(a) State the first law of thermodynamics. Derive the Maxwell relation

$$\left(\frac{\partial S}{\partial V}\right)_{T}=\left(\frac{\partial p}{\partial T}\right)_{V}$$

(b) Consider a thermodynamic system whose energy $E$ at constant temperature $T$ is volume independent, i.e.

$$\left(\frac{\partial E}{\partial V}\right)_{T}=0$$

Show that this implies that the pressure has the form $p(T, V)=T f(V)$ for some function $f$.

(c) For a photon gas inside a cavity of volume $V$, the energy $E$ and pressure $p$ are given in terms of the energy density $U$, which depends only on the temperature $T$, by

$$E(T, V)=U(T) V, \quad p(T, V)=\frac{1}{3} U(T)$$

Show that this implies $U(T)=\sigma T^{4}$ where $\sigma$ is a constant. Show that the entropy is

$$S=\frac{4}{3} \sigma T^{3} V$$

and calculate the energy $E(S, V)$ and free energy $F(T, V)$ in terms of their respective fundamental variables.