Ideal gas Helmholtz energy function

From equations

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

and

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.A\right.=-k_{B}T\ln Q_{NVT}}

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =-k_BT\left( -N\ln N +N + N\ln N - N\ln \Lambda^3 \rho \right)}

one arrives at

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=Nk_BT\left(\ln \Lambda^3 \rho -1 \right)}