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Problems and Solutions on Thermodynamics and Statistical Mechanics

U.S.T. of China Physics Coaching Class, Yung-Kuo Lim

Chapter 2

Statistical Physics - all with Video Answers

Educators

Chapter Questions

18:36

Problem 1

A classical harmonic oscillator of mass $m$ and spring constant $k$ is known to have a total energy of $E$, but its starting time is completely unknown. Find the probability density function, $p(x)$, where $p(x) d x$ is the probability that the mass would be found in the interval $d x$ at $x$.

Ren Jie Tuieng
Ren Jie Tuieng
Numerade Educator
04:23

Problem 2

Suppose there are two kinds of E. coli (bacteria), "red" ones and "green" ones. Each reproduces faithfully (no sex) by splitting into half, red $\rightarrow$ red + red or green $\rightarrow$ green + green, with a reproduction time of 1 hour. Other than the markers "red" and "green", there are no differences between them. A colony of $5,000{ }^{\text {"red" }}$ and 5,000 "green" E. coli is allowed to eat and reproduce. In order to keep the colony size down, a predator is introduced which keeps the colony size at 10,000 by eating (at random) bacteria.
(a) After a very long time, what is the probability distribution of the number of red bacteria?
(b) About how long must one wait for this answer to be true?
(c) What would be the effect of a $1 \%$ preference of the predator for eating red bacteria on (a) and (b)?

Linda Winkler
Linda Winkler
Numerade Educator

Problem 3

(a) What are the reduced density matrices in position and momentum spaces?
(b) Let us denote the reduced density matrix in momentum space by $\phi\left(\mathbf{p}_1, \mathbf{p}_2\right)$. Show that if $\phi$ is diagonal, that is,
$$
\phi\left(\mathbf{p}_1, \mathbf{p}_2\right)=f\left(\mathbf{p}_1\right) \delta_{\mathbf{p}_1, \mathbf{p}_2},
$$
then the diagonal elements of the position density matrix are constant.

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46:10

Problem 4

(a) Consider a large number of $N$ localized particles in an external magnetic field H. Each particle has spin 1/2. Find the number of states accessible to the system as a function of $M_{\mathrm{n}}$, the $z$-component of the total spin of the system. Determine the value of $M_s$ for which the number of states is maximum.
(b) Define the absolute zero of the thermodynamic temperature. Explain the meaning of negative absolute temperature, and give a concrete example to show how the negative absolute temperature can be reached.

Jack Hou
Jack Hou
Numerade Educator
02:41

Problem 5

There is an one-dimensional lattice with lattice constant $a$ as shown in Fig. 2.1. An atom transits from a site to a nearest-neighbor site every $\tau$ seconds. The probabilities of transiting to the right and left are $p$ and $q=1-p$ respectively.
FIGURE CAN'T COPY.
(a) Calculate the average position $\bar{x}$ of the atom at the time $t=N \tau$, where $N \gg 1$;
(b) Calculate the mean-square value $\overline{(x-\bar{x})^2}$ at the time $t$.

Ajay Singhal
Ajay Singhal
Numerade Educator
08:28

Problem 6

(a) Give the definition of entropy in statistical physics.
(b) Give a general argument to explain why and under what circumstances the entropy of an isolated system $A$ will remain constant, or increase. For convenience you may assume that $A$ can be divided into subsystems $B$ and $C$ which are in weak contact with each other, but which themselves remain in internal thermodynamic equilibrium.

Andrew Eddins
Andrew Eddins
Emory University
08:28

Problem 7

Give Boltzmann's statistical definition of entropy and present its physical meaning briefly but clearly. A two-level system of $N=n_1+n_2$ particles is distributed among two eigenstates 1 and 2 with eigenenergies $E_1$ and $E_2$ respectively. The system is in contact with a heat reservoir at temperature $T$. If a single quantum emission into the reservoir occurs, population changes $n_2 \rightarrow n_2-1$ and $n_1 \rightarrow n_1+1$ take place in the system. For $n_1 \gg 1$ and $n_2 \gg 1$, obtain the expression for the entropy change of
(a) the two level system, and of
(b) the reservoir, and finally
(c) from (a) and (b) derive the Boltzmann relation for the ratio $n_1 / n_2$.

Andrew Eddins
Andrew Eddins
Emory University
03:28

Problem 8

Consider a system composed of a very large number $N$ of distinguishable atoms, non-moving and mutually non-interacting, each of which has only two (non-degenerate) energy levels: $0, \varepsilon>0$. Let $E / N$ be the mean energy per atom in the limit $N \rightarrow \infty$.
(a) What is the maximum possible value of $E / N$ if the system is not necessarily in thermodynamic equilibrium? What is the maximum attainable value of $E / N$ if the system is in equilibrium (at positive temperature, of course)?
(b) For thermodynamic equilibrium, compute the entropy per atom, $S / N$, as a function of $E / N$.

Chai Santi
Chai Santi
Numerade Educator
46:10

Problem 9

Consider a system of $N$ non-interacting particles, each fixed in position and carrying a magnetic moment $\mu$, which is immersed in a magnetic field $H$. Each particle may then exist in one of the two energy states $E=0$ or $E=2 \mu H$. Treat the particles as distinguishable.
(a) The entropy, $S$, of the system can be written in the form $S=$ $k \ln \Omega(E)$, where $k$ is the Boltzmann constant and $E$ is the total system energy. Explain the meaning of $\Omega(E)$.
(b) Write a formula for $S(n)$, where $n$ is the number of particles in the upper state. Crudely sketch $S(n)$.
(c) Derive Stirling's approximation for large $n$ :
$$
\ln n!=n \ln n-n
$$
by approximating $\ln n!$ by an integral.
(d) Rewrite the result of (b) using the result of (c). Find the value of $n$ for which $S(n)$ is maximum.
(e) Treating $E$ as continuous, show that this system can have negative absolute temperature.
(f) Why is negative temperature possible here but not for a gas in a box?

Jack Hou
Jack Hou
Numerade Educator
16:28

Problem 10

A solid contains $N$ magnetic atoms having spin $1 / 2$. At sufficiently high temperatures each spin is completely randomly oriented. At sufficiently low temperatures all the spins become oriented along the same direction (i.e., Ferromagnetic). Let us approximate the heat capacity as a function of temperature $T$ by
$$
C(T)= \begin{cases}c_1\left(\frac{2 T}{T_1}-1\right) & \text { if } T_1 / 2<T<T_1 \\ 0 & \text { otherwise }\end{cases}
$$
where $T_1$ is a constant. Find the maximum value $c_1$ of the specific heat (use entropy considerations).

Dr. Rajveer Singh
Dr. Rajveer Singh
Numerade Educator

Problem 11

The elasticity of a rubber band can be described in terms of a onedimensional model of polymer involving $N$ molecules linked together endto-end. The angle between successive links is equally likely to be $0^{\circ}$ or $180^{\circ}$.
(a) Show that the number of arrangements that give an overall length of $L=2 m d$ is given by
$$
g(N, m)=\frac{2 N!}{\left(\frac{N}{2}+m\right)!\left(\frac{N}{2}-m\right)!},
$$
where $m$ is positive .
Indicate clearly the reasoning you used to get this result.
(b) For $m \ll N$, this expression becomes
$$
g(N, m) \approx g(N, 0) \exp \left(-2 m^2 / N\right) .
$$

Find the entropy of the system as a function of $L$ for $N \gg 1, L \ll N d$.
(c) Find the force required to maintain the length $L$ for $L \ll N d$.
(d) Find the relationship between the force and the length, without using the condition in (c), i.e., for any possible value of $L$, but $N \gg 1$.
FIGURE CAN'T COPY.

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02:06

Problem 12

Consider a one-dimensional chain consisting of $n \gg 1$ segments as illustrated in the figure. Let the length of each segment be $a$ when the long dimension of the segment is parallel to the chain and zero when the segment is vertical (i.e., long dimension normal to the chain direction). Each segment has just two states, a horizontal orientation and a vertical orientation, and each of these states is not degenerate. The distance between the chain ends is $n x$.
(a) Find the entropy of the chain as a function of $x$.
(b) Obtain a relation between the temperature $T$ of the chain and the tension $F$ which is necessary to maintain the distance $n x$, assuming the joints turn freely.
(c) Under which conditions does your answer lead to Hook's law?
FIGURE CAN'T COPY.

Averell Hause
Averell Hause
Carnegie Mellon University

Problem 13

Consider an idealization of a crystal which has $N$ lattice points and the same number of interstitial positions (places between the lattice points where atoms can reside). Let $E$ be the energy necessary to remove an atom from a lattice site to an interstitial position and let $n$ be the number of atoms occupying interstitial sites in equilibrium.
(a) What is the internal energy of the system?
(b) What is the entropy $S$ ? Give an asymptotic formula valid when $n \gg 1$ ?
(c) In equilibrium at temperature $T$, how many such defects are there in the solid, i.e., what is $n$ ? (Assume $n \gg 1$.)

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01:07

Problem 14

(a) Explain Boltzmann statistics, Fermi statistics and Bose statistics, especially about their differences. How are they related to the indistinguishability of identical particles?
(b) Give as physical a discussion as you can, on why the distinction between the above three types of statistics becomes unimportant in the limit of high temperature (how high is high?). Do not merely quote formulas.
(c) In what temperature range will quantum statistics have to be applied to a collection of neutrons spread out in a two-dimensional plane with the number of neutrons per unit area being $\sim 10^{12} / \mathrm{cm}^2$ ?

Chai Santi
Chai Santi
Numerade Educator
02:11

Problem 15

(a) State the basic differences in the fundamental assumptions underlying Maxwell-Boltzman (MB) and Fermi-Dirac (FD) statistics.
(b) Make a rough plot of the energy distribution function at two different temperatures for a system of free particles governed by MB statistics and one governed by FD statistics. Indicate which curve corresponds to the higher temperature.
(c) Explain briefly the discrepancy between experimental values of the specific heat of a metal and the prediction of MB statistics. How did FD statistics overcome the difficulty?

Dominador Tan
Dominador Tan
Numerade Educator
01:16

Problem 16

State which statistics (classical Maxwell-Boltzmann; Fermi-Dirac; or Bose-Einstein) would be appropriate in these problems and explain why (semi-quantitatively):
(a) Density of $\mathrm{He}^4$ gas at room temperature and pressure.
(b) Density of electrons in copper at room temperature.
(c) Density of electrons and holes in semiconducting Ge at room temperature (Ge band-gap $\approx 1$ volt).

Zhuxi Luo
Zhuxi Luo
Numerade Educator
02:32

Problem 17

Show that $\lambda=\exp (\mu / k T)=n V_Q$ for an ideal gas, valid where $\lambda \ll 1$; here $\mu$ is the chemical potential, $n$ is the gas density and
$$
V_Q=\left(h^2 / 2 \pi m k T\right)^{3 / 2}
$$
is the quantum volume. Even if you cannot prove this, this result will be useful in other problems.

Ajay Singhal
Ajay Singhal
Numerade Educator
01:55

Problem 18

A long, thin (i.e., needle-shaped) dust grain floats in a box filled with gas at a constant temperature $T$. On average, is the angular momentum vector nearly parallel to or perpendicular to the long axis of the grain? Explain.

Dheeraj Sharma
Dheeraj Sharma
Numerade Educator
05:14

Problem 19

A cubically shaped vessel 20 cm on a side constains diatomic $\mathrm{H}_2$ gas at a temperature of 300 K . Each $\mathrm{H}_2$ molecule consists of two hydrogen atoms with mass of $1.66 \times 10^{-24} \mathrm{~g}$ each, separated by $\sim 10^{-8} \mathrm{~cm}$. Assume that the gas behaves like an ideal gas. Ignore the vibrational degree of freedom.
(a) What is the average velocity of the molecules?
(b) What is the average velocity of rotation of the molecules around an axis which is the perpendicular bisector of the line joining the two atoms (consider each atom as a point mass)?
(c) Derive the values expected for the molar heat capacities $C_p$ and $C_v$ for such a gas.

Kyle Godbey
Kyle Godbey
Numerade Educator
02:37

Problem 20

The circuit shown is in thermal equilibrium with its surroundings at a temperature $T$. Find the classical expression for the root mean square current through the inductor.
FIGURE CAN'T COPY.

Ajay Singhal
Ajay Singhal
Numerade Educator
09:17

Problem 21

Energy probability. Find and make careful sketch of the probability density, $\rho(E)$, for the energy $E$ of a single atom in a classical non-interacting monatomic gas in thermal equilibrium.

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
01:56

Problem 22

Suppose that the energy of a particle can be represented by the expression $E(z)=a z^2$ where $z$ is a coordinate or momentum and can take on all values from $-\infty$ to $+\infty$.
(a) Show that the average energy per particle for a system of such particles subject to Boltzmann statistics will be $\bar{E}=k T / 2$.
(b) State the principle of equipartition of energy and discuss briefly its relation to the above calculation.

Suzanne W.
Suzanne W.
Numerade Educator
07:57

Problem 23

A system of two energy levels $E_0$ and $E_1$ is populated by $N$ particles at temperature $T$. The particles populate the energy levels according to the classical distribution law.
(a) Derive an expression for the average energy per particle.
(b) Compute the average energy per particle vs the temperature as $T \rightarrow 0$ and $T \rightarrow \infty$.
(c) Derive an expression for the specific heat of the system of $N$ particles.
(d) Compute the specific heat in the limits $T \rightarrow 0$ and $T \rightarrow \infty$.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
16:57

Problem 24

Consider a glass in which some fraction of its constituent atoms may occupy either of two slightly different positions giving rise to two energy levels $\Delta_i>0$ and $-\Delta_i$ for the $i$ th atom.
(a) If each participating atom has the same levels $\Delta$ and $-\Delta$, calculate the contribution of these atoms to the heat capacity. (Ignore the usual Debye specific heat which will also be present in a real solid.)
(b) If the glass has a random composition of such atoms so that all values of $\Delta_i$ are equally likely up to some limiting value $\Delta_0>0$, find the behavior of the low temperature heat capacity, i.e., $k T \ll \Delta_0$. (Definite integrals need not be evaluated provided they do not depend on any of the parameters.)

Dr. Rajveer Singh
Dr. Rajveer Singh
Numerade Educator
10:22

Problem 25

The three lowest energy levels of a certain molecule are $E_1=0, E_2=$ $\varepsilon, E_3=10 \varepsilon$. Show that at sufficiently low temperatures (how low?) only levels $E_1, E_2$ are populated. Find the average energy $E$ of the molecule at temperature $T$. Find the contributions of these levels to the specific heat per mole, $C_v$, and sketch $C_v$ as a function of $T$.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
08:49

Problem 26

Given a system of two distinct lattice sites, each occupied by an atom whose spin $(s=1)$ is so oriented that its energy takes one of three values $\varepsilon=1,0,-1$ with equal probability. The atoms do not interact with each other. Calculate the ensemble average values $\bar{U}$ and $\bar{U}^2$ for the energy $U$ of the system, assumed to be that of the spins only.

Amit Srivastava
Amit Srivastava
Numerade Educator
07:57

Problem 27

Obtain the temperature of each system:
(a) $6.0 \times 10^{22}$ atoms of helium gas occupy 2.0 litres at atmospheric pressure. What is the temperature of the gas?
(b) A system of particles occupying single-particle levels and obeying Maxwell-Boltzmann statistics is in thermal contact with a heat reservoir at temperature $T$. If the population distribution in the non-degenerate energy levels is as shown, what is the temperature of the system?
$$
\begin{array}{cc}
\text { Energy (eV) } & \text { population } \\
30.1 \times 10^{-3} & 3.1 \% \\
21.5 \times 10^{-3} & 8.5 \% \\
12.9 \times 10^{-3} & 23 \% \\
4.3 \times 10^{-3} & 63 \%
\end{array}
$$
(c) In a cryogenic experiment, heat is supplied to a sample at the constant rate of 0.01 watts. The entropy of the sample increases with time as shown in the table. What is the temperature of the sample at $t=500 \mathrm{sec}$ ?
$$
\begin{array}{lllllllll}
\text { Time: } & 100 & 200 & 300 & 400 & 500 & 600 & 700 & (\mathrm{sec}) \\
\text { Entropy: } & 2.30 & 2.65 & 2.85 & 3.00 & 3.11 & 3.20 & 3.28 & (\mathrm{~J} / \mathrm{K})
\end{array}
$$

Eduard Sanchez
Eduard Sanchez
Numerade Educator
10:15

Problem 28

Assume that the reaction $\mathrm{H} \rightleftharpoons \mathrm{p}+\mathrm{e}$ occurs in thermal equilibrium at $T=4000 \mathrm{~K}$ in a very low density gas (no degeneracy) of each species with overall charge neutrality.
(a) Write the chemical potential of each gas in terms of its number density $[\mathrm{H}],[\mathrm{p}]$, or [e]. For simplicity you may ignore the spectrum of excited bound states of H and consider only the ground state. Justify this assumption.
(b) Give the condition for thermal equilibrium and calculate the equilibrium value of $[\mathrm{e}]$ as a function of $[\mathrm{H}]$ and $T$.
(c) Estimate the nucleon density for which the gas is half-ionized at $T$ $=4000 \mathrm{~K}$. (Note that this is an approximate picture of the universe at a redshift $z=10^3$.)

