Each quadratic degree of freedom in the energy contributes $\frac{1}{2}{k}_{B}T$ to the average energy.

For a monatomic ideal gas: 3 translational DOF $\to$ $⟨E⟩=\frac{3}{2}{k}_{B}T$

Diatomic molecules

A diatomic molecule has additional rotational modes. The principal axes of a diatomic molecule:

• 2 rotational DOF with $I>0$ (axes perpendicular to the bond)
• 1 axis with $I=0$ along the bond — does not contribute (quantum effect: energy spacing too large to excite at typical temperatures)

Total DOF: 3 translational + 2 rotational = 5

Molar heat capacity

At constant volume, $C_V=\frac{\partial ⟨E⟩}{\partial T}$ per mole. Using $k_B\to R$ (per mole):

• Monatomic: $C_V=\frac{3}{2}R$
• Diatomic (no vibration): $C_V=\frac{5}{2}R$
• Diatomic (with vibration): $C_V=\frac{7}{2}R$ — vibrational mode adds 2 DOF (kinetic + potential)

Vibration only activates at high temperatures. At room temperature, diatomic gases behave as if $C_V=\frac{5}{2}R$.

Connection to partition function

For a diatomic gas, the partition function factorizes:

Average energy from partition function:

Each independent quadratic DOF contributes a factor of $\sqrt{\pi /\beta }$ to $Z$, and $\frac{1}{2}{k}_{B}T$ to $⟨E⟩$ — recovering equipartition.

Quantum reason for low-temperature disagreement

Equipartition is a classical result — it assumes energy is continuous. Quantum mechanics says each mode has discrete energy levels. For a vibrational mode:

A mode contributes $\frac{1}{2}{k}_{B}T$ only if it can be thermally excited, i.e. $k_{B}T\gg \hslash \omega$. If $k_{B}T\ll \hslash \omega$, the system lacks enough energy to reach the first excited state — the mode is frozen out and contributes nothing.

This defines a characteristic temperature for each mode:

• $T\ll \mathrm{\Theta }$: mode frozen, no contribution to $C_V$
• $T\gg \mathrm{\Theta }$: mode fully active, classical equipartition recovered

For diatomic molecules, ${\mathrm{\Theta }}_{\mathrm{vib}}\gg {\mathrm{\Theta }}_{\mathrm{rot}}\gg {\mathrm{\Theta }}_{\mathrm{trans}}$, which is why modes activate at different temperatures. This is why $C_V$ is experimentally observed to rise in steps as $T$ increases, rather than being constant as classical equipartition predicts.