the equipartition theorem
Energy shares out evenly. In thermal equilibrium, each independent way a molecule can store energy (each degree of freedom) receives, on average, the same amount: (1/2) k_B T. Nature does not play favourites among the different modes of motion.
Precisely, each quadratic degree of freedom contributes (1/2) k_B T to the average energy per molecule, or (1/2) R T per mole. A monatomic gas has 3 translational degrees of freedom, giving average energy (3/2) k_B T; a diatomic gas adds 2 rotational ones for (5/2) k_B T; each vibrational mode adds 2 more, since it stores both kinetic and potential energy.
A caveat worth stating honestly: this is a classical result. At low temperatures quantum effects freeze out some degrees of freedom, so rotations and then vibrations stop contributing, which lowers heat capacities and makes classical equipartition fail. That temperature dependence of heat capacity is one of the early clues that led to quantum theory.
A monatomic gas has molar heat capacity (3/2)R and a diatomic gas (5/2)R, the difference coming exactly from the two rotational degrees of freedom.
Each degree of freedom gets an average of (1/2) k_B T.
Equipartition is a classical idealization; it works well at ordinary temperatures but overcounts the active modes when quantum freezing sets in at low temperature.