What does the theorem of equipartition of energy state about molecules in thermal equilibrium?

The theorem of equipartition of energy states that in thermal equilibrium, the average energy associated with each independent degree of freedom of motion for molecules is equal. This means that for every degree of freedom, such as translational or rotational motion, the molecules in the system will have the same average energy spread across these different modes of energy storage.

In a system where molecules are in thermal equilibrium, they can possess different forms of energy, such as kinetic energy from motion (both translational and rotational) and potential energy due to intermolecular forces. However, the key point of the equipartition theorem is that this energy is equally averaged per degree of freedom. For instance, in three-dimensional space, each molecule contributes ( \frac{1}{2}kT ) (where ( k ) is the Boltzmann constant and ( T ) is the temperature in Kelvin) for each translational degree of freedom (x, y, and z directions). This results in the idea that the energy is averaged equally across the different ways in which the molecules can move.

This theorem helps to explain the behavior of gases and other systems at a molecular level, linking temperature and energy in a way that characterizes the distribution of speeds and motion among particles