { + dQ h Q 2 and Q 2 + dQ 2 • • • Qn and Q „ + dQ M . + +After these systems have moved for the same time t, according to their equations of motion, +the coordinates and momenta will lie between the li mits + + +(pi and pi + dp 1, p 2 and p 2 + dp 2 • • • p h and p M + dp M + +(44) < + +Igi and <71 + dqi, q 2 and q 2 + d £ji+1 — Xl ‘ “ Xu) + +then it follows from the theorem on functional determinants that: + +dx\dxi • • • dx2n = ©d(• • • dfa, + +where + + +7fca^4e«toiZc.a/ + + + + +dh dh +dxi dx\ + + + +% % +^2 + + +000 • • • 100 • • • +000 • • • 010 • • • + + +100 • • • 000 • • • +010 • • • 000 • • • + + +Now since permutation of two horizontal rows changes the sign of the determinant, we +have + + +0 = (- 1 )' + + +100 ■ • • + +010 •• • + + + +If one sets + +^1 = Pi) Xi = Pi • 2!u+l — PXft+i — Pi • • • , + +then + +h = P\ t it m Pi • • • &+i = Pit fc»+2 ■ Pi • • *) + +hence + + +dpidpz • • • dprfPidPi • • • dPn += {-\ydP\dPi • • • dP^dpidpt • • • dp^f + +and it follows from (47) that + +dpidpi • • • dpndqidqt • • • + += (- \yDdP\dPi • • • dP^dpidpt • • • dp„ + + +24 + + +Ma£4cjvi(i£icaZ + + +hence, according to Equation (51), + + + + + + + +dp\dpi • • • dp^dqidqt • • • dq^ + += MPidPi • • • dP^dpidpi • • • dp M , + +which, in conjunction with Equation (49), gives Equation (52) with the correct sign. + + +27. On the introduction of new variables in a product of differentials. + + +Equation (52) is the fundamental equation for the following. Before I proceed to its +application, I shall mention a difficulty that often arises in the theory of definite integrals, +and is not always completely clarified. + +I consider the most general case. Let there be given n arbitrary functions 2 1 . 2?, ••• > +of then independent variablesjq, x 2 , ...,x n , the functions being single-valued and +continuous in a specified region. Conversely, let thex be single-valued continuous +functions of the If we set + + +ah + +5^2 + +c 1 + +5Xi + +dxi + +dXi + +ah + +dh + +ah + +6 X 2 + +dXi + +5X2 + +ah + +ah + +ah + +az„ + +az„ + +ax. + + +then the differentials are related by the equation + + + +The meaning of this equation is perfectly clear if is only a function of .iq, 2 2 is only a +function of x 2 , etc. If then only X| changes by dx\ and all the other 2 remain constant, then +only 2 i will change by df \ , and all the other 2 will remain constant. Likewise a definite +increment d^ 2 °f ^2 will correspond to a definite increment dx 2 of x 2 , and so forth. +Equation (53) will then give the relation between the increments of x and the increments of +£ + + +24 + + +aV^lc. 7fca^4e«tai2c.a/ + + + + +If A'| runs through all its possible values between .iq and x | + dx |, while x 2 has a constant +value somewhere between Aq and x 2 +dx 2 , and l ik ewise all the other a: have constant +values, then in general not only but also all the other <; will change at the same time. +Likewise, in general all the t, will change when x 2 runs through all values between x 2 and +x 2 +dx 2 , while all the other a: remain constant; and each 2 will experience in general a +completely different increment in the second case. We have therefore to consider n different +increments of none of them is equal to the quantity denoted by in Equation (53). +Neither does one obtain Equation (53) by assuming that d ^ means the largest increase that +can experience when each variation of a | between x | and Aq +dx\ is combined with +each variation of Aq between Aq and x 2 +dx 2 , and each such pair of values of x [ and Aq is +combined with each variation of Aq between Aq and x 2 +dx 3, and so forth. + +In order to present clearly the meaning of Equation (53) in the general case, we must +examine the matter more closely. This equation in general has a meaning only when it +refers to the transformation of a definite integral which is to be extended over a certain set +of values of all the x, to another integral in which the variables 2, replace the x. We shall use +the following notation. Let the value of each of the variables x be given; the corresponding +values of all the £ are thereby determined. We call them the values of £ corresponding to +the given x. A region G of values of the x means the aggregate of the set of values of these +variables bounded in the following way: we first include all values of Aq that lie between +two arbitrary given li mits and j-j. Those values of Aq that lie between these l im its are to + +be associated with all values of the second variable x 2 that he between arbitrary given l im its +X® and jd, where however and jd can be continuous functions of the values of A'| that + +are to be associated with the value ofAq. Likewise, each pair of values of.vj and Aq +satisfying the above conditions will be associated with ah values of Aq that lie between r: + +* V + +and , where q-® and can be continuous functions of x | andAq, and so forth. 2 It is +then well known what one means by the definite integral + + + +Xi • • • x n )dxidx 2 • • • dx n + + +extended over the entire region G. This region of values is then said to be of in finitesimal +extent with respect to ah n dimensions —or briefly, n-fold infinitesimal—when the +difference - jr® is infinitesimal, and also the difference jd _ ^.0 j s infinitesimal for all + +values of Aq, the difference q-* ~ .p® is infinitesimal for all pairs of values of Aq and x 2 , and + +so forth. When n = 2, Aq andAq can be represented as the coordinates of a point in the +plane; each region of values then corresponds to a bounded pail of the surface in the plane; +for n = 3, each region of values can be represented by a bounded volume in space. + +Each set of values of the x that lies in the region G corresponds to a set of values of the +£. The region g of the £, which corresponds to the region G of the x, means the aggregate +of all sets of values of the £ that correspond to all sets of values of the x lying in the region + + +24 + + +7fca^4e«tai2c.a/ + + + + + +G. + +According to the way we have formulated our definitions, Jacobi’s theorem on +functional determinants can be expressed in the following completely unambiguous way. + +Let there be given an arbitrary single-valued continuous function of the independent +variables q, x 2 ,... , x n . We denote it by/iq. x 2 ,... , x n ). If we express therein q, x 2 , ..., x n +in terms of F 2 ,..., £ n , then the functional, x 2> •••> x n) transformed to F (^, $ 2 ,..., f n ) +so that we have identically + +f{x i, x 2 • • • x n ) = F({i, (2 * • • &»)• + +However, it is by no means the case that + +* • * f(x l, Xi • • • Xn)dX\dX2 • • 'dx n + += J J ’ * * {2 ’ * * 1^2 * * ‘ + +when the former integral is extended over an arbitrary region G of the x, and the latter over +the corresponding region g of the A If we again denote the functional determinant + +dh flf, + +dll dli + +dh dfc +di2 dx 2 + + +by D. then the definite integral extended over the region G, + +• • -}(x h a: 2 • • • x n )dx\dxi • • • cfx„, + +is always equal to the definite integral extended over the corresponding region g: 3 + + + + +7fca^4e«taiZc.a/ + + + + + +If G is infinitesimal, then, if the values of £ in question are continuous functions of the x, +the region g is also infinitesimal, and the values of the functions / and F, as well as the +functional determinant, will be considered constant throughout the region. Since the values +of the two functions are the same, one can divide through by these values and it follows +that : 4 + + + +Indeed, Kirchhoff writes the equation in this form. 5 It is customary to write (as we did in +the previous paragraph) simply + +1 + +— dfadh • • • dt n = dx\dxi • • • dx n . + + +Here dx\ dxj ... dx n means strictly the /z-t'old integral of this quantity over an arbitrary n-fold +infinitesimal region G, and d^yd^ ■■■ dc, n means the n-fold integral of the latter quantity +over the corresponding region g. Since the theorem will only be applied to the calculation +of definite integrals extended over a finite region, and these can always be decomposed into +an infinite number of integrals extended over infinitesimal regions, one always obtains the +correct result if he writes the equations as follows : + +1 + +— dkdh • • • dk = dx\dx 2 • • • dx n + +D + +I, {:•••£»)= }(x 1, Ij • • • *„), + +hence + +1 + +F(( i, (i ■ ■ • •••#„= f(x u Xi ■ ■ • xjdxidxi ■ ■ ■ dx„ + + +24 + + +7fca^4e«tai2c.a/ + + +and hence finally + + + + +The first of these equations has the following meaning. Each n-fold definite integral +extended over all .x; can be decomposed into infinitely many integrals over n-fold +infinitesimal regions. If one wishes to introduce the T as new variables of integration, then +in each of the latter, and hence also in the entire region of integration, he has to replace the +product dxydx 2 ... dx n by + + + +§28. Application to the formulas of §26. + +If one wishes to use these more correct expressions in §26, then, instead of saying “for a +certain system the initial values of the coordinates and momenta he between + +Pi and Pi + dP\ • • • Q M and Q M + + +he must say instead, “each initial value lies in the 2/./-fold in finitesimal region + + + +Instead of saying “then at time t the values he between /q and p\ +dp |, ... , q u and q^+dq ,” +he must use the corresponding expression, “they he in the corresponding region + + + +Here the integration over the entire appropriate region is indicated by a single integral sign +for the sake of brevity. The region g corresponding to the region G includes all +combinations of values that the variables assume after the time t (considered constant), if +they initiahy have a set of values lying inside the region G. Ah the conclusions of the +preceding section will then remain vahd, except that in place of the simple product of + + +25 + + +Pi£^ie 7fca^4e«tai2c.a/ + + + +differentials there will always occur the integral of this product of differentials over a region +infinitesimal on all sides. Equation (52) then mns, in this more precise formulation: + + + +One therefore sees that the conclusion is not in the least changed, except that one has an +integral sign in front of each differential expression, expressing the integration over a +corresponding infinitesimal region. + +If an example is needed, one can think of the x as the spatial polar coordinates r, 6,

+ + +Ss/da a i da + +-(—I-h + +a \ds a dSi +St 8a t a + += i+- = ~~’ + +r a a t + + +a n da +a ds n / + + + + +C hs/da da i + +dCn\~ + +(63) + +II + ++ + ++ + +i + +i + +— + + + +a \ds ds i + +dsj J + + +The prime always means that the value corresponds to .v+<&, while the initial values are + + +25 + + +7fca^4e«tai2c.a/ + + + + + +those given in Equation (58). Hence one can write Equation (62) in the form: + + + +Similarly, just as we go from 5 to s+ris, we can go from s+<& to s+26s, and so forth, and +also from s - Ss to 5 . We shall denote all values corresponding to s+25s, starting from the +initial values (58), by two primes. + +Let g" be the region that includes all values of the dependent variables corresponding to +initial values lying in G, after s+26s, and let + + + +be the integral of the product of differentials of all dependent variables over the region g". +Then one finds, in the same way that was used to obtain Equation (64), that + + + +Since a similar equation holds for all preceding and following increments of s, one has in +general: + + + +Here cr 0 and r 0 denote the values of a and r for ,v = 0; \clS | clS 2 ... dS n is the integral of the +product of the differentials of all dependent variables extended over the region G. + +One sees at once that Equation (55) is that special case of Equation (65) which one +obtains when s means the time and sq, s 2 , ..., s n are the generalized coordinates p\, p 2 , ... , +p„ and momenta c/\, q 2 ,... , q„ of an arbitrary mechanical system. If in particular, as in §25, +L and V are the kinetic and potential energies of the mechanical systems, and if one sets +L+V = E, then the Lagrange equations for the mechanical system run as follows : 8 + + +25 + + +7fca^4e«tai2c.a/ + + + +( 66 ) + + +dpi dE dqi dE + +dt dqi dt dpi + +The symbol cl has the same meaning that 5 had previously. We have to specialize our +previous formulas to the case n = 2/./, + + +Here the integral on the left is to be extended over g |, and that on the right over G j, +while E is constant. Equation (80) therefore has the following meaning : we consider many +systems, for all of which the energy has the same value, while the values of the variables +(78) lie initially in the (2// - l)-fold infinitesimal region G \, and the missing momentum +variable c/y is determined by E. For all these systems, the energy will have the same value E +after time t, but the region filled by the values of the variables (78) after time t will be +denoted by gj and will be the region corresponding to G 1 after time t. Equation (80) +always holds then, if one extends the integral on the right over G\, and that on the left over +8l- + + +32. Ergoden.* + +We now imagine again an enormously large number of mechanical systems, all of + + +26 + + +7fca^4e«tai2c.a/ + + + +which have the same properties described earlier. The total energy E will have the same +value for all of them. On the other hand, the initial values of the coordinates and momenta +will have different values for different systems. Let + + +}(p i, Vi • • • p f , 9t • • • t)dp t • • • dp^ dq, + +be the number of systems for which, at time I, the variables (78) he between the limits + +Pi and pi + dpi • • • p M and p M + dp M , „ 0 = /(pi • • • p„, 92' •' % 0 ,) + + +in which the variables p, q can have any arbitrary values but must be the same on both +sides. Therefore one can also denote them by the corresponding capital letters, so that +Equation (86) takes the foim: + + +(87) /(Pi • • • P„, & • • • Q„ t ) = /(Pi ■ • • P„ Qi • • ■ Qn, 0). + +Using the last equation, Equation (85) becomes 13 + + +P,7(P>, P 2 ■ + + +• Pn, Qi ' • ' Qn, t) = Pl7(Pl, P2 • • • P», 92 • * • % <)• + + +Since the function f no longer contains the time, it is better to omit t from the function sign +and to write : + + +Pi ■ • • K Qi--- Qn) = + += Pl7(Pl, P2 • • • P», 92 • • • 9»)- + + +Here, Py,P 2 , ... . P u , Qi, ... ,Q^ are completely arbitrary initial values;pj, p 2 > ••• > Pm + + +26 + + +7fca^4e«tai2c.a/ + + + +> <7u are the values of the coordinates and momenta that a system would acquire, +stalling from these initial values, after an arbitrary time t. + +Thus if we imagine a system S, stalling from some initial values of the coordinates and +momenta, then in the course of its motion the coordinates and momenta will take various +different values. The coordinates and momenta are therefore functions of the initial values +and of the time. But in general there will be certain functions of the coordinates and +momenta (known as invariants) that have constant values during the entire motion: for +example, for a free system these would include the velocity components of the center of +gravity, and the components of the total angular momentum. Now suppose that we + + +substitute into the expression ©//(Dj p 2 + + +* * > ? 2 , • * * , q») + +first the initial values and then the values of the coordinates and momenta at later times. In +order that the distribution be stationary, it is necessary and sufficient that the value of p.7 +remain unchanged—or in other words, P.7 should contain only those functions of the + + +coordinates and momenta that remain constant throughout the entire motion of a system and +depend only on the initial values, but not on the elapsed time. Thus P.7 should be a + + +function only of the invariants. + +The simplest case of a stationary state distribution is obtained by setting + + +Pl7(P.) P2, • • ‘ , Pc, ?2, • ' + + + +equal to a constant; then + + + +is the number of systems for which the variables (78) he in the region over which the +integration is to be extended. I once allowed myself to call the distribution of states among +an infinite number of systems described by this formula an ergodic one. + + +§33. Concept of the momentoid. + +The state distribution mentioned at the conclusion of the preceding section will be +considered further in the following, and indeed we shall introduce other variables in place +of the momenta. + +The kinetic energy L of a system is a homogeneous quadratic function of the momenta; +therefore + + +2 2 + +2 L - duqi + 022*72 * * * + 2ai2<7i - + +-v'2^43/03 \ ^ / + + +If one also performs the integration overr 3 according to Equation (104), then it follows +that: + + +(105) \ + + +J. = + + +2 V' 1 + + + + +2 2 + +■■ • ■ ■ + +uj aj + + + +X + + +dp41 __ U,-—) dr 4 . + + +V2A4/a4 \ + + +From this result we can find the integral of (97) by leaving out the last differential, +performing the other integrations exactly, as we have already done in the expression for J K , +and then setting K = - 1. Then it follows that + + +27 + + +7fca^4e«tai2c.a/ + + + + +If one denotes the last expression by j, then the mean value of in all systems + +for which the coordinates he between the l im its (94) is + + +<*/[ + +2 + + +vU »l a » ct/n + + + + + + +Evaluation of the integral yields + +Of* + +2 + + + +A, _ E - V + + +If one allows k to be arbitrary in Equation (105) and performs ah the integrations, then it +follows that: + + +7fca^4e«taiZc.a/ + + + + +By means of the last two formulas the integrations over the r in all the previous +expressions can be written down immediately, and dN \, dNj, and will be + +calculated in closed form. To perform the integrations over the p, a knowledge of the +potential function V would of course be required. One obtains, for example, for the +probability that, for a system that satisfies the conditions (94), r u lies between r u and +f/j+dfjj, the value + + + +If one sets = X> then + + + +hence the probability that, for a system satisfying the conditions (94), r„ is positive and + + +\(XpT* ik hes between x and x+dx, is: + + +27 + + +^ufuc. 7fca^4e«tai2c.a/ + + + + + + + +(A, - x )^- !),s dx + +2A ( 7 2m 7* + + +Since for negative r f/ an equal value of is equally probable, the probability that + +iaA lies between x and x+dx for either positive or negative r LI is equal to + + + + +(A, - dx +y/x + + +( m - 2)/2 + + +Here r u may be any arbitrary momentoid. If /,/ is very large, and one sets A u = /rf, then the +above expression approaches the li mit + + + +dx + +e -*m ___ . + +y/2^x + + +From the general formulas one finds, furthermore: + + +(109) + + +ot\T? a\J\ E - V + +2 2J-i n u + + +in agreement with Equation (105a). Since the same holds of course for the pail of the +kinetic energy corresponding to the other momentoids, it follows that: + + +2 + + +( 110 ) + + +<* 1>1 Ot\Ti + + +! + + + + +2 + + +2 + + +Thus however the limits (94) may be chosen, the following theorem will always hold in the +case of our assumed (ergodic) distribution of states: we pick out of all systems those for + + +2 7 + + + + +which the coordinates lie between the limits (94). We denote by the kinetic energy + +corresponding to some one momentoid, and calculate the average of its values for all +specified systems at some time t. This average always comes out to be the same for all +times and all values of the index i. It is equal to the //th pail of the energy E - V, which has +the form of a kinetic energy in this case. + +The integration over the coordinates can of course only be indicated formally, and one +finds for the mean value of f° r all systems, for all values of the index i, + + + +Of course this equality of the mean values of the kinetic energy corresponding to each +momentoid has only been proved for the assumed (ergodic) distribution of states. This +distribution is certainly a stationary one. In general there can and will be other stationary +distributions for which this theorem does not hold. + +In the special case that V is a homogeneous quadratic function of the coordinates, as L is +of the momenta, the integration over coordinates can be performed by the same method as +that used for the momenta. Then one obtains from Equation (103) + +E + +(Ilia) 7 = + +2 + +if one determines the additive constant in the potential in such a way that V vanishes when +all the material points are at their rest positions. + +Before I proceed to the application of these theorems to the theory of gases with +polyatomic molecules, I will first adduce a completely general argument which is irrelevant +from a mathematical viewpoint, but calls on experimental evidence; it may perhaps justify +the supposition that the significance of these theorems is not restricted to the theory of +polyatomic gas molecules. + +§35. General relationship to temperature equilibrium. + +We now consider an arbitrary warm body as a mechanical system obeying the laws we +have deduced up to now—in other words, as a system of atoms, or molecules, or rather +some kind of constituents whose positions are determined by generalized coordinates. + + +2 7 + + +7fca^4e«tai2c.a/ + + + +Experience shows that whenever a body has the same thermal energy and is subjected +to the same external conditions, it eventually comes to the same state, no matter what its +initial state may have been. In the sense of the mechanical view of nature, it thus happens +that only certain average values—the mean kinetic energy of a molecule in a finite pail of +the body, the momentum that a molecule transports on the average through a finite surface +in a finite time, and so forth—are accessible to observation. However, these average values +are the same in by far the greatest number of possible states. We call each state that has this +particular average value a probable state. + +Thus if the initial state is not a probable state, the body will coot] pass over to a probable +state if the external conditions remain fixed, and it will persist in this state during further +observations, so that although its state changes progressively and occasionally (within a +very long time period exceeding all possibility of observation) it will indeed deviate +considerably from a probable state, nevertheless it gives the appearance of having attained a +stationary final state, since all observable mean values remain fixed. + +The mathematically most complete method would be to take account of the initial +conditions from which a given warm body happened to evolve to a particular thermal state, +which then persists for a long time. However, since the mean values will always be the +same no matter what the initial state may have been, we can also obtain the same mean +values if we imagine that instead of a single warm body an infinite number are present, +which are completely independent of each other and, each having the same heat content +and the same external conditions, have stalled from all possible initial states. We thus obtain +the correct average values if we consider, instead of a single mechanical system, an infinite +number of equivalent systems, which stalled from arbitrary different initial conditions. Now +these mean values must be the same at all times, as is certainly the case if the average state +of the aggregate of all systems remains stationary, and the state that we consider should not +be an individual singular state, but rather all possible states must be included. + +These conditions are satisfied if we imagine infinitely many mechanical systems, among +which there is initially a state distribution such as the one we called ergodic in §32. We saw +there that this state distribution is stationary, and that it includes all possible states consistent +with the given kinetic energy. + +There is therefore a certain probability that the mean values found in §34 are valid not +only for the aggregate of systems but also for the stationary final state of each individual +warm body, and that in particular in this case the equality of the mean kinetic energy +corresponding to each momentoid is the condition of temperature equilibrium between the +different parts of the warm body. That the condition of temperature equilibrium of warm +bodies has a very simple mechanical meaning independent of their initial state will thereby +be made probable, in that compression, expansion, displacement, etc. of the individual parts +does not affect this equilibrium. + +If we substitute for our general system a system foimed from two different gases +separated by a solid heat-conducting dividing wall (which is clearly a special case of the +general system considered earlier) then we can interpret one of the r as the velocity +component of a molecule multiplied by its mass. According to Equation (110) the mean +kinetic energy of the center of gravity of a molecule must be equal for both gases, whence + + +2 7 + + +7fca^4e«toi2c.a/ + + + +Avogadro’s law follows. + +This mean kinetic energy must be equal to the average kinetic energy corresponding to +an arbitrary momentoid which determines the molecular motion of any body in thermal +equilibrium with the gas. Hence, if we use a perfect gas as thermomet-ric substance, the +increment of kinetic energy corresponding to each such momentoid must be equal to the +temperature increment multiplied by a constant that is the same for all momentoids. The +heat present in the form of kinetic energy of molecular motion in any such body would +therefore be equal to the product of the absolute temperature and the number of +momentoids determining the molecular motion, multiplied by a constant that is the same for +all bodies and all temperatures. + +If we substitute for one of the mechanical systems a pure gas with compound molecules, +which is again a special case, then it follows that for each molecule the mean kinetic energy +of the center of gravity must be equal to three times the mean kinetic energy corresponding +to any one of the momentoids determining the internal motion of the molecules. We shall +derive this theorem (insofar as it concerns gases) in another way in the following sections. + +We can assign six of the momentoids r of a system subjected only to internal forces to +the three components of total momentum and the three components of total angular +momentum, referred to three perpendicular axes. For ergodic systems, the mean kinetic +energy corresponding to each of these is equal to that for any other momentoid, and hence +is vanishingly small when the system consists of many atoms. Our considerations are +therefore relevant to the case of nonrotating bodies at rest, subject to internal forces only. + +Just as we have restricted ourselves in §32 to systems in which the energy has the same +value, we can still further restrict ourselves to systems in which other quantities that are +constant during the entire motion of a system have the same values, for example the +velocity components of the center of gravity or the components of total angular momentum, +as long as the system is subject only to internal forces. One then has to introduce the +differentials of these quantities in place of the differentials of momenta, just as we +introduced the energy differential in §31. One thus obtains other stationary state +distributions which are not ergodic. The corresponding theorems are not necessarily of no +mechanical interest; however, we shall not go into them here, since we shall not need them +for the sequel. 