The theorem of equipartition of energy states that in thermal equilibrium, all degrees-of-freedom that give quadratic energy dependence have an average energy of $\frac{1}{2} k_{B} T$ and therefore contributes $\frac{1}{2} k_{B}$ to the heat capacity of the system.Examples of degrees-of-freedom with quadratic energy dependence: linear velocity components ( $E_{\text{kin}} = \frac{1}{2} mv^{2}_{x}$), angular velocity components ($E_{\text{kin}} = \frac{1}{2} mv^{2}_{x}$), potential energy of harmonic oscillator ($E_{\text{pot}} = \frac{1}{2} mv^{2}_{x}$), kinetic energy of harmoic oscillator ($E_{\text{kin}} = \frac{1}{2} v^{2}_{x}$).

This theorem came from classical statistical mechanics, and will breakdown if quantum effects are strong.

**Translational kinetic energy**

Each translational degree of freedom (in the x,y and z directions) has an average energy of $\frac{1}{2} k_{B} T$, so a particle in 3-D space has an average kinetic energy of $\frac{3}{2} k_{B} T$.

**Rotational kinetic energy**

Each rotational degree of freedom (in the x,y and z axes) has an average energy of $\frac{1}{2} k_{B} T$.

**Vibrational Energy**

Each vibrational mode has an average energy of $k_{B} T$ due to contributions from both the kinetic and potential energy components.

Next: Counting the number of modes