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Ideal gas Helmholtz energy function

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From equations

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


\left.A\right.=-k_B T \ln Q_{NVT}

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)