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 by equipartition theorem. Thus , while follows by differentiating the ideal-gas enthalpy . Consequently the adiabatic index is and the specific internal energy is . 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.
