the equipartition theorem
A beautifully democratic rule of classical thermal physics: at temperature T, every independent way of storing energy that appears as a square in the energy gets, on average, the very same share, exactly (1/2) k_B T. Translation, rotation, the stretch of a spring, each such quadratic slot receives an equal slice of the thermal pie.
Precisely, in classical statistical mechanics each quadratic degree of freedom in the Hamiltonian, each term of the form (1/2) m v_x^2 or (1/2) k x^2, contributes on average (1/2) k_B T to the internal energy. A monatomic ideal gas has three translational quadratic terms, so <E> = (3/2) N k_B T and C_V = (3/2) N k_B. A one-dimensional harmonic oscillator has two such terms, kinetic and potential, giving k_B T each, and 3N oscillators in a solid give 3 N k_B T, the Dulong-Petit law.
It explains ideal-gas heat capacities and the Dulong-Petit law almost effortlessly, but its failures are historically decisive. It is purely classical and breaks down when k_B T falls below the spacing of the quantum energy levels: vibrational and rotational modes freeze out, so a diatomic gas's C_V rises in steps with temperature, and the classical equipartition of the electromagnetic field gave the ultraviolet catastrophe, resolved only by quantization. So it is an honest idealization, valid only in the high-temperature, classical limit.
A diatomic gas such as N2 at room temperature has three translational plus two rotational active quadratic modes, giving C_V = (5/2) R; its vibrational mode is frozen out because its quantum spacing exceeds k_B T.
Which modes are active, and thus C_V, depends on temperature, contradicting naive equipartition.
Equipartition is a classical result and counts quadratic terms in the energy, not merely mechanical degrees of freedom; quantum freezing-out of modes is why real heat capacities depend on temperature.