14 + + +* The reader accustomed to modern notation should note that Boltz-mann uses p for position and q +for momentum, not the other way around. + +1 Jacobi, Vorlesung. iib. Dynamik, 19th lecture, Equation 4, p. 146. + +* For references see the footnotes for §29. + +2 Exceptions to the continuity must be limited to individual points. + +3 Naturally we have similarly + + +2a + + +7fca^4e«ta£2c.a/ + + + + +where + + +JJ * ’ 1 F(th $2 * • • £n)$l, »•• • *.) + +and which are not interacting with any other molecules be: + +Aie~ 2 hE 'dP\ • • • dQn . + +Here A j is a constant that is different for the different kinds of molecules, while h is a +constant that has the same value for all kinds of molecules. Let Ey be the value of the sum +of the kinetic energy of a molecule and the potential energy of the intramolecular and +external forces acting on the molecule at the initial time. The negative partial derivatives of +the potential energy function with respect to the coordinates give the components of the +force, so that E | represents the total energy of a molecule, whose value remains constant as +long as the molecule does not interact with others. + +The number of molecules of the first kind not interacting with others, for which the +variables (112) (defined in the previous section) initially lie in a 2/r-fold in finitesimal region +G that includes the value + + +2a + + +IPuJia*. 7fca^4e«ta£2c.a/ + + + +Pi, Pf • • P n Qu Qi ■ • • Q„ + + +(114) + +is therefore + + + +dN i = A\e~ 2hEl + + +i + + + +where the integration is to be extended over the region G. The center of gravity should +have enough room to move around in G so that, although all variables are confined within +very narrow limits, the expression (115) is still a very large number. + +When a molecule of the first kind moves under the influence of internal and external +forces, without interacting with other molecules, and the variables (112) stall from the +initial values (114), then after time t they will have the values + +(116) Pi,P 2 •••«„. + +These will be the actual values of these variables, whereas (112) only gives the names of +the variables. Let e j be the value of the total energy at time t, so that according to the +conservation of energy principle + +(117) fi = E\. + +If, furthermore, all molecules for which the values of the variables (112) initially fill the +region G move without interacting with other molecules, then the values of these variables +after time t will fill a region which we call the region g. It includes of course the values +(116). + +If there were no interaction between the molecules at all, then the molecules whose +variables he in G at time zero must be the same ones whose variables lie in g at time t. If we +denote the number of the latter by dn\, then dn | would be equal to the expression (115), +hence + + +dn\ = A\e ~ 2hE » + +But according to Equation (55), + +j iPi'-dQ, + + +JV. + +-/*■ + + +dQr + +■ ■ dq„ + + +where the latter integration is to be extended over the region g, corresponding after time t to +the region G. Taking account of this fact, and of Equation (117), one finds: + + +2a + + +7fca^4e«tai2c.a/ + + + + +dn\ = Aie~ 2ht[ + + + + +This expression differs from (115) only in that the values of the variables (116) appear in +places of the values (114), e 1 in place of Ey and g in place of G. However, since Equation +(115) should be valid for any values of the variables and any regions enclosing them, (118) +must also represent the number of molecules of the first kind for which the values of the +variables (112) were initially in g. Hence the number of molecules of the first kind for +which the values of the variables (112) he in g has not changed during this time. Since, +finally, the region G and hence also the region g are chosen completely arbitrarily, this must +hold for any arbitrary region. In other words, the number of molecules for which the +variables (112) lie in any arbitrary region does not change during an arbitrary time 1. The +distribution of states remains stationary, as long as only the intramolecular motion is +considered. + + +38. On the possibility that the states of a very large number of molecules can +actually lie within very narrow limits. + + +We have assumed up to now that the regions G and g are very narrowly bounded, and +yet at the same time we have assumed that the values of the variables for a very large +number of molecules lie within these regions. If there are no external forces, this involves +no difficulty. For then all points within the entire gas, when chosen as the position of the +center of gravity of a molecule, are equivalent. The region + + + +dP\dPidP%, + + +within which the center of gravity of a molecule lies does not then need to be infinitesimal, +but rather it can be chosen arbitrarily large, since indeed we may set it equal to the entire +volume of the container, and this can be chosen arbitrarily large. It is only the region within +which the other variables /; 4 , ... , q, t are enclosed—which we call symbolically the region +G/T—which must be (2/r - 3)-fold infinitesimal. + +We have therefore two quantities, one of which (namely the region T) can be chosen +arbitrarily large, whereas the other (G/T) has to be made very small; and there is no relation +between the size of these two regions. Indeed, the differential dp^ ... dq LI expresses the fact +that we can choose G/T to be as small as we wish. However, for any particular such +choice, we can choose T so large than a large number of molecules will always lie in G. + +However, if external forces are present, then there is an upper l im it to the size of the +region E In particular, this region must be chosen so small that the external force can be +considered constant inside it. Then G and g are to be considered 2/r-fold very small; and the +condition that the number of molecules for which the values of the variables he within one + + +28. + + +% J u3uc- 7fca^4e«tai2c.a/ + + + +of these regions must be very large can be satisfied only if the number of molecules in unit +volume is infinite in the mathematical sense. Hence the satisfaction of the above conditions +in this case remains merely an ideal; yet we still expect agreement with experience, for the +following reasons. + +In the molecular theory we assume that the laws of the phenomena found in nature do +not essentially deviate from the l im its that they would approach in the case of an infinite +number of infinitesimally small molecules. This assumption was already made in Part I, for +reasons given in §6. It is indispensable for any application of the in finitesimal calculus to +molecular theory; indeed, without it, our model which strictly deals always with a large +finite number, would not be applicable to apparently continuous quantities. This +assumption will seem best justified to those who have carefully considered experiments for +the direct proof of the atomic constitution of matter. Even in the smallest neighborhood of +the tiniest particles suspended in a gas, the number of molecules is already so large that it +seems futile to hope for any observable deviation, even in a very small time, from the l im its +that the phenomena would approach in the case of an infinite number of molecules. + +If we accept this assumption, then we should also obtain agreement with experience by +calculating the l im it that the laws of the phenomena would approach in the case of an +infinitely increasing number and decreasing size of the molecules. In calculating the latter +l im it, we again have in fact two quantities, which can independently be made arbitrarily +small: the size of the volume element, and the dimensions of the molecules. For any given +choice of the former, we can always choose the latter so small that each volume element +still contains very many molecules, whose properties are closely defined within the given +narrow li mits. + +If, with Kirchhoff, one interprets the expressions (115) and (118) as simply statements +of probabilities, then one can allow them to be fractions or even very small quantities; yet +one thereby loses their perspicuousness. We shall come back to this point at the end of the +book (§92). + +§39. Treatment of collisions of two molecules. + +Up to now we have not considered the interactions of two molecules, and we still have +to seek the conditions under which the initial distribution of states will not be altered by +collisions. For this purpose we must seek the probability of the occurrence of groups of +several molecules. We shall first restrict ourselves to the case that the simultaneous +interaction of more than two molecules occurs so extraordinarily seldom that it is +completely negligible. We can then limit ourselves to the consideration of molecule pairs. + +The number of molecules of the first kind for which the variables (112) initially he in the +region G enclosing the values (114), when none of the molecules are assumed to interact +with each other, will again be given by Equation (115). + +Similarly the coordinates and momenta that determine the position and state of a +molecule belonging to another type (called the second) will be denoted by + + +2a + + +l ? lc . 7fca^4e«tai2c.a/ + + + + +Pm+1j Ph-2 * * ' P/H-'j 9l * ' ‘ ?<*+•'• + + + +We shall ignore other kinds of molecules for the present. Nevertheless, the extension of our +results to simultaneous interactions of several molecules does not offer any difficulty, +though the expressions would become more complicated. + +The number of molecules of the second kind for which the variables (119) initially he in +a region H enclosing the values + +( 120 ) P t+l ■ ■ • for which one again obtains the result (122). + +Since we are at present ignoring the case where more than two molecules interact +simultaneously, we have to consider only all molecule pairs that are initially interacting. We +first consider a pair in which one molecule belongs to the first kind and the other to the +second. The number of such pairs initially interacting, for which the positions and velocities +lie in a 2(/i+v)-fold in fin itesimal region J, will be given by the expression + + + +dN'n = + + + + + +This region J will include certain given values of the variables (112) and (119), which we +call as before P\ ... and Bu+i ••• Q +v , and as before we call these the values (114) and +(120), although of course they do not agree numerically with the values thus denoted +earlier, since now there is an interaction where there was none before. In Equation (123) +the integration is to be extended over the region J. T is the value of the potential function of +the interaction force—i.e., the interaction that occurs during this time between the +constituents of the two molecules. The additive constant in ¥ is to be chosen such that this +function vanishes for all distances of the molecules at which there is no interaction. We +denote by p^+\, p u+ 2 - and p u+3 the differences of the centers of gravity of the two +molecules. + +For any molecule pair considered, the position of the center of gravity of the first +molecule will be equally likely to be at any point within a pail of the volume of the +container, as long as this pail is so small that the external forces can be considered constant +within it. + + +40. Proof that the distribution of states assumed in §37 will not be changed + + +29 + + +PtthLe 7fca^4e«tai2c.a/ + + + +by collisions. + + +The formula (123) is to be considered as the most general one, which also includes +(122), since if the two molecules are not interacting initially, T = 0 and the region J +decomposes into two separate regions G and H, so that (123) reduces to (122). + +A formula similar to (123) will likewise hold if the two interacting molecules belong to +the same kind. + +We now allow an arbitrary time t to elapse, which however should be so short that one +may neglect the possibility that a molecule interacts more than once with another one +during this time. + +If, for a pair of un li ke molecules, the position of the first is initially (114) and the +position of the second is (120), then after a time t has elapsed, these same variables will +have the values + + + +Pi • ■ ■ % P*+1 • • • 4 and i. One thereby obtains again just the expression (127), +regardless of whether there is any interaction at time zero or at time 1 or within this interval. +However, since the region /, and hence also the region i determined by /, are completely +arbitrary, we see that for an arbitrarily chosen region the number of molecule pairs for +which the values of the variables lie within the region is the same at the initial time and at +time 1. The state distribution therefore remains stationary when one takes account of +collisions. + +One sees at once that he can apply completely analogous considerations to molecule +pairs in which both molecules are of the same kind ; and that the same considerations can +be extended to the case where more than two kinds of gas are present in the container. + +Up to now we have chosen the time t so small that we could ignore molecules that +interacted with others twice during this time. But, since we saw that exactly the same +distribution of states holds at time t as at time zero, the same method of reasoning can be +applied again to another time interval of length t, over and over again. One therefore sees +that the state distribution must remain stationary. Also, our assumption that in calculating +the probability of a particular kind of encounter of two molecules, the two events that the +two molecules are found in certain states can be considered independent, must also be true +at all later times. For, according to our assumptions, each molecule moves past a very large +number of molecules between successive interactions, so that the state of the gas at the +place where the molecule experiences one interaction is completely independent of the state +of the gas at the place of its previous interaction, and is determined only by the laws of +probability. Naturally one has to remember that laws of probability are just that. The +possibility of fluctuations hardly comes into consideration; yet when the number of +molecules is finite, the probability of a fluctuation, while very small, is not zero; it can +actually be calculated numerically in any particular case by the laws of probability, and it +vanishes only for the limiting case of an infinite number of molecules. + + +§41. Generalizations. + +We have still imposed a restriction on ourselves by the assumption that the case when +more than two molecules interact plays no role. However, one perceives that this restriction +was made only in order to simplify the proof, whose validity is completely independent of +it. Likewise, just as we have discussed the probability of occurrence of certain molecule + + +29 . + + +“PiLh-d. 7fca^4e«tai2c.a/ + + + +pairs, we can also try to calculate the probability of occurrence of groups of three and more +molecules, and it will be found that such interactions of three or more molecules do not +change the law that the state distribution represented by a formula similar to (123) is a +stationary one. Also, the effect of the wall (not previously mentioned) cannot disturb the +stationary character of this state distribution, in the case where the molecules rebound +directly from it as if it were another identical gas. Any other assumed property of the wall +would of course require new calculations. Yet it is manifest that even then, if the container +is large enough, its effect would not extend into the interior. + +Of course we have not yet proved that the state distribution expressed by Equation (123) +is the only possible stationary one under all circumstances. Indeed such a general proof +cannot be given, since in fact there are other special state distributions which can lik ewise +be stationary. Such cases would for example, be encountered if all the gas molecules +consisted of material points that all originally move in a plane or in a straight line, and the +wall is everywhere perpendicular to this plane or line. But these are special distributions in +which all variables take only a relatively small number of their possible values, whereas +Equation (123) provides a distribution for which all variables take all their possible values. + +It appears scarcely conceivable that there could be any other distributions which are +stationary and for which all variables can run through all their possible values. In addition, +there is a complete analogy between the state distribution represented by Equation (123) +and that for a gas of monatomic molecules. This analogy has a definite foundation. + +Just as in the game of Lotto, any particular quintuple is not a hair less probable than the +quintuple 12345; the latter is distinguished from the others only by having a definite +sequential property lacking in the others. Likewise, the most probable state distribution has +this property only because it shares the same observable average value with by far the +greatest number of equally possible state distributions. 1 Hence that state distribution is the +most probable which, without altering this mean value, allows the greatest number of +permutations of the individual values among the individual molecules. I have already +shown in Pari I, §6, how the mathematical condition for this property leads to Maxwell’s +distribution in the case of monatomic gas molecules. Without going into that further, I will +remark that the validity of the considerations established there is in no way l im ited to the +case of monatomic molecules; on the contrary, similar’ considerations can also be used in +the case of compound molecules. In this case the momentoids corresponding to the +generalized coordinates play exactly the same role that was played by the velocity +components of the center of gravity in the case of monatomic molecules; the potential +function of the internal and external forces plays the same role as was played earlier by the +potential of the external forces alone, so that we obtain Equation (123) directly as a +generalization of the formulas found in Part I. + +That the formula (123) is the only one corresponding to thermal equilibrium, we shall +try to make probable on several grounds in Chapter VII; we shall also give a direct proof in +one of the simplest special cases. At this point, however, in order not to exhaust ourselves +by too long a sojourn with abstract matters, we shall be content with the arguments here +advanced in favor of Equation (123), and draw from it the most important consequences. + + +293 _ + + +7fca^4e«tai2c.a/ + + + +42. Mean value of the kinetic energy corresponding to a momentoid. + + +We shall next consider a mixture of several gases, none of which is considered to be +partly dissociated. At any time the number of molecules interacting with each other will be +vanishingly small compared to the rest, and it is therefore permitted to take account of only +the noninteracting molecules in calculating average values. + +If we introduce instead of the momenta q±, r/ 2 , ..., q, t the corresponding momentoids r 1? +r 2 , ... , r LI , then the number of molecules for which the coordinates and momentoids lie in +any region K enclosing the values + + + +Pi, P 2 • • • p„ ri • • • r„ + + +is given by the expression + + + + + + +which is valid for any kind of gas in the container, since the determinant for the +transformation from the variables q to the variables r is equal to 1. The constant h must +have the same value for all gases present in the same container. The constant A, however, +can have a different value for each kind of gas. e is the sum of the kinetic energy and the +potential energy of intramolecular and external forces of a molecule; the potential function +will be called V. + +For the kinetic energy of a molecule one has, as we saw, the expression + +2 2 2 2 + +L = K«in + aft • • • + + + +-W+Sar*) + +e + + + +where the single integral sign indicates an integration over all possible values of the +differentials. + +If one integrates over r, in both numerator and denominator, then he can bring outside +the r t integral in both cases a factor not depending on r- t . These factors cancel in numerator +and denominator. The integral over r ( - in the numerator is + + + +2 + +2 + + + +while that in the denominator is + + +2 f e-* oiri dr;. + + +In order to find the limits of integration, we recall that for the velocity p' each coordinate +may go through all values from -oo to + oo. The r are linear functions of the p' and can +therefore run through all values from -oo to +oo likewise. These are therefore the l im its of +integration for r- t , and one obtains + + + +as can be seen by integrating by parts in the first integral, or by calculating both integrals +using Equation (39), §7 of Pari I. One can now put the factor ~fi in front of all the integral +signs in the numerator, and the factor 2 in the denominator. The expressions multiplying +these factors are the same in both numerator and denominator, so one can cancel them and +obtain : + + +29 + + +7fca^4e«tai2c.a/ + + + +( 134 ) + + +_ M + +L = — +4 h + + +1 _ +— a,rf +2 + + +1 + +4 h + + +T means the average value of the total kinetic energy of a molecule of the kind considered. +Hence the kinetic energy corresponding to each momentoid has the same value on the +average, and indeed this value is equal for all kinds of gas, since h has the same value for +all kinds of gas. Similarly, as in Part I, §19, this theorem can also be extended to gases that +are in theimal equilibrium with each other by means of a heat-conducting partition. + +Since we have everywhere integrated over each p and r independently, and in general +we have always considered the p as independent variables, we have continually assumed +that there is no relation between the generalized coordinates /q, p 2 , .... /?„. p is therefore the +number of independent variables required to determine the absolute position of all +constituents of a molecule in space, as well as their positions relative to each other. One +calls p the number of degrees of freedom of the molecule, conceived as a mechanical +system. + +One can always choose three of the r to be the three velocity components of the center +of gravity of a molecule in the three coordinate directions, since the total kinetic energy of a +system is always the sum of the kinetic energies of the motion of the center of gravity and +of the motion relative to the center of gravity.