Energy equation

The energy equation is given, in classical thermodynamics, by

${\frac{\partial U}{\partial V}|}_T=T{\frac{\partial p}{\partial T}|}_V-p$

and in statistical mechanics it is obtained via the thermodynamic relation

$U=\frac{\partial (A/T)}{\partial (1/T)}$

and making use of the Helmholtz energy function and the canonical partition function one arrives at

$\frac{U^{\mathrm{e}\mathrm{x}}}{N}=\frac{\rho }{2}\underset{0}{\overset{\mathrm{\infty }}{\int }}\mathrm{\Phi }(r)\mathrm{g}(r)4\pi r^2\mathrm{d}r$

where $\mathrm{\Phi }(r)$ is a two-body central potential, $U^{\mathrm{e}\mathrm{x}}$ is the excess internal energy per particle, and $\mathrm{g}(r)$ is the radial distribution function.