João Gabriel Alencar Caribé
João Gabriel Alencar Caribé
Numerade Educator
02:14

Problem 29

A piece of metal can be considered as a reservoir of electrons; the work function (energy to remove an electron from the metal) is 4 eV . Considering only the is orbital (which can be occupied by zero, one, or two electrons) and knowing that the hydrogen atom has an ionization energy of 13.6 eV and an electron affinity of 0.6 eV , determine for atomic hydrogen in chemical equilibrium at $T=300 \mathrm{~K}$ in the vicinity of a metal the probabilities of finding $\mathrm{H}^{+}, \mathrm{H}^0$ and $\mathrm{H}^{-}$. Give only one significant figure.

What value of the work function would give equal probabilities to $\mathrm{H}^0$ and $\mathrm{H}^{-}$?

Suzanne W.
Suzanne W.
Numerade Educator

Problem 30

The potential energy $V$ between the two atoms ( $\left.m_{\mathrm{H}}=1.672 \times 10^{-24} \mathrm{~g}\right)$ in a hydrogen molecule is given by the empirical expression
$$
V=D\left\{e^{-2 a\left(r-r_0\right)}-2 e^{-a\left(r-r_0\right)}\right\} .
$$
where $r$ is the distance between the atoms.
$$
D=7 \times 10^{-12} \mathrm{erg} \text {, }
$$
$$
\begin{aligned}
& a=2 \times 10^8 \mathrm{~cm}^{-1} \\
& r_0=8 \times 10^{-9} \mathrm{~cm}
\end{aligned}
$$

Estimate the temperatures at which rotation $\left(T_{\mathrm{R}}\right)$ and vibration $\left(T_{\mathrm{V}}\right)$ begin to contribute to the specific heat of hydrogen gas. Give the approximate values of $C_v$ and $C_p$ (the molar specific heats at constant volume and at constant pressure) for the following temperatures:
$$
T_1=25 \mathrm{~K}, T_2=250 \mathrm{~K}, T_3=2500 \mathrm{~K}, T_4=10000 \mathrm{~K} .
$$
Neglect ionization and dissociation.

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Problem 31

Derive an expression for the vibrational specific heat of a diatomic gas as a function of temperature. (Let $\hbar \omega_0 / k=\theta$ ). For full credit start with an expression for the vibrational partition function, evaluate it, and use the result to calculate $C_{\text {vib }}$.
Describe the high and low $T$ limits of $C_{\text {vib. }}$.

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04:22

Problem 32

A one-dimensional quantum harmonic oscillator (whose ground state energy is $\hbar \omega / 2$ ) is in thermal equilibrium with a heat bath at temperature $T$.
(a) What is the mean value of the oscillator's energy, $\langle E\rangle$, as a function of $T$ ?
(b) What is the value of $\Delta E$, the root-mean-square fluctuation in energy about $\langle E\rangle$ ?
(c) How do $\langle E\rangle$ and $\Delta E$ behave in the limits $k T \ll \hbar \omega$ and $k T \gg \hbar \omega$

Stanley Enemuo
Stanley Enemuo
Numerade Educator
22:23

Problem 33

Consider a system of $N_0$ non-interacting quantum mechanical oscillators in equilibrium at temperature $T$. The energy levels of a single oscillator are
$$
E_m=(m+1 / 2) \gamma / V \quad \text { with } m=0,1,2 \ldots \text { etc. }
$$
( $\gamma$ is a constant, the oscillators and volume $V$ are one dimensional.)
(a) Find $U$ and $C_v$ as functions of $T$.
(b) Sketch $U(T)$ and $C_v(T)$.
(c) Determine the equation of state for the system.
(d) What is the fraction of particles in the $m$-th level?

Jack Hou
Jack Hou
Numerade Educator
06:59

Problem 34

The molecules of a certain gas consist of two different atoms, each with zero nuclear spin, bound together. Measurements of the specific heat of this material, over a wide range of temperatures, give the graph shown below.
FIGURE CAN'T COPY.
(The values marked on the vertical scale correspond to the height of the curve in each of the "plateau" regions.)
(a) Account for each of the different results found in the temperature regions: above $T_3$; between $T_2$ and $T_3$; between $T_1$ and $T_2$; below $T_1$,
(b) Given that the first excited state of the rotational spectrum of this molecule is at an energy $k T_e$ above the ground rotational state, and $T_e=64 \mathrm{~K}$, calculate from basic theory the rotational contribution to the specific heat capacity of this gas at 20 K at 100 K , at 300 K .

Keshav Singh
Keshav Singh
Numerade Educator
01:45

Problem 35

The quantum energy levels of a rigid rotator are
$$
\varepsilon_j=j(j+1) h^2 / 8 \pi^2 m a^2,
$$
where $j=0,1,2, \ldots \quad$ The degeneracy of each level is $g_j=2 j+1$.
(a) Find the general expression for the partition function, and show that at high temperatures it can be approximated by an integral.
(b) Evaluate the high-temperature energy and heat capacity.
(c) Find the low-temperature approximations to $z, U$ and $C_v$.

John Nicolle
John Nicolle
Numerade Educator
01:45

Problem 36

The quantum energy levels of a rigid rotator are
$$
\varepsilon_j=j(j+1) h^2 / 8 \pi^2 m a^2,
$$
where $j=0,1,2, \ldots, m$ and $a$ are positive constants. The degeneracy of each level is $g_j=2 j+1$.
(a) Find the general expression for the partition function $z_0$.
(b) Show that at high temperatures it can be approximated by an integral.
(c) Evaluate the high-temperature energy $U$ and heat capacity $C_v$.
(d) Also, find the low-temperature approximations to $z_0, U$ and $C_v$.

John Nicolle
John Nicolle
Numerade Educator
14:33

Problem 37

The energy levels of a three-dimensional rigid rotor of moment of inertial $I$ are given by
$$
E_{J, M}=\hbar^2 J(J+1) / 2 I,
$$
where $J=0,1,2, \ldots ; M=-J,-J+1, \ldots, J$. Consider a system of $N$ rotors:
(a) Using Boltzmann statistics, find an expression for the thermodynamical internal energy of the system.
(b) Under what conditions can the sum in part (a) be approximated by an integral? In this case calculate the specific heat $C_v$ of the system.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
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Problem 38

Consider a heteronuclear diatomic molecule with moment of inertia I. In this problem, only the rotational motion of the molecule should be considered.
(a) Using classical statistical mechanics, calculate the specific heat $C(T)$ of this system at temperature $T$.
(b) In quantum mechanics, this system has energy levels
$$
E_j=\frac{\hbar^2}{2 I} j(j+1), j=0,1,2, \ldots .
$$

Each $j$ level is $(2 j+1)$-fold degenerate. Using quantum statistics, derive expressions for the partition function $z$ and the average energy $\langle E\rangle$ of this system, as a function of temperature. Do not attempt to evaluate these expressions.
(c) By simplifying your expressions in (b), derive an expression for the specific heat $C(T)$ that is valid at very low temperatures. In what range of temperatures is your expression valid?
(d) By simplifying your answer to (b), derive a high temperature approximation to the specific heat $C(T)$. What is the range of validity of your approximation?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
03:01

Problem 39

At the temperature of liquid hydrogen, 20.4 K , one would expect molecular $\mathrm{H}_2$ to be mostly (nearly $100 \%$ ) in a rotational state with zero angular momentum. In fact, if $\mathrm{H}_2$ is cooled to this temperature, it is found that more than half is in a rotational state with angular momentum $\hbar$. A catalyst must be used at 20.4 K to convert it to a state with zero rotational angular momentum. Explain these facts.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
05:14

Problem 40

A gas of molecular hydrogen $\mathrm{H}_2$, is originally in equilibrium at a temperature of $1,000 \mathrm{~K}$. It is cooled to 20 K so quickly that the nuclear spin states of the molecules do not change, although the translational and rotational degrees of freedom do readjust through collisions. What is the approximate internal energy per molecule in terms of temperature units K ?

Note that the rotational part of the energy for a diatomic molecule is $A l(l+1)$ where $l$ is the rotational quantum number and $A \sim 90 \mathrm{~K}$ for $\mathrm{H}_2$. Vibrational motion can be neglected.

Kyle Godbey
Kyle Godbey
Numerade Educator

Problem 41

The graph below shows the equilibrium ratio of the number of orthohydrogen molecules to the number of parahydrogen molecules, as a function of the absolute temperature. The spins of the protons are parallel in orthohydrogen and antiparallel in parahydrogen.
(a) Exhibit a theoretical expression for this ratio as a function of the temperature.
(b) Calculate the value of the ratio for 100 K , corresponding to the point $P$ on the graph. The separation of the protons in the hydrogen molecule is $0.7415 \AA$.
FIGURE CAN'T COPY.

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Problem 42

In hydrogen gas at low temperatures, the molecules can exist in two states: proton spins parallel (orthohydrogen) or anti-parallel (parahydrogen). The transition betwen these two molecular forms is slow. Experiments performed over a time scale of less than a few hours can be considered as if we are dealing with two separate gases, in proportions given by their statistical distributions at the last temperature at which the gas was allowed to come to equilibrium.
(a) Knowing that the separation between protons in a hydrogen molecule is $7.4 \times 10^{-9} \mathrm{~cm}$, estimate the energy difference between the ground state and the first excited rotational state of parahydrogen. Use degrees Kelvin as your unit of energy. Call this energy $k \theta_0$, so that rrors in (a) do not propagate into the other parts of the question.
(b) Express the energy difference between the ground and first excited rotational states of orthohydrogen, $k \theta_1$, in terms of $k \theta_0$. In an experiment to measure specific heats, the gas is allowed to come to equilibrium at elevated temperature, then cooled quickly to the temperature at which specific heat is measured. What will the constant-volume molar specific heat be at:
(c) temperatures well above $\theta_0$ and $\theta_1$, but not high enough to excite vibrational levels?
(d) temperatures much below $\theta_0$ and $\theta_1$ [include the leading temperature-dependent term]?
(e) $T=\theta_0 / 2$ ?

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01:01

Problem 43

Molecular hydrogen is usually found in two forms, orthohydrogen ("parallel" nuclear spins) and parahydrogen ("anti-parallel" nuclear spins).
(a) After coming to equilibrium at "high" temperatures, what fraction of $\mathrm{H}_2$ gas is parahydrogen (assuming that each variety of hydrogen is mostly in its lowest energy state)?
(b) At low temperatures orthohydrogen converts mostly to parahydrogen. Explain why the energy released by each converting molecule is much larger than the energy change due to the nuclear spin flip.

Narayan Hari
Narayan Hari
Numerade Educator

Problem 44

A ${ }^7 \mathrm{~N}_{14}$ nucleus has nuclear spin $I=1$. Assume that the diatomic molecule $\mathrm{N}_2$ can rotate but does not vibrate at ordinary temperatures and ignore electronic motion. Find the relative abundances of the ortho- and para-molecules in a sample of nitrogen gas. (Ortho $=$ symmetric spin state; Para $=$ antisymmetric spin state). What happens to the relative abundance as the temperature is lowered towards absolute zero? (Justify your answers!)

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03:28

Problem 45

(a) Write down a simple expression for the internal part of the partition function for a single isolated hydrogen atom in very weak contact with a reservoir at temperature $T$. Does your expression diverge for $T=0$, for $T \neq 0$ ?
(b) Does all or part of this divergence arise from your choice of the zero of energy?
(c) Show explicitly any effects of this divergence on calculations of the average thermal energy $U$.
(d) Is the divergence affected if the single atom is assumed to be confined to a box of finite volume $L^3$ in order to do a quantum calculation of the full partition function? Explain your answer.

Chai Santi
Chai Santi
Numerade Educator
00:05

Problem 46

The average kinetic energy of the hydrogen atoms in a certain stellar atmosphere (assumed to be in thermal equilibrium) is 1.0 eV .
(a) What is the temperature of the atmosphere in Kelvins?
(b) What is the ratio of the number of atoms in the second excited state $(n=3)$ to the number in the ground state?
(c) Discuss qualitatively the number of ionized atoms. Is it likely to be much greater than or much less than the number in $n=3$ ? Why?

John Johnson
John Johnson
Numerade Educator
03:28

Problem 47

A monatomic gas consists of atoms with two internal energy levels: a ground state of degeneracy $g_1$ and a low-lying excited state of degeneracy $g_2$ at an energy $E$ above the ground state. Find the specific heat of this gas.

Chai Santi
Chai Santi
Numerade Educator
02:58

Problem 48

Consider a system which has two orbital (single particle) states both of the same energy. When both orbitals are unoccupied, the energy of the system is zero; when one orbital or the other is occupied by one particle, the energy is $\varepsilon$. We suppose that the energy of the system is much higher, say infinitely high, when both orbitals are occupied. Show that the ensemble average number of particles in the level is
$$
\langle N\rangle=\frac{2}{2+e^{(\epsilon-\mu) \tau}} .
$$

Suzanne W.
Suzanne W.
Numerade Educator
04:25

Problem 49

(a) State the Maxwell-Boltzmann energy distribution law. Define terms. Discuss briefly an application where the law fails.
(b) Assume the earth's atmosphere is pure nitrogen in thermodynamic equilibrium at a temperature of 300 K . Calculate the height above sea-level at which the density of the atmosphere is one half its sea-level value.

CA
Chi-Chung Ai
Numerade Educator
03:43

Problem 50

A circular cylinder of height $L$, cross-sectional area $A$, is filled with a gas of classical point particles whose mutual interactions can be ignored. The particles, all of mass $m$, are acted on by gravity (let $g$ denote the gravitational acceleration, assumed constant). The system is maintained in thermal equilibrium at temperature $T$. Let $c_v$ be the constant volume specific heat (per particle). Compute $c_v$ as a function of $T$, the other parameters given, and universal parameters. Also, note especially the result for the limiting cases, $T \rightarrow 0, T \rightarrow \infty$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
05:34

Problem 51

Ideal monatomic gas is enclosed in cylinder of radius $a$ and length $L$. The cylinder rotates with angular velocity $\omega$ about its symmetry axis and the ideal gas is in equilibrium at temperature $T$ in the coordinate system rotating with the cylinder. Assume that the gas atoms have mass $m$, have no internal degrees of freedom, and obey classical statistics.
(a) What is the Hamiltonian in the rotating coordinates system?
(b) What is the partition function for the system?
(c) What is the average particle number density as a function of $r$ ?

Eric Mockensturm
Eric Mockensturm
Numerade Educator
08:14

Problem 52

Find the particle density as a function of radial position for a gas of $N$ molecules, each of mass $M$, contained in a centrifuge of radius $R$ and length $L$ rotating with angular velocity $\omega$ about its axis. Neglect the effect of gravity and assume that the centrifuge has been rotating long enough for the gas particles to reach equilibrium.
FIGURE CAN'T COPY.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:13

Problem 53

Suppose that a quantity of neutral hydrogen gas is heated to a temperature $T . T$ is sufficiently high that the hydrogen is completely ionized, but low enough that $k T / m_e c^2 \ll 1$ ( $m_e$ is the mass of the electron). In this gas, there will be a small density of positrons due to processes such as $\mathrm{e}^{-}+\mathrm{e}^{-} \leftrightarrow \mathrm{e}^{-}+\mathrm{e}^{-}+\mathrm{e}^{-}+\mathrm{e}^{+}$or $\mathrm{e}^{-}+\mathrm{p} \leftrightarrow \mathrm{e}^{-}+\mathrm{p}+\mathrm{e}^{-}+\mathrm{e}^{+}$in which electron-positron pairs are created and destroyed.

For this problem, you need not understand these reactions in detail. Just assume that they are reactions that change the number of electrons and positrons, but in such a way that charge is always conserved.
Suppose that the number density of protons is $10^{10} / \mathrm{cm}^3$. Find the chemical potentials for the electrons and positrons. Find the temperature at which the positron density is $1 / \mathrm{cm}^3$. Find the temperature at which it is $10^{10} / \mathrm{cm}^3$.

Farhanul Hasan
Farhanul Hasan
Numerade Educator
06:50

Problem 54

Consider a rigid lattice of distinguishable spin $1 / 2$ atoms in a magnetic field. The spins have two states, with energies $-\mu_0 H$ and $+\mu_0 H$ for spins up $(\uparrow)$ and down ( $\downarrow$ ), respectively, relative to $\mathbf{H}$. The system is at temperature $T$.
(a) Determine the canonical partition function $z$ for this system.
(b) Determine the total magnetic moment $M=\mu_0\left(N_{+}-N_{-}\right)$of the system.
(c) Determine the entropy of the system.

Stanley Enemuo
Stanley Enemuo
Numerade Educator
05:07

Problem 55

A paramagnetic system consists of $N$ magnetic dipoles. Each dipole carries a magnetic moment $\mu$ which can be treated classically. If the system at a finite temperature $T$ is in a uniform magnetic field $H$, find
(a) the induced magnetization in the system, and
(b) the heat capacity at constant $H$.

