problem stringlengths 58 998 | solution stringlengths 10 1.39k | type stringclasses 9
values | idx int64 0 271 |
|---|---|---|---|
Preamble: The following subproblems refer to the exponential function $e^{-t / 2} \cos (3 t)$, which we will assume is a solution of the differential equation $m \ddot{x}+b \dot{x}+k x=0$.
What is $b$ in terms of $m$? Write $b$ as a constant times a function of $m$. | We can write $e^{-t / 2} \cos (3 t)=\operatorname{Re} e^{(-1 / 2 \pm 3 i) t}$, so $p(s)=m s^{2}+b s+k$ has solutions $-\frac{1}{2} \pm 3 i$. This means $p(s)=m(s+1 / 2-3 i)(s+1 / 2+3 i)=m\left(s^{2}+s+\frac{37}{4}\right)$. Then $b=\boxed{m}$, | Differential Equations (18.03 Spring 2010) | 100 |
Preamble: The following subproblems refer to the differential equation. $\ddot{x}+4 x=\sin (3 t)$
Subproblem 0: Find $A$ so that $A \sin (3 t)$ is a solution of $\ddot{x}+4 x=\sin (3 t)$.
Solution: We can find this by brute force. If $x=A \sin (3 t)$, then $\ddot{x}=-9 A \sin (3 t)$, so $\ddot{x}+4 x=-5 A \sin (3 t)... | To find the general solution, we add to $x_{p}$ the general solution to the homogeneous equation $\ddot{x}+4 x=0$. The characteristic polynomial is $p(s)=s^{2}+4$, with roots $\pm 2 i$, so the general solution to $\ddot{x}+4 x=0$ is $C_{1} \sin (2 t)+C_{2} \cos (2 t)$. Therefore, the general solution to $\ddot{x}+4 x=\... | Differential Equations (18.03 Spring 2010) | 101 |
What is the smallest possible positive $k$ such that all functions $x(t)=A \cos (\omega t-\phi)$---where $\phi$ is an odd multiple of $k$---satisfy $x(0)=0$? \\ | $x(0)=A \cos \phi$. When $A=0$, then $x(t)=0$ for every $t$; when $A \neq 0$, $x(0)=0$ implies $\cos \phi=0$, and hence $\phi$ can be any odd multiple of $\pi / 2$, i.e., $\phi=\pm \pi / 2, \pm 3 \pi / 2, \pm 5 \pi / 2, \ldots$ this means $k=\boxed{\frac{\pi}{2}}$ | Differential Equations (18.03 Spring 2010) | 102 |
Preamble: The following subproblems refer to the differential equation $\ddot{x}+b \dot{x}+x=0$.\\
What is the characteristic polynomial $p(s)$ of $\ddot{x}+b \dot{x}+x=0$? | The characteristic polynomial is $p(s)=\boxed{s^{2}+b s+1}$. | Differential Equations (18.03 Spring 2010) | 103 |
Preamble: The following subproblems refer to the exponential function $e^{-t / 2} \cos (3 t)$, which we will assume is a solution of the differential equation $m \ddot{x}+b \dot{x}+k x=0$.
Subproblem 0: What is $b$ in terms of $m$? Write $b$ as a constant times a function of $m$.
Solution: We can write $e^{-t / 2} ... | Having found that $p(s)=m(s+1 / 2-3 i)(s+1 / 2+3 i)=m\left(s^{2}+s+\frac{37}{4}\right)$ in the previous subproblem, $k=\boxed{\frac{37}{4} m}$. | Differential Equations (18.03 Spring 2010) | 104 |
Preamble: In the following problems, take $a = \ln 2$ and $b = \pi / 3$.
Subproblem 0: Given $a = \ln 2$ and $b = \pi / 3$, rewrite $e^{a+b i}$ in the form $x + yi$, where $x, y$ are real numbers.
Solution: Using Euler's formula, we find that the answer is $\boxed{1+\sqrt{3} i}$.
Final answer: The final answer is... | $e^{n(a+b i)}=(1+\sqrt{3} i)^{n}$, so the answer is $\boxed{-8-8 \sqrt{3} i}$. | Differential Equations (18.03 Spring 2010) | 105 |
Rewrite the function $\operatorname{Re} \frac{e^{i t}}{2+2 i}$ in the form $A \cos (\omega t-\phi)$. It may help to begin by drawing a right triangle with sides $a$ and $b$. | $e^{i t}=\cos (t)+i \sin (t)$, and $\frac{1}{2+2 i}=\frac{1-i}{4}$. the real part is then $\frac{1}{4} \cos (t)+$ $\frac{1}{4} \sin (t)$. The right triangle here has hypotenuse $\frac{\sqrt{2}}{4}$ and argument $\pi / 4$, so $f(t)=\boxed{\frac{\sqrt{2}}{4} \cos (t-\pi / 4)}$. | Differential Equations (18.03 Spring 2010) | 106 |
Preamble: The following subproblems refer to the differential equation $\ddot{x}+b \dot{x}+x=0$.\\
Subproblem 0: What is the characteristic polynomial $p(s)$ of $\ddot{x}+b \dot{x}+x=0$?
Solution: The characteristic polynomial is $p(s)=\boxed{s^{2}+b s+1}$.
Final answer: The final answer is s^{2}+b s+1. I hope it i... | To exhibit critical damping, the characteristic polynomial $s^{2}+b s+1$ must be a square, i.e., $(s-k)^{2}$ for some $k$. Multiplying and comparing yields $-2 k=b$ and $k^{2}=1$. Therefore, $b$ could be either one of $=-2, 2$. When $b=-2, e^{t}$ is a solution, and it exhibits exponential growth instead of damping, so ... | Differential Equations (18.03 Spring 2010) | 107 |
Find the general (complex-valued) solution of the differential equation $\dot{z}+2 z=e^{2 i t}$, using $C$ to stand for any complex-valued integration constants which may arise. | Using integrating factors, we get $e^{2 t} z=e^{(2+2 i) t} /(2+2 i)+C$, or $z=\boxed{\frac{e^{2 i t}}{(2+2 i)}+C e^{-2 t}}$, where $C$ is any complex number. | Differential Equations (18.03 Spring 2010) | 108 |
Preamble: Consider the first-order system
\[
\tau \dot{y}+y=u
\]
driven with a unit step from zero initial conditions. The input to this system is \(u\) and the output is \(y\).
Derive and expression for the settling time \(t_{s}\), where the settling is to within an error \(\pm \Delta\) from the final value of 1. | Rise and Settling Times. We are given the first-order transfer function
\[
H(s)=\frac{1}{\tau s+1}
\]
The response to a unit step with zero initial conditions will be \(y(t)=1-e^{-t / \tau}\). To determine the amount of time it take \(y\) to settle to within \(\Delta\) of its final value, we want to find the time \(t_... | Dynamics and Control (2.003 Spring 2005) | 109 |
Preamble: Consider the first-order system
\[
\tau \dot{y}+y=u
\]
driven with a unit step from zero initial conditions. The input to this system is \(u\) and the output is \(y\).
