The molar heat capacity at constant volume ($C_{V,m}$) of helium ($ ext{He}$) is invariant with tem

Question

The molar heat capacity at constant volume ($C_{V,m}$) of helium ($	ext{He}$) is invariant with temperature and remains $3R/2$. However, for hydrogen gas ($	ext{H}_2$), $C_{V,m}$ is $3R/2$ at very low temperatures, increases to $5R/2$ at moderate temperatures, and exceeds $5R/2$ at very high temperatures. Which of the following correctly explains the observed temperature dependence of $C_{V,m}$ for $	ext{H}_2$ compared to $	ext{He}$?

Professor PC Thomas Classes · Answer

By the equipartition theorem, each active translational and rotational degree of freedom contributes $0.5R$ to $C_{V,m}$, while each active vibrational mode contributes $1.0R$. Being monoatomic, $	ext{He}$ only possesses 3 translational degrees of freedom at all temperatures ($C_{V,m} = 1.5R$). Being diatomic, $	ext{H}_2$ has 3 translational, 2 rotational, and 1 vibrational degree of freedom. At very low temperatures ($< 100	ext{ K}$), rotational and vibrational modes are frozen ($C_{V,m} = 1.5R$). At moderate temperatures ($\approx 300	ext{ K}$), rotational modes are fully excited ($C_{V,m} = 1.5R + R = 2.5R$). At very high temperatures ($> 1000	ext{ K}$), the vibrational mode also becomes active, raising $C_{V,m}$ beyond $2.5R$ toward $3.5R$.