Ideal gas Helmholtz energy function

From equations

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

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

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

for the Helmholtz energy function, one has

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

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

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

using Stirling's approximation

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

one arrives at

$A=N{k}_{B}T(\ln {\mathrm{\Lambda }}^3\rho -1)$

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