Subproblem 0: Derive and expression for the settling time \(t_{s}\), where the settling is to within an error \(\pm \Delta\) from the final ... | The \(10-90 \%\) rise time \(t_{r}\) may be thought of as the difference between the \(90 \%\) settling time \((\Delta=0.1)\) and the \(10 \%\) settling time \((\Delta=0.9)\).
\[
t_{r}=t_{\Delta=0.1}-t_{\Delta=0.9}
\]
Therefore, we find \(t_{r}=\boxed{2.2 \tau}\). | Dynamics and Control (2.003 Spring 2005) | 110 |
Preamble: For each of the functions $y(t)$, find the Laplace Transform $Y(s)$ :
$y(t)=e^{-a t}$ | This function is one of the most widely used in dynamic systems, so we memorize its transform!
\[
Y(s)=\boxed{\frac{1}{s+a}}
\] | Dynamics and Control (2.003 Spring 2005) | 111 |
Preamble: For each Laplace Transform \(Y(s)\), find the function \(y(t)\) :
Subproblem 0: \[
Y(s)=\boxed{\frac{1}{(s+a)(s+b)}}
\]
Solution: We can simplify with partial fractions:
\[
Y(s)=\frac{1}{(s+a)(s+b)}=\frac{C}{s+a}+\frac{D}{s+b}
\]
find the constants \(C\) and \(D\) by setting \(s=-a\) and \(s=-b\)
\[
\begin... | First, note that the transform is
\[
\begin{aligned}
Y(s) &=\frac{s}{\frac{s^{2}}{\omega_{n}^{2}}+\frac{2 \zeta}{\omega_{n}} s+1} \\
&=s \cdot \frac{\omega_{n}^{2}}{s^{2}+2 \zeta \omega_{n} s+\omega_{n}^{2}}
\end{aligned}
\]
We will solve this problem using the property
\[
\frac{d f}{d t}=s F(s)-f(0)
\]
therefore
\[
\b... | Dynamics and Control (2.003 Spring 2005) | 112 |
A signal \(x(t)\) is given by
\[
x(t)=\left(e^{-t}-e^{-1}\right)\left(u_{s}(t)-u_{s}(t-1)\right)
\]
Calculate its Laplace transform \(X(s)\). Make sure to clearly show the steps in your calculation. | Simplify the expression in to a sum of terms,
\[
x(t)=e^{-t} u_{s}(t)-e^{-1} u_{s}(t)-e^{-t} u_{s}(t-1)+e^{-1} u_{s}(t-1)
\]
Now take the Laplace transform of the first, second and fourth terms,
\[
X(s)=\frac{1}{s+1}-\frac{e^{-1}}{s}-\mathcal{L} e^{-t} u_{s}(t-1)+\frac{e^{-1} e^{-s}}{s}
\]
The third term requires some ... | Dynamics and Control (2.003 Spring 2005) | 113 |
Preamble: You are given an equation of motion of the form:
\[
\dot{y}+5 y=10 u
\]
Subproblem 0: What is the time constant for this system?
Solution: We find the homogenous solution, solving:
\[
\dot{y}+5 y=0
\]
by trying a solution of the form $y=A \cdot e^{s, t}$.
Calculation:
\[
\dot{y}=A \cdot s \cdot e^{s \cdot ... | Steady state implies $\dot{y} = 0$, so in the case when $u=10$, we get $y=\boxed{20}$. | Dynamics and Control (2.003 Spring 2005) | 114 |
A signal \(w(t)\) is defined as
\[
w(t)=u_{s}(t)-u_{s}(t-T)
\]
where \(T\) is a fixed time in seconds and \(u_{s}(t)\) is the unit step. Compute the Laplace transform \(W(s)\) of \(w(t)\). Show your work. | The Laplace Transform of \(x(t)\) is defined as
\[
\mathcal{L}[x(t)]=X(s)=\int_{0}^{\infty} x(t) e^{-s t} d t
\]
therefore
\[
\begin{aligned}
W(s) &=\int_{0}^{\infty} e^{-s t} d t-\left(\int_{0}^{T} 0 d t+\int_{T}^{\infty} e^{-s t} d t\right) \\
&=-\left.\frac{1}{s} e^{-s t}\right|_{0} ^{\infty}-\left(0+-\left.\frac{1}... | Dynamics and Control (2.003 Spring 2005) | 115 |
Preamble: Assume that we apply a unit step in force separately to a mass \(m\), a dashpot \(c\), and a spring \(k\). The mass moves in inertial space. The spring and dashpot have one end connected to inertial space (reference velocity \(=0\) ), and the force is applied to the other end. Assume zero initial velocity an... | \[
\begin{aligned}
m \ddot{x}_{m} &=u_{s}(t) \\
\dot{x}_{m}=v_{m} &=\int_{-\infty}^{t} \frac{1}{m} u_{s}(t) d t=\boxed{\frac{1}{m} t} \\
\end{aligned}
\] | Dynamics and Control (2.003 Spring 2005) | 116 |
Preamble: For each of the functions $y(t)$, find the Laplace Transform $Y(s)$ :
Subproblem 0: $y(t)=e^{-a t}$
Solution: This function is one of the most widely used in dynamic systems, so we memorize its transform!
\[
Y(s)=\boxed{\frac{1}{s+a}}
\]
Final answer: The final answer is \frac{1}{s+a}. I hope it is correc... | \[
Y(s)=\boxed{\frac{s+\sigma}{(s+\sigma)^{2}+\omega_{d}^{2}}}
\] | Dynamics and Control (2.003 Spring 2005) | 117 |
Preamble: For each of the functions $y(t)$, find the Laplace Transform $Y(s)$ :
Subproblem 0: $y(t)=e^{-a t}$
Solution: This function is one of the most widely used in dynamic systems, so we memorize its transform!
\[
Y(s)=\boxed{\frac{1}{s+a}}
\]
Final answer: The final answer is \frac{1}{s+a}. I hope it is correc... | \[
Y(s)=\boxed{\frac{\omega_{d}}{(s+\sigma)^{2}+\omega_{d}^{2}}}
\] | Dynamics and Control (2.003 Spring 2005) | 118 |
Preamble: Consider the mass \(m\) sliding horizontally under the influence of the applied force \(f\) and a friction force which can be approximated by a linear friction element with coefficient \(b\).
Formulate the state-determined equation of motion for the velocity \(v\) as output and the force \(f\) as input. | The equation of motion is
\[
\boxed{m \frac{d v}{d t}+b v=f} \quad \text { or } \quad \frac{d v}{d t}=-\frac{b}{m} v+\frac{1}{m} f
\] | Dynamics and Control (2.003 Spring 2005) | 119 |
Preamble: Consider the rotor with moment of inertia \(I\) rotating under the influence of an applied torque \(T\) and the frictional torques from two bearings, each of which can be approximated by a linear frictional element with coefficient \(B\).
Subproblem 0: Formulate the state-determined equation of motion for th... | The steady-state angular velocity, when \(T=10\) Newton-meters, and \(I=0.001 \mathrm{~kg}-\mathrm{m}^{2}\), and \(B=0.005 \mathrm{~N}-\mathrm{m} / \mathrm{r} / \mathrm{s}\) is
\[
\omega_{s s}=\frac{T}{2 B}=\frac{10}{2(0.005)}=\boxed{1000} \mathrm{r} / \mathrm{s}
\] | Dynamics and Control (2.003 Spring 2005) | 120 |
Preamble: Consider the mass \(m\) sliding horizontally under the influence of the applied force \(f\) and a friction force which can be approximated by a linear friction element with coefficient \(b\).
Subproblem 0: Formulate the state-determined equation of motion for the velocity \(v\) as output and the force \(f\)... | The steady-state velocity, when \(f=10\) Newtons, and \(m=1000 \mathrm{~kg}\), and \(b=100 \mathrm{~N} / \mathrm{m} / \mathrm{s}\) is
\[
v_{s s}=\frac{f}{b}=\frac{10}{100}=\boxed{0.10} \mathrm{~m} / \mathrm{s}
\] | Dynamics and Control (2.003 Spring 2005) | 121 |
Obtain the inverse Laplace transform of the following frequency-domain expression: $F(s) = -\frac{(4 s-10)}{s(s+2)(s+5)}$.
