PX284 - H4 - ideal gas

ideal gases are classical and indistinguishable
therefore, the partition function:

Z_{N} = \frac{Z_{1}^{N}}{N!} = \frac{1}{N!} {(\frac{V}{λ_{t h}^{3}})}^{N}

where, $λ_{t h} = \frac{h}{\sqrt{2 π m k_{B} T}}$

using stirling's approximation:

\ln Z_{N} = N \ln V - 2 N \ln λ_{t h} - N \ln N + N

previously:

p = - {(\frac{\partial F}{\partial V})}_{T}

where, $F = - k_{B} T \ln Z$

p = k_{B} T {(\frac{\partial \ln Z_{N}}{\partial V})}_{T} = N k_{B} T \frac{\partial \ln V}{\partial V} = N k_{B} T

therefore, the ideal gas law is obtained:

p V = N k_{B} T

it can also be found that:

U = \frac{3}{2} N k_{B} T ⟹ C_{V} = \frac{3}{2} N k_{B} T

this agrees with equipartition
also:

F = N k_{B} T [\ln (\frac{N λ_{t h}^{3}}{V}) - 1]

hence, the sackur-tetrode equation:

S = \frac{U - F}{T} = F = N k_{B} [\frac{5}{2} - \ln (\frac{N λ_{t h}^{3}}{V})]