Ignoring the zero-point energy, show that the par- tition function Z for a gas of photons in volume V is given by ln Z= - V/(pi ² c ³) ∫ 0 [infinity] ω² ln(1-e⁻ ω β )d ω,
(23.68)
and hence, by integrating by parts, that ln Z= V pi ² (k_BT) ³ 45 ³ c³ . (23.69)
Hence show that F = - (4σV T ⁴)/(3c) S = (16σV T ³)/(3c) U = (4σV T ⁴)/c p = (4σ T ⁴)/(3c) (23.70) (23.71) (23.72) (23.73)
and hence that U = - 3F pV = U / 3 and S = 4U/3T.