$ \bar{p} =\displaystyle\frac{ N }{ V } k_B T $

mp = N * k_B * T / V

$ p V = n R_C T $

p * V = n * R_C * T

$ R_C = N_A k_B $

R_C = N_A * k_B

$ S = N k_B \left(\displaystyle\frac{3}{2}\ln\displaystyle\frac{ U }{ N }+\ln\displaystyle\frac{ V }{ N }+ \ln \gamma \right)$

S = N * k_B *((3/2)*log( U / N )+ log( V / N )+ log( g ))

$ \gamma = B \left(\displaystyle\frac{2 m }{ h ^2}\right)^{3/2}$

g = B *(2* m / h ^2)^3/2

$ S = N k_B \ln\left(\left(\displaystyle\frac{ U }{ N }\right)^{3/2}\displaystyle\frac{ V }{ N } \gamma \right)$

S = N * k_B *ln(( U / N )^(3/2)*( V / N )* gamma )