The equipartition theorem is a principle in classical statistical mechanics that states:
At thermal equilibrium, each independent degree of freedom that appears quadratically in the energy contributes, on average, an equal amount of energy—specifically, (1/2) kT per degree of freedom, where k is the Boltzmann constant and T is the temperature in kelvins.
Degrees of freedom refer to the independent ways in which a system can store energy, such as:
- Translational motion (movement in x, y, z directions),
- Rotational motion (rotation about axes),
- Vibrational motion (bond stretching in molecules).
Key Implications:
- A monatomic ideal gas (with 3 translational degrees) has an average energy of (3/2) kT per atom.
- A diatomic gas has more degrees of freedom (translational + rotational + vibrational), so its total energy per molecule is higher.
- The theorem breaks down at very low temperatures or in systems where quantum effects dominate (some degrees of freedom may be “frozen out”).
Examples:
- Helps predict specific heat capacities of gases and solids.
- Explains why monatomic gases like helium have lower heat capacities than diatomic gases like oxygen.
- Used to understand thermal energy distribution in molecules.
The equipartition theorem provides deep insight into how energy is partitioned across the various motions and interactions within a system at thermal equilibrium.