Obtain the value of $\frac{C_{P}}{C_{V}}$ for a polyatomic gas.

(D) polyatomic gas has $3$ translational,$3$ rotational,and $f$ vibrational degrees of freedom. Thus,the total degree of freedom is $f_{total} = (3 + 3 + f) = (6 + f)$.
The internal energy $U$ is given by the law of equipartition of energy:
$U = \frac{f_{total}}{2} RT = \frac{6+f}{2} RT$.
The molar specific heat at constant volume is $C_{V} = \frac{dU}{dT} = \frac{6+f}{2} R$.
Using Mayer's relation,$C_{P} = C_{V} + R = \left(\frac{6+f}{2} + 1\right) R = \left(\frac{8+f}{2}\right) R$.
Therefore,the ratio $\gamma = \frac{C_{P}}{C_{V}} = \frac{\frac{8+f}{2} R}{\frac{6+f}{2} R} = \frac{8+f}{6+f}$.