Degrees of Freedom and Equipartition of Energy

A molecule can store energy by moving, spinning and vibrating. The law of equipartition says how that energy is shared: ½kT for every degree of freedom.

Builds on: Part 6 · Pressure, Kinetic Energy and Temperature.

Video coming soon

Equipartition

$\frac{1}{2} k T$ per degree of freedom

A degree of freedom is an independent way a molecule can move and store energy. Since $⟨ v_{x}^{2} ⟩ = ⟨ v_{y}^{2} ⟩ = ⟨ v_{z}^{2} ⟩ = \frac{v _{rms}^{2}}{3}$ , each translational direction carries $\frac{1}{2} k T$ . Equipartition extends this to every kind of motion: a molecule with f degrees of freedom has $E = \frac{f}{2} k T$ .

Diatomic

f = 3 + 2 = 5

Monatomic (He, Ne, Ar): f = 3. Rotation stores negligible energy.
Diatomic (H₂, N₂, O₂): 3 translations + 2 rotations perpendicular to the bond. Rotation about the bond axis doesn't count (I ≈ 0): f = 5.
Above about 5000 K the bond vibrates, adding KE + PE: f = 7.
H₂ below about 50 K can't rotate: f = 3 (frozen modes, a quantum effect).

Polyatomic

Linear vs non-linear

Linear (CO₂): 3 + 2 + x. Non-linear (H₂O, CH₄): 3 + 3 + x, with x the active vibrational modes.

A linear CO₂ molecule and a bent H₂O molecule with rotation rings. — CO₂ rotates about two axes, H₂O about three.

Summary

Key formulas

⟨ v_{x}^{2} ⟩ = ⟨ v_{y}^{2} ⟩ = ⟨ v_{z}^{2} ⟩ = \frac{v _{rms}^{2}}{3}

Translational degrees of freedom

\frac{1}{2} m^{'} ⟨ v_{x}^{2} ⟩ = \frac{1}{2} k T

Translational degrees of freedom

E = \frac{f}{2} k T per molecule

Law of equipartition of energy

JEE-style question

Your turn

What is γ = C_P/C_V for a gas of rigid diatomic molecules?

\frac{7}{5}

\frac{5}{3}

\frac{9}{7}

\frac{4}{3}

Show the answer and the traps

f = 5, so γ = 1 + 2/f = 7/5. Option 1.

5/3 is monatomic. 9/7 adds vibration. 4/3 is non-linear polyatomic.

Watch out

Common mistakes

Counting rotation about a diatomic's bond axis.Its moment of inertia is almost zero, so it stores no energy: f = 5, not 6.

Using f = 7 for a diatomic at room temperature.Vibration only switches on at high temperature (around 5000 K).

Practice

Try these

1. Degrees of freedom of rigid CO₂ and rigid NH₃?

CO₂ is linear: 3 + 2 = 5. NH₃ is non-linear: 3 + 3 = 6.

γ = C_{P} / C_{V}

for a gas with f = 6?

$γ = 1 + \frac{2}{f} = \frac{4}{3}$ .

3. Energy of 1 mol of N₂ (rigid) at 300 K?

$\frac{5}{2} R T = 2.5 \times 8.314 \times 300$ ≈ 6.24 × 10³ J.