Ideal gas: Energy

The energy of the ideal gas is given by (Hill Eq. 4-16)

$E=-T^{2}\left.{\frac {\partial (A/T)}{\partial T}}\right\vert _{V,N}=kT^{2}\left.{\frac {\partial \ln Q}{\partial T}}\right\vert _{V,N}=NkT^{2}{\frac {d\ln T^{3/2}}{dT}}={\frac {3}{2}}NkT\equiv {\frac {3}{2}}RT$

where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle R} is the molar gas constant. This energy is all kinetic energy, 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 1/2 kT} per degree of freedom, by equipartition. This is because there are no intermolecular forces, thus no potential energy.

References

1. Terrell L. Hill "An Introduction to Statistical Thermodynamics" 2nd Ed. Dover (1962)