Chapter 12, Statistical Mechanics Video Solutions, Heat and Thermodynamics

Problem 1

A mercury atom moves in a cubical box whose edge is $1 \mathrm{~m}$ long. Its kinetic energy is equal to the average kinetic energy of an atom of an ideal gas at $1000 \mathrm{~K}$. If the quantum numbers $n_{x}, n_{y}$, and $n_{z}$ are all equal to $n$, calculate $n .$

Mayukh Banik

Numerade Educator

02:02

Problem 2

The quantum states available for gas atoms of energy $\epsilon$ in a cubical box of length $L$ correspond to integer values for cach $n_{x}, n_{y}$, and $n_{z}$, according to Eq. (12.1). In a three-dimensional Euclidean space with coordinates $n_{x}, n_{y}$, and $n_{z+}$ each unit volume will contain one quantum state. The total number of quantum states $\boldsymbol{g}^{\prime}$ with energy less than $\epsilon^{\prime}$ is equal to the volume of the positive octant of a sphere of radius $r=L\left(8 m \epsilon_{i}\right)^{1 / 2} / h$
(a) Show that
$$
g^{\prime}=\frac{4 \pi V\left(2 m \epsilon^{\prime}\right)^{3 / 2}}{3 h^{3}}
$$
(b) In a volume of $1 \mathrm{~cm}^{3}$ of helium gas at $300 \mathrm{~K}$ and $1 \mathrm{~atm}$ pressure, $\epsilon^{\prime}$ is about $10^{-5}$ J. Calculate $g^{\prime}$.
(c) Calculate the number $N$ of helium atoms.
(d) Show that $g^{\prime} \gg N$.

Stanley Enemuo

Numerade Educator

11:54

Problem 3

Take the expression for the kinetic energy of a particle in a cubical box and imagine a space defined by the cartesian coordinates $n_{x}, n_{y}$, and $n_{z}$. Note that a single quantum state occupies unit volume in this space.
(a) Setting $n^{2}=n_{x}^{2}+n_{y}^{2}+n_{t}^{2}$, show that the number of quantum states in the small interval $d n$ is $\frac{1}{8}\left(4 \pi n^{2} d n\right)$.
(b) Prove that the number of quantum states $d g_{\epsilon}$ in the energy interval $d \epsilon$ is $\left(2 \pi / h^{3}\right) V(2 m)^{3 / 2} \epsilon^{1 / 2} d \epsilon$
(c) Show that the number of ideal-gas particles $d N_{e}$ occupying these quantum states is given by
$$
d N_{t}=\frac{2 N}{\pi^{1 / 2}} \frac{1}{(k T)^{3 / 2}} \epsilon^{1 / 2} e^{-\epsilon / k T} d \epsilon
$$
(d) Derive the Maxwell's speed-distribution function, that is, Eq. (12.36).

Ren Jie Tuieng

Numerade Educator

02:46

Problem 4

Show that, when $N$ ideal-gas atoms come to equilibrium,
and
$$
\begin{aligned}
&\frac{g_{i}}{N_{i}}=\frac{Z}{N} e^{c_{1} / k T} \\
&\frac{Z}{N}=\frac{(k T)^{5 / 2}}{P}\left(\frac{2 \pi m}{h^{2}}\right)^{3 / 2}
\end{aligned}
$$
Taking $\epsilon_{i}=\frac{3}{2} k T, T=300 K, P=10^{3} \mathrm{~Pa}$, and $m=10^{-26} \mathrm{~kg}_{1}$ calculate $\boldsymbol{g}_{i} / N_{i} .$

Stephen Ho

Numerade Educator

01:15

Problem 5

Consider a function $f$ defined by the relation
$$
f\left(\Omega_{A} \Omega_{B}\right)=f\left(\Omega_{A}\right)+f\left(\Omega_{B}\right)
$$
First, differentiate partially with respect to $\Omega_{B}$, and then with respect to $\Omega_{A}$. Integrate twice to show
$$
f(\Omega)=\text { const. } \ln \Omega+\text { const. }
$$

Narayan Hari

Numerade Educator

08:22

Problem 6

In the case of $N$ distinguishable particles, the number of ways $\Omega$, in which a macrostate defined by $N_{1}$ particles in $g_{1}$ quantum states with energy $\epsilon_{1}, N_{2}$ particles in $g_{2}$ quantum states with energy $\epsilon_{2}, \cdots$, may be achieved, is given by the MaxwellBoltzmann expression

Maria Gabriela Cota Moreira

Numerade Educator

08:28

Problem 7

Given $N$ indistinguishable, quasi-independent particles capable of existing in energy levels $\epsilon_{1}, \epsilon_{2}, \cdots$, with degeneracies $g_{1}, g_{2}, \cdots$, respectively; in any given macrostate in which there are $N_{1}$ particles in energy level $\epsilon_{1}, N_{2}$ particles in energy level $\epsilon_{2}, \cdots$, assume the thermodynamic probability to be given by the Bose-Einstein expression,
$$
\Omega_{\mathrm{BE}}=\frac{\left(g_{1}+N_{1}\right) !\left(g_{2}+N_{2}\right) ! \cdots}{g_{1} ! N_{1} \backslash g_{2} ! N_{2} !}
$$
Using Stirling's approximation and the method of Lagrangian multipliers, render $\ln \Omega_{\mathrm{BE}}$ a maximum, subject to the equations of constraint $\sum N_{i}=N=$ const. and $\sum N_{i} \epsilon_{i}=U=$ const., and show that
$$
N_{i}=\frac{\boldsymbol{g}_{i}}{\lambda e^{-\beta \mathrm{k}_{4}}-1}
$$

