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 by

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\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 central potential, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U^{\rm ex}} is the excess internal energy per particle, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm g}(r)} is the pair distribution function.