Internal energy

The internal energy is given by

$U=-T^{2}\left.{\frac {\partial (A/T)}{\partial T}}\right\vert _{N,V}=k_{B}T^{2}\left.{\frac {\partial \log Z(T)}{\partial T}}\right\vert _{N,V}$

where 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 A} is the Helmholtz energy function 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 Z} is the partition function.