equipartition theorem
/ ee-kwi-par-TISH-un THEER-um /
Picture a generous host dividing a pot of warmth equally among everyone at the table — nobody gets more, nobody less. The equipartition theorem says nature behaves like that host: in a system warm enough to be classical, the thermal energy is shared out evenly, with the same portion handed to every independent way a molecule can store energy.
More precisely, the theorem says each quadratic degree of freedom — each independent direction of motion or each spring-like vibration that stores energy as the square of a coordinate or speed — carries on average one-half of kT of energy, where k is the Boltzmann constant and T the temperature. A monatomic gas atom, moving in three directions, therefore holds three-halves kT; add rotations and vibrations and the share grows accordingly.
Why it matters: equipartition gives quick, surprisingly accurate estimates of heat capacities and average energies without solving anything hard. But it has a famous failure: it is a classical result that ignores quantum spacing. When energy levels are spaced wider than kT — as vibrations often are at room temperature — those modes are 'frozen out,' hold less than their fair share, and equipartition over-counts. That very breakdown helped launch quantum theory.
A monatomic gas like argon has three directions of motion, so equipartition predicts an average energy of three-halves kT per atom and a molar heat capacity of about 12.5 joules per kelvin per mole — a figure that matches experiment beautifully.
Three translational degrees of freedom give argon an energy of three-halves kT per atom.
Equipartition counts only degrees of freedom that store energy quadratically and are 'unfrozen' at the temperature of interest. Vibrational modes are often frozen at room temperature; that is why diatomic gases show their full vibrational heat capacity only when strongly heated.