Use $u(t)$ to denote the unit step function. | Using partial fraction expansion, the above can be rewritten as
\[
F(s) = \frac{1}{s} - \frac{3}{s+2} + \frac{2}{s+5}
\]
Apply the inverse Laplace transform, then we end up with
\[
f(t) = \boxed{(1 - 3e^{-2t} + 2e^{-5t}) u(t)}
\] | Dynamics and Control (2.003 Spring 2005) | 122 |
A signal has a Laplace transform
\[
X(s)=b+\frac{a}{s(s+a)}
\]
where \(a, b>0\), and with a region of convergence of \(|s|>0\). Find \(x(t), t>0\). | Each term of \(X(s)\) can be evaluated directly using a table of Laplace Transforms:
\[
\mathcal{L}^{-1}\{b\}=b \delta(t)
\]
and
\[
\mathcal{L}^{-1}\left\{\frac{a}{s(s+a)}\right\}=1-e^{-a t}
\]
The final result is then
\[
\mathcal{L}^{-1}\{X(s)\}=\boxed{b \delta(t)+1-e^{-a t}}
\] | Dynamics and Control (2.003 Spring 2005) | 123 |
Preamble: For each Laplace Transform \(Y(s)\), find the function \(y(t)\) :
\[
Y(s)=\boxed{\frac{1}{(s+a)(s+b)}}
\] | We can simplify with partial fractions:
\[
Y(s)=\frac{1}{(s+a)(s+b)}=\frac{C}{s+a}+\frac{D}{s+b}
\]
find the constants \(C\) and \(D\) by setting \(s=-a\) and \(s=-b\)
\[
\begin{aligned}
\frac{1}{(s+a)(s+b)} &=\frac{C}{s+a}+\frac{D}{s+b} \\
1 &=C(s+b)+D(s+a) \\
C &=\frac{1}{b-a} \\
D &=\frac{1}{a-b}
\end{aligned}
\]
th... | Dynamics and Control (2.003 Spring 2005) | 124 |
Preamble: Consider the rotor with moment of inertia \(I\) rotating under the influence of an applied torque \(T\) and the frictional torques from two bearings, each of which can be approximated by a linear frictional element with coefficient \(B\).
Formulate the state-determined equation of motion for the angular velo... | The equation of motion is
\[
\boxed{I \frac{d \omega}{d t}+2 B \omega=T} \quad \text { or } \quad \frac{d \omega}{d t}=-\frac{2 B}{I} \omega+\frac{1}{I} T
\] | Dynamics and Control (2.003 Spring 2005) | 125 |
Obtain the inverse Laplace transform of the following frequency-domain expression: $F(s) = \frac{4}{s^2(s^2+4)}$.
Use $u(t)$ to denote the unit step function. | Since $F(s) = \frac{1}{s^2} + \frac{-1}{s^2+4}$, its inverse Laplace transform is
\[
f(t) = \boxed{(t + \frac{1}{2} \sin{2t}) u(t)}
\] | Dynamics and Control (2.003 Spring 2005) | 126 |
Preamble: This problem considers the simple RLC circuit, in which a voltage source $v_{i}$ is in series with a resistor $R$, inductor $L$, and capacitor $C$. We measure the voltage $v_{o}$ across the capacitor. $v_{i}$ and $v_{o}$ share a ground reference.
Calculate the transfer function \(V_{o}(s) / V_{i}(s)\). | Using the voltage divider relationship:
\[
\begin{aligned}
V_{o}(s) &=\frac{Z_{e q}}{Z_{\text {total }}}V_{i}(s)=\frac{\frac{1}{C s}}{R+L s+\frac{1}{C s}} V_{i}(s) \\
\frac{V_{o}(s)}{V_{i}(s)} &=\boxed{\frac{1}{L C s^{2}+R C s+1}}
\end{aligned}
\] | Dynamics and Control (2.003 Spring 2005) | 127 |
Preamble: You are given an equation of motion of the form:
\[
\dot{y}+5 y=10 u
\]
What is the time constant for this system? | We find the homogenous solution, solving:
\[
\dot{y}+5 y=0
\]
by trying a solution of the form $y=A \cdot e^{s, t}$.
Calculation:
\[
\dot{y}=A \cdot s \cdot e^{s \cdot t} \mid \Rightarrow A \cdot s \cdot e^{s t}+5 A \cdot e^{s t}=0
\]
yields that $s=-5$, meaning the solution is $y=A \cdot e^{-5 \cdot t}=A \cdot e^{-t /... | Dynamics and Control (2.003 Spring 2005) | 128 |
Preamble: This problem considers the simple RLC circuit, in which a voltage source $v_{i}$ is in series with a resistor $R$, inductor $L$, and capacitor $C$. We measure the voltage $v_{o}$ across the capacitor. $v_{i}$ and $v_{o}$ share a ground reference.
Subproblem 0: Calculate the transfer function \(V_{o}(s) / V... | $C=\frac{1}{\omega_{n}^{2}L}=\boxed{1e-8}[\mathrm{~F}]$ | Dynamics and Control (2.003 Spring 2005) | 129 |
Preamble: Here we consider a system described by the differential equation
\[
\ddot{y}+10 \dot{y}+10000 y=0 .
\]
What is the value of the natural frequency \(\omega_{n}\) in radians per second? | $\omega_{n}=\sqrt{\frac{k}{m}}$
So
$\omega_{n} =\boxed{100} \mathrm{rad} / \mathrm{s}$ | Dynamics and Control (2.003 Spring 2005) | 130 |
Preamble: Consider a circuit in which a voltage source of voltage in $v_{i}(t)$ is connected in series with an inductor $L$ and capacitor $C$. We consider the voltage across the capacitor $v_{o}(t)$ to be the output of the system.
Both $v_{i}(t)$ and $v_{o}(t)$ share ground reference.
Write the governing differential... | Using Kirchoff Current Law at the node between the inductor and capacitor with the assumed currents both positive into the node gives the following:
\[
\begin{gathered}
i_{L}+i_{C}=0 \\
i_{L}=\frac{1}{L} \int v_{L} d t \\
i_{C}=C \frac{d v_{c}}{d t}
\end{gathered}
\]
The above equation must be differentiated before sub... | Dynamics and Control (2.003 Spring 2005) | 131 |
Write (but don't solve) the equation of motion for a pendulum consisting of a mass $m$ attached to a rigid massless rod, suspended from the ceiling and free to rotate in a single vertical plane. Let the rod (of length $l$) make an angle of $\theta$ with the vertical. Gravity ($mg$) acts directly downward, the system ... | From force balance, we can derive the equation of motion. Choosing the system variable system variable $\theta(t)$ with polar coordinates, we don't need to care about tension on the rod and centrifugal force.
We can use the relation between torque and angular momentum to immediately write down the equation for $\theta(... | Dynamics and Control (2.003 Spring 2005) | 132 |
Preamble: Here we consider a system described by the differential equation
\[
\ddot{y}+10 \dot{y}+10000 y=0 .
\]
Subproblem 0: What is the value of the natural frequency \(\omega_{n}\) in radians per second?
Solution: $\omega_{n}=\sqrt{\frac{k}{m}}$
So
$\omega_{n} =\boxed{100} \mathrm{rad} / \mathrm{s}$
Final answe... | $\omega_{d}=\omega_{n} \sqrt{1-\zeta^{2}}$
So
$\omega_{d}=\boxed{99.9} \mathrm{rad} / \mathrm{s}$ | Dynamics and Control (2.003 Spring 2005) | 133 |
Preamble: Here we consider a system described by the differential equation
\[
\ddot{y}+10 \dot{y}+10000 y=0 .
\]
Subproblem 0: What is the value of the natural frequency \(\omega_{n}\) in radians per second?
Solution: $\omega_{n}=\sqrt{\frac{k}{m}}$
So
$\omega_{n} =\boxed{100} \mathrm{rad} / \mathrm{s}$
Final answe... | $\zeta=\frac{b}{2 \sqrt{k m}}$
So
$\zeta =\boxed{0.05}$ | Dynamics and Control (2.003 Spring 2005) | 134 |
What is the speed of light in meters/second to 1 significant figure? Use the format $a \times 10^{b}$ where a and b are numbers. | $\boxed{3e8}$ m/s. | Relativity (8.033 Fall 2006) | 135 |
Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested.
Subproblem 0: Age of our universe when most He nuclei were formed in minutes:
Solution: \boxed{1} minute.
Final answer: The final answer is 1. I hope it is correct.
Subproblem 1: Age of our universe when hydrog... | \boxed{1e11}. | Relativity (8.033 Fall 2006) | 136 |
Preamble: In a parallel universe, the Boston baseball team made the playoffs.
