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# Difference between revisions of "Ideal gas Helmholtz energy function"

From equations

$Q_{NVT}=\frac{1}{N!} \left( \frac{V}{\Lambda^{3}}\right)^N$
$\left.A\right.=-k_B T \ln Q_{NVT}$

for the Helmholtz energy function, one has

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

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.