The Equipartition Theorem

Core Idea

The equipartition theorem states that each quadratic degree of freedom (each independent way a molecule can store energy) contributes (1/2)kT to the average energy per molecule. Monatomic gases have 3 translational degrees, so U = (3/2)NkT. Diatomic molecules add 2 rotational degrees at room temperature (U = (5/2)NkT) and 2 vibrational degrees at high temperatures. This explains why different gases have different heat capacities and why heat capacity can change with temperature.

How It's Best Learned

Physically interpret each degree of freedom: three independent directions of translation, two rotation axes for a dumbbell-shaped diatomic molecule. Predict Cv for monatomic, diatomic, and triatomic gases and compare to experimental values.

Common Misconceptions

• Vibrational modes are not always active — at room temperature, quantum effects 'freeze out' vibrational degrees in diatomic gases; classical equipartition overestimates Cv in that regime.
• The theorem is statistical and holds only on average over many molecules.

Explainer

From your study of the kinetic theory of gases — specifically rms speed and kinetic energy — you know that the average translational kinetic energy of a molecule in thermal equilibrium is (3/2)kT. This result came from computing ⟨½mv²⟩ = ½m⟨v_x² + v_y² + v_z²⟩ and using the Maxwell-Boltzmann distribution. The equipartition theorem generalizes this: the factor of (3/2) comes from having three independent translational directions, and each one contributes exactly (1/2)kT. The theorem says this is not a coincidence — it is a universal rule that applies to any quadratic term in the energy, regardless of whether it is kinetic or potential.

The precise statement is: for any degree of freedom that appears quadratically in the total energy — of the form ½ax² for any constant a and generalized coordinate x — the thermal average of that term is exactly (1/2)kT, independent of a. "Quadratic" is the key word. A spring has potential energy ½kx² and kinetic energy ½mv²: both are quadratic, so each contributes (1/2)kT to the average energy. A vibrational mode thus contributes (1/2)kT from kinetic energy plus (1/2)kT from potential energy = kT total, compared to (1/2)kT per direction of pure translation. This is why heat capacities differ so dramatically by molecular type.

Applying the theorem: a monatomic gas molecule (e.g., argon) has only three translational degrees of freedom, so its internal energy per molecule is U = 3 × (½kT) = (3/2)kT, and its molar heat capacity at constant volume is C_v = (3/2)R. A diatomic molecule (e.g., N₂) at room temperature adds two rotational degrees (rotation about the two axes perpendicular to the bond axis — rotation about the bond axis has negligible moment of inertia and does not contribute), giving U = (5/2)kT and C_v = (5/2)R. At high temperatures, the two vibrational modes (one kinetic, one potential) activate, pushing C_v toward (7/2)R. These predictions match experiment beautifully in the appropriate temperature ranges.

The important caveat is quantum freezing. The equipartition theorem is a classical result. Quantum mechanics imposes discrete energy levels on each mode, with level spacing ΔE = ħω for vibrations and ΔE ∝ ħ²/I for rotations. If the thermal energy kT is much smaller than ΔE, the mode cannot absorb thermal energy in the small increments that equipartition assumes — it stays in its ground state and contributes essentially zero to the heat capacity. This is why vibrational modes in H₂ are frozen out at room temperature (their ħω is large), while rotational modes in N₂ are not (their ħ²/I is much smaller). The failure of classical equipartition at low temperatures was one of the early clues that classical mechanics was incomplete, and its resolution by quantum statistics was a triumph of early quantum theory.