The following calculations are for EM radiation, for which the internal energy U=Vu, where u=u(T)is the total energy density.

Question

a) Show that (∂S/∂T)_v=V/T(du/dT), and (∂S/∂V)T=(u+P)/T
Note: use the answers in part a in part b and c
b) Using Stefan-Boltzmann law, u=aT^4, where a is a constant, integrate the expression for (∂S/∂T)_v, and obtain an expression for S in terms of a, T and V.
Hint: use Planck’s statement of the 3rd law of thermodynamics: lim_T→0 S=0 to evaluate the constant of integration.
c) Assuming P=bT^n, where b and n are constants, use the equality ∂^2S/∂T∂V=∂^2S/∂V∂T to obtain values for b and n. Note: b has the same units as a.