= Solution
Set $k_B=1$. After the <Gaussian integral> over momenta, the <canonical partition function> factors as
$$
Z=Z_{\rm kin}\frac1{N!}\int_{V^N}e^{-U(\mathbf x_1,\ldots,\mathbf x_N)/T}\,d^{3N}x,
$$
whereas the <ideal gas> has $Z_{\rm ideal}=Z_{\rm kin}V^N/N!$. Hence
$$
\frac Z{Z_{\rm ideal}}
=\frac1{V^N}\int_{V^N}e^{-U/T}\,d^{3N}x
=1+\frac1{V^N}\int_{V^N}(e^{-U/T}-1)\,d^{3N}x.
$$
Using the <Helmholtz free energy> $F=-T\log Z$ therefore gives
$$
\boxed{F=F_{\rm ideal}-T\log\left\{1+\frac1{V^N}\int_{V^N}(e^{-U/T}-1)\,d^{3N}x\right\}}.
$$
Back to article page