- The product of half the total mass of a +molecule and the mean square of one of these velocity components of its center of gravity- +is then the mean kinetic energy corresponding to this momentoid; according to Equation +(134), it has the value }- for each of the coordinate directions. The sum of the three + +average kinetic energies for the three coordinate directions is however equal to the product +of half the total mass of the molecule and the mean square velocity of its center of gravity. +This latter product we shall call the mean kinetic energy of the motion of the center of +gravity, or of the progressive motion of the molecule, and denote it by Therefore + + + +.3 _ _ + +S = —> S:L = 2:n. +4 h + + +The mean kinetic energy of the motion of the center of gravity of a molecule is therefore +the same for any gases in theimal equilibrium with each other. From this there follows, as +we saw in §7 of Part I, the Boyle-Charles-Avogadro law, which therefore appears to have +a kinetic foundation for gases with compound molecules as well. 1 + +§43. The ratio of specific heats, k . + +We shall now assume for a moment that only one kind of gas is present in the container, +and that the external forces can be ignored, so that the intramolecular and intermolecular +interaction forces, and the opposing pressure of the wall of the container against the gas, are + + +29 + + +Pi£h;e 7fca^4e«tai2c.a/ + + + +the only forces that need be considered. + +As in Pail I, § 8 , we denote by CIQ 2 the amount of heat used to raise the kinetic energy of +motion of the centers of gravity of the molecules, and by clQ^ the heat used to increase the +kinetic and potential energy of the intramolecular motion, when the gas experiences a +temperature increase dT. The ratio dQ^ldQ 2 will be denoted by as in Pail I, § 8 . The heat +will always be measured in mechanical units. + +The quantity clQ 3 can be decomposed naturally into two parts: a part dQ$, which is used +to increase the kinetic energy of intramolecular motion, and a part clQ g, which is used to +increase the value of the potential function of the force acting between the constituents of a +molecule (intramolecular force). + +We have denoted by £ the mean kinetic energy of the motion of the center of gravity of +a molecule. If n is the total number of molecules in the gas, then the total kinetic energy of +the progressive motion of the molecules is nS ; hence dQ 3 — ndS Since we have + +denoted by the total mean kinetic energy of a molecule, then £_^ is the mean + +kinetic energy of intramolecular motion of a molecule. The kinetic energy of the +intramolecular motion of all molecules in the gas is therefore — S) or ’ according to + +Equation (135), (p /*•$ — J whence it follows that + + + +If we denote the average value of the potential for a molecule by y, then + +(137) dQ, = ndV. + +The latter quantity cannot be calculated unless we make some special assumption about +V. We shall therefore simply set + + +dQt = edQi + +in order not to restrict the degree of generality, and we then obtain: + +dQ e + dQ e n + +dQ 1 3 + +The ratio of specific heats of the gas k will therefore be, according to Equation (56) of § 8 , +Pail I, + + + +29 . + + +7fca^4e«tai2c.a/ + + + +(138) + + +2 + + +k= 1 +- + +M + 3e + +§44. Value of k for special cases. + +If the molecules are single material points, then they have no other motion besides the +motion of the center of gravity; hence e = 0. To determine their position in space, three +rectangular coordinates are sufficient; hence /./ = 3, k = 1 ^ . + +Now suppose that the molecules are smooth undeformable elastic bodies; then no +variation of the potential of the intramolecular force is permitted; hence e = 0. + +If each molecule is constructed absolutely symmetrically with respect to its center of +gravity—or, still more generally, if it has the form of a sphere whose center of gravity +coincides with its midpoint—then indeed each molecule can make arbitrary rotations +around an arbitrary axis passing through its midpoints; but the velocity of this rotation +cannot be altered in any way by collisions between molecules. If all molecules were +initially not rotating, then they would remain so for all time. On the other hand, if they were +initially rotating, then each molecule would retain its rotation independently of all the +others, although this rotation would exert no observable action. + +Of the variables determining the position of a molecule, only the three coordinates of the +center of gravity come into consideration for collisions, and one has again 4 + +n = 3, k = If. + +It is otherwise when the molecules are absolutely smooth undeformable elastic bodies +that have either the foim of solids of revolution differing from the spherical form, or the +form of spheres in which the center of gravity does not coincide with the midpoint. In the +former case, it will be assumed that either their mass is completely symmetrically arranged +around the axis of rotation, or this axis is at least a principal axis of inertia, that the center of +gravity lies on it, and that the moment of inertia of the molecule with respect to each line +passing through the center of gravity perpendicular to the axis of rotation is the same. If the +bodies are spheres with eccentric centers of gravity, the moments of inertia of the molecule +with respect to any line through the center of gravity perpendicular to the line connecting +the center of gravity and the midpoints must be the same. Then, only the rotation about the +axis of rotation will be without effect on the collisions. All other rotations will be changed +by collisions, so that their kinetic energy must be in theimal equilibrium with the kinetic +energy of progressive motion. + +Five variables are now needed to determine the position of a molecule in space: the +three coordinates of its center of gravity and two angles determining the position of its axis +of rotation in space. Hence/r = 5 and since & is again zero, one has k = 1.4.* If the +molecules are absolutely flat undeformable bodies, not constructed by either of the above +methods, then their rotation about all possible axes will be modified by collisions. Then, to + + +29 . + + +7fca^4e«tai2c.a/ + + + +determine the position of a molecule, three angles determining the total rotation around the +center of gravity are necessary as well as the three coordinates of its center of gravity, and +one gets + + +M = 6, k = If + + +§45. Comparison with experiment. + +It is remarkable that for mercury vapor—whose molecules have long been considered +monatomic on chemical grounds— according to the researches of Kundt and Warburg,* k +is in fact very nearly equal to the value obtained for simple molecules, If. Also for helium, +neon, argon, metargon [xenon], and krypton, Ramsay found almost the same value of k. t +The limited chemical activity of these gases is l ik ewise consistent with their being +monatomic. + +For many gases with very simply constructed compound molecules (possibly for all +those for which a variation of a with temperature has not yet been verified) observation +gives values of k that lie very near to the two other ones that we found, 1.4 and ^ + +Of course the problem is still far from being solved. For many gases, n has yet smaller +values; moreover, Wiillner found that often—and indeed just for these latter gases —k +varies strongly with temperature.]: Our theory predicts that k should vary with temperature +as soon as the potential of the intramolecular forces, V, becomes important; yet it is easy to +see that much remains to be done on the theory of specific heats. + +If the molecules are spheres filled with mass symmetrically around their midpoints, then +of course there is no possibility that they can be set into rotation by collisions, nor that any +initial rotation can be lost. Nevertheless it is improbable that such molecules would remain +rotationless throughout all eternity, or that they would always preserve the same amount of +rotation. It seems more likely that they possess this property only to a very close +approximation, so that their rotational state does not noticeably change during the time in +which the specific heat is determined, even though over a long period of time rotation will +be equilibrated with other molecular motions, so slowly that such energy exchanges escape +our observation. + +Similarly one can assume that in gases for which n = 1.4, the constituents of the +molecule are by no means connected together as absolutely undeformable bodies, but rather +that this connection is so intimate that during the time of observation these constituents do +not move noticeably with respect to each other, and later on their thermal equilibrium with +the progressive motion is established so slowly that this process is not accessible to +observation. In any case, for air at temperatures where noticeable heat begins to be radiated, +it is found that in addition to the five variables determining the state of a molecule, another +one begins to participate in the thermal equilibrium during the time of observation, so that k +varies with temperature and becomes smaller than 1.4; the same must be tine for all other +gases. + +Naturally, because of the obscurity of the nature of all molecular processes, all + + +2 QQ + + +IRuJuc- 7fca^4e«tai2c.a/ PA.ylic.1 + + + +hypotheses about them must be expressed with the greatest caution. The hypotheses +proposed here would be con fir med experimentally if it were to be shown that, for any gas +for which tz varies with temperature, observations extended over a longer period of time +give a smaller value of k than those of shorter duration. + +In spite of the appearance of a potential function for intramolecular forces, k remains +independent of temperature in the case that the constituents of the molecule have fixed rest +positions relative to each other, and that the force acting on them when they move away +from these rest positions is a linear function of the distance. If then X is the number of +variables on which the relative positions of the constituents of a molecule depend, the +coordinates can always be chosen such that the potential function takes the form: + +Kte + + • • • to). + +The quantity i&P should then be the intramolecular potential energy corresponding to +the coordinate pj. Its mean value can be calculated just as we calculated the mean value of +the quantity above, and one obtains likewise the value JL for each i. Hence + +7 = v/ih, 5:7 = 3:X. + +Since these equations are completely similar to Equations (135), then 5 + +dQi = JXdQj, hence e = JX, + +2 + +K = 1+ - + +M + X + +As an example we consider a molecule consisting of two simple material points, or of +two absolutely smooth spheres filled with mass distributed completely symmetrically +around their midpoints. These should exert no force on each other at a certain distance, but +at greater distances they attract and at smaller distances they repel each other, the force +being proportional to the change of distance in both cases. Then this distance is the only +coordinate determining the relative positions, hence X = 1 . + +Five other coordinates are needed to determine the absolute position in space. The total +number p of the p, or the number of degrees of freedom, is then six, and therefore + +If = 1.2857 + +Treatment of further special cases would not be difficult, but it seems to me to be +superfluous as long as more comprehensive experimental data are not available. + + +3Q + + +iPu+LC- 7fca^4e«toi2c.a/ + + +46. Other mean values. + + + +In the previous sections we have calculated the average kinetic energy for a momentoid +obtained by averaging over all values for all molecules of a certain kind in the container. +This mean value does not change if we impose at the same time restrictions on one or more +of the coordinates, for example when we take only those values of all molecules of this +kind whose centers of gravity he within an arbitrarily small rcgion jjjc /P \ dP^dP^'. the +temperature is the same everywhere in the gas. This theorem, which is self-evident in the +absence of external forces, also holds if external forces of any kind are present. + +Nor does this mean value change when we include in the averaging only those +molecules for which some other coordinates are restricted to lie within some arbitrary/ li mits. +The mean value in question will again be given by Equation (133); except that the +integration is to be extended not over all values of the coordinates but only over the +specified region, which is of course the same in both numerator and denominator. Then just +as in §42 the entire factor independent of the integration over dr t comes out in front of the +integral sign, and after performing the integration over dr t in both numerator and + +denominator, one can divide through by this factor, so that one obtains again the value — + +4 h + +for the integral. + +If the a are constant then, as Equation (132) shows, the relative probability that the +value of any momentoid lies between any l im its whatever and that it lies between any other +l im its is completely independent of the position of the molecule in space, and of the relative +positions of its constituents. We shall call this theorem, which we shall use later, the S- +theorem. It refers to the case where the r are quantities proportional to the velocity +components of material points or the angular velocities of a rigid body around its principal +moments of inertia. + +For those kinds of gas to which Equation (129) pertains, one finds the numberin' of +molecules for which the values of the coordinates lie between the limits (130), when the +values of the momenta are not subjected to any restrictive conditions, by integrating +Equation (129) over all the r from - oo to + oo. One obtains in this way + + +Air* 1 * + +(139) dn' - - e~ nv dpidp 2 • • • dp^. + +/i m/2 \/^i«2 • * • + +The mean value ^ of the potential function is jVdn'/jdn', where the integrations are to be +extended over all possible values of the coordinates. + +The number of molecules for which the coordinates he in any one /./-fold infinitesimal +region is then in the same ratio to the number of molecules for which they are in a lik ewise +p-fold in finitesimal region F' without any restriction on the values of the momenta, as + + +3Q + + +^Pi l * lc - 7fca^4e«toi2c.a/ + + + + + + + +where the letters without primes indicate the values for the region F, and those with primes +indicate those for F'\ the first integral is over the former region, and the second one over the +latter region. + +When there are no intramolecular and external forces, and the molecules consist purely +of simple material points, then this is the ratio of the product of the volumes of all volume +elements available to the material points in the region F, to the analogous product for the +region F’. Then the exponentials are equal to 1, so that + + + +\(x{ +y rdpi'-- dp^y + + +/ + + +This formula and Equation (140) are nothing but generalizations—remarkable for their +simplicity and symmetry—of Equation (167), Part I, i.e., the completely trivial formula for +the barometric measurement of height, according to which the numbers of molecules found +in unit volume at different heights z behave like + +g-2 hmgt — Q-gtlrT + + +30 + + +~*Puii.c- 7fca^4e«tai2c.a/ + + + +Since this formula permits one to calculate the pressure of the saturated vapor and the law +of dissociation (cf. §60 and §§62-73) it must be regarded as one of the fundamental +formulas of gas theory. + +If, other things being equal, the molecule pairs exert no forces on each other, either in D +or D', then one has the same expression as before for the quotient dN'/dN, except that he +has to set ip = xfJ = 0. If it is possible to compute the ratio dN'/dN for this latter ease, then +one can obtain the corresponding value when there is some interaction by multiplying it by +Likewise, if there is no interaction in D either time, and there is an interaction in +D' one time but not the other, then one should multiply by e - 1 ' . One can also call dN'/dN +the relative probability of the two events, that for a molecule pair the values of the variables +lie in the region I)' or the region D. + +A similar theorem holds, as can easily be shown, for the interaction of more than two +molecules. The relative probability of two configurations is times larger when + +interactions occur than when they are absent, where ip and iff are the values of the +potentials of the interaction for the two configurations. + + +1 The criterion for equal possibility is provided by Liouville’s theorem. + +2 Cf. Boltzmann, Vorlesungen iiberdie Principe der Mechanik, Part I, §64, p. 208. + +3 We shall think of a particular given solid or liquid as an aggregate of n material points, which +therefore have 3 n degrees of freedom, perhaps just the 3n rectangular coordinates. If it is surrounded by +a larger gas mass, then it can be considered in certain respects as a single gas molecule, and the law found +in the text can be applied to it. The total kinetic energy is therefore 3n/4/z. If the temperature is increased +in such a way that 1/4/? increases by d(U4h), then the total heat, measured in mechanical units, that must +be supplied to raise the mean kinetic energy isdQ, = 3nd(\/4h). This heat, per unit of mass and +temperature increment, is what Clausius calls the true specific heat. It is invariable in all states and forms +of aggregation. It would be the total specific heat if the body were a gas dissociated into its atoms. For +all bodies it is proportional to the number of atoms in the body. The total specific heat is also +proportional to this number (Dulong-Petit law of chemical elements, or Neumann’s law for +compounds) if the heat increment dQ , used to perform internal work is in a constant ratio to that used to +increase the kinetic energy, dQ ,. This is always the case if the internal forces acting on each atom are +proportional to its distance from its rest position, or still more generally if they are linear functions of its +coordinate variations. Then the force function V is a homogeneous quadratic function of the +coordinates, just as the kinetic energy L is such a function of the momenta. The integrations in the +formula for can then be performed in the same way used to obtain Equation (134), and one finds dQf = +dQj. (Cf. Eq. [111a] at the end of §45.) The total heat capacity is then twice the true monatomic gas +value. The assumed law of action of internal molecular forces holds approximately for most solid +bodies. For such bodies, whose heat capacity is smaller than half that predicted by the Dulong-Petit law +(e.g., diamond) one must assume that the motions related to certain parameters come into equilibrium +with the others so slowly that they do not contribute to the specific heats determined by experiment. (Cf. +§35. For the case that the molecule makes approximately pendulum-like motions, see Boltzmann, Wien. +Ber.53, 219 [1866]; 56, 68b (1867); 63, 731 (1871); Richarz, Ann. Physik [3] 48, 708 [1893]; +Staigmuller, Ann. Physik [3] 65,670 [ 1898].) + +4 In general, one obtains in these two cases all the formulas developed in Part I for monatomic +molecules directly from Equation (118), whether external forces are absent (§7) or present (§19). These +formulas are therefore only special cases of (118). + +* This explanation of the anomalous specific heats of diatomic gases was first proposed by +Boltzmann in December 1876 (Wien. Ber. 74,553 [1877]), and independently in April 1877 by R. H. M. + + +30 : + + +7fca^4e«toi2c.a/ + + + + +Bosanquet (Phil. Mag. 15] 3,271 [1877]). + +* Kundt and Warburg, Ann. Physik [2] 157,353 (1876). + +t Ramsay, C. R. Paris 120, 1049 (1895). Ramsay and Collie, Proc. R. S. London 60, 206 (1896). +Rayleigh and Ramsay, Phil. Trans. 186 , 187 (1896). Ramsay and Travers, Proc. R. S. London 63, 405 +(1898). + +t A. Wullner, Ann. Physik [3] 4 , 321 (1878). + +5 Cf. Staigmtiller, Ann. Physik [3] 65 , 655 (1898) and footnote 2, §42. + + +3Q + + +7fca^4e«tai2c.a/ + + + +CHAPTER V + + +Derivation of van der Waal’s equation by means of the virial concept. + +§48. Specification of the point at which van der Waals’ mode of reasoning +requires improvement. + +In deriving van der Waals’ equation in Chapter I, we followed the method that he +himself first used, which is characterized by the greatest simplicity and perspicuousness. It +was already remarked (footnote 1, §7) that it is perhaps not completely free of objection. + +The first assumption subject to doubt is the one that we made in §3 and also later on: +that in the entire container, as well as in the cylinder lying near the boundary of the space— +which we called there the cylinder y—each volume element is equally probable as a +position for the midpoint of a molecule, regardless of its distance from another molecule. + +If there are no forces except the collision forces, the correctness of this assumption +follows directly from Equation (140) , in which V is then constant, so that it therefore +predicts that the average number of molecules will be the same in each equal volume +element. + +On the other hand, the van der Waals cohesion force will produce a denser arrangement +of the molecules in the interior of the fluid than in the immediate neighborhood of the wall. +Van der Waals did not take account of this effect, either in deriving the expression for +collisions at the wall or in calculating the dependence of the term a/v 2 on the density of the +gas. In both cases, however, correct treatment of the volume elements at the boundary is +essential; the quantities being calculated vanish more strongly as the number of particles at +the surface becomes small compared to those inside. Thus one cannot, as with the formulas +of this chapter, improve the accuracy of the results arbitrarily by making the volume large +compared to the surface. + +Van der Waals calculated the correction to the Boyle-Charles law due to the finite +extension of the rigid core of the molecule, as was explained in Chapter I, just as if the +cohesion force were not present. He then calculated the additional term in the external +pressure due to the cohesion forces as if the molecules were vanishingly small. Since the +validity of this procedure might be doubted, we shall give a second deduction of the van +der Waals formula from the theory of the virial (to which van der Waals has l ik ewise +already related it) against which this objection cannot be made. This second derivation +shows that van der Waals’ conclusion is completely correct. Yet we cannot of course +obtain by the exact method precisely the reciprocal of v - b which occurs in van der Waals’ +equation, which van der Waals himself has called inexact; on the contrary we obtain an +infinite series in powers of b/v. + +§49. More general concept of the virial. + + +3(1 + + +7fca^4e«toi2c.a/ ~Ph-y-&XjcS- + + + +The concept of the virial was introduced into gas theory by Clausius.* Let there be +given an arbitrary number of material points. Let in be the mass of one point, and let . 17 ,, +> 7 j, zj v Cj v uj v v h , W/ } be its rectangular coordinates, its velocity, and its velocity components +along the coordinate axes, respectively, at some time !. Let %, (h be the components of +the total force acting on this material point at the same time. The force should have the +property that all the material points can move under its influence for an arbitrarily/ long time, +without any of their coordinates or velocity components increasing without bound. The +initial conditions should be such that this can actually be true. No matter how long a time of +motion may be chosen, the absolute value of any coordinate or velocity component must +remain smaller than a fixed finite quantity, which has the value E for the coordinates and +the value e for the velocity components. Such motions—of which all molecular motions +giving rise to thermal phenomena in a body of finite extent are clearly examples—we shall +call stationary. + +Now let G be the value of any quantity at a particular time t; then we shall call the +quantity + + +1 + +T + + + +Gdt = G + + +the time average of G during the time of motion r. + +By virtue of the equations of motion of mechanics, we have : + +duh + +m h — = fc. + +dt + + +Hence + + +U 2 + +— {m k XhUh) = m h u k + Xkh- +dt + +If one multiplies this equation by dt, integrates over an arbitrary time (from 0 to r) and +finally divides by r, then it follows that: + +— - — r x 0 0 + +m h ul + xfa = — (i»u» - x k u k ), + +T + +where the values at time r are characterized by the upper index r, and the values at time +zero by the upper index 0. By value of the stationary character of the motion + + +3(1 + + +7fca^4e«tai2c.a/ + + + + +is smaller than 2m h E e . If one allows the time of the entire motion, r, to increase beyond +any l im it, then 2m h E e remains finite; hence the expression 2m h EJx approaches zero. If +one takes the mean value for a sufficiently long time, then : + + +m h ui + Xkh = 0. + + +Similar equations are obtained for all coordinate directions and all material points. If one +adds them all up, it follows that: + + +(143) Y1 nihCff + ^(x*{a + + Zhth) = 0. + + +1 Y 2 + + +is the kinetic energy L of the system. The expression + + +Yh (®*{* y^h + thin) + + +Clausius calls the virial of the forces acting on the system. Therefore the above equation +says that twice the time average of the kinetic energy is equal to the negative time average +of the virial of the system during a very long time. + +We now assume that between any two material points m^ and m^, whose distance is r hk , +a force fhk( r hk) acts > n !hc direction of r hk , which we call the internal force. It has a positive +sign when it is repulsive, and a negative sign when it is attractive. Moreover, on each +material point m/ 2 an external force acts, which results from causes lying outside the system +and whose components along the coordinate directions we denote by X h , Y jv and Z h . Then: + +X\ - X2 X\~ Xi + +{l = XlH-/l2^12) H- fu{ r n) + * • •• + +T 12 ^ 13 + +One sees easily 1 that then Equation (143) becomes: + +(144) 2L 4* Yj ( x t$h "I" y^Yh + ZkZh) + Y1 Y1 r Mcfhk(Thk) = 0. + + +The first addend is twice the time average of the kinetic energy of the entire system. The +second is the external virial, and the third the internal virial. The two virials will be denoted +by W a and W- r so that Equation (144) reduces to + +(145) 21 + W„ + IF i = 0. + + +50. Virial of the external pressure acting on a gas. + + +3Q + + +7fca^4e«tai2c.a/ + + + +We consider as a special case a gas in equilibrium, whose molecules, just as explained +in Chapter I, behave according to van der Waals’ assumptions. Let the gas be enclosed in +any container of volume V; it consists of n similar - molecules of mass m and diameter a, and +the mean square velocity of a molecule is q1 . Then: + +( 146 ) 2 L = Yj m h c l ~ nmc2 * + +There are no external forces other than the pressure on the container, whose intensity on +unit surface will be p. The container has the form of a parallelepiped of edges a, /?, y, of +which three adjacent ones will be chosen as the x, y, and z axes. The two lateral surfaces +having the same surface area /3y will have abscissas zero and a respectively. The pressure +forces pf3y and - pfiy