Monatomic gas

A monatomic gas consists of individual atoms or ions rather than molecules. In the classical dilute ideal-gas regime, its three translational degrees of freedom give $U=3N{k}_{B}T/2$ by equipartition theorem. Thus $C_V=3N{k}_B/2$, while $C_P-C_V=N{k}_B$ follows by differentiating the ideal-gas enthalpy $H=U+PV=5N{k}_{B}T/2$. Consequently the adiabatic index is $\gamma =C_P/C_V=5/3$ and the specific internal energy is $e=3P/(2\rho )$. The same translational result applies to a fully ionized mixture when the particle numbers are fixed. This excludes additional excitation, ionization and radiation contributions to the heat capacity.