Manny Relativirez hits the ball and starts running towards first base at speed $\beta$. How fast is he running, given that he sees third base $45^{\circ}$ to his left (as opposed to straight to his left before he started running)? Assume tha... | Using the aberration formula with $\cos \theta^{\prime}=-1 / \sqrt{2}, \beta=1 / \sqrt{2}$, so $v=\boxed{\frac{1}{\sqrt{2}}c}$. | Relativity (8.033 Fall 2006) | 137 |
Preamble: In the Sun, one of the processes in the He fusion chain is $p+p+e^{-} \rightarrow d+\nu$, where $d$ is a deuteron. Make the approximations that the deuteron rest mass is $2 m_{p}$, and that $m_{e} \approx 0$ and $m_{\nu} \approx 0$, since both the electron and the neutrino have negligible rest mass compared w... | Use the fact that the quantity $E^{2}-p^{2} c^{2}$ is invariant. In the deutron's rest frame, after the collison:
\[
\begin{aligned}
E^{2}-p^{2} c^{2} &=\left(2 m_{p} c^{2}+E_{\nu}\right)^{2}-E_{\nu}^{2} \\
&=4 m_{p}^{2} c^{4}+4 m_{p} c^{2} E_{\nu}=4 m_{p} c^{2}\left(m_{p} c^{2}+E_{\nu}\right)
\end{aligned}
\]
In the l... | Relativity (8.033 Fall 2006) | 138 |
Preamble: In a parallel universe, the Boston baseball team made the playoffs.
Subproblem 0: Manny Relativirez hits the ball and starts running towards first base at speed $\beta$. How fast is he running, given that he sees third base $45^{\circ}$ to his left (as opposed to straight to his left before he started runnin... | Using the doppler shift formula, $\lambda^{\prime}= \boxed{\lambda_{\text {red}} / \sqrt{2}}$. | Relativity (8.033 Fall 2006) | 139 |
Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested.
Subproblem 0: Age of our universe when most He nuclei were formed in minutes:
Solution: \boxed{1} minute.
Final answer: The final answer is 1. I hope it is correct.
Subproblem 1: Age of our universe when hydrog... | \boxed{10} Gyr. | Relativity (8.033 Fall 2006) | 140 |
How many down quarks does a tritium ($H^3$) nucleus contain? | \boxed{5}. | Relativity (8.033 Fall 2006) | 141 |
How many up quarks does a tritium ($H^3$) nucleus contain? | \boxed{4}. | Relativity (8.033 Fall 2006) | 142 |
Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested.
Age of our universe when most He nuclei were formed in minutes: | \boxed{1} minute. | Relativity (8.033 Fall 2006) | 143 |
Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested.
Subproblem 0: Age of our universe when most He nuclei were formed in minutes:
Solution: \boxed{1} minute.
Final answer: The final answer is 1. I hope it is correct.
Subproblem 1: Age of our universe when hydrog... | \boxed{8} minutes. | Relativity (8.033 Fall 2006) | 144 |
Preamble: Give each of the following quantities to the nearest power of 10 and in the units requested.
Subproblem 0: Age of our universe when most He nuclei were formed in minutes:
Solution: \boxed{1} minute.
Final answer: The final answer is 1. I hope it is correct.
Subproblem 1: Age of our universe when hydrog... | \boxed{400000} years. | Relativity (8.033 Fall 2006) | 145 |
Potassium metal can be used as the active surface in a photodiode because electrons are relatively easily removed from a potassium surface. The energy needed is $2.15 \times 10^{5} J$ per mole of electrons removed ( 1 mole $=6.02 \times 10^{23}$ electrons). What is the longest wavelength light (in nm) with quanta of su... | \includegraphics[scale=0.5]{set_02_img_00.jpg}
\nonessentialimage
$I_{p}$, the photocurrent, is proportional to the intensity of incident radiation, i.e. the number of incident photons capable of generating a photoelectron.
This device should be called a phototube rather than a photodiode - a solar cell is a photodiode... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 146 |
Preamble: For red light of wavelength $(\lambda) 6.7102 \times 10^{-5} cm$, emitted by excited lithium atoms, calculate:
Subproblem 0: the frequency $(v)$ in Hz, to 4 decimal places.
Solution: $c=\lambda v$ and $v=c / \lambda$ where $v$ is the frequency of radiation (number of waves/s).
For: $\quad \lambda=6.7102 \... | $\lambda=6.7102 \times 10^{-5} cm \times \frac{1 nm}{10^{-7} cm}= \boxed{671.02} cm$ | Introduction to Solid State Chemistry (3.091 Fall 2010) | 147 |
What is the net charge of arginine in a solution of $\mathrm{pH} \mathrm{} 1.0$ ? Please format your answer as +n or -n. | \boxed{+2}. | Introduction to Solid State Chemistry (3.091 Fall 2010) | 148 |
Preamble: For red light of wavelength $(\lambda) 6.7102 \times 10^{-5} cm$, emitted by excited lithium atoms, calculate:
Subproblem 0: the frequency $(v)$ in Hz, to 4 decimal places.
Solution: $c=\lambda v$ and $v=c / \lambda$ where $v$ is the frequency of radiation (number of waves/s).
For: $\quad \lambda=6.7102 \... | $\bar{v}=\frac{1}{\lambda}=\frac{1}{6.7102 \times 10^{-7} m}=1.4903 \times 10^{6} m^{-1}= \boxed{1.4903e4} {cm}^{-1}$ | Introduction to Solid State Chemistry (3.091 Fall 2010) | 149 |
Determine the atomic weight of ${He}^{++}$ in amu to 5 decimal places from the values of its constituents. | The mass of the constituents $(2 p+2 n)$ is given as:
\[
\begin{array}{ll}
2 p= & 2 \times 1.6726485 \times 10^{-24} g \\
2 n= & 2 \times 16749543 \times 10^{-24} g
\end{array}
\]
The atomic weight (calculated) in amu is given as:
\[
\begin{aligned}
&\frac{6.6952056 \times 10^{-24} g}{1.660565 \times 10^{-24} g} / amu ... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 150 |
Preamble: Determine the following values from a standard radio dial.
Subproblem 0: What is the minimum wavelength in m for broadcasts on the AM band? Format your answer as an integer.
Solution: \[
\mathrm{c}=v \lambda, \therefore \lambda_{\min }=\frac{\mathrm{c}}{v_{\max }} ; \lambda_{\max }=\frac{\mathrm{c}}{v_{\... | \[
\mathrm{c}=v \lambda, \therefore \lambda_{\min }=\frac{\mathrm{c}}{v_{\max }} ; \lambda_{\max }=\frac{\mathrm{c}}{v_{\min }}
\]
\[
\lambda_{\max }=\frac{3 \times 10^{8}}{530 \times 10^{3}}=\boxed{566} m
\] | Introduction to Solid State Chemistry (3.091 Fall 2010) | 151 |
Determine the wavelength of radiation emitted by hydrogen atoms in angstroms upon electron transitions from $n=6$ to $n=2$. | From the Rydberg relationship we obtain:
\[
\begin{aligned}
&\frac{1}{\lambda}=\bar{v}=R\left(\frac{1}{n_{i}^{2}}-\frac{1}{n_{f}^{2}}\right)=1.097 \times 10^{7}\left(\frac{1}{36}-\frac{1}{4}\right)=(-) 2.44 \times 10^{6} \\
&\lambda=\frac{1}{v}=\frac{1}{2.44 \times 10^{6}}=4.1 \times 10^{-7} {~m}=0.41 \mu {m}=\boxed{41... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 152 |
Preamble: Determine the following values from a standard radio dial.
Subproblem 0: What is the minimum wavelength in m for broadcasts on the AM band? Format your answer as an integer.