respectively will act on these surfaces, in the direction of the positive +abscissa axis. For these two lateral surfaces together the sum X has the value - pocfdy += —pV. Since the equality holds also for the two other coordinate directions, we have for the +entire gas: + + +( x hXh + yhYh + hZh) - - 3pF. + +Since the pressure does not change with time, this is also the mean value of the same +quantity; hence it is the external virial W a . + +The same equation can easily be derived for a container of any shape. Let dto be a +surface element of the projection to of the surface of the container on the yz plane, and K be +the cylinder erected on dto, perpendicular to it and extended to infinity on both sides. This +cylinder cuts from the container surface a series of surface elements c/0 1 , d0 2 , . . . whose +abscissas are jtq, x 2 , • • • and whose normals drawn into the interior of the gas are N\ , N 2 , ■ ■ +. The x component of the pressure force acting on c/0| is: + +j)do\ cos (N ix) = pdu. + +The same x component has, for the surface element d0 2 , the value + +pdo cos ( N2X) = - pdui + +and so forth. The sum Y. extends over all surface elements lying within the cyhnder K. +and therefore has the value: + +- pdo)(xo - X\ + X\ - Xi + • • •). + +The factor of the quantity - p is just the volume cut out by the cyhnder from the interior of +the container. The sum YxhXp extended over the entire gas, is found by integrating this +expression over all surface elements da) of the entire projection to, whereby one obtains the +product of the total volume V of the gas and the quantity —p. Since the same considerations + + +3Q + + +7fca^4e«toi2c.a/ + + + +are applicable also to the y and z axes, it follows that: + +(147) Yj i x hXh + VhYh + ZhZh) = - 3pV = W a . + +§51. Probability of finding the centers of two molecules at a given distance. + +The internal virial will consist of two parts, of which the first, W'i , arises from the +forces acting during the collision of two molecules, and the second, wy, from the +attractive force assumed by van der Waals. + +In order to find Wl we denote (as earlier) by a the diameter of a molecule, and call a +sphere of radius a circumscribed around a molecule its covering sphere, so that the volume +of the covering sphere is eight times as large as the volume of the molecule itself. The +midpoint of a second molecule cannot come closer to the midpoint of our molecule than a +distance cr, and we shall first calculate the probability that the center of a particular +molecule is at a distance between a and a+S (where 6 is infinitesimally small compared to +cr) from one of the other molecules. In order to have a precise name we call this other +molecule the remaining molecule. + +In order to have a concept of probability as free from objection as possible, we imagine +that the same gas is present infinitely many times (N times) in identical containers at +different positions in space. Our specified molecule will in general be in a different place in +each of these N gases. Of all the N gases, let the remaining molecule in iV) gases be in very +nearly the same place relative to the container. N\ is then very small compared to N, but it +should still always be a very large number. The influence of the wall on the interior can in +any case be made small by making the container large, and in the interior the van der Waals +cohesion forces acting on a molecule from all directions will cancel out. Hence, according +to Equation (140) , all positions in the container will be equally probable for the center of +the specified molecule in all these N\ gases. Let N 2 be the number of gases in which the +center of the specified molecule is at a distance between a and a+S from the remaining +molecule. Then the ratio of N\ to N 2 will be the same as the ratio of the total space +available to the center of the specified molecule in one of the N \ gases to the space in +which this center must be found in order that its distance from the center of one of the +remaining molecules may he between a and a+S. The latter space we shall call the +favorable space. + +Since in ah N\ gases the center of each of the remaining molecules has a given position, +and since the center of the specified molecule cannot come nearer to it than the distance a , +we obtain the available space for the center of the specified molecule in one of these gases +by subtracting from the total volume V of the gas the volume of the covering spheres of all +remaining molecules, viz., the quantity + + +4ir(n - 1 ) a and increases +beyond all l im its as soon as r becomes smaller than a. From now on we shall denote by r a +distance slightly less than cr. If the repulsion first began at distances a little less than r, then +the number of molecule pairs for which the distance of centers lay between r and r + 6 +would, according to Equation (150), be equal to: + +ImVh / + +— ( 1+ + + + +Sxnr 3 ^ + +~mr + + +where we can replace a by r in Equation (150) since the latter is only infinitesimally +different from the former. We still have to calculate how the number of pairs will be +decreased by the repulsive force. If in Equation (142) one replaces the p by the rectangular +coordinates of the molecule centers, then he finds that the number of systems for which +these he in a certain volume c/0 1 , d0 2 , ... is proportional to e~ 2l ‘ vo r/0 1 c/0 2 . . ., where V 0 is +the potential energy function whose negative derivative with respect to a coordinate is the +force acting to increase that coordinate. For our molecule pair, V 0 is j usl a function of r, and +indeed it is equal to the negative integral of J{r)dr. As soon as the distance of centers of two +molecules is equal to or greater than a, the repulsion stops, and the potential energy then +has the same value that it has at infinity, which we denote by F( qo). The value of the +potential at r is denoted by F(r). + +The probability that the distance of centers of two molecules lies between r and r+S, in +the absence of ah repulsive forces, is to the probability that it lies between the same limits, +when repulsive forces are present, as + +g-2W(oo).g-2W(r) + + +and for the number of molecule pairs for which the distance of centers is between r and r+S +one finds instead of (151) the expression + + + +2 im 2 r 2 5 / + +— 1 + + + +gJH [/(«•)-? (r)] + + +F 0 = F(r) = + + + +31 . + + +7fca^4e«toi2c.a/ + + +Since + + + +therefore + + + +f(r)dr. + + +Since moreover S in Equation (152) represents an infinitesimal increase of r, we can denote +it by dr following general usage, and Equation (152) then becomes + + + +2 jn 2 r 2 dr + +V + + + + +g-2 hf T f(r)dr + + +If we multiply this expression by the virial rf( r) of the molecule pair in question, and +integrate over all colliding molecules, we obtain the total virial Wi arising from the + +forces acting during collisions. Therefore if tr - e is the smallest permitted distance of +approach of two molecules when they impinge on each other with an enormous velocity, +then one obtains: + + + + +r 3 /(r)dre~ 2 ^ /(r)dr . + + +Since r is always infinitesimally different from 7 + + +According to Equation (21) we now set c J = 3rT Furthermore, V/nm = v = 1/p, so that +the above equation becomes + + + + +rT ( b bb\ +—( 1 H-f- —) + +i' \ v 8rv + + +9 9 + +If one neglects quantities of order b /v , the right-hand side is identical with the +expression rTKv - b) found by van der Waals. But there is already a discrepancy in the +terms of order b tv . Van der Waals himself noted that his equation cannot be valid for +arbitrary v, since it predicts that the pressure becomes infinite at v = b, whereas actually the +pressure does not become infinite until v is much smaller than b. + + +§54. Alternatives to van der Waals’ formulas. + +Our present considerations teach us that the expression for the pressure given by van der +Waals does not agree with the theoretical one for small values of v, as soon as one takes +account of terms of order/? 2 A’ 2 . Since the theoretical determination of terms of still higher +order would be extraordinarily difficult,* one can try to replace van der Waals’ equation by +one that agrees with theory at least as far as terms of order A 2 /v 2 . We saw, moreover, that +one can also determine theoretically the smallest permitted value of v, for which the +pressure becomes infinite. It is (cf. §6), since for approximately this value of v the + +molecules are as densely packed as possible, and any further decrease of v must make the +molecules interpenetrate. One can therefore construct an equation of state for which p +becomes infinite at this value of v. + +In order to depart as little as possible from the form of van der Waals’ equation, we shall + + +31. + + +^Pu+Lc. 7fca^4e«toi2c.a/ + + + + +write the equation of state in the following form, where x, y, and z are numbers to be +chosen suitably : + + + +a\ ( yb zb\ + +p + —)(v - xb) = rT[\+ — + — ). + +V \ V V 2 + + +We then have the advantage that for given p and T we obtain a third degree equation for +v. If we set y = 1 -x, Z = $ — X-, then for small values of b/v the terms of order b~/v~ +agree with those found theoretically. If we set + + + +1 2 7 + +x = — > hence y = — > z = — > +3 3 24 + + +then the condition that/? becomes infinite for y = is also satisfied. Since all our + +considerations are only approximate, it would perhaps be more rational not just to choose +these values for x, y, and z, but rather to assign to them values that yield the best possible +agreement with observations. + +If one does not wish to add any factors to rT in the numerator, then a quadratic function +of b/v in the denominator, such as a law of the form + + + +rT / xb +— 1 + - + + + +V \ V + + + +would hardly be very advisable; for then p, if it is to become infinite at all, and if it is also to +be correct to first order in b/v, would have to become infinite for a value of v that is greater +than or equal to ^ 5 + +The latter occurs if one sets + + + + + +It would be better to set + + +a rT + +(158a) p + — =-- + +V 2 v - w + + +and to choose for e a transcendental or algebraic function of higher degree, which is nearly + + +31 . + + +7fca^4e«tai2c.a/ + + + +equal to 1 for large v, and for \) = is nearly equal to J, and also agrees as well as +possible with experimental data. I owe Equation (158a) to an oral communication of van +der Waals (cf. also the work of Kamerlingh-Onnes cited in footnote 1, §60). Moreover, van +der Waals has given us such a valuable tool that it would cost us much trouble to obtain by +the subtlest deliberations a formula that would really be any more useful than the one that +van der Waals found by inspiration, as it were. + +A more general way to bring the original van der Waals formula into better agreement +with experience would consist in treating the expressions a/v~ and v - b in it as empirically +chosen functions of volume and temperature instead of constants, or, quite generally, to +seek in place of a/v 2 and v - b the functions best fitting the observations. Of course these +functions must be chosen so that the theorems on critical quantities and on liquefaction do +not come out qualitatively different. Clausius and Sarrau have modified the van der Waals +formula in this way.* Even though they have been guided by theoretical ideas (Clausius +especially seems to have in mind taking account of the combination of molecules into larger +complexes), their equations have more the character of empirical approximations, which I +will not go into further, although I would not wish to disparage their practical usefulness. + +§55. Virial for any arbitrary law of repulsion of the molecules. + +By means of Equation (153) we can also calculate by the same method the quantity +Wi for the case where the molecules do not behave like elastic spheres, but rather like +material points that exert an arbitrary central repulsive force /O') on each other during +collisions. Since the time during which two colliding molecules act on each other can then +no longer be neglected, the method by which we derived Equation (150) from (149) is no +longer correct; however, the latter formula is still correct as a first approximation. + +The number of pairs of molecules whose distance lies between r and r+dr is therefore +2mr rdr/v y as soon as no force acts at the distance r. The modification of this number by +the action of repulsive forces was previously found according to the general formula (142). +This formula remains applicable here, and hence as soon as there is a repulsive force at the +distance r, the number of pairs of molecules at a distance between r and r+dr is +(corresponding to Eq. [153]) equal to: + +2 im 2 r 2 dr r + +r g-2 hj r f(r)dr + +V + +Each such molecule pair provides a contribution rf( r) to the virial wi . The sum of all +these contributions is therefore + +2m 2 + +(159) WI = — + +v + + + +31 . + + +7fca^4e«toi2c.a/ + + + +where C, is the closest allowed distance of approach of two molecules, and a is the distance +at which they cease to interact. However, one can integrate from zero to infinity instead of +from ( to a, since for r<( the exponential factor is zero, while for r>a,J(r)=0. + +If one sets /( r) = K/r^ as in Chapter HI of Part I, then of course the present assumptions +are not strictly satisfied, since properly speaking all molecules are continually repelling each +other; but this repulsion decreases so rapidly with increasing distance that the deviations +thereby produced in our formulas are probably completely insignificant. Hence we obtain: + + + + +We shall introduce the distance of closest approach of two molecules if one were held fixed +and the other approached it with a velocity whose square is equal to the mean square +velocity of a molecule (cf. Pail I, §24). However, we shall denote this distance here by a +instead of s. Then : + + +K/2mc 2 = \hK = it follows moreover that + +V + +X = —-1 + +V2?r a 2 nft + + +or, on substituting this value (162) for [3 and expanding in powers of blv. + + + +This is therefore the value of the mean free path, one order of magnitude more exact (with +respect to blv) than that given in Part I, § 10. + +We can now easily find the mean virial of all forces acting in collisions from Equations +(161) and (168). For each of the collisions whose number is given by Equation (168) , the +component 7 of the relative velocity g in the direction of the line of centers has the value 7 += g cos 9: each of these collisions contributes therefore the term mag cos d to the sum +(161). If one multiplies this by the expression (168), then he finds the contribution of all +these co ll isions to the sum (161). If one then integrates over all possible values, he obtains +finally the total contribution to the sum, and hence, according to Equation (161) , the +quantity W! . Finally, however, one must divide by 2, since otherwise he would have + + +7fca^4e«toi2c.a/ + + + +counted each collision twice—once when the velocity of one molecule lies between c and +c+clc and then again when the velocity of the other molecule lies between c and c+dc. +Hence: + + +laWmfi r rl2 + +W! =- sin t? cos 2 + +27 J 0 + + +X + + + +sin e de. + + +If one substitutes for g the value (167) and recalls that: + + + +then he obtains the value already found, + + + +1 + + +I. + + +x + + +c'V(c')dc' + + + +Wj = 2wdn 2 mcy/3V. + + +The corrected value (171) for the mean free path was first given by Clausius. 1 The +additional term of order b/v in the Boyle-Charles law was first calculated by the foregoing +method by H. A. Lorentz; the additional teim of order b /v was calculated by Jager and +van der Waals; 4 the value found by Jager agrees with the one calculated here, but the one +found by van der Waals does not.* + + +§59. More exact calculation of the space available for the center of a +molecule. + +Let there be present, in a container of volume V, a total of n similar molecules, which +we consider as spheres of diameter a. Then we can find the space available for the center +of another molecule introduced into the container, given the positions of all n molecules, by +subtracting from the total volume V the space occupied by the n molecules: T = +47nur’/3=2Gb (cf. Eq. [148] ). As before, m is the mass of a molecule, mn = G is the total +mass of the gas, and + + +We 7fca^4e«toi2ea/ lPl±y. i .Ze l + + + +2tt(t 3 + +6 = — + +3m + +as in Equation (20) is half the sum of the covering spheres of all molecules found in unit +mass of the gas. Here the term of order T~/V , corresponding to the effect of +interpenetration of the covering spheres of two molecules, is omitted. We shall now +calculate this term, although we shall still leave out terms of order Y IV . + +Let Z be the sum of the volumes of all those parts of the covering spheres of molecules +that he within the covering spheres of any other molecule, so that we have to set: + +(172) D = V - 2Gb + Z. + + +The case that the covering spheres of two molecules overlap occurs each time that their +centers have a distance between cf and 2 a. Let x be such a distance. The covering spheres +are spheres of radius a, which are concentric with the molecule in question. If the centers +of two molecules are at a distance x, then the total space that belongs simultaneously to the +covering spheres of both molecules has the form of two spherical sections of height o - x/2. +Such a spherical section has a volume + + +K = IT + + + + +2a* ah X*\ + +.1 2~ + 24/' + + +We shall construct, for each molecule, a concentric spherical shell of inner radius x and +outer radius x+clx. The sum of the volumes of these spherical shells, 4rnixrdx, has the same +ratio to the total volume V of the gas that the number dn x of molecules whose centers are at +a distance between x and x+dx from another one has to the total number n of molecules. +Therefore : + + +4 rnWdx + + +Here terms of order Ydn x iV have been omitted, but one can easily convince himself that in +the final result these would only give terms of order T 3 /V 3 . + +The number of pairs of molecules for which the distance of centers lies between x and +x+dx is i dn x . Since, for each such molecule pair, two spherical sections of volume K he +within the covering spheres, ah these molecule pairs provide a contribution Kdn z to Z, and +we find the quantity Z itself by integrating these contributions from x = a to x = 2 a. One +obtains in this way + + + + +We 7fca^4e«toi2ea/ “Pity. i .Ze l + + + + + + + +T*nV + +T~ + + +17 W + +16 ~T' + + +17 (W + +(173) Z) = F — 2G6 H— — • + +16 V + + +60 . + + +Calculation of the pressure of the saturated vapor from the laws of +probability. 1 + + +Now let the liquid and vapor phase of a substance be in contact with each other at a +particular temperature T. The total mass of the liquid pail will be Gt, and its total volume +Vf the mass of the vapor pail will be G ? , and it fills the total volume V„, so that vy = VJGf ., +v g = VJG g are the specific volumes or reciprocals of the densities pj and p g of the liquid +and the vapor. + +If one now introduces a molecule into the space in which the two phases are present, +then according to Equation (173) the space available for this molecule within the liquid is + + +17 Gib- + +K/-2C/6 + -—, +16 V, + + +while the space in the vapor is + + +V, - 2 G,b + - + + +17 G,V + + +16 7. + + +The ratio of these volumes would be—if the van der Waals cohesion force did not exist— +the ratio of the probabilities that, for given positions of all other molecules, the last one lay +within the liquid or the vapor. This ratio is (on account of the action of the van der Waals +cohesion force) to be multiplied by e~ 2h ^: c~ 2lr, ‘^, where ipf and ip g are the values of the +potential of the van der Waals cohesion force for a molecule that finds itself in the liquid or +the vapor, respectively. If one determines the constant so that ip = 0 for infinite separation, +then - ipf is the work needed to bring a molecule of mass m, under the action of the van der +Waals cohesion force, out of the interior of the liquid and remove it to a large distance. In +§24 we found the expression 2 mapf= 2ma/iy for this work, while apj- was the total work of +separation for all molecules in unit mass. Likewise, + +- = 2map 0 = 2 ma/v 0 . + + +Wte, 7fca^4e«taizc.a/ TRk.y l i .Zc. 1 + + + +Taking account of the van der Waals force, we find that the ratio of the probability that the +last molecule is in the liquid to the probability that it is in the gas is: + + +17 GV\ / 17 G 2 b\ + +V, -2 G,b + -]: V, - 2 G„b +- + +16 V,} \ 16 v, ) + + +In the equihbrium state this must also be equal to the ratio of m to n g , or, if one multiphes m +and n„ by m, equal to GJG„. If one writes out the proportion, he obtains immediately + + +v, + + +17 b 2 + + + +i 17 b\ + +[ vi - 2b H-1 e 4 '" M| «/»i, + +\ 16 !;// + + +Now according to Equations (21) and (135) 2 (cf. also Part I, Eq. [44] ), 2 h = llmrT, if r is +the gas constant of the vapor at high temperature and low density. If one takes the +logarithms, expands in powers of b and retains terms of order b , then + + + + + + + +l--2b + + +v 0 + + +2 a + + +L v f + + + + + + +Naturally this formula can hardly be expected to give more than qualitative agreement, +since the assumption that b is small compared to v is incorrect for liquids. + +If we introduce the Celsius temperature t, consider pj to be constant and large compared +to p g , and assume that the vapor obeys the Boyle-Charles law, i.e., pv g = rT, then there +follows from Equation (174) an equation of the form + +1 + +_ ftUe+Dl) + +A + Bt + + + +—an equation that has some practical usefulness, though with a somewhat different +meaning of the constants A, B , C, and D. + +We can also calculate the pressure of the saturated vapor from the condition found in +§16, that the two shaded areas of Figure 2 of that section must be equal. In this figure, the +abscissa OJ\ is the specific volume ry of the liquid, the abscissa OG \ is the specific volume +v g of the vapor, while the ordinates J\J = G | G are equal to the corresponding saturation + + +33 , + + +7fca^4e«tai2c.a/ + + + +pressure. The equality of the shaded areas requires that the rectangle JJ \ G | GJ = p(v g - ry) +must be equal to the surface area fl'pdv , which is bounded above by the curve JCHDG, + +below by the abscissa axis, and on the left and right by the ordinates J\J and G | G. One +therefore obtains: + + + +If one starts from the van der Waals equation + +rT a + +(177) P- -- + +v - 0 V 2 + +as a basis, then it follows on performing the integration that: + +Vg “ b / 1 + +(178) p{v g - vj) = rTl -- + a — + +Vf-b \v g + +T and the constants a , b, and r are to be considered as given. +v g follow from Equation (178) and the two conditions: that vj- is the smallest and v g the +largest root of Equation (177) . If one again takes for the vapor the Boyle-Charles law pv g += rT , neglecting b and vy compared to v g , and p g compared to pj (the latter is considered to +be a linear function of the temperature), then he obtains for the saturation pressure an +expression of the form (175) again; yet Equation (178) by no means agrees exactly with +Equation (174). + +This would not be expected, since Equation (177) is only provisory, and is not an exact +consequence of the conditions of the problem. + +On the contrary, one must obtain precisely Equation (174) if he uses instead of Equation +(177) the equation + + + +The three unknowns p, vy, and + + + +1 b 5b +p = rT[ — + — + — +v v 2 8y 3 / + + +a + +V 2 + + +which, on neglecting the terms containing powers of b higher than the second, exactly +satisfies the conditions of the problem. + +In fact, it then follows from (176), on performing the integration, that + + +33 , + + +Uuc. 7fca^4e«tai2c.a/ + + + + +Since now v g as well as Vf must satisfy Equation (179) , one can calculate pv g by +substituting v = v g m this equation, and pvj- by setting v = Vf. Subtraction of the two values +yields + + + +which, together with Equation (180) yields exactly Equation (174). + +§61. Calculation of the entropy of a gas satisfying van der Waals’ +assumptions, using the calculus of probabilities. + +I will now indicate briefly how one can calculate, according to the principles developed +in Part I, §§8 and 19, the entropy of a gas for which the space fi ll ed by the molecules is not +vanishingly small compared to the entire volume of the gas, and in which also the van der +Waals cohesion force acts. The van der Waals cohesion force does not change the velocity +distribution among the molecules, but only causes them to draw closer together. Like +gravity, it has no direct influence on the entropy, so that the dependence of the entropy on +the temperature for such a gas will be obtained just as in the sections cited for an ideal gas; +in the present case only a correction for the finite size of the molecules is needed. + +The expression found in Part I, §8, for the entropy—which we shall call S in the +following—can easily be put in the form 1 + +S = RMin = RMl(v n T 3 " n ). + +If M is the mass of a hydrogen atom, then R is the gas constant of dissociated hydrogen, +hence it is twice the gas constant of ordinary hydrogen. The exponent of T must be + + +7fca^4e«taiZc.a/ + + + +3 n 3 n + +— (1 + 0) instead of — +2 2 + + +when internal motions take place in the molecule, for which the variation of the sum of +mean kinetic energy and potential energy stands in a constant ratio f}\ 1 to the mean +progressive kinetic energy. Ifis a function of temperature, then + + +3 n + + + +dT + +~T + + +must take the place of lT inl2 . + +S is the entropy per unit mass, so that n is the number of molecules in unit mass; v is the +volume of unit mass. + +If one interprets S as a probability expression, then the quantity v n occurring therein has +the following meaning: it represents the ratio of the probability that all n molecules are +simultaneously in volume v, to the probability of some standard configuration, for example +one in which the first molecule is in a definite space of volume 1, the second in a +completely different space of volume 1, and so forth. This quantity is the only one that is +changed when we take into account the extension of the molecules, and indeed there +occurs in its place the probability W of the simultaneous occurrence of the following events: +the first molecule is found in v, and also the second, third, fourth, etc. are also found there +(not, as in §60, simply the probability that one additional molecule is found in v). + +The entire volume v is available for the center of the first molecule. The ratio of the +probability that it is there to the probability that it is in the given space of volume 1 is thus v. +In calculating the probability that the center of the second molecule is simultaneously in the +space v, we have to subtract the volume of the covering sphere of the first molecule, +iira* =■ from v. If there are already v molecules in the space v, then the space available +for the center of a (v+l)th molecule is, according to Equation (173) : + + + +17M 2 + +v - 2 mb +- + +lb + + +This expression is therefore equal also to the ratio of the probability that the (v+l)th +molecule finds itself in the space v to the probability that it finds itself in another space of +volume v completely separate from the other space. Hence the product + + + +- 2 mb + + + +\7vWW + + +lb + + +33 + + +7fca^4e«ta£2c.a/ + + + +represents the ratio of the following probabilities: the probability that all n molecules +simultaneously are in the space v, and the probability that each of them lies in a separate +space of volume unity. 