Equipartition of Energy

The equipartition theorem is a deep result of classical statistical mechanics: each independent quadratic term in the molecular energy has an average value of $\tfrac{1}{2} k_B T$ in thermal equilibrium.

Concept

For any energy term of the form $\alpha q^2$ (where $q$ is a coordinate or velocity component and $\alpha$ is a constant) in the Hamiltonian, the average value at temperature $T$ is

$\langle \alpha q^2\rangle = \tfrac{1}{2} k_B T$

Examples of such quadratic terms:

• Translational kinetic energy: $\tfrac{1}{2} m v_x^2, \tfrac{1}{2} m v_y^2, \tfrac{1}{2} m v_z^2$ — three terms.
• Rotational kinetic energy: $\tfrac{1}{2} I \omega^2$ — one term per rotational axis.
• Vibrational energy: $\tfrac{1}{2} m \dot u^2 + \tfrac{1}{2} k u^2$ — two terms per mode (KE and PE).

Total average energy per molecule

$\langle E\rangle = \frac{f}{2} k_B T$

where $f$ is the total count of quadratic terms.

Internal energy of $n$ moles

$U = n N_A \langle E\rangle = \frac{f}{2} n R T$

Molar heat capacity at constant volume

$C_v = \left(\frac{\partial U}{\partial T}\right)_V = \frac{f}{2} R$

And via Mayer's relation, $C_p = C_v + R$.

Computing $C_v, C_p$ for common gases

• Monatomic ($f = 3$): $C_v = \tfrac{3}{2} R$, $C_p = \tfrac{5}{2} R$, $\gamma = 5/3 \approx 1.67$.
• Rigid diatomic ($f = 5$): $C_v = \tfrac{5}{2} R$, $C_p = \tfrac{7}{2} R$, $\gamma = 7/5 = 1.4$.
• Diatomic with vibration ($f = 7$): $C_v = \tfrac{7}{2} R$, $\gamma = 9/7 \approx 1.29$.
• Polyatomic non-linear ($f = 6$): $C_v = 3 R$, $\gamma = 4/3 \approx 1.33$.

Limits of validity

Equipartition is purely classical. It fails when energy quanta become comparable to $k_B T$:

• Vibrational modes have large quanta ($\hbar\omega \gg k_B T$ at room temperature for most diatomics) and are "frozen out".
• At very low $T$, even rotational modes freeze, and H$_2$ shows $C_v \to \tfrac{3}{2} R$.
• Solids: Dulong–Petit law $C \approx 3 R$ per mole comes from equipartition (3 translational + 3 vibrational positions = 6 quadratic terms... wait, 3 atoms vibrate with KE + PE, giving 6 terms, hence $3R$).

Worked Example

Q: Compute the internal energy of 2 moles of oxygen gas at 300 K, treating it as a rigid diatomic.

A: $f = 5$. $U = (5/2) n R T = (5/2)(2)(8.314)(300) \approx 12471$ J $\approx 12.5$ kJ.

Q2: A solid has 6 quadratic terms per atom (3 KE + 3 PE for vibrations). What is its molar heat capacity?

A: $C \approx \tfrac{6}{2} R = 3 R \approx 24.9$ J/(mol·K). This is the Dulong–Petit value, reasonable for many metals at room temperature.

Common Confusions

• Equipartition counts quadratic terms, not just coordinates. Vibrations contribute 2 per mode (KE + PE).
• It is a classical result — fails at low temperatures where quantum effects matter.
• For $C_p$, use $C_p = C_v + R$ (per mole), not $C_p = (f/2 + 1) R$ derived separately.

Key Takeaways

• Each quadratic energy term contributes $\tfrac{1}{2} k_B T$ on average.
• $C_v = (f/2) R$, $C_p = (f/2 + 1) R$, $\gamma = (f+2)/f$.
• Equipartition gives the law of Dulong–Petit for solids ($\approx 3R$) and the heat capacities of all ideal gases at moderate temperatures.