Chapter 27, Fluctuations Video Solutions, Thermodynamics: A complete undergraduate course

Problem 1

Show that, for a rigid system in thermal equilibrium with a reservoir,
$$
e^{S_{\mathrm{wat}} / k_{\mathrm{B}}} \propto e^{-F / k_{\mathrm{B}} T}
$$
and for a flexible system in equilibrium with a pressure and temperature reservoir,
$$
e^{s_{\mathrm{wit}} / k_{\mathrm{B}}} \propto e^{-G / k_{\mathrm{B}} T}
$$
where, as usual, $S_{\text {tot }}$ is the total entropy of reservoir and system, and $F, G, T$ are properties of the system alone. Comment on the physical implications.

Jack Hou

Numerade Educator

02:53

Problem 2

Consider joint fluctuations in temperature and volume for a system in equilibrium with a pressure and temperature reservoir. Show that $\partial^2 A / \partial V \partial T=0$ and obtain equations (27.9) and (27.15).

Willis James

Numerade Educator

00:51

Problem 3

Discuss physically why the temperature and volume fluctuations considered in question 27.2 are uncorrelated, whereas in general the internal energy and volume fluctuations are correlated.

James Stenhouse

Numerade Educator

02:30

Problem 4

Obtain (27.17), as follows. Throughout this exercise, we take $N$ to be constant. By considering $U=U(T, V)$, show that
$$
\left.\frac{\partial U}{\partial T}\right|_\alpha=C_V+\left.\left.\frac{\partial U}{\partial V}\right|_T \frac{\partial V}{\partial T}\right|_\alpha,
$$
where $\alpha=p / T$. By considering $V=V(T, p)$, show that
$$
\left.\frac{\partial V}{\partial T}\right|_\alpha=\left.\frac{\partial V}{\partial T}\right|_p+\left.\left.\frac{\partial V}{\partial p}\right|_T \frac{\partial p}{\partial T}\right|_\alpha .
$$
From this, obtain
$$
\left.\frac{\partial V}{\partial T}\right|_\alpha=-\left.\left.\frac{1}{T} \frac{\partial V}{\partial p}\right|_T \frac{\partial U}{\partial V}\right|_T .
$$

Nick Johnson

Numerade Educator

01:00

Problem 5

Obtain equations (27.25) and (27.26).

Raj Bala

Numerade Educator

02:46

Problem 6

Show that equation (27.6) can be written
$$
\Delta U^2=\left.\frac{\partial^2 \ln Z}{\partial \beta^2}\right|_{V, N},
$$
where $\beta=1 / k_{\mathrm{B}} T$ and $Z=e^{-\beta F}$. This result is useful in statistical mechanics, because for many systems the partition function $Z$ can be obtained from (14.95).

Lucas Finney

Numerade Educator