Specific Heat With Vibration

When a diatomic molecule's vibrational mode becomes fully active at high temperature, it contributes two additional degrees of freedom—one kinetic and one potential—bringing the total to seven. By equipartition, the molar specific heat at constant volume is $C_v=\frac{f}{2}R=\frac{7}{2}R$. Numerically, $C_v=3.5\times 8.314=29.1$ J/mol\cdotK. Vibrational modes are quantised and stay frozen until the thermal energy $k_{B}T$ becomes comparable to the spacing between vibrational levels, which is why measured $C_v$ rises in steps as temperature increases. The value 20.8 J/mol\cdotK equals $\frac{5}{2}R$, the rigid diatomic case without vibration. The value 12.5 J/mol\cdotK equals $\frac{3}{2}R$, valid only for a monatomic gas. The value 37.4 J/mol\cdotK corresponds to $\frac{9}{2}R$, which over-counts the vibrational contribution. This applies the NCERT law of equipartition, noting that each vibrational mode counts twice because it stores both kinetic and potential energy in the oscillating bond. The corresponding ratio of specific heats falls toward $\gamma =9/7\approx 1.29$ once vibration is fully active, since extra storage modes always lower $\gamma$. As a plausibility check, real diatomic gases show $C_v$ rising from about $\frac{5}{2}R$ toward $\frac{7}{2}R$ as temperature increases and vibrations switch on, so the value 29.1 J/mol\cdotK is physically reasonable for the fully excited diatomic case described.