The equipartition theorem

A single point mass moving in three dimensions has three independent ways to store kinetic energy — motion along x, along y, along z — each contributing ½mv² to the total, one term per direction. The equipartition theorem generalises this observation into a precise rule: at thermal equilibrium, every independent quadratic degree of freedom — every separate way a system can store energy that enters as the square of some variable — carries the same average thermal energy, ½kT, regardless of what that degree of freedom physically is.

For a single structureless (monatomic) gas molecule, only the three translational directions count, giving ³⁄₂kT of kinetic energy per molecule on average — exactly the result kinetic theory derives directly. A diatomic molecule like N₂ has that same translational motion, but it can also rotate about two independent axes perpendicular to the bond (rotation about the bond axis itself carries essentially no moment of inertia and does not count), and each of those adds another ½kT. At ordinary temperatures a diatomic gas therefore carries ⁵⁄₂kT per molecule — more internal energy at the same temperature than a monatomic gas, purely because it has more independent ways to store thermal motion.

This is not a minor bookkeeping detail; it directly predicts each gas's molar heat capacity, since more degrees of freedom mean more energy must be supplied to raise the temperature by a given amount. At sufficiently high temperatures, diatomic and polyatomic molecules also begin vibrating, adding yet more degrees of freedom (each vibrational mode actually contributes a full kT, since it stores energy in both a kinetic and a potential quadratic term at once) and raising the heat capacity further — which is exactly why measured heat capacities climb with temperature in a way the simplest kinetic theory, counting only translation, cannot explain on its own.