2 This expression has to occur in S instead of v”, when we take +account of the finite extension of the molecules. Therefore the entropy of unit mass is + + +r3n C dT / 17 + +S = m — I (1 + 0) — + £l(»-2nm6 + — + +12 J T £J \ 16 + + +17 vW + + +V + + +Here r is the gas constant of our substance in states that are sufficiently similar to the ideal +gas state, so that rm = RM. + +If one expands the logarithm in powers of b and neglects, as usual, powers of b higher +than the second, it follows that: + + +f3n r dT rz? / 17 + +(i+» T + + +Since moreover we assume that n is large compared to 1, we can set: + +/ 17 vVb 2 \ 2 mb 15 vWb 2 + +l[v - 2 mb H-) = Iv - + +16 v ) v 16y 2 + + +17 + + +so that we obtain, since nm = 1, + + +r3 n r dT b + + +6 56 s + + +16d 2 J + + +Since the partial derivative of TS with respect to v at constant temperature is equal to the +pressure due to molecular collisions alone, we find for this pressure, in agreement with the +earlier result, the value + + +/ 1 b 5iA + +\ V V 2 8 dv ' + +The calculation of terms with higher powers of b in this expression can be done most easily +by the same method, taking account of these terms in the expressions for S and IT . 3 + +If the molecules are not spherical, but behave like solid bodies, then the probability that +the center of the (v+l)’th molecule lies in the volume v is still given by an expression of the + + + +33 , + + +7fca^4e«toi2c.a/ + + + +form + + +v 2 m 2 + + +v k m k + + +v - C\vm - c 2 + + +• • • “ Ck + + +M 1 + + +We assume that the series development yields + + +(182) + + +l[ v - C\vm • • • - c* + + +v k m k + + + + += Iv - + + +26 \vm + + +36 2 ^ 2 m 2 + + +v 1 + + +tr -1 / v + +(k + l)6*j>*m* + +■ - ■ ■ ■ i ■. • • • + +kv k + + +then + + +S + + +r3n r dT + +HrJ «+# + + +T + +2bim + + +r-n+1 / + ++ £ (to- + +r-0 \ V + + +k +1 6jt^m fc +k v k + + +r 3 r dT + + +^ 6i 6 2 6* + +t; 2y 2 to* + + +Hence the pressure due simply to molecular collisions will be + +w>.*(L + h.... + ±.. + +dv \ v v 2 v Hl + +and the total external pressure acting on the gas will be : + +/1 6i 6 2 6* \ a + +(183) p = rT (- + - +--- + +\v v 2 v* y* +l / v 2 + + +33 + + +7fca^4e«ta£2c.a/ + + + +If we substitute this into the equation + + +vk + + +-»/) = p*, + +J »/ + + +then we obtain, as a condition that vapor and liquid can coexist: + + +P(v„ ~ v,) = rT + + +1 1 + +I--M-- + + +L v, + + +vi + + +k/l 1, + +T T + T)'" + + +k \v + + +; »/ + +which, on repeated apphcation of Equation (183), can also be written : + + + +( 184 ) + + + +v B + + +1 1 + + +L v, + + +h v f + + +36 2 /1 1 + + +2 \vl + + +v + + +/ + + + +In agreement with this, one now obtains by the same method by which we previously +obtained Equation (174) the following condition for equilibrium of liquid and vapor: + + + += Z ( Vn - C\ ~ + + +which, taking account of Equation (182), again yields Equation (184). + +The following additions to Chapter I, which van der Waals communicated orally to me +after that chapter was printed, may be inserted here. + +1. He never explicitly made the assumption, explained in §2, that the attractive force of + + +33 + + +7fca^4e«tai2c.a/ + + + +the molecules decreases so slowly with increasing distance that it is constant at distances +large compared to the average separation of two neighboring molecules; and he believes +such a force law to be improbable. Nevertheless I cannot obtain an exact foundation for his +equation of state without this assumption. + +2. If one conceives the bounding curve JKG of the two-phase region (Fig. 3, §17) to be +a parabola or circular arc in the immediate neighborhood of the point K, then he sees that +JN will be more nearly equal to NK, the closer N is to K, if N remains continually on the +line KK \. Hence if a substance has exactly the critical volume, and is heated at constant +volume, then at the moment when the meniscus vanishes, the volume of the liquid part will +be exactly equal to the volume of the vapor part. On the other hand, if the volume differs +slightly from the critical volume, then the meniscus always moves progressively a great +distance from the middle of the tube containing the substance, before it vanishes. + +According to the experiments of Kuenen,* gravity plays an important role in causing +deviations from the theoretical behavior. + + +* R. Clausius, Sitzber. Niederrhein Ges. (Bonn) 114 (1870), reprinted in Ann. Physik [2] 141, 124 +(1870). For further references see S. G. Brush, Amer. J. Phys. 29,593 (1961), footnote 14. + +1 The simplest way is to calculate the virial of the force acting between any two material points, as +well as that of the external force, separately, and then to recall that the virial of several forces is the sum +of the virials of the forces taken individually, since the Cj, are contained linearly in the expression +for the virial. + +* For the coefficient of b 3 /v 3 (known as the fourth virial coefficient) see Boltzmann, Verslagen +Acad. Wet. Amsterdam [4] 7, 477 (1899); P. Ehrenfest, Wien. Ber. 112, 1107 (1903); H. Happel, Ann. +Physik [4] 21, 342 (1906); R. Majumdar, Bull. Calcutta Math. Soc. 21, 107 (1929). The fifth virial +coefficient was estimated by the Monte Carlo method by M. N. and A. W. Rosenblueth, J. Chem. Phys. +22, 881 (1954). A more accurate value will probably be available very soon. Modern methods for +calculating such coefficients are reviewed by G. E. Uhlenbeck and G. W. Ford in Studies in Statistical +Mechanics, Vol. I (Amsterdam: North-Holland, 1962), and J. E. Mayer ,Handbuch der Physik, XII, 73 +(Berlin: Springer, 1958). From the recent work of Groeneveld, Phys. Letters 3,50 (1962), it appears that +even if one could calculate all the coefficients in the virial series for the pressure, he would not thereby +gain much information about the equation of state at high densities, since the radius of convergence of +the series is rather small. + +* Clausius, Ann. Physik [3] 9, 337 (1880); Sarrau, C. R. Paris 101, 941 (1885). See Partington, +Treatise on Physical Chemistry (London, 1949), Vol. I, pp. 660-729. + +* f e~ x *dx = ir(i) + +Jo + +t H. A. Lorentz, Ann. Physik [3] 12,127,660(1881). + +2 Clausius ,Kinetische Gastheorie, Vol. 3 of Mechanische Wdnnetheorie (Vieweg, 1889-1891), p. +65. + +3 Lorentz, Ann. Physik [3] 12,127,660(1881). + +* Jager, Wien. Ber. 105 ,15 (16 Jan. 1896). + +4 Van der Waals, Verslagen Acad. Wet. Amsterdam [4] 5,150 (31 October 1896). + +* Van der Waals later accepted the Boltzmann-Jager value as correct, after his son (J. D. van der +Waals, Jr.) showed that the method used in ref. 4 was incorrect. For references to this dispute see Brush, +Am. J. Phys. 29,593 (1961), esp. footnotes 31-41. + +5 The same problem has been treated by Kamerlingh-Onnes, Arch. Neerl. 30, §7, p. 128 (1881). + + +33 + + +7fca^4e«tai2c.a/ + + + + +6 In particular, ^ = £ mc *. + +7 In Part I, §8, n is the number of molecules in unit volume; hence Qn is the total number of +molecules of a gas in volume Q, which we have denoted here by n. + +s Naturally terms of order £> 3 are omitted, and it should also be recalled that v is large compared to 1 +in all terms except those that are vanishingly small. + +9 The expression found here for S would not of course be found comparable with that given in §21 +for the entropy, since the calculation of the latter assumed the exact validity of the van der Waals +equation. However, we find exactly the same expression for the entropy when in Eq. (38) of §21, from +which one finds + + +fdQ + +J T + +we substitute instead of Eq. (22) + + +V + + + +the equation of state + + + + +which forms the basis of our present calculations. + +* See J. P. Kuenen, Comm. Phys. Lab., Leiden, No. 17 (1895), and other works cited therein. + + +33 , + + +7fca^4e«tai2c.a/ + + + +CHAPTER VI + + +Theory of dissociation. + +§62. Mechanical picture of the chemical affinity of monovalent similar atoms. + +I have once previously treated the problem of the dissociation of gases, on the basis of +the most general possible assumptions, which of course I had to specialize at the end. 1 +Since here I prefer perspicuousness to generality, I shall make special assumptions that are +as simple as possible. The reader must not misunderstand the following and perhaps believe +that I hold the opinion that chemical attraction acts precisely according to the laws of the +force assumed here. These laws are rather to be considered the simplest, most perspicuous +possible picture of forces which have a certain similarity to chemical forces, and hence in +the present case can be substituted for them with a certain degree of approximation. + +I will first consider the simplest case of dissociation, for which the dissociation of iodine +vapor can serve as a prototype. At not too high temperatures, all molecules consist of two +iodine atoms; as the temperature increases, however, more and more molecules decompose +into single atoms. We explain the existence of molecules composed of two atoms (the +double atom) by an attractive force acting between the atoms, which we call chemical +attraction. The facts of chemical valence make it probable that the chemical attraction is by +no means simply a function of the distance of centers of the atoms; on the contrary it must +be associated with a relatively small region on the surface of the atom. Moreover, it is only +with the latter assumption, and not the former, that one can obtain a picture of gas +dissociation corresponding to reality. + +Both for the sake of simplicity of calculation, and because of the monovalency of +iodine, we assume that the chemical attraction exerted on one iodine atom by another is +effective only in a space small compared to the size of the atom, which we shall call the +sensitive region. This region will lie on the external surface of the atom and will be firmly +connected to it. The line drawn from the center of the atom to a particular point of the +sensitive region (e.g., its midpoint, or its center of gravity in the purely geometrical sense) +we call the axis of this atom. Only when two atoms are situated so that their sensitive +regions are in contact, or partly overlap, will there be a chemical attraction between them. +We then say that they are chemically bound to each other. They can touch each other at +any other place on the surface without chemical attraction occurring. The sensitive region +shall be such a + + +33 . + + +7fca^4e«ta£2c.a/ + + + + +Fig. 4 + +small part of the entire surface of the atom that the possibility that the sensitive regions of +three atoms can be in contact, or partly overlapping, is completely excluded. For the +following calculation it is not necessary to assume that the atoms have the form of spheres. +Since this is the simplest assumption, however, we shall make it. Let ff be the diameter of +the spherical atom. + +We consider a particular atom; let it be represented by the circle M of Figure 4. Let A be +its center. The shaded region a will be the sensitive region. We need not immediately +exclude the case that this lies partly inside the atom, but we draw it as if it were completely +outside, as one must naturally assume if one imagines the atom to be completely +impenetrable. If a second atom M\ is chemically bound to the first, then the sensitive region +/3 of the second atom must partially overlap the space a, or at least touch it. We shall again +construct the covering sphere of the first atom (a sphere of center A and radius a), which is +indicated in the figure by the circle D. We construct on the surface of the covering sphere +D a space (the critical space, the shaded space to in the figure) with the property that the +sensitive regions a and /3 can never overlap or touch each other unless the center B of the +second atom lies in this critical space to or on its boundary. The converse is not true. If the +center B of the second atom lies in the critical space to, then that atom can still be rotated in +such a way that the sensitive regions a and /? are widely separated from each other. + +In order to define precisely the position that the second atom must have relative to the +first if the two are to be chemically bound, we shall next construct in the critical space to a +volume element cl to. At the same time, to will be the total volume of the critical space. +Further, we shall imagine that a concentric sphere of radius 1 is rigidly connected to the +first atom; this sphere is indicated by the circle E Figure 4. If the second atom is chemically +bound to the first, then the axis of the second atom must not make too great an angle with +the line BA, since otherwise the sensitive regions a and /? would separate from each other. +The line drawn from the point A parallel to and in the same direction as the axis of the +second atom should strike the spherical surface £ in a point which we shall always call the +point A. Since the sphere E is rigidly connected to the first atom, the position of the axis of +the second atom relative to the first is completely determined by this point A, and we can +now construct, for each volume element dto of the critical space, a surface section A on the + + +34 + + +IRuJig. 7fca^4e«tai2c.a/ + + + +spherical surface E with the following property. If A lies within or on the boundary of the +surface section A, then, as soon as the center of the second atom lies within or on the +boundary of the volume element cla>, the two sensitive regions a and /? will interpenetrate +or touch each other. But as soon as A lies outside A, the two sensitive regions fi and /? will +likewise he outside each other. This surface section A will of course in general be of a +different size according to the position of the volume element da> within the critical space to, +and also will have a different position on the sphere E. Now if the center of the second +atom lies within or on the boundary of any volume element dco of the critical space, and A +lies within or on the boundary of a surface element c/A of that area A which corresponds to +the volume element dto, then the second atom will be chemically bound to the first—i.e., +the two atoms will actively attract each other. The work needed to bring them from this +position to a distance at which they no longer interact noticeably with each other will be +denoted by /. This quantity can in general be different according to the position of the +volume element dto within the critical space, and also according to the position of c/A on the +corresponding surface A. + +§63. Probability of chemical binding of an atom with a similar one. + +Now suppose that a identical atoms are present in a container of volume V at a pressure +p and absolute temperature T. Let the mass of an atom be m | and that of all atoms am j = +G. We pick out one of the atoms. The others we call again the remaining atoms. For a +moment we imagine that the gas is present in infinitely many (N) equivalent but spatially +separated containers at the same temperature and pressure. In each of these N gases, let n | +of the remaining atoms not be bound to the other remaining atoms, while 2 n 2 of the +remaining atoms are bound to another one, so that they form /z 2 double atoms. We now +ask, in how many of the N gases will the specified atom be chemically bound to one of the +other atoms, and in how many will this not be the case. + +We consider first only one of the N gases. Since we have excluded chemical binding of +three atoms, the specified atom, if it is bound at all, can only be united with one of the +not otherwise bound atoms of this gas. + +We therefore draw, as in Figure 4, the covering sphere and the concentric sphere E of +radius 1 for each of these atoms; on each of these covering spheres will be found +somewhere the critical space to. In each of the critical spaces corresponding to all ri\ atoms, +we draw the volume element dto, which has exactly the same position relative to the atom +in question as the element dto of Figure 4 with respect to the atom drawn there, and on each +spherical surface E we draw a surface element c/A, which l ik ewise has the same position +relative to the atom in question as has that surface element c/A in Figure 4. If the center of +the specified atom now finds itself in any one of the volume elements dto, and moreover the +point A is in the surface element c/A of the surface A (or its boundary) corresponding +thereto, then it is chemically bound to another atom, and indeed is at a completely +determined position relative to another atom, so that the quantity denoted by % has a definite +value. + +If that attractive force which we call the chemical attraction were not present, then the + + +34 + + +mJuc. 7fca^4e«tai2c.a/ + + + +probability that the center of the specified molecule is in one of the volume elements doj +would be in the same ratio to the probability w that it is in the arbitrary space Q in the gas— +which is neither a part of the covering sphere of a remaining atom, nor contains one of the +spaces oj —as n | cla/.Q. The space Q is, according to what has been said, constructed so that +the center of the specified atom can be in each point of it without being chemically bound. +The probability wo that not only does the center of the specified molecule lie in one of the +volume elements d)/& + +The specified atom is also not chemically bound when its center is in a volume element +of a critical space but the point A does not he on the corresponding surface A, since then it +is rotated so that the sensitive areas do not overlap. For the probability of this latter event +one finds, by analogy with Equation (185), the expression + +dwd\ i + +- ) + +4t + +where, however, for each volume clement cloj, dk \ means an element of the spherical +surface E that does not lie on the surface A corresponding to da>. Again one is to integrate +over ah surface elements that satisfy this condition and over all volume elements dco. The +exponential is now omitted, of course, since in none of the positions now considered does +the attractive force act. In each of the A gases, therefore, the total probability that the +specified atom is not chemically bound is: + +( C C dwd\ A w + +V - Gb - Thu + ni J J ——j— • + +The three quantities Gb, ri \ to, and n | Wdcod/JAn are completely independent of the +chemical attractive forces. The first represents the deviation from the Boyle-Charles law +considered by van der Waals, due to the finite extension of the molecules. Since the critical +space is very small compared to the covering sphere, the second and third of the above +three quantities are small compared to the first. We shall neglect all three quantities +compared to V, since we shall calculate the dissociation of a gas that otherwise has the +same properties as an ideal gas, so that the deviations from the Boyle-Charles law from +causes other than dissociation can be neglected. By the same token we can also forget all +terms added to the expression in parentheses in Equation (189) in the search for greater +accuracy. Hence Equation (189) reduces to + + + +34 + + +‘Pu+lc- 7fca^4e«toi2c.a/ + + + +Vw + +(190) w* = - + +ft + +On the other hand, we may not consider the quantity k (given by Eq. [186] ) to be small +compared to V, since on account of the great intensity of chemical forces the exponential +has a very large value. Noticeable dissociation occurs only when the exponential e 2hx is of +the same order of magnitude as N/n | oj, and hence/: and V are of the same order of +magnitude. From Equations (187) and (190) it follows that + +w 6 :w 3 = V:njk. + +If we now return to the N identical gases, and assume that in Nt, of them the specified +atom is chemically bound, and in N (l it is not bound, then we have also: + +Nt'.Ni = Wi'.Wi = V\njc. + +Since we could just as well have specified any other atom, this must also be the ratio of the +number of unbound atoms, ri \ to the number of bound ones, 2n 2 , in the equilibrium state. +Therefore + +ni:2n 2 = V:nik } + +hence + +(191) n\k = 2 n 2 V. + + +64. Dependence of the degree of dissociation on pressure. + + +In determining the two numbers nj and n 2 , we have of course imagined that one +molecule, which we called the specified one, is excluded. However, since these numbers +are very large compared to 1, Equation (191) is also valid when n | means the total number +of all unbound atoms (simple molecules) in the gas, and rz 2 means the number of all double +atoms (compound molecules). Since a is the total number of all molecules in the gas, one +has moreover n 1 +2n 2 = a. Whence it follows that: + + + + +T 2 Va +4fc 2 + k + + +We denote by G the total mass of gas, and by the mass of one atom, so that a = +G!m\. a!G\ = 1/m| is the number of dissociated and chemically bound atoms together in + + +34 + + +7fca^4e«toi2c.a/ + + + +unit mass. We again denote by v=V/G the specific volume, i.e., the volume of unit mass of +the partially dissociated gas at the given temperature and pressure, and by q = n \ la the +degree of dissociation, i.e., the ratio of the number of chemically unbound (dissociated) +atoms to the total number of atoms. Finally we set + + +(193) + + +K + + +k l r r +mi miJ J + + +e n * + + +dud\ +4 7T + + +Then the above equation reduces to : + + +(194) + + +q = ~m + + + +4 K t + K + + +For the sake of orientation, we make the following remark: if two single atoms happen to +collide in such a way that their sensitive regions interpenetrate, then because of the +smallness of these regions and the large relative velocities of the atoms, which are +accelerated by the chemical forces, the time during which the sensitive regions overlap will +in most cases be small compared to the average time between successive collisions of a +molecule. The energy of the double atom is so large that the two atoms can separate from +each other again. (We refer to this case as virtual chemical binding.) + +The number of these virtually bound atoms is in any case vanishingly small compared to +a, since they always remain together only a very short time. They can therefore make only +a vanishingly small contribution to n 2 , when n 2 is not very small compared to a. Only +when the kinetic energy of the motion of the centers of gravity of the atoms is transformed +into internal energy of the atoms (e.g., rotation around their axis, or internal motion) can +there be, in the case when the atoms are not solid spheres, a somewhat longer interaction. +(First kind of proper chemical binding.) On the other hand, if a third single atom or double +atom intervenes while the sensitive regions of the other twx) atoms are overlapping, the +energy can be lowered so much that it no longer suffices to separate the two atoms again, +so that they must remain together, at least until another collision occurs. (Second kind of +proper chemical binding.) In all cases where our calculation gives the result that the number +of double atoms, n 2 , is not vanishingly small compared to a, numerous double atoms must +remain united with each other for long times. The principal advantage of our general +formula is just that it permits us to calculate the number of chemically bound atom pairs +without having to take special account of the processes of creation and dissolution of these +pairs. In any case, out of the n 2 double atoms found by calculation—in case this number is +not very small—all except a vanishingly small number are united with each other for a +longer time, so that they are to be considered molecules in the sense of gas theory. + +To calculate the pressure we must therefore proceed as if we had a mixture of two +gases. The molecule of one of these gases is a single atom, while the molecule of the other +is a pair of two atoms. The total pressure of any arbitrary gas mixture is: + + +34 + + +7fca^4e«tai2c.a/ + + + +p = — (nimic* + n 2 m 2 <^ • • •), + +where m ., m 2 , ... are the masses, and . . . the mean square velocities of the + +of, 02 + +centers of gravity of the various kinds of molecules. ri\ is the total number of gas molecules +of the first kind, n 2 that of the second kind, and so forth (cf. Pail I, §2, Eq. [8] ). Moreover, +if M is the mass of a molecule of the normal gas and C 2 its mean square velocity at the +same temperature T; p/ } = m h !M is the atomic weight of one of the other gases, if the +molecular weight of the noimal gas is set equal to 1 ; then: + +m\c\ = m 2 Cj = • • • = MC 2 = 3 MRT = 3/2 h, + +hence + +MRT + +(195) p = —~ (ni + n 2 + • • •)• + + +In our special case, + + +2 n 2 = a - n h n 3 = n 4 • • • = 0, + +hence + + + +a + n\ + +2V + + +ami M + +MRT = (l + q) - RT + +2F mi ' + + +therefore, since v = V/am \, + + + + +If one substitutes the value (194) for q here, then the pressure p is obtained as a function of +the specific volume v and the temperature T. K is still a function of temperature, which will +be discussed later. In fact, direct observation gives the relation between p, v, and T. The +chemist is usually accustomed to state the degree of dissociation q as a function of p and T, +however. One can accomplish this by writing Equation (191) in the form: + + +34 + + +f J zLfLG. 