Problem 3.25 (Schroeder’s Intro to Thermal Physics)

Problem 3.25

In Problem 2.18 you showed that the multiplicity of an Einstein solid containing N oscillators and q energy units is approximately

$\Omega (N,q) \approx \Bigl( \dfrac{q+N}{q} \Bigr)^q \Bigl( \dfrac{q+N}{N} \Bigr)^N$

(a) Starting with this formula, find an expression for the entropy of an Einstein solid as a function of $N$ and $q.$ Explain why the factors omitted from the formula have no effect on the entropy, when $N$ and $q$ are large.

(b) Use the result of part (a) to calculate the temperature of an Einstein solid as a function of its energy. (The energy is $U = q \epsilon ,$ where $\epsilon$ is a constant.) Be sure to simplify your result as much as possible.

(c) Invert the relation you found in part (b) to find the energy as a function of temperature, then differentiate to find a formula for the heat capacity.

(d) Show that, in the limit $T \rarrow \infty ,$ the heat capacity is $C = Nk.$ (Hint: When $x$ is very small, $e^x \approx 1 + x$ .) Is this the result you would expect? Explain.

(e) Make a graph (possibly using a computer) of the result of part (c). To avoid awkward numerical factors, plot $C/Nk$ vs. the dimensionless variable $t = kT / \epsilon,$ for $t$ in the range from 0 to about 2. Discuss your prediction for the heat capacity at low temperature, comparing to the data for lead, aluminum, and diamond shown in Figure 1.14. Estimate the value of $\epsilon,$ in electron-volts, for each of those real solids.

(f) Derive a more accurate approximation for the heat capacity at high temperatures, by keeping terms through $x^3$ in the expansions of the exponentials and then carefully expanding the denominator and multiplying everything out. Throw away terms that will be smaller than $(\epsilon /kT)^2$
2 in the final answer. When the smoke clears, you should find $C = Nk[1 - \frac{1}{12}(\epsilon / kT)^2]$