Carlos Henrique De Lima
Carlos Henrique De Lima
Numerade Educator
05:07

Problem 56

Consider a gas of spin $1 / 2$ atoms with density $n$ atoms per unit volume. Each atom has intrinsic magnetic moment $\boldsymbol{\mu}$ and the interaction between atoms is negligible.
Assume that the system obeys classical statistics.
(a) What is the probability of finding an atom with $\boldsymbol{\mu}$ parallel to the applied magnetic field $\mathbf{H}$ at absolute temperature $T$ ? With $\boldsymbol{\mu}$ anti-parallel to $\mathbf{H}$ ?
(b) Find the mean magnetization of the gas in both the high and low temperature limits?
(c) Determine the magnetic susceptibility $\chi$ in terms of $\mu$.

Carlos Henrique De Lima
Carlos Henrique De Lima
Numerade Educator
46:10

Problem 57

A material consists of $n$ independent particles and is in a weak external magnetic field $H$. Each particle can have a magnetic moment $m \mu$ along the magnetic field, where $m=J, J-1, \ldots,-J+1,-J, J$ being an integer, and $\mu$ is a constant. The system is at temperature $T$.
(a) Find the partition function for this system.
(b) Calculate the average magnetization, $\bar{M}$, of the material.
(c) For large values of $T$ find an asymptotic expression for $\bar{M}$.

Jack Hou
Jack Hou
Numerade Educator
01:10

Problem 58

Two dipoles, with dipole moments $\mathbf{M}_1$ and $\mathbf{M}_2$, are held apart at a separation $R$, only the orientations of the moments being free. They are in thermal equilibrium with the environment at temperature $T$. Compute the mean force $\mathbf{F}$ between the dipoles for the high temperature limit $\frac{M_1 M_2}{k T R^3} \ll$ 1. The system is to be treated classically.

Remark: The potential energy between two dipoles is:
$$
\phi=\frac{\left(3\left(\mathbf{M}_1 \cdot \mathbf{R}\right)\left(\mathbf{M}_2 \cdot \mathbf{R}\right)-\left(\mathbf{M}_1 \cdot \mathbf{M}_2\right) R^2\right)}{R^5} .
$$

Narayan Hari
Narayan Hari
Numerade Educator
01:45

Problem 59

The molecule of a perfect gas consists of two atoms, of mass $m$, rigidly separated by a distance $d$. The atoms of each molecule carry charges $q$ and $-q$ respectively, and the gas is placed in an electric field $\varepsilon$. Find the mean polarization, and the specific heat per molecule, if quantum effects can be neglected.
State the condition for this last assumption to be true.

Salamat Ali
Salamat Ali
Numerade Educator
05:34

Problem 60

The response of polar substances (e.g., $\mathrm{HCl}, \mathrm{H}_2 \mathrm{O}$, etc) to applied electric fields can be described in terms of a classical model which attributes to each molecule a permanent electric dipole moment of magnitude $p$.
(a) Write down a general expression for the average macroscopic polarization $\bar{p}$ (dipole moment per unit volume) for a dilute system of $n$ molecules per unit volume at temperature $T$ in a uniform electric field $E$.
(b) Calculate explicitly an approximate result for the average macroscopic polarization $\bar{p}$ at high temperatures $(K T \gg p E)$.

Derrick Danso
Derrick Danso
Numerade Educator
08:49

Problem 61

The entropy of an ideal paramagnet in a magnetic field is given approximately by
$$
S=S_0-C U^2,
$$
where $U$ is the energy of the spin system and $C$ is a constant with fixed mechanical parameters of the system.
(a) Using the fundamental definition of the temperature, determine the energy $U$ of the spin system as a function of $T$.
(b) Sketch a graph of $U$ versus $T$ for all values of $T(-\infty<T<\infty)$.
(c) Briefly tell what physical sense you can make of the negative temperature part of your result.

Amit Srivastava
Amit Srivastava
Numerade Educator
01:44

Problem 62

Consider a system of $N$ non-interacting particles ( $N \gg 1$ ) in which the energy of each particle can assume two and only two distinct values, 0 and $E(E>0)$. Denote by $n_0$ and $n_1$ the occupation numbers of the energy levels 0 and $E$, respectively. The fixed total energy of the system is $U$.
(a) Find the entropy of the system.
(b) Find the temperature as a function of $U$. For what range of values of $n_0$ is $T<0$ ?
(c) In which direction does heat flow when a system of negative temperature is brought into thermal contact with a system of positive temperature? Why?

Dominador Tan
Dominador Tan
Numerade Educator

Problem 63

A system of $N$ identical spinless bosons of mass $m$ is in a box of volume $V=L^3$ at temperature $T>0$.
(a) Write a general expression for the number of particles, $n(E)$, having an energy between $\varepsilon$ and $\varepsilon+d \varepsilon$ in terms of their mass, the energy, the temperature, the chemical potential, the volume, and any other relevant quantities.
(b) Show that in the limit that the average distance, $d$, between the particles is very large compared to their de Broglie wavelength (i.e., $d \gg$ $\lambda)$ the distribution becomes equal to that calculated using the classical (Boltzmann) distribution function.
(c) Calculate the 1st order difference in average energy between a system of $N$ non-identical spinless particles and a system of $N$ identical spinless bosons when $d \gg \lambda$. For both systems the cubical box has volume $\dot{V}=L^3$ and the particles have mass $m$.

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05:33

Problem 64

Consider a quantum-mechanical gas of non-interacting spin zero bosons, each of mass $m$ which are free to move within volume $V$.
(a) Find the energy and heat capacity in the very low temperature region. Discuss why it is appropriate at low temperatures to put the chemical potential equal to zero.
(b) Show how the calculation is modified for a photon (mass $=0$ ) gas.

Prove that the energy is proportional to $T^4$.
Note: Put all integrals in dimensionless form, but do not evaluate.

Mahnoor Amin
Mahnoor Amin
Numerade Educator
05:33

Problem 65

A gas of $N$ spinless Bose particles of mass $m$ is enclosed in a volume $V$ at a temperature $T$.
(a) Find an expression for the density of single-particle states $D(\varepsilon)$ as a function of the single-particle energy $\varepsilon$. Sketch the result.
(b) Write down an expression for the mean occupation number of a single particle state, $\bar{n}_{\varepsilon}$ as a function of $\varepsilon, T$, and the chemical potential $\mu(T)$. Draw this function on your sketch in part (a) for a moderately high temperature, that is, a temperature above the Bose-Einstein transition. Indicate the place on the $\varepsilon$-axis where $\varepsilon=\mu$.
(c) Write down an integral expression which implicitly determines $\mu(T)$. Referring to your sketch in (a), determine in which direction $\mu(T)$ moves as $T$ is lowered.
(d) Find an expression for the Bose-Einstein transition temperature, $T_c$, below which one must have a macroscopic occupation of some singleparticle states. Leave your answer in terms of a dimensionless integral.
(e) What is $\mu(T)$ for $T<T_c$ ?

Describe $\bar{n}(\varepsilon, T)$ for $T<T_c$ ?
(f) Find an exact expression for the total energy, $U(T, V)$ of the gas for $T<T_c$. Leave your answer in terms of a dimensionless integral.

Mahnoor Amin
Mahnoor Amin
Numerade Educator

Problem 66

(a) In quantum statistical mechanics, define the one-particle density matrix in the r-representation where $\mathbf{r}$ is the position of the particle.
(b) For a system of $N$ identical free bosons, let
$$
\rho_1(\mathbf{r})=\frac{1}{V} \sum_{\mathbf{k}}\left\langle N_k\right\rangle e^{i \mathbf{k} \cdot \mathbf{r}},
$$
where $\left\langle N_k\right\rangle$ is the thermal averaged number of particles in the momentum state $\mathbf{k}$. Discuss the limiting behavior of $\rho_1(\mathbf{r})$ as $r \rightarrow \infty$, when the temperature $T$ passes from $T>T_c$ to $T<T_c$, where $T_c$ is the Bose-Einstein condensation temperature. In the case $\lim _{r \rightarrow \infty} \rho_1(r)$ approaches zero, can you describe how it approaches zero as $r$ becomes larger and larger?

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08:57

Problem 67

Consider a gas of non-interacting, non-relativistic, identical bosons. Explain whether and why the Bose-Einstein condensation effect that applies to a three-dimensional gas applies also to a two-dimensional gas and to a one-dimensional gas.

Mahnoor Amin
Mahnoor Amin
Numerade Educator
02:35

Problem 68

Consider a photon gas enclosed in a volume $V$ and in equilibrium at temperature $T$. The photon is a massless particle, so that $\varepsilon=p c$.
(a) What is the chemical potential of the gas? Explain.
(b) Determine how the number of photons in the volume depends upon the temperature.
(c) One may write the energy density in the form
$$
\frac{\bar{E}}{\bar{V}}=\int_0^{\infty} \rho(\omega) d \omega
$$
Determine the form of $\rho(\omega)$, the spectral density of the energy.
(d) What is the temperature dependence of the energy $\bar{E}$ ?

Manish Jain
Manish Jain
Numerade Educator
01:31

Problem 69

(a) Show that for a photon gas $p=U / 3 V$.
(b) Using thermodynamic arguments (First and Second Laws), and the above relationship between pressure and energy density, obtain the dependence of the energy density on the temperature in a photon gas.

Manish Jain
Manish Jain
Numerade Educator
View

Problem 70

Consider a cubical box of side $L$ with no matter in its interior. The walls are fixed at absolute temperature $T$, and they are in thermal equilibrium with the electromagnetic radiation field in the interior.
(a) Find the mean electromagnetic energy per unit volume in the frequency range from $\omega$ to $\omega+d \omega$ as a function of $\omega$ and $T$. (If you wish to start with a known distribution function - e.g., Maxwell-Boltzmann, Planck, etc. - you need not derive that function.)
(b) Find the temperature dependence of the total electromagnetic energy per unit volume. (Hint: you do not have to actually carry out the integration of the result of part (a) to answer this question.)

Victor Salazar
Victor Salazar
Numerade Educator
03:21

Problem 71

A historic failure of classical physics is its description of the electromagnetic radiation from a black body. Consider a simple model for an ideal black body consisting of a cubic cavity of side $L$ with a small hole in one side.
(a) Assuming the classical equipartition of energy, derive an expression for the average energy per unit volume and unit frequency range (RayleighJeans' Law). In what way does this result deviate from actual observation?

Ahmad Reda
Ahmad Reda
Numerade Educator

Problem 72

Electromagnetic radiation following the Planck distribution fills a cavity of volume $V$. Initially $\omega_i$ is the frequency of the maximum of the curve of $u_i(\omega)$, the energy density per unit angular frequency versus $\omega$. If the volume is expanded quasistatically to 2 V , what is the final peak frequency $\omega_f$ of the $u_f(\omega)$ distribution curve? The expansion is adiabatic.

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01:06

Problem 73

A He-Ne laser generates a quasi-monochromatic beam at $6328 \AA$. The beam has an output power of $1 \mathrm{mw}\left(10^{-3}\right.$ watts), a divergence angle of $10^{-4}$ radians, and a spectral linewidth of $0.01 \AA$. If a black body with an area of $1 \mathrm{~cm}^2$ were used to generate such a beam after proper filtering, what should its temperature be approximately?

Narayan Hari
Narayan Hari
Numerade Educator
04:05

Problem 74

(a) Show that the number of photons in equilibrium at temperature $T$ in a cavity of volume $V$ is $N=V(k T / \hbar c)^3$ times a numerical constant.
(b) Use this result to obtain a qualitative expression for the heat capacity of a photon gas at constant volume.

Narayan Hari
Narayan Hari
Numerade Educator
02:34

Problem 75

As you know, the universe is pervaded by 3 K black body radiation. In a simple view, this radiation arose from the adiabatic expansion of a much hotter photon cloud which was produced during the big bang.
(a) Why is the recent expansion adiabatic rather than, for example, isothermal?
(b) If in the next $10^{10}$ years the volume of the universe increases by a factor of two, what then will be the temperature of the black body radiation? (Show your work.)
(c) Write down an integral which determines how much energy per cubic meter is contained in this cloud of radiation. Estimate the result within an order of magnitude in joules per (meter) ${ }^3$.

Manish Jain
Manish Jain
Numerade Educator
08:00

Problem 76

Our universe is filled with black body radiation (photons) at a temperature $T=3 \mathrm{~K}$. This is thought to be a relic, of early developments following the "big bang".
(a) Express the photon number density $n$ analytically in terms of $T$ and universal constants. Your answer should explicitly show the dependence on $T$ and on the universal constants. However, a certain numerical cofactor may be left in the form of a dimensionless integral which need not be evaluated at this stage.
(b) Now estimate the integral roughly, use your knowledge of the universal constants, and determine $n$ roughly, to within about two orders of magnitude, for $T=3 \mathrm{~K}$.

EC
Emma C
Numerade Educator
02:38

Problem 77

We are surrounded by black body photon radiation at 3 K . Consider the question of whether a similar bath of thermal neutrinos might exist.
(a) What kinds of laboratory experiments put the best limits on how hot a neutrino gas might be? How good are these limits?
(b) The photon gas makes up $10^{-6}$ of the energy density needed to close the universe. Assuming the universe is no more than just closed, what order of magnitude limit does this consideration place on the neutrino's temperature?
(c) In a standard big-bang picture, what do you expect the neutrino temperature to be (roughly)?

Manish Jain
Manish Jain
Numerade Educator
02:34

Problem 78

Imagine the universe to be a spherical cavity, with a radius of $10^{28} \mathrm{~cm}$ and impenetrable walls.
(a) If the temperature inside the cavity is 3 K , estimate the total number of photons in the universe, and the energy content in these photons.
(b) If the temperature were 0 K , and the universe contained $10^{80}$ electrons in a Fermi distribution, calculate the Fermi momentum of the electrons.

Manish Jain
Manish Jain
Numerade Educator
02:34

Problem 79

An $n$-dimensional universe.
In our three-dimensional universe, the following are well-known results from statistical mechanics and thermodynamics:
(a) The energy density of black body radiation depends on the temperature as $T^\alpha$, where $\alpha=4$.
(b) In the Debye model of a solid, the specific heat at low temperatures depends on the temperature as $T^\beta$, where $\beta=3$.
(c) The ratio of the specific heat at constant pressure to the specific heat at constant volume for a monatomic ideal gas is $\gamma=5 / 3$.

Derive the analogous results (i.e., what are $\gamma, \alpha$ and $\beta$ ) in the universe with $n$ dimensions.

Manish Jain
Manish Jain
Numerade Educator
03:10

Problem 80

(a) Suppose one carries out a measurement of the specific heat at constant volume, $C_v$, for some solid as a function of temperature, $T$, and obtains the results:
$$
\begin{array}{cc}
T & C_{\mathrm{v}} \text { (arbitrary units) } \\
1000 \mathrm{~K} & 20 \\
500 \mathrm{~K} & 20 \\
40 \mathrm{~K} & 8 \\
20 \mathrm{~K} & 1
\end{array}
$$
Is the solid a conductor or an insulator? Explain.
(b) If the displacement of an atom about its equilibrium position in a harmonic solid is denoted by $U$, then the average displacement squared is given by
$$
\left\langle U^2\right\rangle=\frac{\hbar^2}{2 M} \int_0^{\infty} \frac{d \varepsilon}{\varepsilon} g(\varepsilon)[1+2 n(\varepsilon)],
$$
where $M$ is the mass of the atom, $g(\varepsilon)$ is a suitably normalized density of energy states and $n(\varepsilon)$ is the Bose-Einstein occupation factor for phonons of energy $\varepsilon$. Assuming a Debye model for the density of states:
$$
\begin{array}{ll}
g(\varepsilon)=9 \varepsilon^2 /\left(\hbar \omega_D\right)^3 & \text { for } \varepsilon<\hbar \omega_D, \\
g(\varepsilon)=0 & \text { for } \varepsilon>\hbar \omega_D,
\end{array}
$$
where $\omega_D$ is the Debye frequency, determine the temperature dependence of $\left\langle U^2\right\rangle$ for very high and very low temperatures. Do your results make sense?

Anand Jangid
Anand Jangid
Numerade Educator
05:03

Problem 81

Graphite has a layered crystal structure in which the coupling between the carbon atoms in different layers is much weaker than that between the atoms in the same layer. Experimentally it is found that the specific heat is proportional to $T$ at low temperatures. How can the Debye theory be adapted to provide an explanation?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator

Problem 82

One Dimensional Debye Solid.
Consider a one dimensional lattice of $N$ identical point particles of mass $m$, interacting via nearest-neighbor spring-like forces with spring constant $m \omega^2$. Denote the lattice spacing by $a$. As is easily shown, the normal mode eigenfrequencies are given by
$$
\omega_k=\omega \sqrt{2(1-\cos k a)}
$$
with $k=2 \pi n / a N$, where the integer $n$ ranges from $-N / 2$ to $+N / 2(N \gg$ 1). Derive an expression for the quantum mechanical specific heat of this system in the Debye approximation. In particular, evaluate the leading non-zero terms as functions of temperature $T$ for the two limits $T \rightarrow \infty$,

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Problem 83

A one dimensional lattice consists of a linear array of $N$ particles ( $N \gg$ 1) interacting via spring-like nearest neighbor forces. The normal mode frequencies (radians $/ \mathrm{sec}$ ) are given by
$$
\omega_n=\bar{\omega} \sqrt{2(1-\cos (2 \pi n / N))},
$$
where $\bar{\omega}$ is a constant and $n$ an integer ranging from $-N / 2$ to $+N / 2$. The system is in thermal equilibrium at temperature $T$. Let $c_v$ be the constant "volume" (length) specific heat.
(a) Compute $c_v$ for the regime $T \rightarrow \infty$.
(b) For $T \rightarrow 0$
$$
c_v \rightarrow A \omega^{-\alpha} T^\gamma,
$$
where $A$ is a constant that you need not compute. Compute the exponents $\alpha$ and $\gamma$.
The problem is to be treated quantum mechanically.

