Starting with the density of electromagnetic radiation modes in $\mathbf{k}$-space, determine the energy $E$ of black-body radiation in a box of volume $V$ at temperature $T$.

Using the first law of thermodynamics show that

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

By using this relation determine the pressure $P$ of the black-body radiation.

[You are given the following:

(i) The mean number of photons in a radiation mode of frequency $\omega$ is $\frac{1}{e^{\hbar \omega / T}-1}$,

(ii) $1+\frac{1}{2^{4}}+\frac{1}{3^{4}}+\cdots=\frac{\pi^{4}}{90}$,

(iii) You may assume $P$ vanishes with $T$ more rapidly than linearly, as $T \rightarrow 0$. ]