1. The black body temperature (2.9) is defined as, i.e.
T(S,V,N_1,……N_r)≡\frac{\partial U(S,V,N_1,…,N_r)}{\partial S}.
T(S,V)= \frac{\partial U(S,V)}{\partial S}\biggl(\frac{3c}{16\sigma } \biggr)^{1/3} S^{1/3}V^{-1/3}.
When inverting this relation, we obtain the radiation entropy S (T, V) as a function of
the temperature T and of the volume V, i.e.
S (T, V) =\biggl(\frac{16\sigma}{3c } \biggr) T^{3}V.
When substituting this result into the expression for the internal energy of the radiation U(S, V), we find,
U=\frac{4\sigma}{c}T^4V.
The free energy F (T, V) is obtained by Legendre transformation (4.22) of the internal energy U(S, V) with respect to the entropy S. Using the two previous equations this transformation is written explicitly as, i.e.
F = U − TS
F(T,V)=U-TS=-\frac{4\sigma}{3c}T^4V.
2. For the black body radiation, the entropy defined by (4.26) is found to be,
S(T,V,\left\{N_A\right\} )=-\frac{\partial F(T,V,\left\{N_A\right\})}{\partial T}.
S(T,V)=-\frac{\partial F(T,V)}{\partial T}=\frac{16\sigma}{3c}T^3V.
When inverting this relation, we obtain the radiation temperature T (S, V) as a function of the entropy S and of the volume V, i.e.
T(S,V)= \biggl(\frac{3c}{16\sigma } \biggr)^{1/3} S^{1/3}V^{-1/3}.
When substituting for T in the radiation free energy F (T, V), we find,
F=- \frac{1}{4}\biggl(\frac{3c}{16\sigma } \biggr)^{1/3} S^{4/3}V^{-1/3}.
The internal energy U(S, V) is obtained by Legendre transformation (4.22) of the free energy F (T, V) with respect to the temperature T. Using the two previous equations, this transformation is written explicitly as,
F = U − TS
U(S,V)=F+ST= \frac{3}{4}\biggl(\frac{3c}{16\sigma } \biggr)^{1/3} S^{4/3}V^{-1/3}.
3. According to definition (2.10), the black body radiation pressure p (S, V) is expressed in terms of S and V as,
p(S,V,N_1,……N_r)≡-\frac{\partial U(S,V,N_1,…,N_r)}{\partial V}.
p(S,V)=–\frac{\partial U(S,V)}{\partial V}= \frac{1}{4}\biggl(\frac{3c}{16\sigma } \biggr)^{1/3} S^{4/3}V^{-4/3}.
According to definition (4.27), the black body radiation pressure p (T, V) is expressed in terms of T and V as,
p(T,V,\left\{N_A\right\} )=-\frac{\partial F(T,V,\left\{N_A\right\})}{\partial V}.
p(T,V)=-\frac{\partial F(T,V)}{\partial V}=\frac{4\sigma}{3c }T^4.