Equipartition of Energy
½kT for every way a molecule can store energy.
Lesson at a glanceShow the key points
- Degrees of freedom: independent squared energy terms; 3 monatomic, 5 diatomic (2 rotations), 7 with vibration.
- Equipartition: average ½kT per degree of freedom.
- Monatomic: Cv = 3R/2, Cp = 5R/2, γ = 1.67. Diatomic: Cv = 5R/2, Cp = 7R/2, γ = 1.40; vibrating: 9/7 = 1.29.
- Experiment matches; diatomic molecules don't vibrate at room temperature; CO₂, H₂O, CH₄ act like f = 6 (CO₂ partly vibrating).
Count the ways a molecule can move and you can predict the heat capacity of a gas, before measuring anything.
Degrees of freedom
A point-like molecule has kinetic energy ½mvₓ² + ½mv_y² + ½mv_z²: three independent terms. That suits monatomic gases.
A diatomic molecule is like a dumbbell. Besides moving, it can rotate about the two axes perpendicular to its length: ½I_xω_x² + ½I_yω_y². (Rotation about its own axis contributes nothing, since point atoms have no moment of inertia about it.) That makes 5 terms. If the atoms also vibrate along the bond, add ½μv² + ½kr²: 7 terms.
The equipartition theorem
For a monatomic gas this gives (3/2)kT per molecule, as kinetic theory found. For n moles with f degrees of freedom, U = n(f/2)RT, so
Measured at 15°C (Cᵥ in J K⁻¹ mol⁻¹, γ):
Red line: equipartition prediction fR/2.
| Gas type | f | Cᵥ (J K⁻¹ mol⁻¹) | Cₚ | γ |
|---|---|---|---|---|
| Monatomic | 3 | 12.5 | 20.8 | 1.67 |
| Diatomic, rotating | 5 | 20.8 | 29.1 | 1.40 |
| Diatomic, also vibrating | 7 | 29.1 | 37.4 | 1.29 |
Measured values agree well: He and Ar have Cᵥ ≈ 12.5 and γ ≈ 1.67; H₂, O₂, N₂ and CO have Cᵥ ≈ 20.8 and γ ≈ 1.40, so ordinary diatomic molecules don’t vibrate at room temperature. CO₂, H₂O and CH₄ behave as if f = 6.
Find the internal energy of 2.0 g of helium (4.0 g/mol) at 300 K.
Find the internal energy of 2.0 g of nitrogen (28 g/mol) at 273 K.
- 1Degrees of freedom: independent energy terms; 3 monatomic, 5 rigid diatomic, 7 vibrating diatomic.
- 2Equipartition: ½kT per degree of freedom.
- 3Cᵥ = fR/2, Cₚ = Cᵥ + R, γ = 1 + 2/f: 1.67 monatomic, 1.40 diatomic.
- 4Diatomic gases don't vibrate at room temperature; CO₂, H₂O, CH₄ act as f ≈ 6.
Check yourself
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