Past exam of the physics course of the year 4 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / Section B / B2 / c

Write $b=\pi k_B^2/(3\hslash c)$, so $U=bL{T}^2$. At fixed length, $dS=C_{L}dT/T$, giving $S=2bLT$ with the entropy constant fixed by the unique photon vacuum. The thermal Helmholtz free energy and generalized pressure follow:The entropy constant can also be checked from the exact discrete photon partition function; the continuum approximation should not be extrapolated blindly below its level-spacing regime. In one dimension $P=U/L$, whereas the isotropic three-dimensional relation is $p=U/(3V)$. Both follow from the number of spatial directions carrying momentum flux.