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04:20

Problem 84

Given the energy spectrum
$$
\varepsilon_p=\left[(p c)^2+m_0^2 c^4\right]^{1 / 2} \rightarrow p c \text { as } p \rightarrow \infty .
$$
(a) Prove that an ultrarelativistic ideal fermion gas satisfies the equation of state $p V=E / 3$, where $E$ is the total energy.
(b) Prove that the entropy of an ideal quantum gas is given by
$$
S=-k \sum_i\left[n_i \ln \left(n_i\right) \mp\left(1 \pm n_i\right) \ln \left(1 \pm n_i\right)\right]
$$
where the upper (lower) signs refer to bosons (fermions).

Eduard Sanchez
Eduard Sanchez
Numerade Educator
07:40

Problem 85

Consider an ideal quantum gas of Fermi particles at a temperature $T$.
(a) Write the probability $p(n)$ that there are $n$ particles in a given single particle state as a function of the mean occupation number, $\langle n\rangle$.
(b) Find the root-mean-square fluctuation $\left\langle(n-\langle n\rangle)^2\right\rangle^{1 / 2}$ in the occupation number of a single particle state as a function of the mean occupation number $\langle n\rangle$. Sketch the result.

Sheh Lit Chang
Sheh Lit Chang
University of Washington

Problem 86

In a perfect gas of electrons, the mean number of particles occupying a single-particle quantum state of energy $E_i$ is:
$$
N_i=\frac{1}{\exp \left[\left(E_i-\mu\right) / k T\right]+1} .
$$
(a) Obtain a formula which could be used to determine $\mu$ in terms of the particle density $n$ and various constants.
(b) Show that the expression above reduces to the Maxwell-Boltzmann distribution in the limit $n \lambda^3 \ll 1$, where $\lambda$ is the thermal de Broglie wavelength.
(c) Sketch $N_i$ versus $E_i$ for $T=0 \mathrm{~K}$ and for $T=\mu / 5 \mathrm{~K}$. Label significant points along both axes.

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03:10

Problem 87

Suppose that in some sample the density of states of the electrons $D(\varepsilon)$ is a constant $D_0$ for energy $\varepsilon>0(D(\varepsilon)=0$ for $\varepsilon<0)$ and that the total number of electrons is equal to $N$.
(a) Calculate the Fermi potential $\mu_0$ at 0 K .
(b) For non-zero temperatures, derive the condition that the system is non-degenerate.
(c) Show that the electronic specific heat is proportional to the temperature, $T$, when the system is highly degenerate.

Anand Jangid
Anand Jangid
Numerade Educator

Problem 88

Consider a system of $N$ "non-interacting" electrons $/ \mathrm{cm}^3$, each of which can occupy either a bound state with energy $\varepsilon=-E_d$ or a free-particle continuum with $\varepsilon=\frac{p^2}{2 m}$. (This could be a semiconductor like Si with $N$ shallow donors $/ \mathrm{cm}^3$.)
(a) Compute the density of states as a function of $\varepsilon$ in the continuum.
(b) Find an expression for the chemical potential in the low temperature limit.
(c) Compute the number of free electrons (i.e., electrons in the continuum) as a function of $T$ in the low temperature limit.

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08:25

Problem 89

(a) For a system of electrons, assumed non-interacting, show that the probability of finding an electron in a state with energy $\Delta$ above the chemical potential $\mu$ is the same as the probability of finding an electron absent from a state with energy $\Delta$ below $\mu$ at any given temperature $T$.
(b) Suppose that the density of states $D(\varepsilon)$ is given by
$$
D(\varepsilon)=\left\{\begin{array}{lr}
a\left(\varepsilon-\varepsilon_g\right)^{1 / 2}, & \varepsilon>\varepsilon_g, \\
0, & 0<\varepsilon<\varepsilon_g, \\
b(-\varepsilon)^{1 / 2}, & \varepsilon<0,
\end{array}\right.
$$
as shown in Fig. 2.19,
FIGURE CAN'T COPY.
and that at $T=0$ all states with $\varepsilon<0$ are occupied while the other states are empty. Now for $T>0$, some states with $\varepsilon>0$ will be occupied while some states with $\varepsilon<0$ will be empty. If $a=b$, where is the position of $\mu$ ? For $a \neq b$, write down the mathematical equation for the determination of $\mu$ and discuss qualitatively where $\mu$ will be if $a>b$ ? $a<b$ ?
(c) If there is an excess of $n_d$ electrons per unit volume than can be accommodated by the states with $\varepsilon<0$, what is the equation for $\mu$ for $T=0$ ? How will $\mu$ shift as $T$ increases?

Ben Nicholson
Ben Nicholson
Numerade Educator
03:02

Problem 90

(a) Calculate the magnitude of the Fermi wavevector for $4.2 \times 10^{21}$ electrons confined in a box of volume $1 \mathrm{~cm}^3$.
(b) Compute the Fermi energy (in eV ) for this system.
(c) If the electrons are replaced by neutrons, compute the magnitude of the Fermi wavevector and the Fermi energy.

Stanley Enemuo
Stanley Enemuo
Numerade Educator
01:26

Problem 91

Calculate the average energy per particle, $\varepsilon$, for a Fermi gas at $T=0$, given that $\varepsilon_F$ is the Fermi energy.

Chai Santi
Chai Santi
Numerade Educator
06:08

Problem 92

Derive the density of states $D(\varepsilon)$ as a function of energy $\varepsilon$ for a free electron gas in one-dimension. (Assume periodic boundary conditions or confine the linear chain to some length $L$.) Then calculate the Fermi energy $\varepsilon_F$ at zero temperature for an $N$ electron system.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
05:07

Problem 93

Consider a Fermi gas at low temperatures $k T \ll \mu(0)$, where $\mu(0)$ is the chemical potential at $T=0$. Give qualitative arguments for the leading value of the exponent of the temperature-dependent term in each of the following quantities: (a) energy; (b) heat capacity; (c) entropy; (d) Helmholtz free energy; (e) chemical potential. The zero of the energy scale is at the lowest orbital.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
02:46

Problem 94

Derive an expression for the chemical potential of a free electron gas with a density of $N$ electrons per unit volume at zero temperature ( $T=$ 0 K ). Find the chemical potential of the conduction electrons (which can be considered as free electrons) in a metal with $N=10^{22}$ electrons $/ \mathrm{cm}^3$ at $T=0 \mathrm{~K}$.

Penny Riley
Penny Riley
Numerade Educator
05:36

Problem 95

$D(E)$ is the density of states in a metal, and $E_F$ is the Fermi energy. At the Fermi energy $D\left(E_F\right) \neq 0$.
(a) Give an expression for the total number of electrons in the system at temperature $T=0$ in terms of $E_F$ and $D\left(E_F\right)$.
(b) Give an expression of the total number of electrons in the system at $T \neq 0$ in terms of the chemical potential $\mu$ and $D(E)$.
(c) Calculate the temperature dependence of the chemical potential at low temperatures, i.e., $\mu \gg k T$.
(Remember: $\int_{-\infty}^{+\infty} \frac{x e^x}{\left(e^x+1\right)^2} d x=\frac{\pi^2}{3}$.)
FIGURE CAN'T COPY.

Keshav Singh
Keshav Singh
Numerade Educator
04:33

Problem 96

For Na metal there are approximately $2.6 \times 10^{22}$ conduction electrons/ $\mathrm{cm}^3$, which behave approximately as a free electron gas. From these facts,
(a) give an approximate value (in eV ) of the Fermi energy in Na ,
(b) give an approximate value for the electronic specific heat of Na at room temperature.

Kyle Godbey
Kyle Godbey
Numerade Educator
06:08

Problem 97

The electrons in a metallic solid may be considered to be a threedimensional free electron gas. For this case:
(a) Obtain the allowed values of $k$, and sketch the appropriate Fermi sphere in $k$-space. (Use periodic boundary conditions with length $L$ ).
(b) Obtain the maximum value of $k$ for a system of $N$ electrons, and hence an expression for the Fermi energy at $T=0 \mathrm{~K}$.
(c) Using a simple argument show that the contribution the electrons make to the specific heat is proportional to $T$.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
01:24

Problem 98

Sketch the specific heat curve at constant volume, $C_v$, as a function of the absolute temperature, $T$, for a metallic solid. Give an argument showing why the contribution to $C_v$ from the free electrons is proportional to $T$.

Ajay Singhal
Ajay Singhal
Numerade Educator
03:02

Problem 99

(a) Derive a formula for the maximum kinetic energy of an electron in a non-interacting Fermi gas consisting of $N$ electrons in a volume $V$ at zero absolute tempcrature.
(b) Calculate the energy gap between the ground state and first excited state for such a Fermi gas consisting of the valence electrons in a $100 \AA$ cube of copper.
(c) Compare the energy gap with $k T$ at 1 K .

The mass density and atomic weight of copper are $8.93 \mathrm{~g} / \mathrm{cm}^3$ and 63.6 respectively.

Stanley Enemuo
Stanley Enemuo
Numerade Educator
02:02

Problem 100

(a) For a degenerate, spin $\frac{1}{2}$, non-interacting Fermi gas at zero temperature, find an expression for the energy of a system of $N$ such particles confined to a volume $V$. Assume the particles are non-relativistic.
(b) Given such an expression for the internal energy of a general system (not necessarily a free gas) at zero temperature, how does one determine the pressure?
(c) Hence calculate the pressure of this gas and show that it agrees with the result given by the kinetic theory.
(d) Cite, and explain briefly, two phenomena which are at least qualitatively explained by the Fermi gas model of metals, but are not in accord with classical statistical mechanics. Cite one phenomenon for which this simple model is inadequate for even a qualitative explanation.

Stanley Enemuo
Stanley Enemuo
Numerade Educator
02:40

Problem 101

The free-electron model of the conduction electrons in metals seems naive but is often successful. Among other things, it gives a reasonably good account of the compressibility for certain metals. This prompts the following question. You are given the number density $n$ and the Fermi energy $\varepsilon$ of a non-interacting Fermi gas at zero absolute temperature, $T=0 \mathrm{~K}$. Find the isothermal compressibility
$$
\kappa=-\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_T,
$$
where $V$ is volume, $p$ is pressure.

Hubert Agamasu
Hubert Agamasu
Numerade Educator
02:39

Problem 102

Fermi gas. Consider an ideal Fermi gas whose atoms have mass $m=$ $5 \times 10^{-24}$ grams, nuclear spin $I=\frac{1}{2}$, and nuclear magnetic moment $\mu=$ $1 \times 10^{-23} \mathrm{erg} /$ gauss. At $T=0 \mathrm{~K}$, what is the largest density for which the gas can be completely polarized by an external magnetic field of $10^5$ gauss? (Assume no electronic magnetic moment).

Narayan Hari
Narayan Hari
Numerade Educator
03:10

Problem 103

State and give a brief justification for the leading exponent $n$ in the temperature dependence of the following quantities in a highly degenerate three-dimensional electron gas:
(a) the specific heat at constant volume;
(b) the spin contribution to the magnetic moment $M$ in a fixed magnetic field $H$.

Anand Jangid
Anand Jangid
Numerade Educator
06:21

Problem 104

Take a system of $N=2 \times 10^{22}$ electrons in a "box" of volume $V=$ $1 \mathrm{~cm}^3$. The walls of the box are infinitely high potential barriers. Calculate the following within a factor of five and show the dependence on the relevant physical parameters:
(a) the specific heat, $C$,
(b) the magnetic susceptibility, $\chi$,
(c) the pressure on the walls of the box, $p$,
(d) the average kinetic energy, $\left\langle E_{\mathbf{k}}\right\rangle$.

Kai Chen
Kai Chen
Princeton University

Problem 105

An ideal gas of $N \operatorname{spin} \frac{1}{2}$ fermions is confined to a volume $V$. Calculate the zero temperature limit of (a) the chemical potential, (b) the average energy per particle, (c) the pressure, (d) the Pauli spin susceptibility. Show that in Gaussian units the susceptibility can be written as $3 \mu_{\mathrm{B}}^2 N / 2 \mu(0) V$, where $\mu(0)$ is the chemical potential at zero temperature. Assume each fermion has interaction with an external magnetic field of the form $2 \mu_0 H S_z$, where $\mu_{\mathrm{B}}$ is the Bohr magneton and $S_z$ is the $z$-component of the spin.

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01:41

Problem 106

Consider a Fermi gas model of nuclei.
Except for the Pauli principle, the nucleons in a heavy nucleus are assumed to move independently in a sphere corresponding to the nuclear volume $V$. They are considered as a completely degenerate Fermi gas. Let $A=N$ (the number of neutrons) $+Z$ (the number of protons), assume $N=$ $Z$, and compute the kinetic energy per nucleon, $E_{\text {kin }} / A$, with this model.
The volume of the nucleus is given by $V=\frac{4 \pi}{3} R_0^3 A, R_0 \approx 1.4 \times 10^{-13} \mathrm{~cm}$. Please give the result in MeV .

Yuva S
Yuva S
Numerade Educator
09:34

Problem 107

At low temperatures, a mixture of ${ }^3 \mathrm{He}$ and ${ }^4 \mathrm{He}$ atoms form a liquid which separates into two phases: a concentrated phase (nearly pure ${ }^3 \mathrm{He}$ ), and a dilute phase (roughly $6.5 \%{ }^3 \mathrm{He}$ for $T \leq 0.1 \mathrm{~K}$ ). The lighter ${ }^3 \mathrm{He}$ floats on top of the dilute phase, and ${ }^3 \mathrm{He}$ atoms can cross the phase boundary (see Fig. 2.23).

The superfluid ${ }^4 \mathrm{He}$ has negligible excitation, and the thermodynamics of the dilute phase can be represented as an ideal degenerate Fermi gas of particles with density $n_{\mathrm{d}}$ and effective mass $m^*$ ( $m^*$ is larger than $m_3$, the mass of the bare ${ }^3 \mathrm{He}$ atom, due to the presence of the liquid ${ }^4 \mathrm{He}$, actually $m^*=2.4 m_3$ ). We can crudely represent the concentrated phase by an ideal degenerate Fermi gas of density $n_{\mathrm{c}}$ and particle mass $m_3$.
(a) Calculate the Fermi energies for the two fluids.
(b) Using simple physical arguments, make an estimate of the very low temperature specific heat of the concentrated phase $c_c\left(T, T_{\mathrm{F}_{\mathrm{c}}}\right)$ which explicitly shows its functional dependence on $T$ and $T_{\mathrm{Fc}}$ (where $T_{\mathrm{Fc}}$ is the Fermi temperature of the concentrated phase, and any constants independent of $T$ and $T_{F C}$ need not be determined). Compare the specific heats of the dilute and concentrated phases.
(c) How much heat is required to warm each phase from $T=0 \mathrm{~K}$ to $T ?$
FIGURE CAN'T COPY.
(d) Suppose the container in the figure is now connected to external plumbing so that ${ }^3 \mathrm{He}$ atoms can be transferred from the concentrated phase to the dilute phase at a rate of $N$, atoms per second (as in a dilution refrigerator). For fixed temperature $T$, how much power can this system absorb?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
06:50

Problem 108

A white-dwarf star is thought to constitute a degenerate electron gas system at a uniform temperature much below the Fermi temperature. This system is stable against gravitational collapse so long as the electrons are non-relativistic.
(a) Calculate the electron density for which the Fermi momentum is one-tenth of the electron rest mass $\times c$.
(b) Calculate the pressure of the degenerate electron gas under these conditions.

Mahnoor Amin
Mahnoor Amin
Numerade Educator
06:50

Problem 109

A white dwarf is a star supported by the pressure of degenerate electrons. As a simplified model for such an object, consider a sphere of an ideal gas consisting of electrons and completely ionized $\mathrm{Si}^{28}$, and of constant density throughout the star. (Note that the assumption of a constant density is inconsistent with hydrostatic equilibrium, since the pressure is then also constant. The assumption that the gas is ideal is also not really tenable. These shortcomings of the model are, however, not crucial for the issues which we wish to consider.) Let $n_{\mathrm{i}}$ denote the density of the silicon ions, and let $n_e=14 n_{\mathrm{i}}$ denote the electron density. (The atomic number of silicon is 14).
(a) Find the relation between the mean kinetic energy $\bar{E}_e$ of the electrons and the density $n_e$, assuming that the densities are such that the electrons are "extremely relativistic," i.e., such that the rest energy is negligible compared with the total energy.
(b) Compute $\bar{E}_e$ (in MeV ) in the case that the (rest mass) density of the gas equals $\rho=10^{\circ} \mathrm{g} / \mathrm{cm}^3$. Also compute the mean kinetic energy $\bar{E}_{\mathrm{i}}$ of the silicon ions in the central region of the dwarf, assuming that the temperature is $10^8 \mathrm{~K}$ and assuming that the "ion gas" can be regarded as a Maxwell-Boltzmann gas, and hence convince yourself that $\bar{E}_{\mathrm{e}} \gg \bar{E}_1$.
(c) If $M$ is the mass of the star, and if $R$ is its radius, then the gravitational potential energy is given by
$$
U_{\mathrm{G}}=\frac{3 G M^2}{5 R} .
$$

In the case in which the internal energy is dominated by extremely relativistic electrons (as in part (b) above), the virial theorem implies that the total internal energy is approximately equal to the gravitational potential energy. Assuming equality, and assuming that the electrons do not contribute significantly to the mass of the star, show that the stellar mass can be expressed in terms of fundamental physical constants alone. Evaluate your answer numerically and compare it with the mass of the sun, $2 \times 10^{30} \mathrm{~kg}$. (It can be shown that this is approximately the maximum possible mass of a white dwarf.)