Andrew Eddins

Emory University

08:28

Problem 8

Assume the same system as in Prob. $12.7$, except that the thermodynamic probability is given by the Fermi-Dirac expression,
$$
\Omega_{\mathrm{FD}}=\frac{g_{1} ! g_{2} ! \ldots}{N_{1} !\left(\boldsymbol{g}_{1}-N_{1}\right) ! N_{2} !\left(\boldsymbol{g}_{2}-N_{2}\right) ! \cdots}
$$
Using Stirling's approximation and the method of Lagrangian multipliers, render $\ln \Omega_{F D}$ a maximum, subject to the equations of constraint $\sum N_{i}=N=$ const, and $\sum N_{i} \epsilon_{i}=U=$ const., and show that
$$
N_{i}=\frac{\boldsymbol{g}_{i}}{\lambda e^{-\delta k_{1}}+1}
$$

Andrew Eddins

Emory University

03:37

Problem 9

Given a gaseous system of $N_{\mathrm{A}}$ indistinguishable, weakly interacting diatomic molecules:
(a) Each molecule may vibrate with the same frequency $v$, but with an energy $\epsilon_{i}$, given by
$$
\epsilon_{i}=\left(\frac{1}{2}+i\right) h v \quad(i=0,1,2, \cdots)
$$
Show that the vibrational partition function $Z_{y}$ is
$$
Z_{v}=\frac{e^{-h v / 2 k T}}{1-e^{-h v / k T}}
$$
(b) Each molecule may rotate, and the rotational partition function $Z$, has the same form as that for translation, except that the volume $V$ is replaced by the total solid angle $4 \pi$, the mass is replaced by the moment of inertia 1, and the exponent $\frac{3}{2}$ (referring to three translational degrees of freedom) is replaced by $\frac{2}{2}$
since there are only two rotational degrees of freedom. Write the rotational partition function.
(c) Taking into account translation, vibration, and rotation, calculate the Helmholtz function.
(d) Calculate the pressure.
(e) Calculate the internal energy.
(f) Calculate the molar heat capacity at constant volume.

Adriano Chikande

Numerade Educator

00:37

Problem 10

Defining the average speed (w) by the equation
show that
$$
\begin{aligned}
&\langle w\rangle=\frac{1}{N} \int_{0}^{\infty} w d N_{w}, \\
&\langle w\rangle=\sqrt{\frac{8 k T}{\pi m}} .
\end{aligned}
$$

Mohit Sharma

Numerade Educator

02:53

Problem 11

(a) In Fig. 12-5, let $w_{m}$ be the value of $w$ at which $d N_{w} / d w$ is a maximum. Calculate $w_{m}$
(b) Choose a new variable $x=w / w_{m}$, and calculate $d N_{x} / N d x .$ What is the maximum value of $d N_{x} / N d x ?$

Gregory Higby

Numerade Educator

01:19

Problem 12

(a) Calculate $\langle 1 / w\rangle$ and compare the result with $1 /\langle w\rangle$.
(b) Show that the number of particles striking a unit area of a wall per unit time is equal to
$$
\frac{P}{\sqrt{2 \pi m k T}}
$$

Prem Bijarniya

Numerade Educator

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Problem 13

The Doppler broadening of a spectral line increases with the rms speed of the atoms in the source of light. What should give narrower spectral lines: a mercury-198 lamp at $300 \mathrm{~K}$ or a krypton-86 lamp at $77 \mathrm{~K}$ ?

James Kiss

Numerade Educator

07:19

Problem 14

At what temperature is the mean translational kinetic energy of an atom equal to that of a singly charged ion of the same mass which has been accelerated from rest through a potential difference of: $(a) 1 \mathrm{~V} ?(b) 1,000 \mathrm{~V} ?(c) 1,000,000 \mathrm{~V} ?$ (Note:
Neglect relativistic effects.)

Shalini Tyagi

Numerade Educator

02:19

Problem 15

An oven contains cadmium vapor at a pressure of $2.28 \mathrm{~Pa}$ and at a temperature of $550 \mathrm{~K}$. In one wall of the oven there is a slit with a width of $10^{-5} \mathrm{~m}$ and a length of $10^{-2} \mathrm{~m} .$ On the other side of the wall is a very high vacuum. If one assumes that all the atoms arriving at the slit pass through, what is the atomic beam current?

Hailey Tomashek

Numerade Educator

04:39

Problem 16

A vessel of volume $V$ contains a gas that is kept at constant temperature. The gas slowly leaks out of a small hole of area $A$. The outside pressure is so low that no atoms leak back.
(a) Prove that the pressure at any time $t$ is given by
$$
P=P_{0} e^{-k^{\prime} t}
$$
where $P_{0}$ is the initial pressure.

Surendra Kumar

Numerade Educator

02:48

Problem 17

A spherical glass bulb of $0.1 \mathrm{~m}$ radius is maintained at $300 \mathrm{~K}$, except for an appendix with a cross-sectional area of $10^{-4} \mathrm{~m}^{2}$ immersed in liquid nitrogen, as shown in Fig. P12-1. The bulb contains water vapor originally at a pressure of $13.3 \mathrm{~Pa}$. Assuming that every water molecule that enters the appendix condenses on the wall and stays there, find the time required for the pressure to decrease to $1.33 \times 10^{-4} \mathrm{~Pa}$.

Km Neeraj

Numerade Educator

02:13

Problem 18

A vessel partially filled with mercury, and closed except for a hole of area $10^{-7} \mathrm{~m}^{2}$ above the liquid level, is kept at $0^{\circ} \mathrm{C}$ in a continuously evacuated enclosure. After 30 days, it is found that $2.4 \times 10^{-5} \mathrm{~kg}$ of mercury has been lost. What is the vapor pressure of mercury at $0^{\circ} \mathrm{C} ?$

Ajay Singhal

Numerade Educator