Equipartition and molar heat capacities of gases

Take one term in a molecule's energy of the form ½aq², where q is a velocity component, an angular velocity or a displacement, and a is the mass, moment of inertia or stiffness that belongs with it. In equilibrium at temperature T the Boltzmann factor weights each value of q by e^{(−aq²/2kT)}, so the mean of the term is the ratio of two integrals over q: ∫ ½aq² e^{(−aq²/2kT)} dq divided by ∫ e^{(−aq²/2kT)} dq. Substitute s = q√(a/2kT). Both integrals become pure numbers multiplied by powers of 2kT/a, the coefficient a cancels between them, and what survives is ½kT. That is equipartition: the average does not know whether the term described a heavy atom moving slowly or a stiff spring stretched slightly, only that the term was quadratic. At 300 K, ½kT = 2.07 × 10⁻²¹ J, or 12.9 meV. Applied to the three translational terms ½mvₓ², ½mv_y² and ½mv_z² it returns the (3/2)kT of the kinetic model, which checks that the earlier identification of temperature and this Boltzmann average are one statement, not two.

A molecule of N atoms has 3N coordinates, so 3N independent ways to move, and the quadratic terms follow from how those motions are shared out. Three are always translation of the centre of mass. A linear molecule has two rotational terms, ½I₁ω₁² and ½I₂ω₂², about the two axes perpendicular to its bond; rotation about the bond itself has a moment of inertia so small that its first excited level lies far above any kT the molecule survives, so it never holds energy. A non-linear molecule has three rotational terms. Everything left is vibration, 3N − 5 modes if the molecule is linear and 3N − 6 if not, and each vibration counts twice, because a mode carries a kinetic term ½μẋ² and a potential term ½κx² at once. So the classical ceiling is f = 3 for helium, 3 + 2 + 2 = 7 for N₂ or Cl₂, 3 + 2 + 2 × 4 = 13 for linear CO₂ and 3 + 3 + 2 × 3 = 12 for water vapour. Per mole, U = (f/2)RT and C_V = (f/2)R. The count is the model; whether each term is actually awake at the temperature in hand is the question the measurements answer.

Equipartition gives the energy of one molecule; a heat capacity needs the energy of a mole and its slope. A molecule with f active terms holds f × ½kT on average, and in an ideal gas that is all it holds, because the molecules exert no forces on one another between collisions and so share no potential energy. N = nN_A molecules therefore hold U = N(f/2)kT = (f/2)nRT, since N_A k = R, and U depends on T alone. The molar heat capacity at constant volume is its slope, C_V = (1/n) dU/dT = (f/2)R: 12.47 J mol⁻¹ K⁻¹ for f = 3, 20.79 for f = 5, 29.10 for f = 7, or ½k = 6.9 × 10⁻²⁴ J K⁻¹ per active term per molecule. Nothing else enters. Helium and xenon differ in mass by a factor of 33 and share C_V = 12.47 J mol⁻¹ K⁻¹, and N₂ and CO share 20.79 although their moments of inertia differ, because the coefficient of every quadratic term cancelled in the Boltzmann average: a classical heat capacity counts the ways a molecule can hold energy, not how heavy it is. Mass and stiffness return only through freezing out, which decides whether a term is active at all. The constant-pressure numbers follow from the same f, Cₚ = C_V + R and γ = (f + 2)/f, and are derived from the first law in Molar heat capacities and equipartition (UPIII-07.03); here they are simply a second reading of the count.

Measured C_V at 300 K, in J mol⁻¹ K⁻¹: He 12.5, Ar 12.5, N₂ 20.8, O₂ 21.1, Cl₂ 25.6, CO₂ 28.8. Divide each by R and the numbers become counts. The monatomic gases sit at 1.50: three terms, exactly the kinetic model. Nitrogen sits at 2.50 and oxygen at 2.54, so five terms are awake and the two vibrational ones are not. Chlorine at 3.08 lies between the rigid 2.50 and the fully vibrating 3.50: its vibration is partly excited. Carbon dioxide at 3.46 is nowhere near its classical 6.50; of its four vibrational modes only the two low-frequency bends, with θᵥ = 960 K, are substantially active, and together they lift it by about 0.9R. The pattern is systematic: the stiffer and lighter the bond, the higher the temperature needed to excite it, and at 300 K the light, stiffly bound H₂, N₂, CO and O₂ show almost no vibration. A measured heat capacity is therefore a census, taken at that temperature, of which quadratic terms can accept energy, and a measured γ reports the same census through (f + 2)/f.

Equipartition assumes a term can take energy in any amount. A quantum oscillator of frequency ν takes it only in steps of hν, so compare the step with kT through the vibrational temperature θᵥ = hν/k: I₂ 309 K, Cl₂ 805 K, O₂ 2270 K, N₂ 3390 K, H₂ 6330 K. Level n sits nhν above the ground level and carries the Boltzmann weight e^{(−nθᵥ/T)}. The weights form a geometric series summing to 1/(1 − e^(−θᵥ/T)), so the normalised populations are pₙ = (1 − e^(−θᵥ/T))e^{(−nθᵥ/T)}, and the first excited level is populated relative to the ground level by e^(−θᵥ/T): for nitrogen at 300 K, e^(−11.3) = 1.2 × 10⁻⁵, one molecule in 80 000. Weighting nhν by pₙ gives the mode's mean energy, ⟨E_vib⟩ = hν/(e^(θᵥ/T) − 1), which for nitrogen at 300 K is 1.4 × 10⁻⁴ kT instead of the classical kT: a level nobody occupies stores nothing. Chlorine at 300 K, whose step is only 2.7kT high, holds 0.20kT. The correspondence check is the other limit: for T ≫ θᵥ, e^(θᵥ/T) − 1 ≈ θᵥ/T and ⟨E_vib⟩ → kT, the ½kT + ½kT that equipartition assigns to one vibration. Rotation passes the same test with θᵣₒₜ = ħ²/(2Ik), 2.9 K for N₂ and 85 K for H₂, so its steps are tiny against kT wherever nitrogen is a gas and its two terms count in full. The heat capacity obtained by differentiating ⟨E_vib⟩, and the temperature-dependent γ that goes with it, are worked through in Molar heat capacities and equipartition (UPIII-07.03).

The kinetic model gave pV = (2/3)Uₜᵣ with Uₜᵣ the translational energy alone, and temperature was identified through that translational share: ⟨½mv²⟩ = (3/2)kT in every gas, whatever else the molecule can do. A heat capacity counts everything. So when 208 J raises a mole of nitrogen by 10 K at constant volume, only 3/5 of it, 125 J, goes into translation and shows up as a pressure rise; the other 83 J sits in rotation, invisible to the gauge but not to the calorimeter. Two consequences carry forward. For a given heat input a gas with more active terms warms less, so a mole of diatomic gas is a heat reservoir 5/3 the size of a mole of monatomic gas. And in a mixture each species keeps its own count, U = Σ(fᵢ/2)nᵢRT, so the C_V of a mixture is the mole-weighted mean of the components' C_V, while its γ is the ratio of the summed Cₚ to the summed C_V and is not the mole-weighted mean of the γ values. Check every heat-capacity answer twice: the unit must be J mol⁻¹ K⁻¹, and C_V/R must land between 1.5 and the classical f/2.