Mahnoor Amin
Mahnoor Amin
Numerade Educator
View

Problem 110

(a) Given that the mass of the sun is $2 \times 10^{33} \mathrm{~g}$, estimate the number of electrons in the sun. Assume the sun is largely composed of atomic hydrogen.
(b) In a white dwarf star of one solar mass the atoms are all ionized and contained in a sphere of radius $2 \times 10^9 \mathrm{~cm}$. Find the Fermi energy of the electrons in eV .
(c) If the temperature of the white dwarf is $10^7 \mathrm{~K}$, discuss whether the electrons and/or nucleons in the star are degenerate.
(d) If the above number of electrons were contained in a pulsar of one solar mass and of radius 10 km , find the order of magnitude of their Fermi energy.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
01:32

Problem 111

At what particle density does a gas of free electrons (considered at $T=0 \mathrm{~K}$ ) have enough one-particle kinetic energy (Fermi energy) to permit the reaction
$$
\text { proton }+ \text { electron }+0.8 \mathrm{MeV} \rightarrow \text { neutron }
$$
to proceed from left to right? Using the result above estimate the minimum density of a neutron star.

Brian Francisco
Brian Francisco
Numerade Educator
00:09

Problem 112

Assume that a neutron star is a highly degenerate non-relativistic gas of neutrons in a spherically symmetric equilibrium configuration. It is held together by the gravitational pull of a heavy object with mass $M$ and radius $r_0$ at the center of the star. Neglect all interactions among the neutrons. Calculate the neutron density as a function of the distance from the center, $r$, for $r>r_0$.

Stanley Enemuo
Stanley Enemuo
Numerade Educator
00:09

Problem 113

Assume that a neutron star is a highly degenerate non-relativistic gas of neutrons in a spherically symmetric equilibrium configuration. It is held together by the gravitational pull of a heavy object with mass $M$ and radius $r_0$ at the center of the star. Neglect all interactions among the neutrons. Calculate the neutron density as a function of the distance from the center, $r$, for $r>r_0$.

Stanley Enemuo
Stanley Enemuo
Numerade Educator

Problem 114

Consider a box of volume $V$ containing electron-positron pairs and photons in equilibrium at a temperature $T=1 / k \beta$. Assume that the equilibrium is established by the reaction
$$
\gamma \mapsto \mathrm{e}^{+}+\mathrm{e}^{-} .
$$
The reaction does not occur in free space, but one may think of it as catalyzed by the walls of the box. Ignoring the walls except insofar as they allow the reaction to occur, find
(a) The chemical potentials for the fermions.
(b) The average number of electron-positron pairs, in the two limits $k T \gg m_e c^2$ and $k T \ll m_e c^2$. (You may leave your answers in terms of dimensionless definite integrals.)
(c) The neglect of the walls is not strictly permissible if they contain a matter-antimatter imbalance. Supposing that this imbalance creates a net chemical potential $\mu \neq 0$ for the electrons, what is then the chemical potential of the positrons?
(d) Calculate the net charge of the system in the presence of this imbalance in the limit $k T \gg \mu \gg m_{\mathrm{e}} c^2$. (Again, your answer may be left in terms of a dimensionless definite integral.)

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02:02

Problem 115

In the very early stages of the universe, it is usually a good approximation to neglect particle masses and chemical potential compared with $k T$.
(a) Write down the average number and energy densities of a gas of non-interacting fermions in thermal equilibrium under these conditions. (You need not evaluate dimensionless integrals of order 1.)
(b) If the gas expands adiabatically while remaining in equilibrium, how do the average number and energy densities depend on the dimensions of the system?

Assume that the fermions are predominantly electrons and positrons when $T \simeq 10^{11} \mathrm{~K}$ in parts (c) and (d) below.
(c) Is the assumption made in (a) that the particles are non-interacting reasonable? Why? [Hint: What is the average coulomb interaction energy?
Positron charge $=1.6 \times 10^{-10}$ coulomb; Boltzmann's constant $k=1.38 \times$ $\left.10^{-16} \mathrm{erg} / \mathrm{K}\right]$.
(d) If the interaction cross sections in the electron-positron gas are typically of order of magnitude of the Thompson cross section $\sigma_T=8 \pi r_0^2 / 3$ (classical electron radius $r_0=2.8 \times 10^{-13} \mathrm{~cm}$ ), estimate the mean free time between collisions of the particles. If the expansion rate in part (b) $\approx$ $10^4 \mathrm{sec}^{-1}$, is the assumption that the gas remains in equilibrium reasonable? Why?

Stanley Enemuo
Stanley Enemuo
Numerade Educator

Problem 116

Heat Capacity.
The constant volume heat capacity of a system with average energy $\langle E\rangle$ is given by $C_v=\left(\frac{\partial\langle E\rangle}{\partial T}\right)_{N, V}$

Use the canonical ensemble to prove that: $C$ is related to the meansquare fluctuation in the energy as follows:
$$
C_v=\frac{1}{k T^2}\left\langle(E-\langle E\rangle)^2\right\rangle .
$$

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00:26

Problem 117

(a) Give the thermodynamic definition of the Helmholtz free energy $F$, the classical statistical mechanical definition of the partition function $Z_1$ and the relationship between these quantities. Define all symbols.
(b) Using these expressions and thermodynamic arguments show that the heat capacity at consant volume $c_v$ is given by
$$
c_v=k T\left[\frac{\partial^2}{\partial T^2}(T \ln Z)\right]_V .
$$
(c) Consider a classical system that has two discrete total energy states $E_0$ and $E_1$. Find $Z$ and $c_v$.

Stanley Enemuo
Stanley Enemuo
Numerade Educator
13:45

Problem 118

Consider the energy and fluctuation in energy of an arbitrary system in contact with a heat reservoir at absolute temperature $T=1 / k \beta$.
(a) Show that the average energy $\vec{E}$ of the system is
$$
\bar{E}=-\frac{\partial \ln z}{\partial \beta}
$$
where $z=\sum_n \exp \left(-\beta E_n\right)$ sums over all states of the system.
(b) Obtain an expression for $\overline{E^2}$ in terms of the derivatives of $\ln z$.
(c) Calculate the dispersion of the energy, $\overline{(\Delta E)^2}=\overline{E^2}-\bar{E}^2$. pressed in terms of the heat capacity of the system and the absolute temperature.
(e) Use this result to derive an expression for $\widetilde{\Delta E} / \bar{E}$ for an ideal monatomic gas.

Jack Hou
Jack Hou
Numerade Educator
09:34

Problem 119

A useful way to cool $\mathrm{He}^3$ is to apply pressure $P$ at sufficiently low temperature $T$ to a co-existing liquid-solid mixture. Describe qualitatively how this works on the basis of the following assumptions:
(a) The molar volume of the liquid $V_{\mathrm{L}}$ is greater than that of the solid $V_{\mathrm{S}}$ at all temperatures.
(b) The molar liquid entropy is given by
$$
S_{\mathrm{L}}=\gamma R T \quad \text { with } \quad \gamma \sim 4.6 \mathrm{~K}^{-1} .
$$
(c) The entropy of the solid $S_{\mathrm{S}}$ comes entirely from the disorder associated with the nuclear spins $(s=1 / 2)$.
Note: Include in your answer a semi-quantitative graph of the $p-T$ diagram of $\mathrm{He}^3$ at low temperatures (derived using the above information).

Eduard Sanchez
Eduard Sanchez
Numerade Educator
02:40

Problem 120

(a) Describe the third law of thermodynamics.
(b) Explain the physical meaning of negative absolute temperature. Does it violate the third law? Why?
(c) Suggest one example in which the negative temperature can actually be achieved.
(d) Discuss why the negative temperature does not make sense in classical thermodynamics.

Vipender Yadav
Vipender Yadav
Numerade Educator
06:58

Problem 121

Consider a system of two atoms, each having ony 3 quantum states of energies $0, \varepsilon$ and $2 \varepsilon$. The system is in contact with a heat reservoir at temperature $T$. Write down the partition function $Z$ for the system if the particles obey
(a) Classical statistics and are distinguishable.
(b) Classical statistics and are indistinguishable.
(c) Fermi-Dirac statistics.
(d) Bose-Einstein statistics.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
06:58

Problem 122

(a) You are given a system of two identical particles which may occupy any of the three energy levels
$$
\varepsilon_n=n \varepsilon, \quad n=0,1,2,
$$

The lowest energy state, $\varepsilon_0=0$, is doubly degenerate. The system is in thermal equilibrium at temperature $T$. For each of the following cases determine the partition function and the energy and carefully enumerate the configurations.
1) The particles obey Fermi statistics.
2) The particle obey Bose statistics.
3) The (now distinguishable) particles obey Boltzmann statistics.
(b) Discuss the conditions under which Fermions or Bosons may be treated as Boltzmann particles.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
00:26

Problem 123

(a) Give a definition of the partition function $z$ for a statistical system.
(b) Find a relation between the heat capacity of a system and $\frac{\partial^2 \ln z}{\partial \beta^2}$, where $\beta=\frac{1}{k T}$.
(c) For a system with one excited state at energy $\Delta$ above the ground state, find an expression for the heat capacity in terms of $\Delta$. Sketch the dependence on temperature and discuss the limiting behavior for high and low temperatures.

Stanley Enemuo
Stanley Enemuo
Numerade Educator
06:50

Problem 124

Consider a collection of $N$ two-level systems in thermal equilibrium at a temperature $T$. Each system has only two states: a ground state of energy 0 and an excited state of energy $\varepsilon$. Find each of the following quantities and make a sketch of the temperature dependence.
(a) The probability that a given system will be found in the excited state.
(b) The entropy of the entire collection.

Stanley Enemuo
Stanley Enemuo
Numerade Educator
07:57

Problem 125

$N$ weakly coupled particles obeying Maxwell-Boltzmann statistics may each exist in one of the 3 non-degenerate energy levels of energies $-E, 0$, $+E$. The system is in contact with a thermal reservoir at temperature $T$.
(a) What is the entropy of the system at $T=0 \mathrm{~K}$ ?
(b) What is the maximum possible entropy of the system?
(c) What is the minimum possible energy of the system?
(d) What is the partition function of the system?
(e) What is the most probable energy of the system?
(f) If $C(T)$ is the heat capacity of the system, what is the value of $\int_0^{\infty} \frac{C(T)}{T} d T ?$

Eduard Sanchez
Eduard Sanchez
Numerade Educator
01:44

Problem 126

Find the pressure, entropy, and specific heat at constant volume of an ideal Boltzmann gas of indistinguishable particles in the extreme relativistic limit, in which the energy of a particle is related to its momentum by $\varepsilon=\mathrm{cp}$. Express your answer as functions of the volume $V$, temperature $T$, and number of particle $N$.

Dominador Tan
Dominador Tan
Numerade Educator

Problem 127

A vessel of volume $V$ contains $N$ molecules of an ideal gas held at temperature $T$ and pressure $P_1$. The energy of a molecule may be written in the form
$$
E_k\left(p_x, p_y, p_z\right)=\frac{1}{2 m}\left(p_x^2+p_y^2+p_z^2\right)+\varepsilon_k,
$$
where $\varepsilon_k$ denotes the energy levels corresponding to the internal states of the molecules of the gas.
(a) Evaluate the free energy $F=-k T \ln Z$, where $Z$ is the partition function and $k$ is Boltzmann's constant. Explicitly display the dependence on the volume $V_1$.

Now consider another vessel, also at temperature $T$, containing the same number of molecules of an identical gas held at pressure $P_2$.
(b) Give an expression for the total entropy of the two gases in terms of $P_1, P_2, T, N$.
(c) The vessels are then connected to permit the gases to mix without doing work. Evaluate explicitly the change in entropy of the system. Check whether your answer makes sense by considering the special case $V_1=$ $V_2\left(\right.$ i.e., $\left.P_1=P_2\right)$.

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Problem 128

(a) Calculate the partition function $z$ of one spinless atom of mass $M$ moving freely in a cube of volume $V=L^3$. Express your result in terms of the quantum concentration
$$
n_q=\left(\frac{M k T}{2 \pi}\right)^{3 / 2} .
$$

Explain the physical meaning of $n_q$.
(b) An ideal gas of $N$ spinless atoms occupies a volume $V$ at temperature $T$. Each atom has only two energy levels separated by an energy $\Delta$. Find the chemical potential, free energy, entropy, pressure and heat capacity at constant pressure.

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Problem 129

(a) Consider an ideal gas of $N$ particles of mass $m$ confined to a volume $V$ at a temperature $T$. Using the classical approximation for the partition function and assuming the particles are indistinguishable, calculate the chemical potential $\mu$ of the gas.
(b) A gas of $N$ particles, also of mass $m$, is absorbed on a surface of area $A$, forming a two-dimensional ideal gas at temperature $T$ on the surface. The energy of an absorbed particle is $\varepsilon=|\mathbf{p}|^2 / 2 m-\varepsilon_0$, where $\mathbf{p}=\left(\mathbf{p}_x, \mathbf{p}_y\right)$ and $\varepsilon_0$ is the surface binding energy per particle. Using the same approximations and assumptions as in part (a), calculate the chemical potential $\mu$ of the absorbed gas.
(c) At temperature $T$, the particles on the surface and in the surrounding three-dimensional gas are in equilibrium. This implies a relationship between the respective chemical potentials. Use this condition to find the mean number $n$ of molecules absorbed per unit area when the mean pressure of the surrounding three-dimensional gas is $p$. (The total number of particles in absorbed gas plus surrounding vapor is $N_0$ ).

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Problem 130

A simple harmonic one-dimensional oscillator has energy levels $E_n=$ $(n+1 / 2) \hbar \omega$, where $\omega$ is the characteristic oscillator (angular) frequency and $n=0,1,2, \ldots$.
(a) Suppose the oscillator is in thermal contact with a heat reservoir at temperature $T$, with $\frac{k T}{\hbar \omega} \ll 1$. Find the mean energy of the oscillator as a function of the temperature $T$.
(b) For a two-dimensional oscillator, $n=n_x+n_y$, where
$$
E_{n_x}=\left(n_x+\frac{1}{2}\right) \hbar \omega_x, \quad E_{n_y}=\left(n_y+\frac{1}{2}\right) \hbar \omega_y,
$$
$n_x=0,1,2, \ldots$ and $n_y=0,1,2, \ldots$, what is the partition function for this case for any value of temperature? Reduce it to the degenerate case $\omega_x=\omega_y$.
(c) If a one-dimensional classical anharmonic oscillator has potential energy $V(x)=c x^2-g x^3$, where $g x^3 \& c x^2$, at equilibrium temperature $T$, carry out the calculations as far as you can and give expressions as functions of temperature for
1) the heat capacity per oscillator and
2) the mean value of the position $x$ of the oscillator.

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06:59

Problem 131

Consider a dilute diatomic gas whose molecules consist of non-identical pairs of atoms. The moment of inertia about an axis through the molecular center of mass perpendicular to the line connecting the two atoms is $I$. Calculate the rotational contributions to the specific heat and to the absolute entropy per mole at temperature $T$ for the following limiting cases:
(a) $k T \gg \hbar^2 / I$,
(b) $k T \ll \hbar^2 / I$.
Make your calculations sufficiently exact to obtain the lowest order non-zero contributions to the specific heat and entropy.

Keshav Singh
Keshav Singh
Numerade Educator
05:07

Problem 132

An assembly of $N$ fixed particles with spin $\frac{1}{2}$ and magnetic moment $\mu_0$ is in a static uniform applied magnetic field. The spins interact with the applied field but are otherwise essentially free.
(a) Express the energy of the system as a function of its total magnetic moment and the applied field.
(b) Find the total magnetic moment and the energy, assuming that the system is in thermal equilibrium at temperature $T$.
(c) Find the heat capacity and the entropy of the system under these same conditions.

Carlos Henrique De Lima
Carlos Henrique De Lima
Numerade Educator
01:33

Problem 133

Given a system of $N$ identical non-interacting magnetic ions of spin $\frac{1}{2}$, magnetic moment $\mu_0$ in a crystal at absolute temperature $T$ in a magnetic field $B$. For this system calculate:
(a) The partition function $Z$.
(b) The entropy $\sigma$.
(c) The average energy $\bar{U}$.
(d) The average magnetic moment $\bar{M}$, and the fluctuation in the magnetic moment, $\Delta M=\sqrt{\overline{(M-\bar{M})^2}}$.
(e) The crystal is initially in thermal equilibrium with a reservoir at $T=1 \mathrm{~K}$, in a magnetic field $B_i=10,000$ Gauss. The crystal is then thermally isolated from the reservoir and the field reduced to $B_f=100$ Gauss. What happens?

