Blundell Thermal Physics · Problem 23.6

Photons

Physics — Statistical Mechanics · photon-gas-partition-function

Problem

Ignoring the zero-point energy, show that the partition function

Z

for a gas of photons in volume

V

is given by

\ln Z = -\frac{V}{\pi^2 c^3}\int_0^\infty \omega^2 \ln(1 - \mathrm{e}^{-\hbar\omega\beta})\,\mathrm{d}\omega,\tag{23.68}

and hence, by integrating by parts, that

\ln Z = \frac{V\pi^2(k_\mathrm{B}T)^3}{45\hbar^3 c^3}.\tag{23.69}

Hence show that

\begin{aligned} F &= -\frac{4\sigma V T^4}{3c},\tag{23.70}\\ S &= \frac{16\sigma V T^3}{3c},\\ U &= \frac{4\sigma V T^4}{c},\\ p &= \frac{4\sigma T^4}{3c}, \end{aligned}

and hence that

U = -3F

pV = U/3

, and

S = 4U/3T