QCC Notes
CLASS 11 · PHYSICSJEE MAIN × NEETहिंदी
§ 12.4NCERT Class 11 · Physics · Chapter 12

Law of Equipartition of Energy, Degrees of Freedom & Molar Specific Heats

QCC Notes · Class 11 Physics NCERT-aligned Updated 26 Sep 2026 4 min read
Central Theorem: The total internal energy of an ideal gas is distributed equally among all its independent modes of motion (degrees of freedom) in thermal equilibrium.

1. Degrees of Freedom (f)

The degrees of freedom of a dynamical system is defined as the total number of independent coordinates or independent terms in the expression of energy required to completely specify the position and configuration of the system.

1. Monoatomic Gas (He, Ne, Ar, Kr)

A monoatomic molecule behaves as a point mass. It can move along x, y, and z axes (3 translational modes). Moment of inertia about its own center is negligibly small.

f = 3  (Translational only)

2. Diatomic Gas (O2, N2, H2, CO)

At room/moderate temperature, molecules behave as rigid rotators (fixed bond distance). They have 3 translational + 2 rotational modes (rotation about the internuclear axis has negligible moment of inertia).

f = 3 (trans) + 2 (rot) = 5

At high temperatures, vibrational modes become active. Each vibrational mode contributes 2 degrees of freedom (1 kinetic + 1 potential):

f = 3 (trans) + 2 (rot) + 2 (vib) = 7

3. Linear Polyatomic Gas (CO2, C2H2)

Similar to diatomic molecules, they have 3 translational and 2 rotational degrees of freedom:

f = 3 (trans) + 2 (rot) = 5  (rigid)

4. Non-Linear Polyatomic Gas (H2O, NH3, CH4)

Have 3 translational and 3 independent rotational axes:

f = 3 (trans) + 3 (rot) = 6  (rigid)

2. Law of Equipartition of Energy

According to this classical statistical law, for any dynamical system in thermal equilibrium at absolute temperature T, the energy of the system is equally divided among all its degrees of freedom:

Average Energy per degree of freedom per molecule = (1 / 2) · kB · T
  • Average kinetic energy of a molecule having f degrees of freedom: ⟨E⟩ = (f / 2) · kB · T.
  • Total internal energy of 1 mole of gas: U = NA · (f / 2) kB T = (f / 2) · R · T.

3. Molar Heat Capacities (CV, CP) and Adiabatic Ratio (γ)

Using the first law of thermodynamics, molar heat capacity at constant volume is:

CV = dU / dT = (f / 2) · R

From Mayer's relation (CP − CV = R):

CP = CV + R = (f / 2 + 1) · R

The ratio of specific heats (adiabatic index γ) is:

γ = CP / CV = 1 + (2 / f)

4. Summary of Gas Heat Capacities

Gas Type Degrees of Freedom (f) CV CP γ = CP / CV
Monoatomic (He, Ar) 3 (trans) (3/2) R (5/2) R 5/3 ≈ 1.67
Diatomic (Rigid) (O2, N2) 5 (3 trans + 2 rot) (5/2) R (7/2) R 7/5 = 1.40
Diatomic (Non-rigid) (High T) 7 (3 trans + 2 rot + 2 vib) (7/2) R (9/2) R 9/7 ≈ 1.29
Polyatomic (Non-linear, Rigid) 6 (3 trans + 3 rot) 3 R 4 R 4/3 ≈ 1.33

5. Specific Heat Capacities of Solids: Dulong-Petit Law

In a solid crystal lattice, each atom can oscillate in three dimensions about its mean equilibrium position. Each vibrational mode possesses 1 kinetic energy term and 1 potential energy term (2 degrees of freedom). Thus, each atom has 3 × 2 = 6 degrees of freedom.

  • Energy per atom = 6 × (1/2 kB T) = 3 kB T.
  • Internal energy per mole: U = 3 NA kB T = 3 R T.
  • Molar specific heat of a solid:
C = dU / dT = 3 · R ≈ 3 × 8.314 ≈ 24.9 J · mol−1 · K−1 ≈ 6 cal · mol−1 · K−1

This is known as the Dulong-Petit Law, which holds remarkably well for most solids at room temperature.

6. Specific Heat Capacity of Water

A water molecule (H2O) consists of 3 atoms. Treating it as a solid-like structure where each of the 3 atoms has 6 degrees of freedom (or 3 atoms × 3 vibrational modes = 9 modes total):

U = 3 × 3 · R · T = 9 · R · T  ⇒  C = 9 · R ≈ 74.8 J · mol−1 · K−1

This explains the exceptionally high specific heat capacity of water compared to other substances.

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