Solution: \[
\mathrm{c}=v \lambda, \therefore \lambda_{\min }=\frac{\mathrm{c}}{v_{\max }} ; \lambda_{\max }=\frac{\mathrm{c}}{v_{\... | \[
\mathrm{c}=v \lambda, \therefore \lambda_{\min }=\frac{\mathrm{c}}{v_{\max }} ; \lambda_{\max }=\frac{\mathrm{c}}{v_{\min }}
\]
$\lambda_{\min }=\frac{3 \times 10^{8}}{108 \times 10^{6}}=\boxed{2.78} m$ | Introduction to Solid State Chemistry (3.091 Fall 2010) | 153 |
Calculate the "Bohr radius" in angstroms to 3 decimal places for ${He}^{+}$. | In its most general form, the Bohr theory considers the attractive force (Coulombic) between the nucleus and an electron being given by:
\[
F_{c}=\frac{Z e^{2}}{4 \pi \varepsilon_{0} r^{2}}
\]
where Z is the charge of the nucleus ( 1 for H, 2 for He, etc.). Correspondingly, the electron energy $\left(E_{e l}\right)$ is... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 154 |
Preamble: For red light of wavelength $(\lambda) 6.7102 \times 10^{-5} cm$, emitted by excited lithium atoms, calculate:
the frequency $(v)$ in Hz, to 4 decimal places. | $c=\lambda v$ and $v=c / \lambda$ where $v$ is the frequency of radiation (number of waves/s).
For: $\quad \lambda=6.7102 \times 10^{-5} cm=6.7102 \times 10^{-7} m$
\[
v=\frac{2.9979 \times 10^{8} {ms}^{-1}}{6.7102 \times 10^{-7} m}=4.4677 \times 10^{14} {s}^{-1}= \boxed{4.4677} Hz
\] | Introduction to Solid State Chemistry (3.091 Fall 2010) | 155 |
Electromagnetic radiation of frequency $3.091 \times 10^{14} \mathrm{~Hz}$ illuminates a crystal of germanium (Ge). Calculate the wavelength of photoemission in meters generated by this interaction. Germanium is an elemental semiconductor with a band gap, $E_{g}$, of $0.7 \mathrm{eV}$. Please format your answer as $n \... | First compare $E$ of the incident photon with $E_{g}$ :
\[
\begin{aligned}
&\mathrm{E}_{\text {incident }}=\mathrm{hv}=6.6 \times 10^{-34} \times 3.091 \times 10^{14}=2.04 \times 10^{-19} \mathrm{~J} \\
&\mathrm{E}_{\mathrm{g}}=0.7 \mathrm{eV}=1.12 \times 10^{-19} \mathrm{~J}<\mathrm{E}_{\text {incident }}
\end{aligned... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 156 |
What is the energy gap (in eV, to 1 decimal place) between the electronic states $n=3$ and $n=8$ in a hydrogen atom? | \[
\begin{array}{rlr}
\text { Required: } & \Delta {E}_{{el}}=\left(\frac{1}{{n}_{{i}}^{2}}-\frac{1}{{n}_{{f}}^{2}}\right) {K} ; & {K}=2.18 \times 10^{-18} \\
& \text { Or } \bar{v}=\left(\frac{1}{{n}_{{i}}^{2}}-\frac{1}{{n}_{{f}}^{2}}\right) {R} ; & {R}=1.097 \times 10^{7} {~m}^{-1}
\end{array}
\]
(Since only the ener... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 157 |
Determine for hydrogen the velocity in m/s of an electron in an ${n}=4$ state. Please format your answer as $n \times 10^x$ where $n$ is to 2 decimal places. | This problem may be solved in a variety of ways, the simplest of which makes use of the Bohr quantization of the angular momentum:
\[
\begin{aligned}
&m v r=n \times \frac{h}{2 \pi} \quad\left(r=r_{0} n^{2}\right) \\
&m v r_{0} n^{2}=n \times \frac{h}{2 \pi} \\
&v=\frac{h}{2 \pi m r_{0} n}= \boxed{5.47e5} m/s
\end{alig... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 158 |
Preamble: A pure crystalline material (no impurities or dopants are present) appears red in transmitted light.
Subproblem 0: Is this material a conductor, semiconductor or insulator? Give the reasons for your answer.
Solution: If the material is pure (no impurity states present), then it must be classified as a \box... | "White light" contains radiation in wavelength ranging from about $4000 \AA$ (violet) to $7000 \AA$ (deep red). A material appearing red in transmission has the following absorption characteristics:
\includegraphics[scale=0.5]{set_17_img_06.jpg}
\nonessentialimage
Taking $\lambda=6500 \AA$ as the optical absorption edg... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 159 |
Calculate the minimum potential $(V)$ in volts (to 1 decimal place) which must be applied to a free electron so that it has enough energy to excite, upon impact, the electron in a hydrogen atom from its ground state to a state of $n=5$. | We can picture this problem more clearly: an electron is accelerated by a potential, $V x$, and thus acquires the kinetic energy e $x V_{x}\left[=\left(m v^{2}\right) / 2\right.$ which is to be exactly the energy required to excite an electron in hydrogen from $n=1$ to $n=5$.\\
${e} \cdot {V}_{{x}} =-{K}\left(\frac{1}{... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 160 |
Preamble: For light with a wavelength $(\lambda)$ of $408 \mathrm{~nm}$ determine:
Subproblem 0: the frequency in $s^{-1}$. Please format your answer as $n \times 10^x$, where $n$ is to 3 decimal places.
Solution: To solve this problem we must know the following relationships:
\[
\begin{aligned}
v \lambda &=c
\end{... | To solve this problem we must know the following relationships:
\[
\begin{aligned}
m =10^{10} angstrom
\end{aligned}
\]
$\lambda=408 \times 10^{-9} m \times \frac{10^{10} angstrom}{\mathrm{m}}=\boxed{4080} angstrom$ | Introduction to Solid State Chemistry (3.091 Fall 2010) | 161 |
Preamble: Reference the information below to solve the following problems.
$\begin{array}{llll}\text { Element } & \text { Ionization Potential } & \text { Element } & \text { Ionization Potential } \\ {Na} & 5.14 & {Ca} & 6.11 \\ {Mg} & 7.64 & {Sc} & 6.54 \\ {Al} & 5.98 & {Ti} & 6.82 \\ {Si} & 8.15 & {~V} & 6.74 \\ ... | The required data can be obtained by multiplying the ionization potentials (listed in the Periodic Table) with the electronic charge ( ${e}^{-}=1.6 \times 10^{-19}$ C).
\boxed{1.22} J. | Introduction to Solid State Chemistry (3.091 Fall 2010) | 162 |
Light of wavelength $\lambda=4.28 \times 10^{-7} {~m}$ interacts with a "motionless" hydrogen atom. During this interaction it transfers all its energy to the orbiting electron of the hydrogen. What is the velocity in m/s of this electron after interaction? Please format your answer as $n \times 10^x$ where $n$ is to 2... | First of all, a sketch:
\includegraphics[scale=0.5]{set_03_img_00.jpg}
\nonessentialimage
\[
\begin{aligned}
&\text { possibly to } {n}=\infty \text { (ionization), } \\
&\text { depending on the magnitude of } E(h v)
\end{aligned}
\]
let us see: $E(h v)=(h c) / \lambda=4.6 \times 10^{-19} {~J}$
To move the electron fr... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 163 |
Determine the minimum potential in V (to 2 decimal places) that must be applied to an $\alpha$-particle so that on interaction with a hydrogen atom, a ground state electron will be excited to $n$ $=6$. | \[
\Delta {E}_{1 \rightarrow 6}={qV} \quad \therefore {V}=\frac{\Delta {E}_{1 \rightarrow 6}}{{q}}
\]
\[
\begin{aligned}
& \Delta {E}_{1 \rightarrow 6}=-{K}\left(\frac{1}{1^{2}}-\frac{1}{6^{2}}\right)=\frac{35}{36} {K} \\
& {q}=+2 {e} \\
& \therefore \quad V=\frac{35}{36} \times \frac{2.18 \times 10^{18}}{2 \times 1.6 ... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 164 |
Preamble: Reference the information below to solve the following problems.
$\begin{array}{llll}\text { Element } & \text { Ionization Potential } & \text { Element } & \text { Ionization Potential } \\ {Na} & 5.14 & {Ca} & 6.11 \\ {Mg} & 7.64 & {Sc} & 6.54 \\ {Al} & 5.98 & {Ti} & 6.82 \\ {Si} & 8.15 & {~V} & 6.74 \\ ... | The required data can be obtained by multiplying the ionization potentials (listed in the Periodic Table) with the electronic charge ( ${e}^{-}=1.6 \times 10^{-19}$ C).
