Ideal gas: Heat capacity

The heat capacity at constant volume is given by

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 }

At constant pressure 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 = \left. \frac{\partial U}{\partial T} \right\vert_p + p \left.\frac{\partial V}{\partial T} \right\vert_p}

we can see that, just as before, 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 \left. \frac{\partial U}{\partial T} \right\vert_p = \frac{3}{2} R }

and from the equation of state of an ideal gas

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 p \left.\frac{\partial V}{\partial T} \right\vert_p = \frac{\partial }{\partial T} (RT) = R}

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_p = C_v + R = \frac{5}{2} R}

where $R$ 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