Ideal gas Helmholtz energy function

From equations

$Q_{NVT}={\frac {1}{N!}}\left({\frac {V}{\Lambda ^{3}}}\right)^{N}$

for the canonical ensemble partition function for an ideal gas, and

$\left.A\right.=-k_{B}T\ln Q_{NVT}$

for the Helmholtz energy function, one has

$A=-k_{B}T\left(\ln {\frac {1}{N!}}+N\ln {\frac {V}{\Lambda ^{3}}}\right)$

$=-k_{B}T\left(-\ln N!+N\ln {\frac {VN}{\Lambda ^{3}N}}\right)$

$=-k_{B}T\left(-\ln N!+N\ln {\frac {N}{\Lambda ^{3}\rho }}\right)$

using Stirling's approximation

$=-k_{B}T\left(-N\ln N+N+N\ln N-N\ln \Lambda ^{3}\rho \right)$

one arrives at

$A=Nk_{B}T\left(\ln \Lambda ^{3}\rho -1\right)$

where $\Lambda$is the de Broglie thermal wavelength and $k_{B}$ is the Boltzmann constant.