Equipartition Theorem

Equivalent forms

\langle E_{\text{kin}} \rangle = \frac{f}{2} k_B T

C_V = \frac{f}{2} N k_B

\langle \frac{1}{2}mv_x^2 \rangle = \langle \frac{1}{2}mv_y^2 \rangle = \langle \frac{1}{2}mv_z^2 \rangle = \frac{1}{2}k_B T

The theorem's failure at low T was a major hint for quantum mechanics: modes freeze out when k_BT ≪ ℏω.

Symbol	Name	SI unit	Dimension	Range
$f$	Degrees of freedom Number of quadratic terms in the Hamiltonian (3 translational + 2/3 rotational + vibrational)	dimensionless	1	1 – 12
$⟨E⟩$	Mean thermal energyoutput Average total thermal energy of one molecule	J	ML²T⁻²	0 – 1e-19
$T$	Temperature Absolute temperature	K	Θ	10 – 10000

Unit systems

SI:: energy in J, $k_B = 1.380649e-23\,\mathrm{J/K}$
chemistry:: C_V in $J/(mol\cdot K)$ ; multiply by $N_A = 6.02214076e23$
eV:: $k_BT \approx 0.02585\,\mathrm{eV}$ at 300 K

Where it holds

Classical regime k_BT ≫

\hbar \omega for

each mode. Quantum corrections freeze modes at low T. Does NOT apply to quantum-degenerate (Fermi/Bose) systems.

Discovery

Ludwig Boltzmann & James Clerk Maxwell · 1876

Maxwell (1860) and Boltzmann (1876) derived that kinetic energy distributes equally among degrees of freedom. The theorem's failure for diatomic molecules at low T puzzled physicists for 40 years — resolved only by quantum mechanics (frozen-out modes).

Try this

Why does every gas molecule get exactly the same average energy?

The equipartition theorem says each quadratic degree of freedom in the Hamiltonian carries exactly (1/2)k_B T of thermal energy — regardless of the molecule's shape, mass, or interaction. This is why the heat capacity of a diatomic gas (5/2 Nk_B) differs from a monatomic gas (3/2 Nk_B).

Research status: stable

Common misconceptions

The theorem counts QUADRATIC terms, not coordinates. A 1D harmonic oscillator has TWO quadratic terms (kinetic + potential) so contributes k_BT (not k_BT/2) to average energy.

Derivation

For any Hamiltonian quadratic in a coordinate q_i: ⟨

q_i \partial H/\partial q_i

⟩

= \int q_i (\partial H/\partial q_i) e^{-\beta H} d\Gamma / \int e^{-\beta H} d\Gamma

.

Integration by parts gives k_BT regardless of the functional form of H.

Limiting cases

f = 3

(monatomic gas)⟶ ⟨E⟩

= 3k_BT/2

;

C_V = 3Nk_B/2

(Dulong-Petit for noble gases)

f = 5

(rigid diatomic gas)⟶ ⟨E⟩

= 5k_BT/2

;

C_V = 5Nk_B/2 (N_{2}

,

O_{2} at

room T)

k_BT ≪

\hbar \omega

(quantum)⟶ Vibrational modes freeze out; C_V drops below classical prediction

Maxwell-Boltzmann Speed Distribution→Ideal Gas Law→Heat Capacity at Constant Volume→