Ideal gas: Heat capacity

The heat capacity at constant volume is given by

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 C_V = \left. \frac{\partial U}{\partial T} \right\vert_V }

where $U$ is the internal energy. Given that an ideal gas has no interatomic potential energy, the only term that is important is the kinetic energy of an ideal gas, which is equal to 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 3/2 RT} . Thus

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 C_V = \frac{\partial ~ }{\partial T} \left( \frac{3}{2}RT \right) = \frac{3}{2} R }

One has

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 C_p - C_V = \left.\frac{\partial V}{\partial T}\right\vert_p \left(p + \left.\frac{\partial U}{\partial V}\right\vert_T \right) }

for an ideal gas this becomes:

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 \left.C_p -C_V \right.=R}

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 R} is the molar gas constant.

References

1. Donald A. McQuarrie "Statistical Mechanics" (1976) Eq. 1-1
2. Landau and Lifshitz Course of Theoretical Physics Volume 5 Statistical Physics 3rd Edition Part 1 Equation 42.11