Degrees of freedom
The independent ways a molecule can store energy — translation, rotation, vibration — each counted toward its heat capacity.
Definition
A degree of freedom is an independent coordinate in which a system can store energy. For a gas molecule these fall into translational (motion of the centre of mass, always three), rotational (tumbling about axes with appreciable moment of inertia), and vibrational (stretching and bending of bonds, each contributing two quadratic terms, one kinetic and one potential). The count f of energy-bearing quadratic degrees of freedom sets the molecule's heat capacity through equipartition.
A monatomic gas such as helium has three (translational only). A diatomic gas such as nitrogen has five at room temperature — three translational plus two rotational — rising toward seven when vibration becomes active at high temperature. A monatomic solid has six per atom: three kinetic and three potential. The adiabatic exponent γ = (f+2)/f reads the count directly off a gas's behaviour.
Crucially, degrees of freedom switch on and off with temperature. Quantum mechanics requires a minimum energy ħω to excite rotation or vibration; below the corresponding temperature that mode is 'frozen out' and contributes nothing to the heat capacity. This temperature-dependent counting is what produces the stepped heat-capacity curve of hydrogen.