How can specific heat be predicted from the law of equipartition of energy?

(N/A) According to the law of equipartition of energy,each degree of freedom contributes $\frac{1}{2} k_B T$ to the internal energy of a system.
For a gas with $f$ degrees of freedom,the internal energy $U$ is given by $U = f \cdot \frac{1}{2} n R T$.
The molar specific heat at constant volume is $C_V = \frac{dU}{dT} = \frac{f}{2} R$.
The molar specific heat at constant pressure is $C_P = C_V + R = (\frac{f}{2} + 1) R$.
This classical prediction assumes that all degrees of freedom are active at all temperatures.
However,experimental observations show that specific heat varies with temperature,approaching zero as $T \to 0 \ K$,which indicates that degrees of freedom become 'frozen' or ineffective at low temperatures.
This limitation of classical mechanics is explained by quantum mechanics,where a minimum energy threshold is required to excite a degree of freedom.