Can we use the degree of freedom to obtain the specific heat capacity of a gas? Explain.

(A) Yes,we can use the degree of freedom $(f)$ to determine the molar specific heat capacity of a gas.
According to the law of equipartition of energy,the energy associated with each degree of freedom per molecule is $\frac{1}{2} k_{B} T$.
For a gas with $f$ degrees of freedom,the total internal energy $(U)$ for $1$ mole of gas is given by:
$U = f \times \left( \frac{1}{2} k_{B} T \right) \times N_{A} = \frac{f}{2} RT$ (since $k_{B} N_{A} = R$)
The molar specific heat at constant volume $(C_{V})$ is defined as the rate of change of internal energy with respect to temperature:
$C_{V} = \frac{dU}{dT} = \frac{d}{dT} \left( \frac{f}{2} RT \right) = \frac{f}{2} R$
Using Mayer's relation,the molar specific heat at constant pressure $(C_{P})$ is:
$C_{P} = C_{V} + R = \frac{f}{2} R + R = \left( \frac{f}{2} + 1 \right) R$
The ratio of specific heats $(\gamma)$ is:
$\gamma = \frac{C_{P}}{C_{V}} = \frac{(\frac{f}{2} + 1)R}{\frac{f}{2}R} = 1 + \frac{2}{f}$