Penny Riley
Penny Riley
Numerade Educator
05:07

Problem 134

Consider $N$ fixed non-interacting magnetic moments each of magnitude $\mu_0$. The system is in thermal equilibrium at temperature $T$ and is in a uniform external magnetic field $B$. Each magnetic moment can be oriented only parallel or antiparallel to $B$. Calculate:
(a) the partition function,
(b) the specific heat,
(c) the thermal average magnetic moment $(\bar{M})$.
Show that in the high temperature limit the Curie Law is satisfied (i.e., $\chi=d \bar{M} / d B \propto 1 / T)$

Carlos Henrique De Lima
Carlos Henrique De Lima
Numerade Educator

Problem 135

Consider a system of non-interacting spins in an applied magnetic field $H$. Using $S=k(\ln Z+\beta E)$, where $Z$ is the partition function, $E$ is the energy, and $\beta=1 / k T$, argue that the dependence of $S$ on $H$ and $T$ is of the form $S=f(H / T)$ where $f(x)$ is some function that need not be determined.

Show that if such a system is magnetized at constant $T$, then thermally isolated, and then demagnetized adiabatically, cooling will result.

Why is this technique of adiabatic demagnetization used for refrigeration only at very low temperatures?

How can we have $T<0$ for this system? Can this give a means of achieving $T=0$ ?

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46:10

Problem 136

Consider a crystal of $N$ atoms ( $N \sim 10^{23}$ ) with spin quantum numbers $s=\frac{1}{2}$ and $m_s= \pm \frac{1}{2}$. The magnetic moment of the $i$-th atom is $\mu_i=g \mu_B \mathbf{s}_i$, where $g$ is the Lande $g$-factor, and $\mu_{\mathrm{B}}=e \hbar / 2 m c$ is the Bohr magneton. Assume that the atoms do not interact appreciably but are in equilibrium at temperature $T$ and are placed in an external magnetic field $\mathbf{H}=\mathrm{Hz}$.
(a) Show that the partition function is $z=(2 \cosh \eta)^N$ where $\eta=$ $g \mu_{\mathrm{B}} H / 2 k T$.
(b) Find an expression for the entropy $S$ of the crystal (you need only consider the contributions from the spin states). Evaluate $S$ in the strong field $(\eta \gg 1)$ and weak field $(\eta \ll 1)$ limits.
(c) An important process for cooling substances below 1 K is adiabatic demagnetization. In this process the magnetic field on the sample is increased from 0 to $H_0$ while the sample is in contact with a heat bath at temperature $T_0$. Then the sample is thermally isolated and the magnetic field is reduced to $H_1<H_0$. What is the final temperature of the sample?
(d) The magnetization $M$ and susceptibility $\chi$ are defined by $M=$ $\left\langle\sum_{i=1}^N\left(\mu_i\right)_z\right\rangle$ and $\chi=M / H$, respectively. Find expressions for $M$ and $\chi$, and evaluate these expressions in the weak field limit.

Now suppose each atom interacts with each of its nearest $n$ neighbors. To include this interaction approximately we assume that the nearest $n$ neighbors generate a 'mean field' $\bar{H}$ at the site of each atom, where
$$
g \mu_{\mathrm{B}} \bar{H}=2 \alpha\left\langle\sum_{k=1}^N\left(S_k\right)_z\right\rangle,
$$
$\alpha$ is a parameter which characterizes the strength of the interaction.
(e) Use the mean field approximation together with the results of part (d) to calculate the susceptibility $\chi$ in the weak field (i.e., the high temperature) limit. At what temperature, $T_c$, does $\chi$ become infinite?

Jack Hou
Jack Hou
Numerade Educator
06:50

Problem 137

Consider a system of free electrons in a uniform magnetic field $B=B_z$, with the electron spin ignored. Show that the quantization of orbits, in contrast to classical orbits, affect the calculation of diamagnetism in the high temperature limit by calculating:
(a) the degeneracy of the quantized energy levels,
(b) the grand partition function,
(c) the magnetic susceptibility in the high temperature limit.

Stanley Enemuo
Stanley Enemuo
Numerade Educator
00:25

Problem 138

A certain insulating solid contains $N_{\mathrm{A}}$ non-magnetic atoms and $N_{\mathrm{I}}$ magnetic impurities each of which has spin $\frac{3}{2}$. Each impurity spin is free to rotate independently of all the rest. There is a very weak spin-phonon interaction which we can for most purposes neglect completely. Thus the solid and the impurities are very weakly interacting.
(a) A magnetic field is applied to the system while it is held at constant temperature, $T$. The field is strong enough to line up the spins completely. What is the magnitude and sign of the change in entropy in the system as the field is applied?
(b) Now the system is held in thermal isolation, no heat is allowed to enter or leave. The magnetic field is reduced to zero. Will the temperature of the solid increase or decrease? Justify your answer.
(c) Assume the heat capacity of the solid is given by $C=3 N_{\mathrm{A}} k$, where $k$ is the Boltzmann constant. What is the temperature change produced by the demagnetization of Part (b)? (Neglect all effects of possible volume changes in the solid.)

Stanley Enemuo
Stanley Enemuo
Numerade Educator

Problem 139

What is the root-mean-square fluctuation in the number of photons of mode frequency $\omega$ in a conducting rectangular cavity? Is it always smaller than the average number of photons in the mode?

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Problem 140

Consider an adsorbent surface having $N$ sites, each of which can adsorb one gas molecule. This surface is in contact with an ideal gas with chemical potential $\mu$ (determined by the pressure $p$ and the temperature $T$ ). Assuming that the adsorbed molecule has energy $-\varepsilon_0$ compared to one in a free state.
(a) Find the grand canonical partition function (sometimes called the grand sum) and
(b) calculate the covering ratio $\theta$, i.e., the ratio of adsorbed molecules to adsorbing sites on the surface.
[A useful relation is $(1+x)^N=\sum_{N_1} N!x^{N_1} / N_{1}!\left(N-N_1\right)$ !].

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02:21

Problem 141

A zipper has $N$ links. Each link has two states: state 1 means it is closed and has energy 0 and state 2 means it is open with energy $\varepsilon$. The zipper can only unzip from the left end and the sth link cannot open unless all the links to its left $(1,2, \ldots, s-1)$ are already open.
(a) Find the partition function for the zipper.
(b) In the low temperature limit, $\varepsilon \gg k T$, find the mean number of open links.
(c) There are actually an infinite number of states corresponding to the same energy when the link is open because the two parts of an open link may have arbitrary orientations. Assume the number of open states is $g$. Write down the partition function and discuss if there is a phase transition.

Rabeya Zahid
Rabeya Zahid
Numerade Educator
00:26

Problem 142

A system consisting of three spins in a line, each having $s=\frac{1}{2}$, is coupled by nearest neighbor interactions (see Fig. 2.29).
Fig. 2.29.

Each spin has a magnetic moment pointing in the same direction as the spin, $\boldsymbol{\mu}=2 \mu \mathrm{s}$. The system is placed in an external magnetic field $H$ in the $z$ direction and is in thermal equilibrium at temperature $T$. The Hamiltonian for the system is approximated by an Ising model, where the true spin-spin interaction is replaced by a term of the form $J S_z(i) S_z(i+1)$ :
$$
H=J S_z(1) S_z(2)+J S_z(2) S_z(3)-2 \mu H\left[S_z(1)+S_z(2)+S_z(3)\right],
$$
where $J$ and $\mu$ are positive constants.
(a) List each of the possible microscopic states of the system and its energy. Sketch the energy level diagram as a function of $H$. Indicate any degeneracies.
(b) For each of the following conditions, write down the limiting values of the internal energy $U(T, H)$, the entropy $S(T, H)$, and the magnetization $M(T, H)$.
1) $T=0$ and $H=0$,
2) $T=0$ and $0<H \ll J / \mu$,
3) $T=0$ and $J / \mu \ll H$,
4) $J \ll k T$ and $H=0$.
(c) On the basis of simple physical considerations, without doing any calculations, sketch the specific heat at constant field, $C_H(T, H)$ when $H=$ 0 . What is the primary temperature dependence at very high and very low $T$ ?
(d) Find a closed form expression for the partition function $Q(T, H)$.
(e) Find the magnetization $M(T, H)$. Find an approximate expression for $M(T, H)$ which is valid when $k T \gg \mu H$ or $J$.

Stanley Enemuo
Stanley Enemuo
Numerade Educator

Problem 143

Consider a crystalline lattice with Ising spins $s_{\ell}=\boldsymbol{\ell}$ at each site $\boldsymbol{\ell}$. In the presence of an external field $\mathbf{H}=\left(0,0, H_0\right)$, the Hamiltonian of the system may be written as
$$
H=-J \sum_{\ell \in} s_{\boldsymbol{e}} s_{\boldsymbol{e}}-\mu_0 H_0 \sum_{\boldsymbol{\ell}} s_{\boldsymbol{\ell}},
$$
where $J>0$ is a constant and the sum $\sum_{\text {CE }}$ is over all nearest-neighbor sites only (each site has $p$ nearest neighbors).
(a) Write an expression for the free energy of the system at temperature $T$ (do not try to evaluate it).
(b) Using the mean-field approximation, derive an equation for the spontaneous magnetization $m=\left\langle s_0\right\rangle$ for $\mathbf{H}_0=0$ and calculate the critical temperature $T_{\mathrm{c}}$ below which $m \neq 0$.
(c) Calculate the critical exponent $\beta$ defined by $m\left(T, \mathbf{H}_0=0\right) \sim$ const. $\left(1-T / T_c\right)^\beta$ as $T \rightarrow T_c$.
(d) Describe the behavior of the specific heat at constant $\mathbf{H}_0, C\left(\mathbf{H}_0=\right.$ $0)$, near $T=T_c$.

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Problem 144

Consider a gas of hard spheres with the 2-body interaction
$$
\begin{aligned}
V\left(\left|\mathbf{r}_i-\mathbf{r}_j\right|\right) & =0, & & \left|\mathbf{r}_i-\mathbf{r}_j\right|>a, \\
& =\infty, & & \left|\mathbf{r}_i-\mathbf{r}_j\right|<a .
\end{aligned}
$$

Using the classical partition function, calculate the average energy at a given temperature and density (thermodynamics: the internal energy).

On the basis of simple physical arguments, would you expect this same simple answer to also result from a calculation with the quantum mechanical partition function?

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Problem 145

A classical gas of $N$ point particles occupies volume $V$ at temperature $T$. The particles interact pairwise, $\phi\left(r_{i j}\right)$ being the potential between particles $i$ and $j, r_{i j}=\left|\mathbf{r}_i-\mathbf{r}_j\right|$. Suppose this is a "hard sphere" potential
$$
\phi\left(r_{i j}\right)= \begin{cases}\infty, & r_{i j}<a, \\ 0, & r_{i j}>a .\end{cases}
$$
(a) Compute the constant volume specific heat as a function of temperature and specific volume $v=\frac{V}{N}$.
(b) The virial expansion for the equation of state is an expansion of $\frac{p V}{R T}$ in inverse powers of $V$ :
$$
\frac{p V}{R T}=1+\frac{A_1(T)}{V}+\frac{A_2(T)}{V^2}+\ldots .
$$

Compute the virial coefficient $A_1$.

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Problem 146

Consider a classical system of $N$ point particles of mass $m$ in a volume $V$ at temperature $T$. Let $U$ be the total energy of the system, $p$ the pressure. The particles interact through a two-body central potential
$$
\phi\left(r_{i j}\right)=\frac{A}{r_{i j}^n}, \quad A>0, \quad n>0, \quad r_{i j}=\left|\mathbf{r}_i-\mathbf{r}_j\right| .
$$
Notice the scaling property $\phi(\gamma r)=\gamma^{-n} \phi(r)$ for any $\gamma$. From this, and from scaling arguments (e.g. applied to the partition function) show that
$$
U=a p V+b N k T, \quad k=\text { Boltzmann's const. },
$$
where the constants $a$ and $b$ depend on the exponent $n$ in the pairwise potential. Express $a$ and $b$ in terms of $n$.

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Problem 147

(a) Given $\int_{-\infty}^{+\infty} \exp \left(-\alpha x^2\right) d x=\sqrt{\pi / \alpha}$, show that
$$
\int_{-\infty}^{\infty} x^2 e^{-\alpha x^2} d x=\frac{\sqrt{\pi}}{2} \alpha^{-3 / 2}, \quad \int_{-\infty}^{\infty} x^4 e^{-\alpha x^2} d x=\frac{3}{4} \sqrt{\pi} \alpha^{-5 / 2} .
$$
(b) Given that $\frac{\beta}{\sqrt{\alpha}} \ll 1$ and that $a$ is of the order of $\frac{1}{\sqrt{\alpha}}$, show
$$
\int_{-a}^a f(x) e^{-\alpha\left[x^2+\beta x^3 \mid\right.} d x \approx \int_{-a}^a f(x)\left(1-\alpha \beta x^3\right) e^{-\alpha x^2} d x .
$$
(c) Two atoms interact through a potential
$$
U(x)=U_0\left[\left(\frac{a}{x}\right)^{12}-2\left(\frac{a}{x}\right)^6\right],
$$
where $x$ is their separation. Sketch this potential. Calculate the value of $x$ for which $U(x)$ is minimum.
(d) Given a row of such atoms constrained to move only on the $x$ axis, each assumed to interact only with its nearest neighbors, use classical statistical mechanics to calculate the mean interatomic separation $\bar{x}(T)$.

To do this, expand $U$ about its minimum, keeping as many terms as necessary to obtain the lowest order temperature dependence of $\bar{x}(T)$. Assume that $k T \ll U_0$, and in the relevant integrals extend the limits of integration to $\pm \infty$ where appropriate. Explain clearly the justification for extending the limits. Also calculate
FIGURE CAN'T COPY.

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Problem 148

A classical system is described by its Hamiltonian $H$, which is a function of a set of generalized co-ordinates $q_i$ and momenta $p_i$. The canonical equations of motion are
$$
\dot{p}_i=-\frac{\partial H}{\partial q_i}, \quad \dot{q}_i=\frac{\partial H}{\partial p_i} .
$$
Write the equation of continuity for $\rho$, the phase space density, and use it to show that the entropy of this system is constant in time.

Now consider a system whose motion is damped by frictional force. For simplicity, consider a damped harmonic oscillator in one dimension. The equations of motion are
$$
\dot{p}=-k q-\frac{\gamma p}{m}, \quad \dot{q}=\frac{p}{m},
$$
where $m$ is the mass, $k$ is the spring constant, and $\gamma$ is related to the friction, $m, k$ and $\gamma$ being all positive.)

What is the equation of motion for the phase space density $\rho$ ? Show that the entropy is now a decreasing function of time.

Can the last result be reconciled with the second law of thermodynamics?

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06:30

Problem 149

Estimate
(a) The number of molecules in the air in a room.
(b) Their energy, in joules or in ergs, per mole.
(c) What quantity of heat (in joules or in ergs) must be added to warm one mole of air at 1 atm from $0^{\circ} \mathrm{C}$ to $20^{\circ} \mathrm{C}$ ?
(d) What is the minimum energy that must be supplied to a refrigerator to cool 1 mole of air at 1 atm from $20^{\circ} \mathrm{C}$ to $18^{\circ} \mathrm{C}$ ? The refrigerator acts in a cyclic process and gives out heat at $40^{\circ} \mathrm{C}$.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
03:22

Problem 150

Consider a cube, 10 cm on a side, of He gas at STP. Estimate (order of magnitude) the number of times one wall is struck by molecules in one second.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
07:39

Problem 151

Estimate the mean free path of a cosmic ray proton in the atmosphere at sea level.

Donald Albin
Donald Albin
Numerade Educator
01:35

Problem 152

Even though there is a high density of electrons in a metal (mean separation $r \sim 1-3 \AA$ ), electron-electron mean free paths are long. ( $\lambda_{\text {ee }} \sim$ $10^4 \AA$ at room temperature.) State reasons for the long electron-electron collision mean free path and give a qualitative argument for its temperature dependence.

Chai Santi
Chai Santi
Numerade Educator
03:22

Problem 153

Estimate the following:
(a) The mean time between collisions for a nitrogen molecule in air at room temperature and atmospheric pressure.
(b) The number density of electrons in a degenerate Fermi electron gas at $T=0 \mathrm{~K}$ and with a Fermi momentum $p_F=m_e c$.

Lottie Adams
Lottie Adams
Numerade Educator
08:28

Problem 154

A container is divided into two parts by a partition containing a small hole of diameter D. Helium gas in the two parts is held at temperature $T_1=150 \mathrm{~K}$ and $T_2=300 \mathrm{~K}$ respectively through heating of the walls.
(a) How does the diameter $D$ determine the physical process by which the gases come into a steady state?
(b) What is the ratio of the mean free paths $l_1 / l_2$ between the two parts when $D \ll l_1, D \ll l_2$, and the system has reached a steady state?
(c) What is the ratio $l_1 / l_2$ when $D \gg l_1, D \gg l_2$ ?