7fca^4e«toi2c.a/ ~Ph-y-&XjcS- + + + + +Multiplication of this equation by Equation (196) yields + + + +If one substitutes this value into Equation (196), he obtains v as a function of p, T, and K. + +§65. Dependence of the degree of dissociation on temperature. + +There remains still the discussion of the quantity K. If we substitute in Equation (193) +for h its value ~ MRT, it follows that: + +1 r r du)d\ + +(198) if = - - eXlMRT > + +m\J J 4 t + +which in any case is only a function of temperature. At constant temperature, K is therefore +a constant, and Equations (194), (196), and (197) give directly the relation between p and +v, as well as the dependence of the quantity q on p and v, in which is involved only one +new constant K which has to be determined. + +Since Equation (198) contains the temperature under the integral sign, the dependence +of the quantity K on temperature cannot immediately be given in a simple manner. One +must instead make some assumption about the dependence of the function / on the amount +of overlapping of the two sensitive regions. In order not to lose ourselves in vague +hypotheses, we shall discuss only the simplest of all assumptions: that % always has a +constant value when the two atoms are chemically bound at all, i.e., as soon as the two +sensitive regions overlap at all, no matter how much. This would be the case when, at the +instant when the sensitive regions touch, there occurs a strong attraction, equal at all points +on the surfaces of these regions. But this attraction at once drops off to zero as soon as the + + +34 + + +7fca^4e«tai2c.a/ + + + +sensitive regions penetrate further into each other, y would then be the constant work of +separation of the two chemically bound atoms, or, conversely, the work done in chemical +binding by the chemical attractive forces. + +If all the a/G atoms present in unit mass of the gas are initially unbound, and +subsequently combine to form a/2G double atoms, then the amount ayJlG of work would +thereby be done; therefore A = ayllG is the total binding (or also dissociation) heat of unit +mass of the gas, measured in mechanical units, and one has: + + + +x + +2GA + +(199) + +X = 2GA/a, + +— = + +MRT + +aMRT + + +1 + +C C dud\ + +(200) + +J( - e 2/uA IRT_ + +)J 17 + +rriv + + +2Ajui + + +In chemistry one calls the mass 2/r 1 “one molecule.” Therefore 2p| A is the heat of +dissociation of “one molecule.” + +One can easily see that y may not be more than at most logarithmically infinite for any +configuration of the atoms, since the probability of that configuration would otherwise be +an infinity of ordered and would thus be so large that the atoms could never separate. +Whatever^ may be as a function of the positions of the atoms, one will not obtain a +qualitatively different result by replacing y by its mean value for all positions, whereby he +would again arrive at Equation (200) . This equation must certainly therefore provide an +approximation to reality in the general case. + +In the case where y is constant, no intramolecular work will be done by the relative +motion of the chemically bound atoms, as long as they remain bound. The mean kinetic +energy however is always the same at a given temperature, whether the atoms are +chemically bound or not; hence equal increments of mean kinetic energy correspond to +equal increments of temperature, and the specific heat is independent of whether the atoms +are chemically bound or not, as long as y is constant. Of course we mean here the specific +heat before the beginning or after the end of the dissociation; in case the degree of +dissociation changes, the heat of dissociation is not included in the specific heat. + +We shall now introduce for brevity the notation + + + +34 + + +IPuJix*. 7fca^4e«ta£2c.a/ + + + + +where X is the surface section on the sphere E. which the point X may not leave without +breaking the chemical bond, when the center B of the second atom lies within da> in Figure +4; then it follows from Equations (197), (200), (201), (202), and (203) that + + + +If q is given experimentally as a function of p and T, then the two constants a and j can +be determined from this formula. From a there follows at once from Equation (201) the +heat of dissociation—or, if one prefers, the heat of combination A of unit mass of the gas. +From 7 , one can determine the quantity/? by means of Equation (203) . According to +Equation (202) this quantity has a significant molecular meaning. For each volume element +da) of the critical space belonging to an atom, the point A must he on a certain surface +section X of the spherical surface E. in order that chemical binding can occur. We shall now +count not ah of the volume of each volume element da> of a critical space, rather only the +fraction of it which is obtained on multiplying the volume element by XIAtt. We call this +fraction of the total volume element the reduced volume. \Xd + +1970270p u + +i»c + +«,• = 6.300. no. ra, ?,=—— • + +2-617py + +p u is the average pressure used by Deville and Troost in their experiments on the +dissociation of hyponitrous acid (about 755.5 mm of mercury) ; p: is that used in the +experiments of Meier and Crafts on the dissociation of iodine vapor (728 mm). According +to Equation (201) , the heat of dissociation of a molecule (in the chemical, macroscopic +sense) is + +n = 2/i i A = aR. + + + + +This formula is based on mechanical units for heat. If one uses theimal units, it is necessaiy +to multiply by the appropriate conversion factor J. The heat of dissociation of a chemical +molecule in thermal units is therefore + + + +P = aRJ. + + + +We shall use grams, centimeters, and seconds as units for the mechanical quantities. A +weight of 430 kilograms raised one meter produces one k il ocalorie. Hence + + + +cal + +J = - + +430 gr • 100 cm G + + +Here G = 981 cm/sec 2 is the acceleration of gravity, and “cal” means a gram-calorie. The +value of the gas constant r for air is found by substituting the following values in the +equation of state pv = rT for air: + + +T =the absolute temperature 273° corresponding to melting ice, + + +1033 gr-G + +p = the pressure of one atmosphere=- + +cm 2 + +1000 cm 3 +V ~ 1.293 gr + + +For a molecule of air, p 0 is about 28.9 when one sets H = 1, H 2 = 2. Since R = rp, we +have for monatomic hydrogen: + + + +28.9 1033 gr G 1000 cm’ G-cm + +R = ---= 84570- + +273° cm* 1.293 gr 1°C + + +Finally it follows that + + +(208) + + +RJ = + + +28.9 1033 gr G 1000 cm 1 + + +cal + + +273° cm’ 1.293 gr 430grd00cmG +cal + + += 1.967 + + +gr. 1°C. + + +35 + + + + +Hence one obtains for hyponitrous acid : + + + +(209) + + +CgJ + +P„ = aJU = 13920 — • + +gr + + +On dividing by the molecular weight 2/jj = 92 of hyponitrous acid (N 2 O 4 ), one obtains for +the heat of dissociation of one gram of it the value + + +C£ll + +(210) D„ = 151.3 — > + +gr + +which is in good agreement with the direct determination of the heat of dissociation of +hyponitrous acid by Berthelot and Ogier . 5 +For iodine vapor it follows that: + + +cfll crI + +(211) Pj = 28530 — 1 D, = 112.5- + +gr gr + +According to Equations (202) and (203), the sum of the reduced critical spaces of all +atoms in unit mass is + + +1 + +(212) 0 = — fir + +2/xi + +Now we have + +1°C. + +7 “ “ 1970270p„ ' + +p u corresponds to a column of mercury of 755.5 mm height. Hence 6 + + + +1033grC 755.5 1027grG + +■ — — ■ ■ • - ■■ ■ — i. T—' ■■ ■ + +cm 2 760 cm 2 + + +35 + + +7fca^4e«tai2c.a/ + + +Hence + + + +1 G cm 1°C. cm* cm 3 + +(214) A, = - 84570—-- =- + +92 1°C. 1970270 1027 gr G 2200000 gr + +This value is three times as small as the one I found previously, 1 due to the fact that there I +made the improbable assumption that the four oxygen atoms can be arbitrarily exchanged +during the uniting and separation of two NCb groups, whereas here I consider such groups +to be undecomposable, playing exactly the same role as atoms. + +For iodine vapor I found 1 + + +1°C. + +2.617 pj + +The experiment was done likewise under atmospheric pressure (728 mm of mercury on the +average). Furthermore, the molecular weight 2/r 1 of iodine is equal to 253.6. Substitution of +these values yields + + + +cm 3 + + + +Neither for hyponitrous acid nor for iodine vapor is the value of van der Waals’ constant +b known, so that one cannot calculate the space filled by the molecules from the van der +Waals equation. This would also give only the order of magnitude since it treats the +molecules as nearly undefoimable spheres, and makes various other simplifications which +can influence the result. There remains finally Loschmidt’s method of estimation.* We shall +set the density of liquid hyponitrous acid equal to 1.5, and that of solid iodine equal to 5, +corresponding to temperatures below that of melting ice. At these temperatures the vapor +pressure of both substances is very small; for iodine it is of course much smaller than for +hyponitrous acid. But we can leave this out of consideration here, since we only want a +rough order-of-magnitude estimation. We assume completely arbitrarily that in these two +substances two thirds of the total space is filled up by the molecules. Then the space fi ll ed +by a gram of molecules of hyponitrous acid would be 0.44 cm 3 /gr; the sum of the volumes +of the covering spheres of these molecules is eight times as large, viz., 3.55 cnrVgr. For +iodine, the same quantities have the values 0.133 cm 3 /gr and 1.07 cm 3 /gr. Therefore, for +hyponitrous acid the reduced critical space of an N0 2 group, considered as one atom, is +only about one 8-millionth pail of the covering sphere, whereas for iodine it is an eighth or +a ninth pail of the covering spheres. The small dis-sociability of iodine is therefore +predominantly due to the relative size of the critical space compared to the covering sphere, +whereas the difference in the heats of dissociation per gram of the substances is relatively +small. Iodine vapor would therefore be about as easily dissociated as hyponitrous acid + + +35 + + +~*Puii.c- 7fca^4e«tai2c.a/ + + + +when the former is diluted a mi ll ionfold. + + +67. Mechanical picture of the affinity of two dissimilar monovalent atoms. + + +We consider a second simple example. Two kinds of atoms are present in a space V at +temperature T and total pressure p. There are a \ of the first kind and a 2 of the second kind. +Let the mass of an atom of the first kind be m\, and that of an atom of the second kind be +m 2 . Two atoms of the first kind can be combined into a molecule (double atom of the first +kind), likewise two atoms of the second kind (double atom of the second kind). For each of +these bonds, exactly the same rules as those established in the preceding section will hold. +We shall denote all quantities pertaining to double atoms of the first kind by an index 1, and +those to double atoms of the second kind by an index 2 . But in addition, chemical binding +of an atom of the first kind to one of the second kind, forming what we shall call a mixed +molecule, shall be possible. Similar laws will hold for these bonds, and we shall denote the +corresponding quantities by two indices, 1 and 2 . + +In the equilibrium state the following will be present in our gas: first, n | single atoms of +the first kind, and n 2 single atoms of the second kind; second, ri\ \ double atoms of the first +kind and n 22 double atoms of the second kind; third, « 12 mixed molecules. Chemical +combination of more than two atoms will be excluded. The atoms of the first kind shall be +impenetrable spheres of diameter cf \. We shall call a sphere of radius (J\ around the center +of such an atom its covering sphere. To it will be attached the critical space co | for the +interaction with a similar atom; da>i denotes the volume element of this critical space. If the +center of another atom of the first kind does not lie within uq, it is not chemically bound to +the first atom. If its center is within dceq, then chemical binding takes place only if it is in +the reduced volume da>\ —i.e., when the point A 1 lies within a certain surface section A\ of +the spherical surface E concentric with the first atom. Let dk\ be an element of the surface +A |. As before, A | is the point of intersection of a line drawn from the center of the first +atom parallel to the axis of the second, with the spherical surface E. Finally, is again the +work performed when the two atoms come from a great distance to those positions where +the center of the second lies within c/oq and the point A j lies within dk j. + +If one picks out an atom of the first kind, then he can assume that, of the remaining +atoms of the first kind, there will always be n \ which are bound neither to an atom of the +second kind nor to another remaining atom of the first kind. If the specified atom forms a +double atom of the first kind, then it can only be bound to one of these ri\ atoms of the first +kind, since we exclude tri-atomic molecules. The probability of this event is to the +probability that it remains single as kpip..V, where + + + + + +do)\d\i + +4 tt + + +This can be found just as in the previous section. The ratio of these two probabilities must +however be equal to the ratio 2 n 1 1 :n |, whence + + +35 + + +7fca^4e«toi2c.a/ + + + +( 217 ) = 2Vn n - + +Likewise one finds for the atoms of the second kind the equation + +( 218 ) k 2 nl = 2Vrin + + +where all quantities have a similar meaning. Therefore : + + + +We still have to discuss the formation of mixed molecules. Since we assumed that the +atoms of the first kind are impenetrable spheres of diameter a and those of the second +kind are impenetrable spheres of diameter 12 , the two atoms attract only when, as before, a certain point A 12 lies within +some surface clement r/A| 2 of the surface A12 —when the atom is in the reduced volume +element i2 of the critical space belonging to one of the n | unbound atoms of the first kind, and +the point A 12 should lie in some surface element c //.| 2 of the corresponding surface /. | 2 . +The probability that its center lies in a certain volume element doj\ 2 and the point A 12 lies +in a certain surface element d/.\j is, to the probability that for an arbitrary configuration of +its axis its center lies within V, as: + + +g2*Xll + + +rfoi>12 ^02 = faq, n 2i - a( 1 - q). + + +2 ^ 2 i+ 2 ^ii -2y, 0 - y (p is the heat liberated when two water vapor molecules form two +hydrogen molecules and one oxygen molecule. If we denote the heat produced when unit +mass of water is formed from the usual explosive gas by A, then: + + +2x2i + 2xn — 2x20 — X02 + +2 ( 2 mi + m 2 ) + +If one sets + + +2(2/ii + ju 2 )A 8x2iKn + +(226) -= a, -= 7 , + +R 2(2wi + miltyt 0 2 + +then: + +(227) (1 -q) t = q'-e‘' T . + +V + +Furtheimore, according to Equation (195) + +MRT ( q\ RT + +(228) p = (ri2o + fto2 + ^ 21 ) ; = (1 + —-) —-- + +\ \ 2 / f(2/ii n<>) + + +If one ehminates q, then he obtains the relation between p, v, and T. On the other hand, if +one eliminates v, then he obtains for the dependence of the degree of dissociation on +pressure and temperature the following equation: + + +9 + + +(l -?) 2 1 + t + + +RT + + +2 / (2*ii + m)p + + + +The equation between q, p, and v would be obtained by eliminating T from Equations +(227) and (228). + +In order to take account of the bivalence of the oxygen atom, one can assume that two +equivalent sensitive regions he on its surface. The critical space for the formation of HO + + +36 + + +“PiLh-d. 7fca^4e«tai2c.a/ + + + + +from H and O would then be twice as large as that for the formation of H 2 0 from HO and +H. But then the sensitive regions would not have to he directly vis a vis ; or else they must +be able to move on the surface of the molecule, in order to make possible double bonding +of two oxygen atoms. One would obtain a phenomenon at least partly similar to bivalence +by assuming that the critical region of an oxygen atom is not completely covered by the +covering sphere of a single oxygen or hydrogen atom chemically bound to it, so that there +is still room for the chemical binding of another atom. I certainly am not very hopeful that a +more precise formulation of these speculations can be established at the present time; yet +perhaps I may be allowed to adduce here the statement of a great scientist, that these +general mechanical models will aid more than hinder the knowledge of the facts of +chemistry. + +§71. General theory of dissociation. + +We shall now make some remarks on the most general case of dissociation. Let there be +given arbitrarily many atoms of arbitrarily many different kinds. A molecule which contains +a | atoms of the first, b\ atoms of the second, c\ atoms of the third kind, and so forth, will +be denoted symbolically by (a | b \ c | • • •)• The aggregate of C| molecules of the form +(rqZqcq ■ ■ ■), C 2 molecules of form (a 2 Z? 2 c 2 • • •)> C 3 molecules of form (a 3 Z? 3 c 3 • • •) and so +forth can be transformed into r | molecules • • •), T 2 molecules ((Xsfh 72 ' ' 0 . T 3 + +molecules (a 3 /?373 • • •) and so forth. Since both aggregates must contain the same atoms, +we have the following equations + +(CiOi + ^202 + * * • = T\a\ + ^<*2 + • • • I +(229) < + +\Cxbi ■+■ cy>2 4* * * * = ri/?i ■+• r^j + * * * + +We now assume that all possible combinations of our atoms are present in the gas, even if +only in small amounts. We denote by Hiqq... the number of single atoms of the first kind, +by n 2 oo-•• the number of double atoms of the first kind, and so forth. Likewise, let /z () | • + +be the number of single atoms, n 02 o- • • the number of double atoms of the first kind, etc.; let +tti 10 - • • he the number of molecules consisting of one atom of the first kind and one of the +second, and so forth. For simplicity we ignore isomers. A double atom of the first kind can +be formed only when the center of an atom of the first kind lies within the reduced critical +space of another atom of the first kind. Hence, if k 2 qq... is this reduced critical space, and +X 200 .. • is the heat of bonding of a double atom of the first kind, then it follows from the +principles of our theory that + +»ioo...:2njoo... = F: nioo • • .**00 • • .c 2A ** 0# *". + +Likewise it follows that + +Tlioo • • •' 3/1200* •• =: F*.71200 • • ./C 300 • • .e 2 ^* 00 ’ ‘', + + +36 + + +7fca^4e«tai2c.a/ + + + +where X300--- ' s the heat of bonding of a molecule consisting of three atoms of the first +kind, formed from a single and a double atom of the first kind. X300--- ' s the reduced +critical space for this bonding which is available for the single atom in the neighborhood of +the double atom. From the two ratios it follows that: + +3 -2 / 2Wioo • • • + +7&300-.. = 7lioo*>' s the product of all the reduced critical spaces, divided by a\, and ip ai 0 o.. + +. is the heat of bonding of a\ atoms of the first kind with each other. + +Each such molecule will have a definite reduced critical space n a 10 . . . for the +annexation of an atom of the second kind. Let iq. . . be the heat of formation of a +molecule consisting of a | atoms of the first and one atom of the second kind, from its +atoms. Then + + +»oio-..:n 0l io... = 7:n a ,oo...K ( > 1 io...d*V ) -A M -) + +If one attaches still more atoms of the second and third kinds, and so forth, then it follows +finally that + + +°1 *>1 ci + +ftatficx • • • “ WlOO • • •WOIO • • ^001 • • • + + +7 + + +1 —fli——cj— • + + +• / 2 /i^OibjC| • • • + +*ai 6 ici> • .6 } + + +where ^> ai £ c .. is the heat of formation of the molecule (a \ b \ c | • • •) from its atoms, and +K a l&jc* • • * ' s the product of all the critical spaces, divided by ci\ \by !q!... + +Completely analogous expressions follow of course for n aibiC2 ..., n a ^ 7| ..., + +and so forth. Then’s that have a single one, and otherwise all zero indices, can be +el im inated by taking account of Equations (229), whence one obtains: + + + +c 1 c 2 Tx r 2 + +^ a l^l c l' • * ’ * * ^ a l&l7l * ’ '' • X + +y lC ~ lT Ke 2 ^ C ^ a[hl ' ‘ ■ +C ^02&2 , “ + " '~ T ^a\Pl - r ^a 2 jV * • + + + + +K, + + +jC2 + +<* 2^2 + + + +. is the quotient in which occur all the + + +reduced critical spaces of the compounds (ci\b\ ■ ■ •), {ajbi • • •), each with its appropriate C + + +36. + + +7fca^4e«toi2c.a/ + + + +r r p p + +as exponent, and all the factorials (oq!) 1 (J3 ]!) 1 • • • (a 2 !) 2 (a!) 2 • • • in the numerator, and +the critical spaces of the compounds (yx\P\ ■ ■ •), {<^ 2 fi 2 ' ' 0 raised to the powers r 1; T 2 , • • • +as well as the factorials (aq!) Cl (£q!) C| • • • (a 2 l) Cl • • • in the denominator, r| ■ ■ • + + +F 2 ipctnfh. ■ ■ ■ ~ C\ ria,/,, is the heat of reaction which would be liberated if C| + +molecules (ci\b\ ■ ■ •), C 2 molecules (a 2 b 2 • • •) and so forth were changed into r| +molecules (xx\f J >\ ■ ■ •), r 2 molecules (a 2 /L ■ ■ •) etc. Furthermore, + +2C = Ci -f C2 "f * • • f 2r = Ti -(“ F2 ”h • • •. + +We shall now denote by m a ^ b ^ ... the mass of a molecule (a\b\ ■ ■ ■) in the gas-theoretic +sense, and by ju a b ... the mass of a molecule of this substance in the macroscopic sense, +i.e. the quotient m a b .../M. One molecule (yx\b | • • •) in the macroscopic sense therefore +contains 1/M gas-theoretic molecules. Likewise, in the aggregate of C 1 macroscopic +molecules (oq/q • • •), C 2 macroscopic molecules (a 2 b 2 ■ ■ •) etc. there are in all C\/M gas- +theoretic molecules (a\b\ ■ ■ •), C 2 /M gas-theoretic molecules (a 2 b 2 ■ ■ •), etc. Therefore + +~~~ • • + ^2^0262* •• ’j” ' ' + +M + +- r -•••] = n + + +is the heat which would be liberated if this reaction took place withCj, C 2 , • • • +macroscopic molecules and Tj, T 2 , • • • macroscopic molecules of the specified kinds. +Hence one can also write Equation (230) as follows: + + + +Ci c 2 r 1 r, ic-zr. n /rt + +^Olil • • •^'fl2&2 ' * * * * ' • • •'^'CljBi • • • ' ' ' r K 6 + + +This equation holds for any possible reaction. We now consider a special case: only one +kind of reaction is possible in the gas. Let there be initially a times C | (gas-theoretic) +molecules of type (aq/q • • •), likewise a times C 2 molecules of type (a 2 b 2 • • •) and so +forth, and no molecules of types (xx\fi\ ■ ■ ■), (a 2 /ri • • •) etc. At the pressure p and +temperature T let there be only (a - b) x C| molecules of type (ci\b\ ■ ■ •), (a -b) x C 2 of +type (a 2 b 2 • • •) etc., and b x Tj molecules of type (aq/?! • • ■), b x T 2 of type (a 2 ^3 • • •) +etc. present in equilibrium. Then b/a = q is the degree of dissoication. Furthermore, + +n«,t, • • • = 0(1 - q)C h n aib) • * ■ = a(l - q)C t • • •, +n aiSl • • ■ = aqTi, • • • = aqTi • • • . + +Hence Equation (231) takes the form: + + +36 + + +Pu.6.e 7fca^4e«tai2c.a/ + + + +• • • + + +„Ci„c t , ,zc / 0\zr-zc zr t x r, + +Ci Ct • • • (1 - q) =(-) q Vi Ti + + +ne + + +n irt + + +The mass of gas present is a[Cpn axh ^ ... + C^jn a ^^ +volume of unit mass then + + +]. If we again denote by v the + + +V = + + +a[Cim ai b r " + Ctfriaiiy.. + • • • + + +and we shall set + + +Ti r, + +vTiTt + + +7 = + + +C^C£ J • • • [Cim ai 6 r .. + Ctfriaiby.. + • • • ] + + +zr-zc + + +Then the above equation becomes + +(232) (1 - j)*c = 7i)JC-ir 9 jr e n/«r. + + +This equation gives the dependence of the degree of dissociation on the temperature and +specific volume, j and (if the heat of reaction is not otherwise known) U/R are constants to +be determined from experiment. + +If one wishes to introduce the total pressure p instead of v, then he obtains according to +Equation (195) + + + + + + + +MRT + +~V~ + + += [(1 - «)2 C + + += [(1 - ?) 