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 \[A=-k_BT\left(\ln \frac{1}{N!} + N\ln\frac{V}{\Lambda^{3}}\right)\]

\[=-k_BT\left(-\ln N! + N\ln\frac{VN}{\Lambda^3N}\right)\]

\[=-k_BT\left(-\ln N! + N\ln\frac{N}{\Lambda^3 \rho}\right)\]

using Stirling's approximation \[=-k_BT\left( -N\ln N +N + N\ln N - N\ln \Lambda^3 \rho \right)\] one arrives at

\[A=Nk_BT\left(\ln \Lambda^3 \rho -1 \right)\]

where \(\Lambda\)is the de Broglie thermal wavelength and \(k_B\) is the Boltzmann constant.