\boxed{0.822} J. | Introduction to Solid State Chemistry (3.091 Fall 2010) | 165 |
Preamble: For "yellow radiation" (frequency, $v,=5.09 \times 10^{14} s^{-1}$ ) emitted by activated sodium, determine:
Subproblem 0: the wavelength $(\lambda)$ in m. Please format your answer as $n \times 10^x$, where n is to 2 decimal places.
Solution: The equation relating $v$ and $\lambda$ is $c=v \lambda$ where ... | The wave number is $1 /$ wavelength, but since the wavelength is in m, and the wave number should be in ${cm}^{-1}$, we first change the wavelength into cm :
\[
\lambda=5.89 \times 10^{-7} m \times 100 cm / m=5.89 \times 10^{-5} cm
\]
Now we take the reciprocal of the wavelength to obtain the wave number:
\[
\bar{v}=\f... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 166 |
Subproblem 0: In the balanced equation for the reaction between $\mathrm{CO}$ and $\mathrm{O}_{2}$ to form $\mathrm{CO}_{2}$, what is the coefficient of $\mathrm{CO}$?
Solution: \boxed{1}.
Final answer: The final answer is 1. I hope it is correct.
Subproblem 1: In the balanced equation for the reaction between $\ma... | \boxed{0.5}. | Introduction to Solid State Chemistry (3.091 Fall 2010) | 167 |
Preamble: Calculate the molecular weight in g/mole (to 2 decimal places) of each of the substances listed below.
$\mathrm{NH}_{4} \mathrm{OH}$ | $\mathrm{NH}_{4} \mathrm{OH}$ :
$5 \times 1.01=5.05(\mathrm{H})$
$1 \times 14.01=14.01(\mathrm{~N})$
$1 \times 16.00=16.00(\mathrm{O})$
$\mathrm{NH}_{4} \mathrm{OH}= \boxed{35.06}$ g/mole | Introduction to Solid State Chemistry (3.091 Fall 2010) | 168 |
Subproblem 0: In the balanced equation for the reaction between $\mathrm{CO}$ and $\mathrm{O}_{2}$ to form $\mathrm{CO}_{2}$, what is the coefficient of $\mathrm{CO}$?
Solution: \boxed{1}.
Final answer: The final answer is 1. I hope it is correct.
Subproblem 1: In the balanced equation for the reaction between $\ma... | \boxed{1}. | Introduction to Solid State Chemistry (3.091 Fall 2010) | 169 |
Magnesium (Mg) has the following isotopic distribution:
\[
\begin{array}{ll}
24_{\mathrm{Mg}} & 23.985 \mathrm{amu} \text { at } 0.7870 \text { fractional abundance } \\
25_{\mathrm{Mg}} & 24.986 \mathrm{amu} \text { at } 0.1013 \text { fractional abundance } \\
26_{\mathrm{Mg}} & 25.983 \mathrm{amu} \text { at } 0.111... | The atomic weight is the arithmetic average of the atomic weights of the isotopes, taking into account the fractional abundance of each isotope.
\[
\text { At.Wt. }=\frac{23.985 \times 0.7870+24.986 \times 0.1013+25.983 \times 0.1117}{0.7870+0.1013+0.1117}=\boxed{24.310}
\] | Introduction to Solid State Chemistry (3.091 Fall 2010) | 170 |
Preamble: Electrons are accelerated by a potential of 10 Volts.
Determine their velocity in m/s. Please format your answer as $n \times 10^x$, where $n$ is to 2 decimal places. | The definition of an ${eV}$ is the energy gained by an electron when it is accelerated through a potential of $1 {~V}$, so an electron accelerated by a potential of $10 {~V}$ would have an energy of $10 {eV}$.\\
${E}=\frac{1}{2} m {v}^{2} \rightarrow {v}=\sqrt{2 {E} / {m}}$
\[
E=10 {eV}=1.60 \times 10^{-18} {~J}
\]
\[
... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 171 |
Determine the frequency (in $s^{-1}$ of radiation capable of generating, in atomic hydrogen, free electrons which have a velocity of $1.3 \times 10^{6} {~ms}^{-1}$. Please format your answer as $n \times 10^x$ where $n$ is to 2 decimal places. | Remember the ground state electron energy in hydrogen $\left({K}=-2.18 \times 10^{-18} {~J}\right)$. The radiation in question will impart to the removed electron a velocity of $1.3 {x}$ $10^{6} {~ms}^{-1}$, which corresponds to:
\[
\begin{aligned}
&E_{\text {Kin }}=\frac{m v^{2}}{2}=\frac{9.1 \times 10^{-31} \times\le... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 172 |
In the balanced equation for the reaction between $\mathrm{CO}$ and $\mathrm{O}_{2}$ to form $\mathrm{CO}_{2}$, what is the coefficient of $\mathrm{CO}$? | \boxed{1}. | Introduction to Solid State Chemistry (3.091 Fall 2010) | 173 |
Preamble: Electrons are accelerated by a potential of 10 Volts.
Subproblem 0: Determine their velocity in m/s. Please format your answer as $n \times 10^x$, where $n$ is to 2 decimal places.
Solution: The definition of an ${eV}$ is the energy gained by an electron when it is accelerated through a potential of $1 {~... | $\lambda_{p}=h / m v$
\[
\lambda_{p}=\frac{6.63 \times 10^{-34}}{9.11 \times 10^{-34} {~kg} \times 1.87 \times 10^{6} {~m} / {s}}= \boxed{3.89e-10} {~m}
\] | Introduction to Solid State Chemistry (3.091 Fall 2010) | 174 |
Preamble: In all likelihood, the Soviet Union and the United States together in the past exploded about ten hydrogen devices underground per year.
If each explosion converted about $10 \mathrm{~g}$ of matter into an equivalent amount of energy (a conservative estimate), how many $k J$ of energy were released per devic... | $\Delta \mathrm{E}=\Delta \mathrm{mc}^{2}=10 \mathrm{~g} \times \frac{1 \mathrm{~kg}}{1000 \mathrm{~g}} \times\left(3 \times 10^{8} \mathrm{~ms}^{-1}\right)^{2}$ $=9 \times 10^{14} \mathrm{~kg} \mathrm{~m}^{2} \mathrm{~s}^{-2}=9 \times 10^{14} \mathrm{~J}= \boxed{9e11} \mathrm{~kJ} /$ bomb. | Introduction to Solid State Chemistry (3.091 Fall 2010) | 175 |
Preamble: Calculate the molecular weight in g/mole (to 2 decimal places) of each of the substances listed below.
Subproblem 0: $\mathrm{NH}_{4} \mathrm{OH}$
Solution: $\mathrm{NH}_{4} \mathrm{OH}$ :
$5 \times 1.01=5.05(\mathrm{H})$
$1 \times 14.01=14.01(\mathrm{~N})$
$1 \times 16.00=16.00(\mathrm{O})$
$\mathrm{NH}_{... | $\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}: \quad 2 \times 12.01=24.02$ (C)
$6 \times 1.01=6.06(\mathrm{H})$
$1 \times 16.00=16.00(\mathrm{O})$
$\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}: \boxed{46.08}$ g/mole | Introduction to Solid State Chemistry (3.091 Fall 2010) | 176 |
Subproblem 0: In the balanced equation for the reaction between $\mathrm{CO}$ and $\mathrm{O}_{2}$ to form $\mathrm{CO}_{2}$, what is the coefficient of $\mathrm{CO}$?
Solution: \boxed{1}.
Final answer: The final answer is 1. I hope it is correct.
