Explain what is meant by an isothermal expansion and an adiabatic expansion of a gas.

By first establishing a suitable Maxwell relation, show that

$$\left.\frac{\partial E}{\partial V}\right|_{T}=\left.T \frac{\partial p}{\partial T}\right|_{V}-p$$

and

$$\left.\frac{\partial C_{V}}{\partial V}\right|_{T}=\left.T \frac{\partial^{2} p}{\partial T^{2}}\right|_{V}$$

The energy in a gas of blackbody radiation is given by $E=a V T^{4}$, where $a$ is a constant. Derive an expression for the pressure $p(V, T)$.

Show that if the radiation expands adiabatically, $V T^{3}$ is constant.