Problem 3.25. In Problem 2.18 you showed that the multiplicity of an Einstein solid containing / oscillators and q energy units is approximately ?( N. q) = ( 9 + N)" + NN N (a) Starting with this formula, find an expression for the entropy of an Einstein solid as a function of N and q. (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 7 - co, the heat capacity is C = NK. (Hint: When a is very small, e* ~ 1 +r.) 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 O/N vs. the dimensionless variable t = kT/e, for t in the range from 0 to about 2. (f) Derive a more accurate approximation for the heat capacity at high temper- atures, by keeping terms through a" in the expansions of the exponentials and then carefully expanding the denominator and multiplying everything out. Throw away terms that will be smaller than (c/k?') in the final answer. When the smoke clears, you should find C = NA[1 - 12 (4/kT)-]