Subproblem 1: In the balanced equation for the reaction between $\ma... | Molecular Weight (M.W.) of (M.W.) of $\mathrm{O}_{2}: 32.0$
(M.W.) of CO: $28.0$
available oxygen: $32.0 \mathrm{~g}=1$ mole, correspondingly the reaction involves 2 moles of CO [see (a)]:
\[
\mathrm{O}_{2}+2 \mathrm{CO} \rightarrow 2 \mathrm{CO}_{2}
\]
mass of CO reacted $=2$ moles $\times 28 \mathrm{~g} /$ mole $=\bo... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 177 |
Preamble: For "yellow radiation" (frequency, $v,=5.09 \times 10^{14} s^{-1}$ ) emitted by activated sodium, determine:
the wavelength $(\lambda)$ in m. Please format your answer as $n \times 10^x$, where n is to 2 decimal places. | The equation relating $v$ and $\lambda$ is $c=v \lambda$ where $c$ is the speed of light $=3.00 \times 10^{8} \mathrm{~m}$.
\[
\lambda=\frac{c}{v}=\frac{3.00 \times 10^{8} m / s}{5.09 \times 10^{14} s^{-1}}=\boxed{5.89e-7} m
\] | Introduction to Solid State Chemistry (3.091 Fall 2010) | 178 |
For a proton which has been subjected to an accelerating potential (V) of 15 Volts, determine its deBroglie wavelength in m. Please format your answer as $n \times 10^x$, where $n$ is to 1 decimal place. | \[
\begin{gathered}
E_{{K}}={eV}=\frac{{m}_{{p}} {v}^{2}}{2} ; \quad {v}_{{p}}=\sqrt{\frac{2 {eV}}{{m}_{{p}}}} \\
\lambda_{{p}}=\frac{{h}}{{m}_{{p}} {v}}=\frac{{h}}{{m}_{{p}} \sqrt{\frac{2 {eV}}{{m}_{{p}}}}}=\frac{{h}}{\sqrt{2 {eVm_{p }}}}=\frac{6.63 \times 10^{-34}}{\left(2 \times 1.6 \times 10^{-19} \times 15 \times ... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 179 |
Preamble: For light with a wavelength $(\lambda)$ of $408 \mathrm{~nm}$ determine:
the frequency in $s^{-1}$. Please format your answer as $n \times 10^x$, where $n$ is to 3 decimal places. | To solve this problem we must know the following relationships:
\[
\begin{aligned}
v \lambda &=c
\end{aligned}
\]
$v$ (frequency) $=\frac{c}{\lambda}=\frac{3 \times 10^{8} m / s}{408 \times 10^{-9} m}= \boxed{7.353e14} s^{-1}$ | Introduction to Solid State Chemistry (3.091 Fall 2010) | 180 |
Determine in units of eV (to 2 decimal places) the energy of a photon ( $h v)$ with the wavelength of $800$ nm. | \[
\begin{aligned}
E_{(\mathrm{eV})}=\frac{\mathrm{hc}}{\lambda} \times \frac{\mathrm{leV}}{1.6 \times 10^{-19} \mathrm{~J}} &=\frac{6.63 \times 10^{-34}[\mathrm{~s}] \times 3 \times 10^{8}\left[\frac{\mathrm{m}}{\mathrm{s}}\right]}{8.00 \times 10^{-7} \mathrm{~m}} \times \frac{\mathrm{leV}}{1.6 \times 10^{-19} \mathrm... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 181 |
Determine for barium (Ba) the linear density of atoms along the $<110>$ directions, in atoms/m. | Determine the lattice parameter and look at the unit cell occupation.
\includegraphics[scale=0.5]{set_23_img_02.jpg}
\nonessentialimage
Ba: $\quad$ BCC; atomic volume $=39.24 \mathrm{~cm}^{3} / \mathrm{mole} ; \mathrm{n}=2 \mathrm{atoms} /$ unit cell\\
$$
3.924 \times 10^{-5}\left(\mathrm{~m}^{3} / \text { mole }\right... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 182 |
A photon with a wavelength $(\lambda)$ of $3.091 \times 10^{-7} {~m}$ strikes an atom of hydrogen. Determine the velocity in m/s of an electron ejected from the excited state, $n=3$. Please format your answer as $n \times 10^x$ where $n$ is to 2 decimal places. | \[
\begin{aligned}
&E_{\text {incident photon }}=E_{\text {binding }}+E_{\text {scattered } e^{-}} \\
&E_{\text {binding }}=-K\left(\frac{1}{3^{2}}\right) \quad \therefore \frac{hc}{\lambda}=\frac{K}{9}+\frac{1}{2} {mv^{2 }} \quad \therefore\left[\left(\frac{{hc}}{\lambda}-\frac{{K}}{9}\right) \frac{2}{{m}}\right]^{\fr... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 183 |
Preamble: For the element copper (Cu) determine:
the distance of second nearest neighbors (in meters). Please format your answer as $n \times 10^x$ where $n$ is to 2 decimal places. | The answer can be found by looking at a unit cell of $\mathrm{Cu}$ (FCC).
\includegraphics[scale=0.5]{set_23_img_00.jpg}
\nonessentialimage
Nearest neighbor distance is observed along $<110>$; second-nearest along $<100>$. The second-nearest neighbor distance is found to be "a".
Cu: atomic volume $=7.1 \times 10^{-6} \... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 184 |
A line of the Lyman series of the spectrum of hydrogen has a wavelength of $9.50 \times 10^{-8} {~m}$. What was the "upper" quantum state $\left({n}_{{i}}\right)$ involved in the associated electron transition? | The Lyman series in hydrogen spectra comprises all electron transitions terminating in the ground state $({n}=1)$. In the present problem it is convenient to convert $\lambda$ into $\bar{v}$ and to use the Rydberg equation. Since we have an "emission spectrum", the sign will be negative in the conventional approach. We... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 185 |
Determine the diffusivity $\mathrm{D}$ of lithium ( $\mathrm{Li}$ ) in silicon (Si) at $1200^{\circ} \mathrm{C}$, knowing that $D_{1100^{\circ} \mathrm{C}}=10^{-5} \mathrm{~cm}^{2} / \mathrm{s}$ and $\mathrm{D}_{695^{\circ} \mathrm{C}}=10^{-6} \mathrm{~cm}^{2} / \mathrm{s}$. Please format your answer as $n \times 10^x$... | \[
\begin{aligned}
&\frac{D_{1}}{D_{2}}=\frac{10^{-6}}{10^{-5}}=10^{-1}=e^{-\frac{E_{A}}{R}\left(\frac{1}{968}-\frac{1}{1373}\right)} \\
&E_{A}=\frac{R \ln 10}{\frac{1}{968}-\frac{1}{1373}}=62.8 \mathrm{~kJ} / \mathrm{mole} \\
&\frac{D_{1100}}{D_{1200}}=e^{-\frac{E_{A}}{R}\left(\frac{1}{1373}-\frac{1}{1473}\right)} \\
... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 186 |
By planar diffusion of antimony (Sb) into p-type germanium (Ge), a p-n junction is obtained at a depth of $3 \times 10^{-3} \mathrm{~cm}$ below the surface. What is the donor concentration in the bulk germanium if diffusion is carried out for three hours at $790^{\circ} \mathrm{C}$? Please format your answer as $n \tim... | \includegraphics[scale=0.5]{set_37_img_00.jpg}
\nonessentialimage
\[
\begin{aligned}
&\frac{c}{c_{s}}=\operatorname{erfc} \frac{x}{2 \sqrt{D t}}=\operatorname{erfc} \frac{3 \times 10^{-3}}{2 \sqrt{D t}}=\operatorname{erfc}(2.083) \\
&\frac{c}{c_{s}}=1-\operatorname{erf}(2.083), \therefore 1-\frac{c}{c_{s}}=0.9964 \\
&\... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 187 |
Preamble: One mole of electromagnetic radiation (light, consisting of energy packages called photons) has an energy of $171 \mathrm{~kJ} /$ mole photons.
Determine the wavelength of this light in nm. | We know: $E_{\text {photon }}=h v=h c / \lambda$ to determine the wavelength associated with a photon we need to know its energy. $E=\frac{171 \mathrm{~kJ}}{\text { mole }}=\frac{1.71 \times 10^{5} \mathrm{~J}}{\text { mole }} \times \frac{1 \text { mole }}{6.02 \times 10^{23} \text { photons }}$
\[
=\frac{2.84 \times ... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 188 |
Preamble: Two lasers generate radiation of (1) $9.5 \mu {m}$ and (2) $0.1 \mu {m}$ respectively.
