Write down the first law of thermodynamics in differential form applied to an infinitesimal reversible change.

Explain what is meant by an adiabatic change.

Starting with the first law in differential form, derive the Maxwell relation

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

Hence show that

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

For radiation in thermal equilibrium at temperature $T$ in volume $V$, it is given that $E=V e(T)$ and $P=e(T) / 3$. Hence deduce Stefan's Law,

$$E=a V T^{4}$$

where $a$ is a constant.

The radiation is allowed to expand adiabatically. Show that $V T^{3}$ is constant during the expansion.