Law Of Equipartition Of Energy

The principle at work is the law of equipartition of energy, stated in NCERT Class 11, Chapter 13 (Kinetic Theory), which asserts that in thermal equilibrium the total energy of a molecule is shared equally among all its degrees of freedom. Each independent quadratic degree of freedom, whether translational or rotational, receives an average energy of exactly $\frac{1}{2}{k}_{B}T$, where $k_B$ is Boltzmann's constant and $T$ is the absolute temperature. This is why a monatomic gas with three translational degrees has average energy $\frac{3}{2}{k}_{B}T$ per molecule. The option $\frac{3}{2}{k}_{B}T$ is the total for three degrees, not the share per single degree. The option $k_{B}T$ doubles the correct value and would only apply to a vibrational mode counting both kinetic and potential terms. The option $\frac{1}{3}{k}_{B}T$ has no basis in the theory. A plausibility check confirms the value: summing $\frac{1}{2}{k}_{B}T$ over $f$ degrees of freedom reproduces the known internal energy $\frac{f}{2}{k}_{B}T$ per molecule, so the per-degree share of $\frac{1}{2}{k}_{B}T$ is internally consistent.