Eric Mockensturm
Eric Mockensturm
Numerade Educator

Problem 155

Consider the orthogonalized drunk who starts out at the proverbial lamp-post: Each step he takes is either due north, due south, due east or due west, but which of the four directions he steps in is chosen purely randomly at each step. Each step is of fixed length $L$. What is the probability that he will be within a circle of radius $2 L$ of the lamp-post after 3 steps?

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01:41

Problem 156

Estimate how long it would take a molecule of air in a room, in which the air is macroscopically 'motionless' and of perfectly uniform temperature and pressure, to move to a position of distance 5 meters away.

João Gabriel Alencar Caribé
João Gabriel Alencar Caribé
Numerade Educator
01:01

Problem 157

You have just very gently exhaled a helium atom in this room. Calculate how long ( $t$ in seconds) it will take to diffuse with a reasonable probability to some point on a spherical surface of radius $R=1$ meter surrounding your head.

Narayan Hari
Narayan Hari
Numerade Educator
03:15

Problem 158

In an experiment a beam of silver atoms emerges from an oven, which contains silver vapor at $T=1200 \mathrm{~K}$. The beam is collimated by being passed through a small circular aperture.
(a) Give an argument to show that it is not possible, by narrowing the aperture $a$, to decrease indefinitely the diameter of the spot, $D$, on the screen.
(b) If the screen is at $L=1$ meter from the aperture, estimate numerically the smallest $D$ that can be obtained by varying $a$. (You may assume for simplicity that all atoms have the same momentum along the direction of the beam and have a mass of $M_{\mathrm{Ag}}=1.8 \times 10^{-22} \mathrm{~g}$ ).
FIGURE CAN'T COPY.

Suzanne W.
Suzanne W.
Numerade Educator
03:12

Problem 159

The range of the potential between two hydrogen atoms is approximately $4 \AA$. For a gas in thermal equilibrium obtain a numerical estimate of the temperature below which the atom-atom scattering is essentially $S$-wave.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
16:11

Problem 160

Show that a small object immersed in a fluid at temperature $T$ will undergo a random motion, due to collisions with the molecules of the fluid, such that the mean-square displacement in any direction satisfies
$$
\left\langle(\Delta x)^2\right\rangle=T t / \lambda,
$$
where $t$ is the elapsed time, and $\lambda$ is a constant proportional to the viscosity of the fluid.

CA
Chi-Chung Ai
Numerade Educator
04:39

Problem 161

A box of volume $2 V$ is divided into halves by a thin partition. The left side contains a perfect gas at pressure $p_0$ and the right side is initially vacuum. A small hole of area $A$ is punched in the partition. What is the pressure $p_1$ in the left hand side as a function of time? Assume the temperature is constant on both sides. Express your answer in terms of the average velocity $v$.

Surendra Kumar
Surendra Kumar
Numerade Educator

Problem 162

Starting with the virial theorem for an equilibrium configuration show that:
(a) the total kinetic energy of a finite gaseous configuration is equal to the total internal energy if $\gamma=C_p / C_v=5 / 3$, where $C_p$ and $C_v$ are the molar specific heats of the gas at constant pressure and at constant volume, respectively,
(b) the finite gaseous configuration can be in Newtonian gravitational equilibrium only if $C_p / C_v>4 / 3$.

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02:58

Problem 163

A system consists of $N$ very weakly interacting particles at a temperature sufficiently high such that classical statistics are applicable. Each particle has mass $m$ and oscillates in one direction about its equilibrium position. Calculate the heat capacity at temperature $T$ in each of the following cases:
(a) The restoring force is proportional to the displacement $x$ from the equilibrium position.
(b) The restoring force is proportional to $x^3$.

The results may be obtained without explicitly evaluating integrals.

Suzanne W.
Suzanne W.
Numerade Educator
01:31

Problem 164

By treating radiation in a cavity as a gas of photons whose energy $\varepsilon$ and momentum $k$ are related by the expression $\varepsilon=c k$, where $c$ is the velocity of light, show that the pressure $p$ exerted on the walls of the cavity is one-third of the energy density.

With the above result prove that when radiation contained in a vessel with perfectly reflecting walls is compressed adiabatically it obeys the equation
$$
P V^\gamma=\text { constant } .
$$

Determine the value of $\gamma$.

Manish Jain
Manish Jain
Numerade Educator
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Problem 165

Radiation pressure.
One may think of radiation as a gas of photons and apply many of the results from kinetic theory and thermodynamics to the radiation gas.
(a) Prove that the pressure exerted by an isotropic radiation field of energy density $u$ on a perfectly reflecting wall is $p=u / 3$.
(b) Blackbody radiation is radiation contained in, and in equilibrium with, a cavity whose walls are at a fixed temperature $T$. Use thermodynamic arguments to show that the energy density of blackbody radiation depends only on $T$ and is independent of the size of the cavity and the material making up the walls.
(c) From (a) and (b) one concludes that for blackbody radiation the pressure depends only on the temperature, $p=p(T)$, and the internal energy $U$ is given by $U=3 p(T) V$ where $V$ is the volume of the cavity. Using these two facts about the gas, derive the functional form of $p(T)$, up to an unspecified multiplicative constant, from purely thermodynamic reasoning.

Andrew Eddins
Andrew Eddins
Emory University
01:44

Problem 166

A gas of interacting atoms has an equation of state and heat capacity at constant volume given by the expressions
$$
\begin{aligned}
& p(T, V)=a T^{1 / 2}+b T^3+c V^{-2}, \\
& C_v(T, V)=d T^{1 / 2} V+e T^2 V+f T^{1 / 2},
\end{aligned}
$$
where $a$ through $f$ are constants which are independent of $T$ and $V$.
(a) Find the differential of the internal energy $d U(T, V)$ in terms of $d T$ and $d V$.
(b) Find the relationships among $a$ through $f$ due to the fact that $U(T, V)$ is a state variable.
(c) Find $U(T, V)$ as a function of $T$ and $V$.
(d) Use kinetic arguments to derive a simple relation between $p$ and $U$ for an ideal monatomic gas (a gas with no interactions between the atoms, but whose velocity distribution is arbitrary). If the gas discussed in the previous parts were to be made ideal, what would be the restrictions on the constants a through $f$ ?

Manik Pulyani
Manik Pulyani
Numerade Educator
02:32

Problem 167

(a) From simplest kinetic theory derive an approximate expression for the diffusion coefficient of a gas, $D$. For purposes of this problem you need not be concerned about small numerical factors and so need not integrate over distribution functions etc.
(b) From numbers you know evaluate $D$ for air at STP.

Jerrah Biggerstaff
Jerrah Biggerstaff
Numerade Educator
02:57

Problem 168

(a) Show that the ratio of the pressure to the viscosity coefficient gives approximately the number of collisions per unit time for a molecule in a gas.
(b) Calculate the number of collisions per unit time for a molecule in a gas at STP using the result of (a) above or by calculating it from the mean velocity, molecular diameter, and number density.

The coefficient of viscosity for air at STP is $1.8 \times 10^{-4}$ in cgs units. Use values you know for other constants you need.

Christopher Provencher
Christopher Provencher
Numerade Educator
03:51

Problem 169

(a) Assuming moderately dilute helium gas so that binary collisions of helium atoms determine the transport coefficients, derive an expression for the thermal conductivity of the gas.
(b) Estimate the ratio of the thermal conductivity of gaseous ${ }^3 \mathrm{He}$ to that of gaseous ${ }^4 \mathrm{He}$ at room temperature.
(c) Will this ratio become different at a temperature near 2 K ? Why?

Jerrah Biggerstaff
Jerrah Biggerstaff
Numerade Educator
02:09

Problem 170

A certain closed cell foam used as an insulating material in houses is manufactured in such a way that the cells are initially filled with a polyatomic gas of molecular weight $\sim 60$. After several years the gas diffuses out of the foam and is replaced by dry air (mean molecular weight $\sim 30$ ). Assuming that the insulating property arises largely from the thermal conductivity of the gas.

Discuss the factors which influence the thermal conductivity of the gas. For each factor make an argument for whether the insulating ability increases or decreases. What is the overall effect upon the insulating ability?
FIGURE CAN'T COPY.

Averell Hause
Averell Hause
Carnegie Mellon University
00:57

Problem 171

Thermos Bottle.
(a) State and justify how the thermal conductivity of an ideal gas depends on its density at fixed temperature.
(b) A thermos (Dewar) bottle is constructed of two concentric glass vessels with the air in the intervening space reduced to a low density. Why can it act as an insulating container even though the vacuum is not perfect?

David Zhang
David Zhang
Numerade Educator
03:15

Problem 172

Sketch the temperature dependence of the heat conductivity of an insulating solid. State the simple temperature dependencies in limiting temperature ranges and dervie them quantitatively.

Hunza Gilgit
Hunza Gilgit
Numerade Educator
03:15

Problem 173

Express the equilibrium heat flow equation in terms of the heat capacity, excitation or particle velocity, mean free path, and thermal gradient. Discuss the manifestation of quantum mechanics and quantum statistics in the thermal conductivity of a metal.

Hunza Gilgit
Hunza Gilgit
Numerade Educator
05:23

Problem 174

A liquid helium container, shown in Fig. 2.38, contains $1000 \mathrm{~cm}^3$ of liquid helium and has a total wall area of $600 \mathrm{~cm}^2$. It is insulated from the surrounding liquid nitrogen reservoir by a vacuum jacket of 0.5 cm thickness. Liquid helium is at 4.2 K , while liquid nitrogen is at 77 K . If the vacuum jacket is now filled with helium at a pressure of $10 \mu \mathrm{m} \mathrm{Hg}$, estimate how long it will take for all the $1000 \mathrm{~cm}^3$ liquid helium in the container to disappear. (As a crude approximation, we can assume a constant temperature gradient across the vacuum jacket and evaluate the thermal conduction of the He-filled jacket at its mean temperature.
FIGURE CAN'T COPY.

Dading Chen
Dading Chen
Numerade Educator
01:33

Problem 175

Many properties of a gas of atoms can be estimated using a simple model of the gas as an assembly of colliding hard spheres. The purpose of this problem is to derive approximate expressions for a number of coefficients that are used to quantitatively describe various phenomena. For each of the coefficients below state your answer in terms of: $k=$ Boltzmann's constant, $T=$ temperature, $R=$ radius of atom, $m=$ mass of atom, $c=$ heat capacity per gram, $\rho=$ density. You may neglect factors of order unity.
(a) Derive the coefficient of thermal conductivity, $\kappa$ (units: $\mathrm{g} \cdot \mathrm{cm} / \mathrm{s}^3$. $\mathrm{K})$. This occurs in the relation between the heat flux and the temperature gradient.
(b) Derive the coefficient of viscosity, $\eta$ (units: $\mathrm{g} / \mathrm{cm} \cdot \mathrm{s}$ ). This occurs in the relation between the tangential force per unit area and the velocity gradient.
(c) Derive the diffusion coefficient, $D$ (units: $\mathrm{cm}^2 / \mathrm{s}$ ). This characterizes a system containing gases of two species. It relates the time rate of change of the density of one species to its inhomogeneity in density.

Surendra Kumar
Surendra Kumar
Numerade Educator
06:02

Problem 176

The speed of sound $\left(c_s\right)$ in a dilute gas like air is given by the adiabatic compressibility:
$$
c_s^2=\left[\left(\frac{\partial \rho}{\partial p}\right)_a\right]^{-1}=\gamma \frac{k T}{M},
$$
where $M$ is the mean molecular weight, $k$ is Boltzmann's constant and $\gamma$ is the ratio of principal specific heats.
(a) Estimate numerically for air at room temperature,
(1) the speed of sound;
(2) the mean molecular collisions frequency;
(3) the molecular mean free path;
(4) the ratio of mean free path to typical wave length;
(5) the ratio of typical wave frequency to collision frequency. (Use $\nu=300 \mathrm{~Hz}$ as typical wave frequency.)
(b) Use the ratios found above to explain why adiabatic conditions are relevant for sound.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
View

Problem 177

The speed of sound in a gas is calculated as
$$
v=\sqrt{\text { adiabatic bulk modulus/density }} .
$$
(a) Show that this is a dimensionally-correct equation.
(b) This formula implies that the propagation of sound through air is a quasi-static process. On the other hand, the speed of sound for air is about $340 \mathrm{~m} / \mathrm{sec}$ at a temperature for which the rms speed of an air molecule is about $500 \mathrm{~m} / \mathrm{sec}$. How then can the process be quasi-static?

Victor Salazar
Victor Salazar
Numerade Educator

Problem 178

Consider a non-interacting relativistic Fermi gas at zero temperature.
(a) Write down expressions for the pressure and the energy density in the rest frame of the gas. What is the equation of state?
(b) Treating the system as a uniform static fluid, derive a wave equation for the propagation of small density fluctuations, and hence deduce an expression for the velocity of sound in the gas.

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07:32

Problem 179

A beam of energetic ( $>100 \mathrm{eV}$ ) neutral hydrogen atoms is coming through a hole in the wall of plasma confinement device. Describe the apparatus that you would use to measure the energy distribution of these atoms.
FIGURE CAN'T COPY.

Robert Schnibbe
Robert Schnibbe
Numerade Educator
09:25

Problem 180

Write the Maxwell distribution, $P\left(v_x, v_y, v_z\right)$, for the velocities of molecules of mass $M$ in a gas at pressure $p$ and temperature $T$. (If you have forgotten the normalization constant, derive it from the Gaussian integral,
$$
\int_{-\infty}^{\infty} \exp \left(-x^2 / 2 \sigma^2\right) d x=\sqrt{2 \pi} \sigma .
$$

When a clean solid surface is exposed to this gas it begins to absorb molecules at a rate $W$ (molecules $/ \mathrm{s} \cdot \mathrm{cm}^2$ ).

A molecule has absorption probability 0 for a normal velocity component less than a threshold $v_T$, and absorption probability 1 for a normal velocity greater than $v_T$. Derive an expression for $W$.

Alex Mangiapane
Alex Mangiapane
Numerade Educator
07:20

Problem 181

A gas in a container consists of molecules of mass $m$. The gas has a well defined temperature $T$. What is
(a) the most probable speed of a molecule?
(b) the average speed of the molecules?
(c) the average velocity of the molecues?

Shital Rijal
Shital Rijal
Numerade Educator
03:22

Problem 182

Find the rate of wall collisions (number of atoms hitting a unit area on the wall per second) for a classical gas in thermal equilibrium in terms of the number density and the mean speed of the atoms.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
02:00

Problem 183

At time $t=0$, a thin walled vessel of volume $V$, kept at constant temperature, contains $N_0$ ideal gas molecules which begin to leak out through a small hole of area $A$. Assuming negligible pressure outside the vessel, calculate the number of molecules leaving through the hole per unit time and the number remaining at time $t$. Express your answer in terms of $N_0, A, V$, and the average molecular velocity, $\bar{v}$.

Adriano Chikande
Adriano Chikande
Numerade Educator
04:43

Problem 184

A beam of molecules is often produced by letting gas escape into a vacuum through a very small hole in the side of the container confining the gas. The total intensity of the beam is defined as the number of molecules escaping from the hole per unit time. Find the change in total intensity of the beam if:
(a) the area of the hole is increased by a factor of 4 ;
(b) the absolute temperature is increased by a factor of 4 , the pressure being maintained constant;
(c) the pressure in the container is increased by a factor of 4 , the temperature remaining constant,
(d) at the original temperature and pressure, a gas of 4 times the molecular weight of the original gas is used.

A. Elizabeth Hildreth
A. Elizabeth Hildreth
Numerade Educator
04:21

Problem 185

Derive a rough estimate for the mean free path of an air molecule at STP. What is the path length between collisions for
(a) a slow molecule?
(b) a fast molecule?

Niamat Khuda
Niamat Khuda
Numerade Educator
08:11

Problem 186

A simple molecular beam apparatus is shown in Fig. 2.40. The oven contains $\mathrm{H}_2$ molecules at 300 K and at a pressure of 1 mm of mercury. The hole on the oven has a diameter of $100 \mu \mathrm{m}$ which is much smaller than the molecular mean free path. After the collimating slits, the beam has a divergence angle of 1 mrad . Find:
(a) the speed distribution of molecules in the beam;
(b) the mean speed of molecules in the beam;
(c) the most probable speed of molecules in the beam;
(d) the beam power (number of molecules passing through the last collimating split per unit time);
(e) the average rotational energy of $\mathrm{H}_2$ molecules.
FIGURE CAN'T COPY.

Christopher Provencher
Christopher Provencher
Numerade Educator
02:00

Problem 187

Consider a gas at temperature $T$ and pressure $p$ escaping into vacuum through a hole of area $A$ which is in the wall of its container. Assume the radius of the hole is much less than the mean free path for the gas in the container.
(a) Roughly, what is the mass-rate of escape of the gas?
(b) If the gas is a mixture, is the relative mass-rate of escape of a component dependent only upon its relative concentration?

Adriano Chikande
Adriano Chikande
Numerade Educator
03:14

Problem 188

Consider a two-dimensional classical system with Hamiltonian
$$
H=\frac{1}{2 m}\left(P_1^2+P_2^2\right)+\frac{1}{2} \mu^2\left(x_1^2+x_2^2\right)-\frac{1}{4} \lambda\left(x_1^2+x_2^2\right)^2 .
$$

A system of $N$ particles of mass $m$ each is in thermal equilibrium at temperature $T$ within the potential well that appears in the Hamiltonian. $T$ is small enough so that an overwhelming majority of the particles reside within the quadratic part of the well. However, some particles will always possess enough thermal energy to escape from the well by passing over the "top" of the well; in the one-dimensional slice of $V(\mathrm{x})$ shown in Fig. 2.14, this occurs at $x_1=b$, where $b$ can be determined from the above equation.
FIGURE CAN'T COPY.
Calculate the escape rate for particles to leave the well by passing over the top.

