Chapter 13, Thermal Properties of Solids Video Solutions, Heat and Thermodynamics

Problem 1

It was shown in Eq. (13.8) that the partition function of a crystal is given by
$$
Z=\frac{e^{-h v / 2 k T}}{1-e^{-h v / k T}}
$$
If the crystal consists of $N_{\mathrm{A}}$ lattice points:
(a) Calculate the Helmholtz function $A$.
(b) Calculate the pressure $P$.
(c) Calculate the entropy $S$.
(d) Express the zero-point energy in terms of $\Theta_{\mathrm{E}}$.

Stanley Enemuo

Numerade Educator

01:19

Problem 2

Using the Debye approximation, show that the total internal energy $U$ of $N_{\mathrm{A}}$ lattice points $\left(3 N_{\mathrm{A}}\right.$ independent harmonic oscillators) is
$$
U=?_{8}^{9} R \Theta+\frac{9 R T}{(\Theta / T)^{3}} \int_{0}^{\theta / T} \frac{x^{3} d x}{e^{x}-1}
$$
What is the interpretation of the term $\frac{9}{8} R \Theta ?$

Adriano Chikande

Numerade Educator

03:22

Problem 3

Plot the Einstein $c_{V} / 3 R$ curve against $T / \Theta_{\mathrm{E}}$ on the same graph as the Debye $c_{V} / 3 R$ curve against $T / \Theta$, and then compare the two curves.

Monica Miller

Numerade Educator

06:58

Problem 4

A system consists of $N_{\mathrm{A}}$ distinguishable, independent particles, cach of which is capable of existing in only two nondegenerate energy states 0 and $\epsilon$.
(a) What is the partition function?
(b) Calculate the internal energy.
(c) Calculate $c_{\mathcal{V}}$.
(d) Plot $c_{V} / R$ as a function of $k T / \epsilon$ from $k T / \epsilon=0$ to 1 .

Rashmi Sinha

Numerade Educator

02:32

Problem 5

Given a crystal obeying Debye's approximation with $\Theta$ a function of $V$ only, assume the entropy $S$ to be a function of $T / \Theta$. The Gruneisen constant $\Gamma$ is defined by the equation
$$
\Gamma=-\frac{d \ln \Theta}{d \ln V}
$$
(a) Show that
$$
\Gamma=\frac{\beta V}{C_{y} \kappa}
$$
(Hint: Use Maxwell's third relation.)
(b) Calculate $\Gamma$ of $\mathrm{NaCl}$ at a few temperatures using Table $9.5$, and plot $\Gamma$ against $\boldsymbol{T}$.
(c) Show that
$$
\gamma=1+\Gamma \beta T
$$

Ajay Singhal

Numerade Educator

05:20

Problem 6

The partition function of a Debye crystal is
$$
\ln Z=-\frac{9}{(\Theta / T)^{3}} \int_{0}^{\theta / T} x^{2} \ln \left(1-e^{-x}\right) d x
$$
(a) Show that
$$
\ln Z=-3 \ln \left(1-e^{-\theta / T}\right)+\frac{9}{(\Theta / T)^{3}} \int_{0}^{\Theta / T} \frac{x^{3} d x}{e^{x}-1}
$$
(b) Calculate the Helmholtz function.
(c) Show that the equation of state of the crystal is given by
$$
P V+f(V)=\Gamma\left(U-U_{0}\right)
$$
where $U_{0}$ is the zero-point energy.

Suzanne W.

Numerade Educator

02:36

Problem 7

Show that the partition function $Z_{F D}$ for particles of half integral spin, called "fermions," is given by
$$
Z_{\mathrm{FD}}(T, V, \mu)=\prod_{i=0}^{\infty}\left(1+e^{-\left(\sigma_{i}-\mu\right) / k T}\right)^{2},
$$
where the Fermi energy $\epsilon_{F}=\mu$, the chemical potential.

Chai Santi

Numerade Educator

02:36

Problem 8

Show that the partition function $Z_{B E}$ for particles of integral spin, called "bosons," is given by
$$
Z_{\mathrm{BE}}(T, V, \mu)=\prod_{i=0}^{\infty}\left(\frac{1}{1-e^{-\left(\epsilon_{4}-\mu\right) / k T}}\right),
$$
where $\mu$ is the chemical potential.

Chai Santi

Numerade Educator