Limits of classical equipartition

Classical counting predicts a contribution from every quadratic energy term. Gas heat capacities show that this rule cannot be applied at every temperature. The missing distinction is between a mode existing and its energy changing appreciably as the sample warms. Here you will use energy gaps to explain freezing without declaring that a frozen mode has no energy.

At a glance

• Prerequisites: equipartition theorem, heat capacity of gases.
• Outcomes: explain the historical mismatch between classical prediction and experiment.
• Key result: low-temperature mode freezing requires quantum energy spacing.
• Common trap: assuming all degrees of freedom are always active at every temperature.

Compare the heat-capacity plateaus

Classical prediction for a diatomic gas with full mode activation:

But experiments at ordinary temperatures often show values closer to

and at lower temperatures can approach (3/2)R.

The missing rotational and vibrational contributions reflect energy spacings that thermal motion cannot readily excite at low T.

Compare the energy gap with thermal energy

Classical equipartition assumes continuous energy and full thermal accessibility. Quantum mechanics gives discrete mode energies.

For a mode with characteristic spacing Δ E:

• If kBT≪Δ E, excited states are rarely occupied.
• That mode contributes little to internal energy change with T.

Physical interpretation:

• Δ E is the typical quantum energy gap between adjacent levels of that mode.
• Heating changes U(T) mainly by populating higher levels. If thermal energy kBT is too small compared with the gap, the population barely changes, so the mode contributes almost nothing to heat capacity.

So heat capacity depends on temperature through mode activation:

• For ordinary molecular gases in their dilute classical regime, translation remains classical over a range in which rotations can be weakly excited.
• Rotational energy spacings are usually smaller than vibrational spacings, so rotational heat capacity becomes appreciable first.
• These are model regimes, not universal activation temperatures. Molecule, density and phase matter; translation cannot be extrapolated as classical all the way to zero temperature.

A frozen mode still exists. A quantum harmonic vibration can retain zero-point energy hν/2; a temperature-independent offset does not contribute to CV = (∂ U/∂ T)V. Freezing means a small temperature-dependent contribution, not zero total energy. Activation is gradual, so a heat capacity between plateaus is not a fractional count of mechanical coordinates. The classical counting rule describes a high-temperature limit for the relevant mode.

Units check:

• Heat capacity remains in J mol⁻¹K⁻¹; what changes is the effective number of active modes.

Physical sanity check:

• In the low-temperature equilibrium limit, heat capacities tend toward zero rather than retaining a classical equipartition value. A real gas may condense before this limit; the dilute classical gas model cannot be extrapolated unchanged to T = 0.

Heat-capacity plateaus from active modes

Practice set

Separate stored energy from heat capacity

Optional quantitative model: one harmonic vibration

For a quantum harmonic mode with spacing Δ = hν, the non-degenerate energies are Eⱼ = (j + 1/2)Δ for j = 0,1,…. This spectrum is the harmonic-oscillator model in MIT transport-process lecture notes, section 2.3.4. Summing its Boltzmann weights gives the mean energy ⟨E⟩ = Δ/2 + Δ/(e^(Δ/(kBT))-1). The first term is the temperature-independent zero-point contribution. Differentiating at fixed frequency gives the molar contribution of one mode C_(V,m) = Rx²e^x/((e^x-1)²), x = Δ/kBT. These are supplied model results for optional quantitative practice. For x≫1, thermal excitation and heat capacity are small while mean energy approaches Δ/2. For x≪1, the mean energy approaches kBT and the mode’s molar capacity approaches R, matching the two-term classical vibration. Do not use this single-mode formula as the entire gas heat capacity.

Explain what a frozen vibration retains

What to keep

A mode can exist and retain energy while contributing little heat capacity. Compare its spacing with kBT, and distinguish stored energy from dU/dT. Plateaus are limiting approximations within a stated material and phase model.