Stanley Enemuo
Stanley Enemuo
Numerade Educator
03:55

Problem 189

A sealed $\frac{1}{4}$ litre bottle filled with oxygen at a pressure of $10^{-4}$ atmospheres is left on the surface of the moon by an astronaut. At a time when the temperature of the bottle is 400 K the jar develops a leak at a thin part of a wall through a small hole of diameter 2 microns. How will the amount of gas in the bottle depend on time and about how long will it take for the gas to decrease to $\frac{1}{10}$ of its original amount?

Show your work, estimate any constants needed besides Boltzmann's constant. You may assume that the temperature is maintained constant by the sunlight of the lunar day.
$$
k=1.38 \times 10^{-16} \mathrm{erg} / \mathrm{K} .
$$

Surendra Kumar
Surendra Kumar
Numerade Educator
05:51

Problem 190

(a) What fraction of $\mathrm{H}_2$ gas at sea level and $T=300 \mathrm{~K}$ has sufficient speed to escape from the earth's gravitational field? (You may assume an ideal gas. Leave your answer in integral form.)
(b) Now imagine an $\mathrm{H}_2$ molecule in the upper atmosphere with a speed equal to the earth's escape velocity. Assume that the remaining atmosphere above the molecule has thickness $d=100 \mathrm{~km}$, and that the earth's entire atmosphere is isothermal and homogeneous with mean number density $n=$ $2.5 \times 10^{25} / \mathrm{m}^3$ (not a very realistic atmosphere).

Using simple arguments, estimate the average time needed for the molecule to escape. Assume all collisions are elastic, and that the total atmospheric height is small compared with the earth's radius.
Some useful numbers:
$$
\begin{aligned}
M_{\text {earth }} & =6 \times 10^{24} \mathrm{~kg}, \\
R_{\text {earth }} & =6.4 \times 10^3 \mathrm{~km} .
\end{aligned}
$$

Averell Hause
Averell Hause
Carnegie Mellon University
01:28

Problem 191

(a) Consider the emission or absorption of visible light by the molecules of hot gas. Derive an expression for the frequency distribution $F(\nu)$ expected for a spectral line of central frequency $\nu_0$ due to the Doppler broadening. Assume an ideal gas at temperature $T$ with molecular mass $M$. Consider a vessel filled with argon gas at a pressure of 10 Torr ( 1 Torr $=$ 1 mm of mercury) and a temperature of $200^{\circ} \mathrm{C}$. Inside the vessel is a small piece of sodium which is heated so that the vessel will contain some sodium vapor. We observe the sodium absorption line at $5896 \AA$ in light from a tungsten filament passing through the vessel. Estimate:
(b) The magnitude of the Doppler broadening of the line.
(c) The magnitude of the collision broadening of the line.
Assume here that the number of sodium atoms is very small compared to the number of argon atoms. Make reasonable estimates of quantities that you may need which are not given and express your answers for the broadening in angstroms.

Arun Bana
Arun Bana
Numerade Educator
01:22

Problem 192

A gas consists of a mixture of two types of molecules, having molecular masses $M_1$ and $M_2$ grams, and number densities $N_1$ and $N_2$ molecules per cubic centimeter, respectively.

The cross-section for collisions between the two different kinds of molecules is given by $A\left|V_{12}\right|$, where $A$ is a constant, and $V_{12}$ is the relative velocity of the pair.
(a) Derive the average, over all pairs of dissimilar molecules, of the center-of-mass kinetic energy per pair.
(b) How many collisions take place per cubic centimeter per second between dissimilar molecules?

Ajay Singhal
Ajay Singhal
Numerade Educator
03:34

Problem 193

Consider air at room temperature moving through a pipe at a pressure low enough so that the mean free path is much longer than the diameter of the pipe. Estimate the net flux of molecules in the steady state resulting from a given pressure gradient in the pipe. Use this result to calculate how long it will take to reduce the pressure in a tank of 100 litres volume from $10^{-5} \mathrm{~mm}$ of Hg to $10^{-8} \mathrm{~mm}$ of Hg , if it is connected to a perfect vacuum through a pipe one meter long and 10 cm in diameter. Assume that the outgassing from the walls of the tank and pipe can be neglected.

Amany Waheeb
Amany Waheeb
Numerade Educator
06:06

Problem 194

Consider the hydrodynamical flow conditions. The cooling of the gas during expansion can be expressed as follows, $\frac{T_0}{T}=1+\frac{M^2}{3}$, where $T_0$ is the temperature before expansion, $T$ is the temperature after expansion, and $M$ is the ratio of the flow velocity $v$ to the velocity of sound $c$ at temperature $T$.
(a) Derive the above expression.
(b) Derive a corresponding expression for $\frac{p_0}{p}$, and calculate the value of $M$ for a condition where $\frac{p_0}{p}=10^4$.
(c) Calculate the value of $T$ for $\frac{p_0}{p}=10^4$ and $T_0=300 \mathrm{~K}$.
(d) Find the maximum value of $v$ in the limit $T \rightarrow 0$.

Christopher Dzorkpata
Christopher Dzorkpata
Numerade Educator
02:19

Problem 195

The schematic drawing below (Fig. 2.42) shows the experimental set up for the production of a well-collimated beam of sodium atoms for an atomic beam experiment. Sodium is present in the oven $S$, which is kept at the temperature $T=550 \mathrm{~K}$. At this temperature the vapor pressure of sodium is $p=6 \times 10^{-3}$ torr. The sodium atoms emerge through a slit in the wall of the oven. The hole is rectangular, with dimensions 10 mm $\times 0.1 \mathrm{~mm}$. The collimator $C$ has a hole of identical size and shape, and the sodium atoms which pass through $C$ thus constitute the atomic beam under consideration. The atomic mass of sodium is 23 . The distance $d$ in the figure is 10 cm .
Fig. 2.42.
(a) Compute the number $\phi$ of sodium atoms which pass through the slit in $C$ per second.
(b) Derive an expression for the function $D(v)$ which describes the distribution of velocities of the particles in the beam in the sense that $D(v) d v$ is the probability that an atom passing through $C$ has a velocity in the range $(v, v+d v)$.
(c) The region in which the beam propagates must, of course, be a reasonably good vacuum. Estimate (and give answer in torr) just how
good the vacuum ought to be if the beam is to remain collimated for at least 1 meter. $(1$ torr $=1 \mathrm{mmHg})$.
FIGURE CAN'T COPY.

Hailey Tomashek
Hailey Tomashek
Numerade Educator
03:51

Problem 196

An insulated box of volume 2 V is divided into two equal parts by a thin, heat-conducting partition. One side contains a gas of hard-sphere molecules at atmospheric pressure and $T=293 \mathrm{~K}$.
(a) Show that the number of molecules striking the partition per unit area and unit time is $n \bar{v} / 4$.
(b) A small round hole of radius $r$ is opened in the partition, small enough so that thermal equilibrium between the two sides is maintained via heat conduction through the partition. Calculate the pressure and temperature as functions of time in both halves of the box.
(c) Suppose the partition is a non-conductor of heat. Discuss briefly and qualitatively any deviations from the time-dependence of temperature and pressure found in part (b).

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:49

Problem 197

Consider a two-dimensional ideal monatomic gas of $N$ molecules of mass $M$ at temperature $T$ constrained to move only in the $x y$ plane. The usual volume becomes in this case an area $A$, and the pressure $p$ is the force per unit length (rather than the force per unit area).
(a) Give an expression for $f(v) d v$, the total number of molecules with speeds between $v$ and $v+d v$. (Assume that the classical limit is applicable in considering the behavior of these molecules).
(b) Give the equation of state (relating pressure, temperature etc.).
(c) Give the specific heats at constant area (two dimensional analogue of specific heat at constant volume) and at constant pressure.
(d) Derive a formula for the number of molecules striking unit length of the wall per unit time. Express your result in terms of $N, A, T, M$ and any other necessary constants.

Brian Francisco
Brian Francisco
Numerade Educator
08:11

Problem 198

A parallel beam of $\mathrm{Be}(\mathrm{A}=9)$ atoms is formed by evaporation from an oven heated to 1000 K through a small hole.
(a) If the beam atoms are to traverse a 1 meter path length with less than $1 / e$ loss resulting from collisions with background gas atoms at room temperature ( 300 K ), what should be the pressure in the vacuum chamber? Assume a collision cross-section of $10^{-16} \mathrm{~cm}^2$, and ignore collisions between 2 beam atoms.
(b) What is the mean time $(\bar{\tau})$ for the beam atoms to travel one meter? Show how the exact value for $\bar{\tau}$ is calculated from the appropriate velocity distribution. Do not evaluate integrals. Make a simple argument to get a numerical estimate for $\bar{\tau}$.
(c) If the Be atoms stick to the far wall, estimate the pressure on the wall due to the beam where the beam strikes the wall. Assume the density of particles in the beam is $10^{10} / \mathrm{cm}^3$. Compare this result with the pressure from the background gas.
FIGURE CAN'T COPY.

Christopher Provencher
Christopher Provencher
Numerade Educator
05:34

Problem 199

A quantity of argon gas (molecular weight 40 ) is contained in a chamber at $T_0=300 \mathrm{~K}$.
(a) Calculate the most probable molecular velocity.
A small hole is drilled in the wall of the chamber and the gas is allowed to effuse into a region of lower pressure.
(b) Calculate the most probable velocity of the molecules which escape through the hole.

The pressures of the chamber and the region outside the hole are adjusted so as to sustain a hydrodynamic flow of gas through the hole, such that viscous effects, turbulence, and heat exchange with the wall of the hole may be neglected. During this expansion the gas is cooled to a temperature of 30 K .
(c) Calculate the velocity of sound $c$ at the lower temperature.
(d) Calculate the average flow velocity $\bar{v}$ at the lower temperature, and compare the distribution of velocities with the original distribution in the chamber.

Eric Mockensturm
Eric Mockensturm
Numerade Educator

Problem 200

Estimate, to within an order of magnitude, on the basis of kinetic theory the heat conductivity of a gas in terms of its temperature, density, molecular weight, and heat capacity at constant volume. Make your own estimates of collision cross-sections and molecular mean free paths. You may restrict your attention to pressures near atmospheric, temperatures near room temperature and dimensions of the order of centimeters or meters. Do not concern yourself with heat transfer by convection. $\left(k=1.38 \times 10^{-16} \mathrm{erg} / \mathrm{K}\right)$.

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02:30

Problem 201

A propagating sound wave causes periodic temperature variations in a gas. Thermal conductivity acts to remove these variations but it is generally claimed that the waves are adiabatic, that is, thermal conductivity is too slow.

The coefficient of thermal conductivity for an ideal gas from kinetic theory is $k \approx 1.23 C_v \bar{v} l$ where $C_v$ is the heat capacity per unit volume, $\bar{v}$ is the mean thermal speed, and $l$ is the mean free path.

What fraction of the temperature variation $\Delta T$ will be conducted away vs $\lambda$ and what is the condition on $\lambda$ for thermal conductivity to be ineffective?
FIGURE CAN'T COPY.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
01:13

Problem 202

Give a qualitative argument based on the kinetic theory of gases to show that the coefficient of viscosity of a classical gas is independent of the pressure at constant temperature.

Rashmi Gondi
Rashmi Gondi
Numerade Educator

Problem 203

Consider a dilute gas whose molecules of mass $m$ have mean velocity of magnitude $\bar{v}$. Suppose that the average velocity in the $x$-direction $u_x$ increases monotonically with $z$, so that $u_x=u_x(z)$ with $\left|u_x\right| \ll \bar{v}$ and all gradients small. There are $n$ molecules per unit volume and their mean free path is $l$ where $l \gg d$ (molecular diameter) and $l \ll L$ (linear dimension of enclosing vessel).
(a) The viscosity $\eta$ is defined as the proportionality constant between the velocity gradient and the stress in the $x$-direction on an imaginary plane whose normal points in the $z$-direction. Find an approximate expression for $\eta$ in terms of the parameters given.
(b) If the scattering of molecules is treated like that of hard spheres, what is the temperature dependence of $\eta$ ? The pressure dependence? Assume a Maxwellian distribution in both cases.
(c) If the molecular scattering cross section $\sigma \propto E_{\mathrm{cm}}^2$, where $E_{\mathrm{cm}}$ is the center-of-mass energy of two colliding particles, what is the temperature dependence of $\eta$ ? Again assume a Maxwellian distribution.
(d) Estimate $\eta$ for air at atmospheric pressure $\left(10 \mathrm{dyn} / \mathrm{cm}^2\right)$ and room temperature. State clearly your assumptions.

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02:50

Problem 204

Electrical conductivity. Derive an approximate expression for the electrical conductivity, $\sigma$, of a degenerate electron gas of density $n$ electrons $/ \mathrm{cm}^3$ in terms of an effective collision time, $\tau$, between the electrons.

Chai Santi
Chai Santi
Numerade Educator
01:14

Problem 205

Consider a system of charged particles confined to a volume $V$. The particles are in thermal equilibrium at temperature $T$ in the presence of an electric field $E$ in the $z$-direction.
(a) Let $n(z)$ be the density of particles at the height $z$. Use equilibrium statistical mechanics to find the constant of proportionality between $\frac{d n}{d z}$ and $n$.
(b) Suppose that the particles can be characterized by a diffusion coefficient $D$. Using the definition of $D$ find the flux $J_D$ arising from the concentration gradient obtained in (a).
(c) Suppose the particles are also characterized by a mobility $\mu$ relating their drift velocity to the applied field. Find the particle flux $J_\mu$ associated with this mobility.
(d) By making use of the fact that at equilibrium the particle flux must vanish, establish the Einstein relation between $\mu$ and $D$ :
$$
\mu=\frac{e}{k} \frac{D}{T} .
$$

Anand Jangid
Anand Jangid
Numerade Educator

Problem 206

Consider a system of degenerate electrons at a low temperature in thermal equilibrium under the simultaneous influence of a density gradient and an electric field.
(a) How is the chemical potential $\mu$ related to the electrostatic potential $\phi(x)$ and the Fermi energy $E_{\mathrm{F}}$ for such a system?
(b) How does $E_{\mathrm{F}}$ depend on the electron density $n$ ?
(c) From the condition for $\mu$ under thermal equilibrium and the considerations in (a) and (b), derive a relation between the electrical conductivity $\sigma$, the diffusion coefficient $D$ and the density of states at the Fermi surface for such a system.

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Problem 207

Consider a non-interacting Fermi gas of electrons. Assume the electrons are nonrelativistic.
(a) Find the density of states $N(E)$ as a function of energy $(N(E)$ is the number of states per unit energy interval) for the following cases:
1) The particles are constrained to move only along a line of length L.
2) The particles move only on a two dimensional area $A$.
3) The particles move in a three dimensional volume $V$.
(b) In a Fermi electron gas in a solid when $T \ll T_{\mathrm{F}}$ (the gas temperature is much less than the Fermi temperature), scattering by phonons and impurities limits electrical conduction. In this case, the conductivity $\sigma$ can be written as
$$
\sigma=e^2 N\left(E_{\mathrm{F}}\right) D,
$$
where $e$ is the electron charge, $N\left(E_{\mathrm{F}}\right)$ is the density of states, defined above, evaluated at the Fermi energy and $D$ is the electron diffusivity. $D$ is proportional to the product of the square of the Fermi velocity and the mean time, $\tau_\epsilon$, between scattering events ( $D \sim v_{\mathrm{F}}^2 \tau_e$ ).
1) Give a physical argument for the dependence of the diffusivity on $N\left(E_{\mathrm{F}}\right)$.
2) Calculate the dependence of $\sigma$ on the total electron density in each of the three cases listed in part (a). The electron density is the total number of electrons per unit volume, or per unit area, or per unit length, as appropriate.

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01:14

Problem 208

(a) List and explain briefly the assumptions made in deriving the Boltzmann kinetic equation.
(b) The Boltzmann collision integral is usually written in the form
$$
\left(\partial f\left(\mathbf{r}, \mathbf{v}_1, t\right) / \partial t\right)_{\text {coll }}=\int d^3 \mathbf{v}_2 \int d \Omega \sigma(\Omega)\left|\mathbf{v}_1-\mathbf{v}_2\right|\left(f_1^{\prime} f_2^{\prime}-f_1 f_2\right),
$$
where $f_1=f\left(\mathbf{r}, \mathbf{v}_1, t\right), f_2^{\prime}=f\left(\mathbf{r}, \mathbf{v}_2^{\prime}, t\right)$ and $\sigma(\Omega)$ is the differential cross section for the collision $\left(\mathbf{v}_1, \mathbf{v}_2\right) \rightarrow\left(\mathbf{v}_1^{\prime}, \mathbf{v}_2^{\prime}\right)$. Derive this expression for the collision integral and explain how the assumptions come in at various stages.

Anand Jangid
Anand Jangid
Numerade Educator