Energy equation

The energy equation is given, in classical thermodynamics, by

$\left.{\frac {\partial U}{\partial V}}\right\vert _{T}=T\left.{\frac {\partial p}{\partial T}}\right\vert _{V}-p$

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

${\frac {U^{\rm {ex}}}{N}}={\frac {\rho }{2}}\int _{0}^{\infty }\Phi (r)~{\rm {g}}(r)~4\pi r^{2}~{\rm {d}}r$

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