Law of Equipartition of Energy

The Core Idea: Why Energy Spreads Evenly

The kinetic theory gives us a microscopic picture of a gas: a huge number of molecules in ceaseless, random motion. Each molecule can store energy in different ways — it can move (translational kinetic energy), it can rotate (rotational kinetic energy), and its atoms can vibrate back and forth (vibrational kinetic and potential energy). The question is: if you have a gas at a certain temperature, how is the total internal energy shared among all these different forms?

The law of equipartition of energy provides the answer. It states that for a system in thermal equilibrium at temperature $T$, the total energy is equally divided among all the independent ways a molecule can store energy. Each such independent way is called a degree of freedom, and each degree of freedom gets, on average, an energy of $\frac{1}{2}k_BT$ per molecule (or $\frac{1}{2}RT$ per mole).

This is not a theorem that can be proved from Newton's laws alone; it is a result of statistical mechanics (the Boltzmann distribution). But it is one of the most powerful tools for understanding the specific heat capacities of gases.

Degrees of Freedom of a Molecule

Before applying the law, we must count the degrees of freedom for different types of molecules.

1. Monatomic gas (e.g., He, Ar, Ne): A single atom. It can only move in three independent directions (x, y, z). It has 3 translational degrees of freedom. It cannot rotate (a point has no moment of inertia about any axis through it) and does not vibrate (no bonds). So total degrees of freedom, $f = 3$.

2. Diatomic gas (e.g., H₂, O₂, N₂): Two atoms joined by a bond.

• Translational: The centre of mass can move in 3 directions. So 3 translational degrees of freedom.
• Rotational: The molecule can rotate about two axes perpendicular to the bond axis. Rotation about the bond axis itself (the line joining the atoms) has a negligible moment of inertia for a diatomic molecule (the atoms are effectively points on that axis), so it does not store significant energy at ordinary temperatures. So 2 rotational degrees of freedom.
• Vibrational: The atoms can vibrate along the bond. This involves both kinetic energy (of the atoms moving towards and away from each other) and potential energy (of the stretched/compressed bond). Each of these is a quadratic term in the energy expression. So vibration contributes 2 vibrational degrees of freedom (one kinetic, one potential).

At ordinary temperatures (around room temperature), the vibrational modes are usually not excited. The energy required to start a vibration is much larger than $k_BT$. So for most practical purposes at room temperature, a diatomic molecule has $f = 3 + 2 = 5$ degrees of freedom.

3. Triatomic (or polyatomic) gas:

• Linear molecule (e.g., CO₂): All atoms lie on a line.
  • Translational: 3
  • Rotational: 2 (about two axes perpendicular to the molecular axis) …