+ \item Find an expression for the pressure as a function of the free energy
+ $F=E-TS$.
+ Rewrite this equation to express the pressure entirely in terms of
+ the internal energy $E$.
+ Evaluate the pressure by using the harmonic form of the internal energy.
+ {\bf Hint:}
+ Step 2 introduced an integral over the temperature $T'$.
+ Change the integration variable $T'$ to $x=\hbar\omega_s({\bf k})/T'$.
+ Use integration by parts with respect to $x$.
+ \item The normal mode frequencies of a rigorously harmonic crystal
+ are unaffected by a change in volume.
+ What does this imply for the pressure
+ (Which variables does the pressure depend on)?
+ Draw conclusions for the coefficient of thermal expansion.
+ \item Find an expression for $C_p-C_V$ in terms of temperature $T$,
+ volume $V$, the coefficient of thermal expansion $\alpha_V$ and
+ the inverse bulk modulus (isothermal compressibility)
+ $\frac{1}{B}=-\frac{1}{V}\left.\frac{\partial V}{\partial p}\right|_T$.\\
+ $C_p=\left.\frac{\partial H}{\partial T}\right|_p$ is the heat capacity
+ for constant pressure and
+ $C_V=\left.\frac{\partial E}{\partial T}\right|_V$ is the heat capacity
+ for constant volume.