2 C + + + +. aM + +qlT]—RT + +V + + +«sr] + + +RT + + +* * * ” 1 ” • ■) + + +Since C\p a ^^ . . . + C 2 q a9 / ;i ... + ••• is the molecular weight of the undissociated +substance, this agrees with the Boyle-Charles-Avogadro law for q = 0, and gives, when q +is different from zero, the deviations from this law due to dissociation. + +If one e li minates from this and from Equation (232) the quantity q, then he obtains again + + +36 + + +7fca^4e«tai2c.a/ + + + +the relation between/?, v, and T; if one eliminates v from the same equations, then he +obtains the degree of dissociation q as a function of p and T. + +More general formulas would follow from the assumption that an atom of the first kind +can be bound to one of the second, and then this complex can be bonded to another atom +of the first kind, although the atoms of the first kind cannot be bonded to each other alone +(isomerism). + +All these formulas agree with experience, as far as present observations go. + +§72. Relation of this theory to that of Gibbs. + +Gibbs 11 has deduced essentially the same formula from the general principles of +thermodynamics, without referring to the dynamics of molecules. Yet one should not forget +that his deduction is based on the assumption that in a dissociating gas all the constituents +are present independently as individual gases, and energy, entropy, pressure, etc. are simply +additive. This hypothesis is completely clear from the molecular-theory standpoint, since +these different molecules are actually present separate from each other; and in many places +it seems evident that Gibbs has these molecular-theoretic concepts continuously in mind, +even if he does not make use of the equations of molecular mechanics. + +On the other hand, if one takes the modem viewpoint, which has been most sharply +advocated by Mach 12 and Ostwald, 13 that in chemical binding something completely new +is created in place of the constituents, then it has no meaning to assume for example that +during the dissociation of water vapor, one has present simultaneously water vapor, +hydrogen, and oxygen. On the contrary, one must say that at low temperature there is only +water vapor; at intermediate temperature some new substance is present, which finally +becomes oxhydrogen gas [Knallgas] at very high temperatures. + +The assumption that at these intermediate temperatures the energy and entropy of water +vapor and oxhydrogen gas are additive loses any sense; without this assumption, however, +the basic equations of dissociation can be derived neither from the first and second laws of +thermodynamics nor from any energetic principles whatever. One can only consider them +as empirically given. + +There is no question that for the calculation of natural processes the mere equations, +without their foundation, are sufficient; likewise, empirically confirmed equations have a +higher degree of certainty than the hypotheses used in deriving them. But on the other +hand, it appears to me that the mechanical basis is necessary to illustrate the abstract +equations, in the same way that geometrical constructions illuminate algebraic relations. +Just as the latter are not made superfluous by mere algebra, so I believe that one cannot +completely dispense with the intuitive representation of the laws valid for the action of +macroscopic masses provided by molecular dynamics, even if he doubts the possibility of +knowledge of the latter, or indeed the existence of the molecules. A clear understanding is +just as important for knowledge as the establishment of results by laws and formulas. + +It should still be mentioned that we have discussed here only the simplest relations +which would give rise to the so-called theoretical dissociation equilibrium. A deeper +investigation into molecular mechanics can also account for the phenomena which one + + +36 + + +A^u.6.e 7fca^4e«tai2c.a/ + + + +denotes as false chemical equilibrum. 14 The pertinent facts are as follows. At room +temperature, oxhydrogen gas as well as water vapor can exist for an arbitrarily long time, +without one being transformed into the other. All the molecules are bound so strongly that +in the time of observation no dissociation or reaction of any observable amount of the +substance is possible. Of course in a time infinite in the mathematical sense the reaction +would take place.* + +The phenomena of false chemical equilibria are completely analogous to the phenomena +of supercooling and superheating that we have discussed in §15, and have exactly the same +basis. + + +73. The sensitive region is uniformly distributed around the entire atom. + + +We shall now, for comparison, consider the simplest case of dissociation on the basis of +another mechanical picture, which is actually a special case of the one earlier considered. +There are present again equivalent atoms, a in number, with diameter cr. What we called +the sensitive region shall now no longer be limited to a small pail of the surface of the +molecule, but rather shall be uniformly distributed over the entire molecule. The sensitive +region therefore has the foim of a spherical shell concentric with the molecule, whose inner +radius is \ where S is small compared to cr. +Whenever the sensitive regions of two molecules touch or overlap, they will be chemically +bound. The heat of separation, measured in mechanical units, will be equal to a constant, y, +for all these positions. + +The covering sphere is then a sphere of radius cr concentric with the molecule. The +critical space, which coincides with the reduced critical space, is a spherical shell which lies +between the surface of the covering sphere and a concentric spherical surface of radius cr + +5. Each time the center of a second atom lies within this critical space, it is chemically +bound to the first one, and the heat of separation is a constant equal to y. + +Let there be present ri\ simple and n 2 double atoms, so that + +nr.2n 2 = V'Amii - a>, we see that when the variables (237) are kept constant, + +du\ = du, dvi = dv, dw\ = dw. + +Hence + +(251) fiduidv\dwidpi • • • dq v + +is the number of molecules for which the variables (237) and (250) he between the l im its +(239)and + +(252) Mi and Mi + du h v\ and V\ + dv i, Wi and W\ + dw h + +It makes no difference here whether one introduces u \, v\, \ in /) instead of u, v, w, or +keeps the old variables. + +We denote the coordinates of the center of atom A 2 of the second molecule by xj, >’ 2 , Z 2 , +and its velocity components by + +(253) «!, t’2, w 2 + +and the other generahzed coordinates and momenta needed to determine the state of the +second molecule by + +(254) P 1.+1 • • • pp.\.p' f + +Then by analogy with the expression (251), the number of molecules of the second kind for +which the variables (253) and (254) lie between the l im its + +(255) u 2 and u 2 + du h v 2 and v t + dv 2 , w 2 and w 2 + dw 2 , + +(256) p, + i and p,+i + dp (+1 • • • q, +> . and + dq,+,’ + +will be denoted by + +(257) f?duidvidwidp,+i • • • dq f + f ' + +Fohowing the method of Pail I, §3, we can find the number of molecule pairs in which the +first molecule belongs to the first kind and the second to the second kind, and which +interact in time dt in such a way that atom A | of the first molecule collides with atom A 2 of +the second, and that at the instant of collision the following conditions are satisfied. The + + +37 + + +^ufuc. 7fca^4e«tai2c.a/ + + + +variables (250), (237), (253), and (254) shall lie between the limits (252), (239), (255), and +(256), and the line of centers of the two atoms A | and A 2 shall be parallel to one of the lines +lying within an infinitely narrow cone dk. All cases of interaction of two molecules which +take place during the time clt in such a way that all these conditions are satisfied, we call the +specified co ll isions. + +If a is the sum of the radii of the two atoms A | and A 2 , g their relative velocity, and the +latter forms, with the line of centers of the colliding atoms at the instant of collision, an +angle whose cosine is e, then one finds by the method in the section cited, for the number +of specified co ll isions: + +(258) dN = d\dt. + +§77. Form of Liouville’s theorem in the special case considered. + +Since the collisions take place instantaneously, the values of the variables (237) and +(254) do not change during collisions. Also, g, e, and the velocity components 6 , >h ( °f the +common center of gravity of Aj and A 2 have the same values after the collision as before +(cf. Part I, §4). Only the values of U \ v 1 oj\ , n 2 , v 2 , to 2 will be changed. The values of these +quantities after the collision will be denoted by the corresponding capital letters; and for +given values of g and e, the values of the variables u \ , iq oj\ , u 2 , v 2 , o) 2 , when they he +between the l im its (252) and (255) before the collision, will lie afterwards between the +l im its + +(259) Ui and U, + dU h F, and F, + dV h IF, and IF, + dW h + +(260) Vi and U 2 + dU 2 , V 2 and F, + dV 2 , 1F 2 and 1F 2 + dW h + +It can then easily be shown from Equation (52)—or even more generally by very simple +means, as in Pail I, §4—that then + +(261) du\dvidwiduidv^lw2 = dU \dV \dW \dU idV idW ^ + +or + + +^ dU 2 dF, dJF, dU t 5F 2 dW 2 + +£ +-= i + +dU\ dv\ dW\ dU2 dVi dWi + +(Previously the letters A, //, C were used instead of u, v, w, and primed letters were used +instead of capital letters.) + +The proof given in Part I, §4 contains an error, which was pointed out to me by C. H. +Wind 1 and later by M. Segel in Kasan. Therefore I will give this proof again here using a + + +37 + + + + + + +method free of error. + +We introduce instead of u 2 , v 2 , w 2 the components 7, //, C of the velocity of the center of +gravity common to A j and A 2 , if one were to consider these two atoms as one mechanical +system. If m\ and 7719 are their masses, then + + +m\U\ + m 2 u 2 + +mi + m 2 + + +with two similar equations for the other two coordinate directions. From these equations it +follows that, if one leaves the variables U\, vj vtq unchanged and only introduces £, 77 , C for +U 2 , V 2 , w 2 , then + + +(262) duidvidwiduidv2dw2 + + +'mi + mi + + +m 2 + + +3 + +du\dv\dw\d(dlrjd(. + + +In the expression on the right-hand side we introduce now instead of u\, vj, vtq the +variables U \, V\, W \, while we leave f, 77 , C unchanged. It is evident geomehically from +Figure 2, page 39 of Pail I that if the position of the center of gravity remains fixed, the +endpoint of the line which represents in magnitude and direction the velocity of the first +atom before the collision describes a volume element, which is congruent to the volume +element described by the endpoint of that line which represents the velocity of the same +atom after the collision. Hence + + +(263) duidv\dwid^drjd( = dU \dV \dW id&rjdf;. + +Now we introduce for £ 77 , C the variables U 2 , V 2 , W 2 , leaving UV\ W\ unchanged. +Since again we have the equation + + +miUi + mtUi + +i = T + +mi + m 2 + +with two similar equations for the other two coordinate directions, it follows that + + +''mi + m 2 \ 3 + + +m 2 + + +dUidVidWidfadt = dVidVidWidUjVrfWi. + + +From this and Equations (262) and (263) there follows at once the Equation (261) which +was to be proved. + +Since the case considered in Part I, §4, is the special case of the one discussed here in + + +37 + + +7fca^4e«tai2c.a/ + + + +which aside from the atoms A j and A 2 there are no other atoms in the molecules, we see +that our foimer proof has now been completed. + + +78. Change of the quantity H as a consequence of collisions. + + +In §76 we have called a certain kind of collision the specified kind. It is that kind which +takes place between a molecule of the first kind and a molecule of the second kind during +time dt, in such a way that at the instant of the beginning of the interaction, the variables +(250), (237), (253), and (254) lie between the limits (252), (239), (255), and (256), and that +the line of centers of the colliding atoms is, at the instant of collision, parallel to one of the +lines within a given infinitesimal cone dh. For the same collisions, at the instant of the end +of the interaction, the variables (237) and (254) he between the same li mits, but the +variables (250) and (253) he between the li mits (259) and (260). Moreover, g, e, and dk +will not be changed during the co ll ision. + +We now denote as opposite colhsions those cohisions which take place during time dt in +such a way that at the beginning the variables (250) and (253) he between the l im its (259) +and (260) and the other variables he between the same li mits as in the case of the specified +colhsions. + +For opposite colhsions, in order that they can occur at all, the initial relative positions of +the two molecules must be changed so that the second molecule appears displaced relative +to the first by a distance which is exactly equal and oppositely directed to the hne of centers +drawn from atom A j to atom A 2 . For opposite colhsions, conversely, the variables (250) +and (253) wih he between the l im its (252) and (255) at the end of the interaction. + +We now calculate the change experienced by the sum denoted in §74 by H (see Eqs. +[241] and [242]) during the time dt, through the combined effects of specified and opposite +colhsions. Each of the former colhsions will decrease by one the number of molecules of +the first kind for which the variables (250) and (237) he between the li mits (252) and (239), +and hence decrease Hi by lf \. Likewise, the number of molecules of the second kind for +which the variables (253) and (254) he between the l im its (255) and (256), wih decrease by +one, and hence H 2 wih decrease by lf 2 . On the other hand, the same cohision wih increase +by one the number of molecules of the first kind for which the variables (250) and (237) he +between the li mits (259) and (239), and will increase by one the number of molecules of +the first kind for which the variables (253) and (254) he between the li mits (260) and (256). +Hence H\ increases by IF j and H 2 by IF 2 , if we write for brevity F\ and F 2 for/|( U\ V\ +W | p| • • • q v , t) and/ 2 (C/ 2 , V 2 , W 2 , p v +1 • • • d v + v ’, 0- The number of specified cohisions is +given by Equation (258); hence ah the specified colhsions will increase H by + + + +(IF 1 + lF t - Ifi - IfiWgffifi +X dudvidwidudvdwdpi • • • dq H ,>d\dt. + + +Conversely, each of the opposite colhsions decreases by one the number of molecules of + + +37 + + +7fca^4e«toi2c.a/ + + + +the first kind for which the variables (250) and (237) lie between the l im its (259) and (239) +; while the number of molecules for which the same variables lie between the limits (252) +and (239) increases by one. Likewise, the number of molecules of the second kind for +which the variables (253) and (254) he between the li mits (260) and (256) will decrease by +one, and those for which the same variables lie between the limits (255) and (256) will +increase by one. Hence H\ increases by If\-IF\ as a result of opposite collisions, while H 2 +increases by lf 2 - IF 2 , so that H increases by lf\+ lf 2 - IF j - IF 2 . + +The total number of opposite collisions in time dl. by analogy with Equation (258), is: + +G 2 geFiF2dU\dV\dW\dU4V^W2fivi • • • dq v + v ’d\dt, + +or, by Equation (261), + +G^eFiFiduidvidwiduidvidWidpi • • 'dq v + v >d\dt } + +so that ah the opposite collisions increase H by + +(//. + lh ~ IF i ~ lFt)-< Wi + +X duidvidwidudi'du'tdpi • • • dq, + ,’d\dt + + +(Remember that g, e, and dk are unchanged by the collisions.) If one compares this with +the expression (264), then he sees that, combining together the specified and the opposite +collisions, the quantity H experiences the increment + + + +X du\dvidwidutdv4widpi ■ ■ ■ dq, + ,’d\dt. + + +The value of the latter expression is essentially negative. If one integrates over all possible +values of all the differentials except dt, and divides by 2 (since otherwise each co ll ision +would be counted twice, once as a specified collision and again as an opposite collision), +he obtains the total increment of FI during time dl. This is therefore also an essentially +negative quantity, provided there is any noticeable change of H at all. Since the same holds +for all other kinds of molecules, and similarly for collisions of different molecules of the +same kind with each other, we have proved that in this special case the value of H can only +decrease as a result of co ll isions. + +For the stationary state, a continual decrease of H is forbidden, so that for such states the +expression (265) must in general vanish. Therefore the equation + +(266) /i/i - FiFi = 0 + + +must hold for all kinds of molecules, with similar equations for collisions of molecules of + + +3 a + + +7fca^4e«tai2c.a/ + + + +the same kind with each other. + + +§79. Most general characterization of the collision of two molecules.* + +We shall now pass from the special kind of interaction discussed in §76 to the most +general case. + +We denote by s the distance of the centers of gravity of a molecule of the first and a +molecule of the second kind, and we assume that if s is greater than a certain constant b, no +perceptible interaction takes place between the two molecules. A sphere of radius b, whose +center is the center of gravity of a molecule of the first kind, will be called for short the +domain of the molecule in question. Hence we can also say: as soon as the center of gravity +of a molecule of the second kind lies outside the domain of a molecule of the first kind, no +noticeable interaction takes place between the two molecules. Any process whereby the +center of gravity of one of the foimer molecules penetrates the domain of the latter will be +called a co ll ision. + +Of course it is possible that even if the center of a molecule of the first kind does +penetrate the domain of a molecule of the second kind, it goes out again without any +perceptible interaction having occurred, so that the collision does not perceptibly modify +the motion of either colliding molecule. But most of the collisions will in fact produce a +significant modification of the motion of the two molecules. + +Just as in §§75-78, the positions of the constituents relative to the center of gravity, the +rotations around the center of gravity, and the velocities of the parts of a molecule of the +first kind will be characterized by the variables (250) and (237), and for a molecule of the +second kind by the variables (253) and (254). n 1 iq nq will now be the velocity +components of the center of gravity of a molecule of the first kind, and /q, v 2> w 2 those of a +molecule of the second kind. + +We shall call a configuration of the two molecules a critical constellation when the +distance of centers is equal to b. We consider critical constellations which satisfy the +following conditions: for the first molecule the variables (250) and (237) lie between the +limits (252) and (239), for the second molecule the variables (253) and (254) he between +the limits (255) and (256). Finally, the direction of the line of centers shah be parallel to +some line lying within an in fin itesimal cone of aperture dk. The set of these conditions we +shall call: + + +the conditions (267). + +When the center of gravity of the second molecule is moving into the domain of the first +molecule, the critical constellation represents the beginning of a process of interaction of the +two molecules (a collision in the wider sense of the word), and it is then called an initial +constellation. If, on the other hand, the second molecule is leaving the domain of the first at +this instant, it represents the end of a collision (final constellation). Critical constellations for +which the distance of centers of the two molecules attains its minimum at that moment can +be ignored, since they represent at the same time the beginning and end of a collision that + + +3 a + + +'§^uJix*. 7fca^4e«tai2c.a/ + + + +has no effect on the motion of the molecules. + +We denote two constellations as opposite, when the coordinates have the same value in +both, while the velocity components have equal magnitude and opposite sign. We denote +two critical constellations as corresponding to each other, when the coordinates (237) of the +first and (254) of the second molecule and likewise all velocity components have the same +magnitude and sign for both collisions, whereas the coordinates of the center of gravity of +one molecule with respect to a coordinate axis parallel to a fixed axis drawn through the +center of gravity of the other have the same value but opposite signs. The constellation +corresponding to any given constellation can therefore be constructed by leaving the first +molecule fixed and displacing the second molecule-without changing the configuration and +velocities of its constituents-by an amount 2b in the direction of the line drawn from its +center of gravity to the center of gravity of the first molecule. In other words, one +interchanges the positions of the centers of the two molecules without changing their state, +and without rotating them. + +The following is now immediately clear: if we imagine collected together all the initial +constellations and look for all the opposite constellations for each of them, then we obtain +all the final constellations, and conversely. Likewise we obtain all the final constellations if +we look for all the corresponding initial constellations, and the converse again holds. + + +80. Application of Liouville’s theorem to collisions of the most general kind. + + +Now, as before, let the number of molecules of the first kind in the gas, for which at +time t the variables (250) and (237) lie between the li mits (252) and (239), be given by the +expression (251). Likewise, let the number of molecules of the second kind, for which at +time t the variables (253) and (254) lie between the li mits (255) and (256), be given by the +expression (257). If we write the abbreviations + + +for + + +du )i and + + +du\dvidwidj)i • • • dq v and • • • dq , +r » + + +then + +(267a) dN = fiftduidw^kdKdt + +is the number of collisions that take place during time dt in such a way that their initial +constellation is a critical constellation determined by the conditions (267). Here k is the +component of the velocity of the center of gravity of the second molecule relative to that of +the first, in the direction of the line of centers at the beginning of the collision. For the +critical constellations with which all these collisions end, the variables (250) and (237) for +the first molecules shall he between the l im its (259) and (243); the variables (253) and + + +3 a + + +7fca^4e«tai2c.a/ ~Ph-y-&XjcS- + + + +(254) for the other molecule shall lie between the l im its (260) and + + +(268) P Hl and P, +l + dP H i • • • Q,+,’ and Q, + ,< + dQ, + ,' + +and the line of centers of the molecules shall be parallel to a line lying within a cone of +aperture dA. The set of these conditions we call + +the conditions (269). + +We again abbreviate the more complicated, though more precise, terminology of §27. +We write c/O | and r/fE for + +dUidVtdWidPi ■■■dQ, and iUJVJWJP+i • • • dQ n ,. + +and let K be the component of the relative velocity of the centers of gravity of the two +molecules at the end of the collision in the direction of the line of centers at this instant. + +Finally, we denote as before the difference of coordinates of the centers of gravity of the +two molecules (drawn from the first) for the initial constellation by X, //, (, and for the final +constellation by S, H, Z. Then Liouville’s theorem (Eq. [52]) applied to this case mns as +follows: + +(270) d^di}d^d(j}\(ht)2 = d'ZdHdZdttidti*. + +We now replace £, //, C and S, H, Z by polar coordinates, setting: + +£ = $ cos i?, t) = s sin # cos v, f = s sin sin , Z = S sin 0 sin + +Equation (270) then becomes: + +(271) s 2 sin Msddd are the apertures of the cones within which the lines of centers +lie before and after the collision. Since we are denoting the apertures of these cones as +before by dk and dA, we have therefore: + +sin MMp = d\ and sin 0d0d$ = dA. + +We shall also introduce for ds and dS the time differential dt. Let g be the relative +velocity of the two centers of gravity, and let 5 be the line connecting the two centers of +gravity before the collision; then the direction cosines of these two lines will be + + +3 a + + +~*Puii.c- 7fca^4e«tai2c.a/ + + + +Uj -1*1 Vi-Vi 10: -101 . i 11 f + +-, -, - and —, —, — • + +g g g s s s + +The component of relative velocity in the direction of the line s is: + +fc = — [(« 2 - t*i){ + (0: - »i)ij + (l0: - 10,)f], + +S + +The corresponding value of this component of relative velocity after the colhsion will be +denoted by K. Therefore we have + +ds = kdt, dS = Kdt. + +On substituting all these values and recalling that s = b at the beginning as well as the end +of the colhsion, Equation (270) takes the form: + +b 2 kd\dtdwidu)i - b 2 Kd\dtdttid% + +where dt has the same value on the right and left sides of the equation, since t is always +considered constant in Liouvihe’s theorem. If we divide the last equation by b 2 dt, it follows +that: + +( 272 ) kdkdaidu 2 = KdkdQidtii. + +We shall now keep in mind ah the final constellations of those collisions whose number +was denoted by dN in Equation (267a). Furthermore, we construct the constellations +corresponding to these, and denote by dN' the number of collisions that occur during time +dt such that they begin with the corresponding constellations in the manner described. Then + +( 273 ) dN' = FiFiVKdUidildMt, + +where F j and F 2 are abbreviations for + +/i(t/1, Fi, W\, P\ • • • Q, t t) and JiiUi, V 2, W2, P r +i • • • Qr+»', t) + +and one has in general dN =dN', if Equation (266) is satisfied for all collisions. Now in +each of the dN collisions of a molecule of the first kind, a state in which the variables (250) +and (237) he between the limits (252) and (239) is replaced by one in which these variables +lie between the limits (259) and (243). Conversely, in each of the dN' of a molecule of the +first kind, the latter state is replaced by the former; similarly for the second kind of molecule +and for ah other collisions. Hence it follows that the distribution of states is not changed by + + +3 a + + +lc, 7fca^4e«toi2c.a/ + + + +the collisions when Equation (266) is satisfied, and it is easily proved that this equation is in +fact satisfied by the formula (115), so that we have given a second proof that the +distribution of states represented by this formula satisfies the conditions which must be +satisfied by a stationary distribution of states. In order to prove, insofar as this is at all +possible, that it is the only one which satisfies these conditions, we shall again calculate the +change of the quantity H. + +§81. Method of calculation with finite differences. + +In the following we shall need an abstraction which may appear surprising to many, but +which must seem natural tc anyone who clearly understands that the entire symbolism of +the differential and integral calculus is meaningless unless one proceeds first by considering +large finite numbers. + +We shall assume that the molecule can have only a finite number of states, which we +denote by the series 1, 2, 3, and so forth; any arbitrary state can be denoted by 1, any other +state by 2, etc. The present representation is related to that of a continuous series of states in +such a way that one always considers as being the same all states which fill a region such +that they correspond according to Liouville’s theorem. Let (a, b) express symbolically a +critical constellation of two molecules which have the states a and b\ let (b, a) express the +corresponding, and ( -a, -b) the opposite constellation. + +A collision which begins with the constellation (a, b ) and ends with the constellation (c, +d) shall be denoted by + + + +Let w a be the number of molecules in unit volume that have the state a; vv/ ; , etc. have +similar meanings. Let + + +C°c,i-W a -W h + +be the number of collisions in the gas that begin with the constellation (a, b ) and end with +the constellation (c, d); then if dw a means the increment experienced by w a as a result of +collisions during time dt, then + +^Ca,iW t Wy 2Cp ( g1ZJ 0 W) n> + +dt + +where the sums are to be extended over all possible values of the quantities x, y, z., n, p, q. +Now suppose that all the expressions for + + + +3 a + + +7fca^4e«tai2c.a/ + + + +dwi dw 2 +dt dt + +have been written out, and set + +E = W\{lW\ - 1) + W 2 {lw 2 — 1) + • • • , + +Denote by dE the increment of E during time clt as a result of collisions, and substitute in + +dE dw\ dw 2 + +— = —Iwy H- lw 2 + + +dt dt dt + +the above values of + +dwi dw 2 + +- ) -; • • • ; + +dt dt + +l means the natural logarithm. The co ll ision + + + +in which 1, 2, 3, 4 can be any states, (2, 1) and (3, 4) any critical constellations, provides in +the expression for dw j, as well as in that for dwj, the term + +- Ci,iW { w 2 . + +However, to the expressions for + +dwz dwi + +— and — +dt dt + +it contributes a positive term. All these terms contribute to dE/dt the sum + +2,1 + +Ci'iWiWiilWi + Iwt - hi - lw 2 ). + + +3 a + + +^ufuc. 7fca^4e«tai2c.a/ + + +The corresponding co ll ision + + + + +-i.e., that one which has as initial constellation the same constellation (4, 3) which +corresponds to the final constellation (3, 4) of the previously considered collision- +contributes to + +dwz dv)i + +— and — +dt dt + +the term + +ft'* + +C 6l wm, + +and to + +dwi dwt + +— and — +dt dt + + +again two equal positive terms. + +In the same way one can continue with the co ll ision + + + +which corresponds to the co ll ision + + + +> + + +and so forth. + +Since we only have a finite number of states, we must eventually arrive at a co ll ision + + +/k, k - 1 + +w ,y + + +which corresponds to one of the previous ones, and it can be proved that the first co ll ision + + +3 a + + +7fca^4e«tai2c.a/ + + + +for which this occurs must correspond to the co ll ision + + + +For if it corresponded, for example, to the co ll ision + +6, 5\ + +7, 8/’ + + +then (x, y ) and (6, 5) must be corresponding collisions, therefore {x, y ) and (5, 6) would be +identical, and two collisions, one beginning with ( k , k - 1), and the other with (4, 3) would +lead to the same final constellation. But then the initial constellation ( -5, -6) would lead to +the final constellation (-4, -3) as well as to the final constellation ( -k, -k+ 1). Hence the +latter two constellations must be identical, hence + + + +must be identical to + + + +and for the same reason + + +'k - 2, k - 3\ (2, 1' + +must be identical to +k - 1, k ) \3, 4/ + + +Therefore the cycle must have already been closed previously. + +Equation (272) means, in our present notation, that the coefficients + +a.b d,e + +C' d and Cij + + +must be equal to each other, since we have collected together all states for which the +variables fill a region that is equal according to Liouville’s theorem, and called them a +single state. Hence it follows that one can arrange all the terms contained in clE/clt in a cycle +of the form: + + +, 2 . 1 . + + ++W k -iW k {lWi+lWi - lw k . 1 - lw k )}. + + +3 a + + +7fca^4e«tai2c.a/ + + + +If one denotes the expression in square brackets by IX and sets w | w 2 = oc, W 3 W 4 = /?,•• +•, then: + +(274) X = ■ • • oT a . + +Among the numbers a, /?, 7, • • •, there must be at least one, for example 7, which is not +larger than its two neighbors [ J > and 6; then + +(275) X = + +where + +Y = • * • CL 0 ~ a + +has exactly the same foim as X but lacks one term. + +The factor of Y in Equation (275) is equal to 1 if either 7 = /? or 7= 6, but otherwise it is +always less than 1. If one applies the same considerations to Y again and again, then he can +finally reduce X to a product of fractions each of which is less than or equal to 1; they +cannot all be equal to 1 unless all the quantities a, /3, j ■■ ■ are equal to each other. + +Hence the quantity E —whose time derivative reduces to dH/dt on passing to +infinitesimals-can only decrease or remain constant as a result of collisions; and it can +remain constant only if for all co ll isions + + +the equation + + + +w a w b = W c Wd + + +is satisfied. Since for the stationary state E cannot decrease any further, the equation + +w a w b = W c Wd + +must be satisfied for all possible collisions in the stationary state, and on passage to +in fin itesimals this is identical with Equation (266). + + +§82. Integral expression for the most general change of H by collisions. + +If one wishes to avoid the transition from a finite number of states to an infinite number, +but at the same time make use of differentials, he may use the method outlined here. As in + + +3 a + + +7fca^4e«toi2c.a/ + + + +Pail I, §18, and Pail n, §75-78, one finds + + + + + +where the single integral denotes an integration over the differentials contained in da>i, and +the triple integral denotes integration over all the differentials contained in da>\ dojjdA. +d[f\ lf\ d(i)\ means the change experienced by this integral merely as a consequence of the +collisions of molecules of the first and second kinds. The change due to intramolecular +motion is zero. The other quantities have the same meaning as in the preceding sections. +We imagine that for each collision the corresponding one has been constructed, whose +initial constellation therefore corresponds to the final constellation of the first one. We +denote by f { / and j.1' the values which the functions fi and /2 la kc when one substitutes +therein the variables characterizing the state of both molecules at the end of this second +collision; furthermore, we shall once again construct the collision corresponding to this +second collision, and denote by f''' and f/, '' the function values that arise on + +substituting the values of the variables characterizing the final states of the two molecules +for this latter collision, and so forth. + +Then the quantity (d/dl)\f\ c an be brought into the form: + + + +b 2 gdoiidaid^[S\Si(lP 1 + IFt — (A — Ifi) + ++W(!/r + If" - IF 1 - IFf) + + +,+!i'n'(ifi"+in:' -iw -ifn+■■■ + + +If one sets again + + +/ 1/2 - F\Fi - P, j\ ji - 7 etc., + +then the expression in square brackets in (277) will be the natural logarithm of + +(278) p°-l>ylHSr-l . . . + +This quantity has exactly the same foim as the expression (274), except that now the +cycle of quantities a, j3, y ■■ ■ is in general not finite. Nevertheless, if one proceeds far +enough along this series, he will eventually come to a term whose base is again very nearly + + +39 . + + +7fca^4e«tai2c.a/ + + + +equal to a, so that the difference between (278) and an expression terminated at this point +can be made arbitrarily small. As soon as the motion of two molecules is not changed by a +collision, then it can of course happen that one of the quantities a, /?, 7 • • • is equal to its +neighbor. However, as long as we do not choose b to be so large that this is the case for +most of the collisions, then most of these quantities will be completely different from their +neighbors, so that most of the fractions being multiplied in the expression (278) will be +smaller than 1; the same is true of the factors of Y in (275). Hence clH/clt will be negative, +and can be zero only when the condition (266) is satisfied for all collisions. + +§83. Detailed specification of the case now to be considered. + +We have shown in the preceding sections that, for thermal equilibrium in ideal gases +with any kind of compound molecules, Equation (266) must be satisfied for all collisions of +like or unlike molecules. In carrying out the proofs we have excluded external forces, yet +the proofs can still be performed when external forces are permitted. One sees immediately, +moreover, that Equation (266) will be satisfied as soon as the distribution of states is +determined by the formula (118). + +However, the proof that this distribution is the only possible one apparently cannot be +carried out in complete generality, so that one still has to provide a proof in each special +case. Naturally we shall have to leave all these different special cases to monographs; we +can only treat here a rather small number of examples. + +The simplest of these is the following special case. Let there be a mixture of any ideal +gases on which no external forces act. The atoms of the different molecules shall be held +together by arbitrary conservative forces, for which the Lagrange equations hold. The +interaction of two different molecules proceeds as if one atom from each molecule collided +with the other in the same way as elastic, negligibly deformable spheres. + +On account of the negligible deformability, during such a collision neither the position +of the colliding atom nor the positions and velocities of the other atoms will change. But +since every direction in space is equally probable for the velocity of an individual atom +before the collision, one can calculate the probabilities of various kinds of collisions just as +in Part I, §3. + +For the sake of generality we consider a collision in which the two interacting molecules +are of different kinds, and call these the first and second kinds of gas. The same statements +will still be valid when both molecules are actually of the same kind. + +§84. Solution of the equation valid for each collision. + +A particular atom of mass m \ belonging to the first molecule collides with a particular +atom of mass m 2 belonging to the second molecule. We call atoms equivalent to the first +atom the in | -atoms, and all the atoms equivalent to the second atom the m 2 - atoms. Let c\ +and c 2 be the velocities of the two colliding atoms just before the collision, and y\ and j 2 +the velocities just after the collision. The values of c\ and c 2 are completely arbitrary, y. + + +39 . + + +7fca^4e«tai2c.a/ + + + +can take any value lying between the l im its zero and + + + + +m<$\ + +mi + + +but 72 must, according to the conservation of energy, be equal to + + +1 Wl 2 2 + +C2 H-(Ci ~ Yl) + +m 2 + + +since because of the shortness of the duration of the collision, the energy of the molecules +does not change noticeably. + +The number of those m 1 -atoms in the entire gas for which the three components of +velocity of their center in the three coordinate directions he between the limits + +(280) u\ and U\ + du h Vi and v\ + do i, W\ and W\ + dw\ + +while ah other variables determining the state of motion of the molecules can have any +possible values, we denote by + + +f(c\)duidv\dwi + +Since there is no preferred direction in space for the velocity of this atom, the coefficient of +the product of differentials is clearly a function only of c\, and hence will be called /| (c\). +Similarly the number of m 2 -atoms whose velocity components he between the li mits + +(281 u 2 and u 2 + du 2) v 2 and v 2 + dv 2) w 2 and w 2 + dw 2 + +will be denoted by + + +f 2 (c 2 )du 2 dv 2 dw 2 . + +For this colhsion, Equation (266) reduces to + + +(282) /i(ci)/ 2 (c 2 ) - /i(yi)/2 c 2 + — (ci - Yi) j. + +Since this equation must be satisfied for ah possible values of the variables which fulfill +the condition of conservation of energy, it fohows that, as a simple calculation shows (cf. +also Part I, §7), + + +39 . + + +“PiLh-d. 7fca^4e«tai2c.a/ + + + +/i(ci) = 4ie-*”‘ c ‘‘, }i(c t ) = 4 2 e- ta,c > 1 . + +These formulas, together with the condition that all directions of space are equally +probable for the velocities, completely determine the probability of the various velocity +components. If all the atoms of all the molecules can collide with each other, then h must +have the same value for all. Hence the mean kinetic energy is the same for all atoms, and, +as can easily be proved from the equivalence of all directions of velocity, the mean kinetic +energy of progressive motion of the center of gravity is the same for all molecules, and is +equal to the mean kinetic energy of an atom. The coefficients A | andA 2 are constants; +however, they would depend on the other variables determining the state of the molecules +and the limits assumed for these variables, if all values were not allowed for these variables +but only values lying between given l im its. + +A special case of the one considered is that of diatomic molecules whose atoms are solid +spheres connected with rods so that they form a solid system like the so-called gymnastic +dumbbells. 3 If the connecting rods were considered to be elastic, then the atoms could +perform radial vibrations back and forth. However, we can go to the limiting case where +the deformability of the rod becomes zero, hence the amplitude of these vibrations is so +small that, just like the rotation around the line connecting the centers of the atoms, they do +not come into thermal equilibrum with the other motions in the time of observation. + +The result then agrees completely with that obtained earlier, where we found the value +1.4 for the ratio of specific heats. + +Another special case is a molecule consisting of three or more spheres rigidly bound +together. We then have the case for which we previously obtained the value J for the +ratio of specific heats. One could treat this case extensively without much difficulty, finding +the probability of various combinations of values of the coordinates as before. We shall not +discuss these cases further, but instead give an example of the method of treating more +difficult cases. + +§85. Only the atoms of a single type collide with each other. + +Let there be given an ideal gas, all of whose molecules are the same. Each molecule +consists of two different atoms of masses m \ and m 2 (to be called the atoms of the first and +second kinds). The two atoms of a molecule shall behave, with respect to their +intramolecular motion, l ik e material points concentrated at the centers of the atoms, exerting +on each other a force in the direction of their connecting line, which is a function of their +distance. The intramolecular motion will therefore be ordinary central motion. 4 The +interaction of two different molecules is as follows: the two atoms of the first kind collide +l ik e negligibly deform-able elastic spheres, but there is no interaction between atoms of the +second kind, or between an atom of the first kind and an atom of the second kind.* + +On applying the same arguments as before, we obtain for the atoms of the first kind an +equation analogous to Equation (282), from which it follows that: + + +393 _ + + +fPii+uc. Ma£4cjvi(i£icaZ + + + +(283) /i(ci) = ic-*”’ 1 ''’. + +As before, U \, v 1; w | are the velocity components for an atom of the first kind; +f\ (c | )du | dv i dw | is the number of atoms of the first kind for which a \, v\, and vv ] he +between the l im its (280) ; A can still depend on the l im its imposed on the states of the +molecule. + +However, the same method of reasoning cannot be applied to the atoms of the second +kind, since they never collide with atoms of other molecules. We must therefore introduce +the probability of orbits and phases of motion of the central motion. + +§86. Determination of the probability of a particular kind of central motion. + +We have already denoted by c\ and c 2 the absolute velocities of the first and second +atoms at any time. Let p be the distance of centers of the two atoms at the same time; let ci\ +and a 2 be the angles formed by the directions of c\ and C 2 with the line drawn from the first +to the second atom; finally, let /3 be the angle between the two planes passing through the +line p, one with the same direction as c \, the other with the same direction as C 2 . + +The total energy of the molecule is + + + + +2 2 +WiCi mfa +T + + +2 + + +2 + + ++ ^(p), + + +where cp is the potential function of the central force. Twice the angular velocity of m 2 +relative to m\ in the orbital plane is: + +(285) K = pVcJ sin 2 ai + c] sin 2 a 2 - 2cic 2 sin ai sin a 2 cos ft + + +the velocity of the center of gravity of the molecule, multiplied by in | +m 2 , is + +(286) G =v / mJc{+m|c 2 + 2 mim 2 CiC 2 (cos <*i cos a 2 +sin sin a 2 cos ft + + +and its component perpendicular to the orbital plane is + +CiC 2 sin a\ sin a 2 sin 0 + + +(287) H = + + +y/c\ sin 2 ai + c\ sin 2 a 2 - 2 cjC 2 sin a\ sin a 2 cos 0 + + +The number of molecules in unit volume for which K. L, G, H lie between the l im its + + +39 . + + +7fca^4e«tai2c.a/ + + +KmAK + dK, L and L + dL, G&uAG + dG, + +H and H + dH + +will be denoted by + +M.K, L, G, H)dKdLdGdH. + +The number of molecules for which p lies between p and p+c/p is + +dp C pl dp dp + +Q-dKdLdGdH- = MKdLdG dH - • +a J v u + +Here + +dp f pl dp +cr = —: I — +dt J Po a + +is the time elapsed from the perigee to the apogee, and is therefore a given function of K, L. +G, and H; + + +V = $: + + + +dp + + +is therefore a given function of these four quantities. We li mit ourselves to those molecules +for which, first, the line of apses of the path forms with a line in the orbital plane parallel to +a fixed plane an angle between e and e+r/e; second, the two planes through the velocity of +the center of gravity noimal to the orbital plane and parallel to a fixed line T foim an angle +between oj and oj+cIoj; and third, the velocity direction of the center of gravity lies within a +cone of specified direction and in fin itesimal aperture d'k. Then we have to multiply by +d&dwdX\\6Tr. The number of molecules in the gas satisfying all these conditions is +therefore + + +1 + +(288) ^- dKdLdGdH dpdtdudX. + +16tt 3 o- + +If we denote by g and g+dg, h and li+dh, k and k+dk the limits between which the velocity +components of the center of gravity of these molecules relative to fixed rectangular +coordinate axes lie, then + + +39 . + + +7fca^4e«tai2c.a/ + + + +G 2 dGd\ = dgdhdk. + +Now keep g, h, and k fixed, and construct through the center of the first atom a rectangular +coordinate system whose z-axis has the direction of G. Denote the coordinates and velocity +components of the second atom relative to this system by x 3 , y 3 , z 3 , « 3 , v 3 , w 3 , and +transform these six variables into K, L , H, p, e, For this purpose we construct through +the center of the second atom a second coordinate system, with respect to which the +coordinates and velocity components of the second atom shall be called x 4 , y 4 , z 4 , w 4 , v 4 , +w 4 . The 2-axis of the second system shall be perpendicular to the orbital plane, and the x- +axis shall lie in its intersection line with the old xy plane. Then + +H = 6 sint>, + +where 90°-j?is the angle between the two z-axes; hence, since G is constant, + +dH = GcostWtf. + +Finally, we denote the angle between the two x-axes by co, since it differs from the angle +previously so denoted only by an amount which we always now consider constant. We +find: + + +Z{ = x 3 cost?sinw + y% cos t? cos w + z 3 sin t? + +W\ = u% cos t? sin w + v% cos t? cos w + w> 3 sin t?, + +both of which expressions must vanish, since the x 4 y 4 plane is the orbital plane. By means +of these two equations, keeping x 3 , y 3 , « 3 and v 3 constant, one can introduce 3. to in place +of z 3 , w 3 , and find + +cost? + +dzdwi = (jjiUi - XiVi) - dddw. + +sin 8 t? + +Now furthermore + +Xi = x% cos w - y% sin w +t /4 sin t? = Xi sin o> + V* cos w + +and similar equations follow for w 4 , v 4 . Hence it follows that + +yiUi - x&s = sint?(t/ 4 W4 - xm) = K sin# + + +39 + + +Auntie 7fca^4e«tai2c.a/ + + + +and for constant &. to, + + +dx4y\ sin d = dx4Vz } du4v 4 sin d = du4v 3 , + +hence + +dx4y4z4u4v4wz = K cos ddx4y4u4v4$dw. + +Now we denote by a and r the velocity components of the motion of the second atom +relative to the first in the direction of p and perpendicular thereto, as before; then, for +constant x 4 and y 4 , + +dadr = du4vi + +K — pr, L = L„ + — —■— -(a 2 + r J ) + + +2(mi t W 2 ) + +dKdL = - apdadr, + +rrii + mi + +where L„ is the energy of motion of the center of gravity, now considered constant. Finally, +if if) is the angle between p and the last line of apses, + +14 = PCOS(e + f), y t = + + +where ip is a function of p, K, and L. But the last two are now constant, hence + +pdpdt = dx4y\‘ + +Collecting all these together, we see that: + +mi + rri2 K + +dx4y4z4u4v4ws = - dKdLdHdpdwde + +and one sees at once that, if x, y, z are the coordinates of the second atom with respect to a +coordinate system going through the center of the first atom, whose axes are parallel to the +originally chosen completely arbitrary coordinate axes, then lik ewise + + + +39 . + + +7fca^4e«tai2c.a/ + + + +mi + m 2 K + +dxdydzdu 2 dv 2 dw 2 =- dKdLdHdpdwde. + +mim 2 a + + +If we introduce this into the expression (288) and recall that for constant « 2 , v 2 , we have + + +3 + +mi + +dgdhdk =- duidvidw t + +(mi + m 2 ) 3 + +then we find + + +(289) + + +1 + + +mim 2 ¥ + +— dxdydzduidi'idwidu 2 dv2dw2 + + +167t 3 (mi + m 2 ) 4 KG 2 + + +as the number of molecules in unit volume for which the variables jc • • • w 2 lie between x +and x+dx ■ ■ ■ w 2 and w 2 ++j • • • U > + +lie between the li mits + + + +p,+i and p „ + 1 • • • + dp f+2 , p, + » and +Pm+! “f dpf .+2 ■' * q^+n and "I dq^t + + +Now let the li mits (292) and (293) be chosen so that the two molecules are not +interacting but will soon be. We shall call the type of interaction which comes about in this +way a co ll ision with property A. Then (cf. Eq. [123]) + +(296) /i(pi • • • qMVr+i ■ • ■ q^r)dpi • • • dq t+ , + +is the probability that for a molecule pair 10 the variables (291) and (294) lie between the +limits (292) and (295), which we call the probability of a collision with property A, for + + +40 + + +“‘Puh.c- 7fca^4e«tai2c.a/ + + + +brevity. + +From the instant when the variables (291) and (294) of a molecule pair (consisting of +molecule B of the first kind and molecule C of the second kind) first fall within the l im its +(292) and (295), let there elapse a certain time L which is longer than the time during which +two molecules interact in any co ll ision. The values of the variables (291) and (294) for +molecules B and C will lie, at the end of time t, between the li mits + +(297) Pi and Pi + dPi • ■ ■ Q.+, and Q, + , + dQ^,. + +Now at the end of the time denoted above by T we reverse the directions of the +velocities of all the constituents of all the molecules, without changing the magnitude of +these velocities or the positions of the constituents. The system will then pass through the +same sequence of states in the opposite order, which we shall call the inverse process, in +contrast to the originally considered variation of states during time T, which we call the +direct process. + +During the inverse process, it will just as often happen that the variables (291) and (294) +lie between the li mits + + + +P i and Pi + dP\ • • • and P^ v + dP M+ „ + +, Qi and ~ Qi “ dQi • • • — Qn+ t , and Qn+y ~ dQn+t + + +as during the direct process it happened that they lay between the l im its (292) and (295). + +We next assume that in our system two states in which all the coordinates and all +magnitudes of the velocities are the same, but the directions of the latter are opposite, are +equally probable. We shall call this assumption A. It is obviously true when the molecules +are simple material points or solid bodies of arbitrary shape, and in many other cases. +However, in certain cases it needs to be proved. + +Therefore in the inverse process the variables will lie between the l im its (297) as many +times as they he between the li mits (292) and (295) in the direct process. But the inverse +process l ik ewise consists of a very long sequence of states in which the variables can +assume many different values. These states therefore cannot consist exclusively or +predominantly of singular states, but rather for the most pail they must be probable states. +Hence the various mean values must be the same for the inverse process as for the direct +process, and the probability that for a molecule pair the values of the variables lie between +the limits (297) must be given by the expression similar to (296): + +/i(Pi • • • Qp)fi{Pp+i - • • Q»+*)dP i • • • dQt+i + +which according to the aforesaid must be equal to the expression (296). But according to +Liouvihe’s theorem, + + +40 + + +aV^lc. 7fca^4e«tai2c.a/ + + + +dpi * • * d(}n+p — dP i dQn+ V) + +hence one obtains finally the equation + +(299) /i(r, • • • qMP >+1 • • • U>) =/i(Pi • • • QMK+> • • •