Equipartition and Heat Capacities

Why do equal mole samples need different heat inputs?

The previous lesson gave every classical ideal-gas particle the same average translational energy 3 kBT/2 at a fixed temperature. Yet one mole of nitrogen generally needs more heat than one mole of helium for the same temperature rise. The extra energy has places to go: a molecule can rotate and, when thermally excited, vibrate. Heat capacity measures how rapidly total internal energy rises with temperature under the stated constraints.

Degrees of freedom and quadratic energy terms

A free point-like particle requires three coordinates to locate it in space, two if it is constrained to a plane and one if constrained to a line. These are independent translational degrees of freedom. In three dimensions its translational kinetic energy is½m₀vx² +½m₀vy² +½m₀vz². Each term contains the square of one independent velocity component; this is a quadratic energy term.

Classical equipartition generalises this result: each independent thermally active quadratic term in the energy contributes an average½kBT at equilibrium. The theorem itself is stated here rather than proved. “Equal sharing” concerns ensemble averages of energy terms, not identical energy in each molecule at every instant. Its classical use assumes that the relevant modes can actually be thermally excited at the temperature of interest.

A rigid diatomic molecule adds two active rotations

Picture a diatomic molecule as two atoms joined by a bond. It translates along three directions. It can also rotate about two independent axes through its centre of mass perpendicular to the bond. Their kinetic energies are½I₁ω₁² and½I₂ω₂², each adding an average½kBT. In the usual moderate-temperature rigid-diatomic model, rotation about the bond axis is not counted as an active mode: its very small moment of inertia makes the relevant quantum excitation difficult. It is not enough simply to count every imaginable rotation without considering thermal activation.

One vibration stores two half-kBT contributions

A molecule such as CO has a vibrational coordinate, but whether it is thermally active is a separate question. If the bond length can oscillate, let y measure its displacement from equilibrium. Its vibrational energy has a kinetic term½μ(dy/dt)² and a restoring-force potential term½ky². Here μ is the effective mass for that relative vibration and k is a force constant; μ in this formula is not the mole number. The derivative dy/dt means the rate of change of the bond displacement. Both terms are quadratic, so a fully thermally active vibrational mode contributes kBT in total, not½kBT.

At sufficiently low temperature, rotational or vibrational energy levels may be too widely spaced for the corresponding modes to absorb classical equipartition amounts of energy. Vibrations often require higher temperatures than rotations. Thus a diatomic molecule can be well approximated as rigid at room temperature even though it has a bond that can vibrate. Adding every possible vibrational mode with its full kBT regardless of temperature would overpredict heat capacity. The detailed quantum calculation is beyond this chapter.

From internal energy to molar heat capacities

For one mole of ideal gas, internal energy Uₘ depends only on temperature for fixed composition. At constant volume and with no other work modes, Q = ΔU, so molar Cᵥ = dUₘ/dT. The derivative describes energy increase per kelvin; if Uₘ = aRT with constant a, its derivative is aR because dT/dT = 1. A temperature-dependent energy Uₘ is not itself a heat capacity.

At constant pressure, heat also supplies expansion work. For one mole, QP = ΔU + PΔV, while PV = RT gives PΔV = RΔT. Dividing by ΔT gives Cₚ = Cᵥ + R. The ratio γ = Cₚ/Cᵥ is dimensionless. Both the difference and the constant-volume definition are molar statements here; mass-specific capacities use different units and a gas-specific constant R/M.

Monatomic and diatomic predictions

A monatomic gas such as helium or argon has three relevant translational terms, giving Uₘ =3 RT/2. Its Cᵥ is 3 R/2, Cₚ is 5 R/2 and γ is 5/3 ≈1.67. In this simple model no thermally active molecular bond rotations or vibrations are present. The derivative removes T: writing Cᵥ =3 RT/2 would have energy-per-mole units and would be incorrect.

A rigid diatomic gas has five terms: three translation and two rotation. Therefore Uₘ =5 RT/2, Cᵥ =5 R/2, Cₚ =7 R/2 and γ =7/5 =1.40. If one vibrational mode becomes fully active, its two extra quadratic terms raise f to 7. Then Uₘ =7 RT/2, Cᵥ =7 R/2, Cₚ =9 R/2 and γ =9/7. Real transitions between these idealised limits are gradual and depend on the molecule and temperature.

Polyatomic molecules: geometry matters

A nonlinear polyatomic molecule has three active translational and three active rotational terms in the classical rigid model. If s vibrational modes are fully active, each adds kBT, so Uₘ = (3+s) RT, Cᵥ = (3+s) R, Cₚ = (4+s) R and γ = (4+s)/(3+s). Here s counts vibrations while f in the general formula counts quadratic terms; f =6+2 s. Using different symbols avoids confusing a vibration with one energy term.

The distinction between linear and nonlinear is essential. A rigid nonlinear triatomic molecule has Cᵥ ≈3 R and γ≈4/3. A rigid linear triatomic molecule instead has three translational plus two active rotational terms, like the rigid diatomic count before vibrations are added. Therefore “triatomic” alone does not guarantee the 3 R prediction. The prediction below is labelled by its geometry rather than applied indiscriminately to every three-atom molecule.

Read heat-capacity measurements as a test of the model

The measured monatomic values cluster around 3 R/2 and 5 R/2, and the common diatomic values around 5 R/2 and 7 R/2. Water vapour and methane are somewhat above the simplest rigid nonlinear prediction, consistent with additional internal energy storage. The capacity entries are rounded, and the difference column is separately reported, so subtracting the displayed capacity entries need not reproduce every final digit. Small real-gas deviations can also occur. Treat the agreement as evidence for an approximate model, not an assertion that every measured row must be exact at all temperatures.

Other gases, including chlorine and ethane, can show larger capacities than the simplest vibration-free model. Additional thermally excited vibrations can account for more energy storage and a lower γ. Both molecule type and temperature matter; one cannot repair every discrepancy merely by counting every possible vibration as fully active. The comparison motivates checking which modes actually participate.

A solid: three vibrations per atom

An atom in a classical solid vibrates about its equilibrium position in three independent directions. In each direction there is a kinetic and a restoring-force potential term, together averaging kBT. Thus one atom has average vibrational energy 3 kBT, and one mole has Uₘ≈3 RT. In an ordinary solid at atmospheric pressure, volume expansion is small, so PΔV is often negligible compared with ΔU. Heat supplied for a small temperature change is then approximately its internal-energy increase.

The molar values for most listed solids are near 24.9 J mol⁻¹ K⁻¹, despite large differences per kilogram. Different elements have different numbers of atoms per kilogram, making molar comparisons useful. Carbon is a strong room-temperature exception. At low temperatures the classical prediction also fails broadly because not all vibrational modes absorb classical equipartition energy. The measured table therefore illustrates both the usefulness and the limits of the 3 R estimate.

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