Determine the photon energy (in eV, to two decimal places) of the laser generating radiation of $9.5 \mu {m}$. | \[
\begin{aligned}
{E} &={h} v=\frac{{hc}}{\lambda} {J} \times \frac{1 {eV}}{1.6 \times 10^{-19} {~J}} \\
{E}_{1} &=\frac{{hc}}{9.5 \times 10^{-6}} \times \frac{1}{1.6 \times 10^{-19}} {eV}= \boxed{0.13} {eV}
\end{aligned}
\] | Introduction to Solid State Chemistry (3.091 Fall 2010) | 189 |
At $100^{\circ} \mathrm{C}$ copper $(\mathrm{Cu})$ has a lattice constant of $3.655 \AA$. What is its density in $g/cm^3$ at this temperature? Please round your answer to 2 decimal places. | $\mathrm{Cu}$ is FCC, so $\mathrm{n}=4$
\[
\begin{aligned}
&\mathrm{a}=3.655 \AA=3.655 \times 10^{-10} \mathrm{~m} \\
&\text { atomic weight }=63.55 \mathrm{~g} / \mathrm{mole} \\
&\frac{\text { atomic weight }}{\rho} \times 10^{-6}=\frac{N_{\mathrm{A}}}{\mathrm{n}} \times \mathrm{a}^{3} \\
&\rho=\frac{(63.55 \mathrm{~... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 190 |
Determine the atomic (metallic) radius of Mo in meters. Do not give the value listed in the periodic table; calculate it from the fact that Mo's atomic weight is $=95.94 \mathrm{~g} /$ mole and $\rho=10.2 \mathrm{~g} / \mathrm{cm}^{3}$. Please format your answer as $n \times 10^x$ where $n$ is to 2 decimal places. | Mo: atomic weight $=95.94 \mathrm{~g} /$ mole
\[
\rho=10.2 \mathrm{~g} / \mathrm{cm}^{3}
\]
BCC, so $n=2$ atoms/unit cell
\[
\begin{aligned}
&\mathrm{a}^{3}=\frac{(95.94 \mathrm{~g} / \mathrm{mole})(2 \text { atoms/unit cell })}{\left(10.2 \mathrm{~g} / \mathrm{cm}^{3}\right)\left(6.023 \times 10^{23} \text { atoms } /... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 191 |
Preamble: Determine the following values from a standard radio dial.
What is the minimum wavelength in m for broadcasts on the AM band? Format your answer as an integer. | \[
\mathrm{c}=v \lambda, \therefore \lambda_{\min }=\frac{\mathrm{c}}{v_{\max }} ; \lambda_{\max }=\frac{\mathrm{c}}{v_{\min }}
\]
$\lambda_{\min }=\frac{3 \times 10^{8} m / s}{1600 \times 10^{3} Hz}=\boxed{188} m$ | Introduction to Solid State Chemistry (3.091 Fall 2010) | 192 |
Consider a (111) plane in an FCC structure. How many different [110]-type directions lie in this (111) plane? | Let's look at the unit cell.
\includegraphics[scale=0.5]{set_23_img_01.jpg}
\nonessentialimage
There are \boxed{6} [110]-type directions in the (111) plane. Their indices are:
\[
(10 \overline{1}),(\overline{1} 01),(\overline{1} 10),(\overline{1} 0),(0 \overline{1} 1),(01 \overline{1})
\] | Introduction to Solid State Chemistry (3.091 Fall 2010) | 193 |
Determine the velocity of an electron (in $\mathrm{m} / \mathrm{s}$ ) that has been subjected to an accelerating potential $V$ of 150 Volt. Please format your answer as $n \times 10^x$, where $n$ is to 2 decimal places.
(The energy imparted to an electron by an accelerating potential of one Volt is $1.6 \times 10^{-19... | We know: $E_{\text {kin }}=m v^{2} / 2=e \times V$ (charge applied potential) $\mathrm{m}_{\mathrm{e}}=9.1 \times 10^{-31} \mathrm{~kg}$
\[
\begin{aligned}
&E_{\text {kin }}=e \times V=m v^{2} / 2 \\
&v=\sqrt{\frac{2 \mathrm{eV}}{\mathrm{m}}}=\sqrt{\frac{2 \times 1.6 \times 10^{-19} \times 150}{9.1 \times 10^{-31}}}=\b... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 194 |
In a diffractometer experiment a specimen of thorium (Th) is irradiated with tungsten (W) $L_{\alpha}$ radiation. Calculate the angle, $\theta$, of the $4^{\text {th }}$ reflection. Round your answer (in degrees) to 2 decimal places. | $\bar{v}=\frac{1}{\lambda}=\frac{5}{36}(74-7.4)^{2} \mathrm{R} \rightarrow \lambda=1.476 \times 10^{-10} \mathrm{~m}$
Th is FCC with a value of $\mathrm{V}_{\text {molar }}=19.9 \mathrm{~cm}^{3}$
$\therefore \frac{4}{\mathrm{a}^{3}}=\frac{\mathrm{N}_{\mathrm{A}}}{\mathrm{V}_{\text {molar }}} \rightarrow \mathrm{a}=\lef... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 195 |
A metal is found to have BCC structure, a lattice constant of $3.31 \AA$, and a density of $16.6 \mathrm{~g} / \mathrm{cm}^{3}$. Determine the atomic weight of this element in g/mole, and round your answer to 1 decimal place. | $B C C$ structure, so $\mathrm{n}=2$
\[
\begin{aligned}
&a=3.31 \AA=3.31 \times 10^{-10} \mathrm{~m} \\
&\rho=16.6 \mathrm{~g} / \mathrm{cm}^{3} \\
&\frac{\text { atomic weight }}{\rho} \times 10^{-6}=\frac{N_{A}}{n} \times a^{3}
\end{aligned}
\]
\[
\begin{aligned}
&\text { atomic weight }=\frac{\left(6.023 \times 10^{... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 196 |
Preamble: Iron $\left(\rho=7.86 \mathrm{~g} / \mathrm{cm}^{3}\right.$ ) crystallizes in a BCC unit cell at room temperature.
Calculate the radius in cm of an iron atom in this crystal. Please format your answer as $n \times 10^x$ where $n$ is to 2 decimal places. | In $\mathrm{BCC}$ there are 2 atoms per unit cell, so $\frac{2}{\mathrm{a}^{3}}=\frac{\mathrm{N}_{\mathrm{A}}}{\mathrm{V}_{\text {molar }}}$, where $\mathrm{V}_{\text {molar }}=\mathrm{A} / \rho ; \mathrm{A}$ is the atomic mass of iron.
\[
\begin{aligned}
&\frac{2}{a^{3}}=\frac{N_{A} \times p}{A} \\
&\therefore a=\left... | Introduction to Solid State Chemistry (3.091 Fall 2010) | 197 |
Preamble: For the element copper (Cu) determine:
Subproblem 0: the distance of second nearest neighbors (in meters). Please format your answer as $n \times 10^x$ where $n$ is to 2 decimal places.
Solution: The answer can be found by looking at a unit cell of $\mathrm{Cu}$ (FCC).
\includegraphics[scale=0.5]{set_23_im... | $d_{h k l}=\frac{a}{\sqrt{h^{2}+k^{2}+1^{2}}}$
\[
d_{110}=\frac{3.61 \times 10^{-10}}{\sqrt{2}}= \boxed{2.55e-10} \mathrm{~m}
\] | Introduction to Solid State Chemistry (3.091 Fall 2010) | 198 |
Subproblem 0: What is the working temperature for silica glass in Celsius?
Solution: \boxed{1950}.
Final answer: The final answer is 1950. I hope it is correct.
Subproblem 1: What is the softening temperature for silica glass in Celsius?
Solution: \boxed{1700}.
Final answer: The final answer is 1700. I hope it i... | \boxed{900}. | Introduction to Solid State Chemistry (3.091 Fall 2010) | 199 |
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