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 (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 = qe, where e 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→∞, the heat capacity is C = Nk. (Hint: When x is very small, eª ≈ 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/e, 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 €, in electron-volts, for each of those real solids.