If a non-linear polyatomic gas has $f$ vibrational nodes, the value of $\gamma = C_p/C_v$ is ($C_p$ and $C_v$ are specific heats at constant pressure and volume respectively)

For a non-linear polyatomic gas, the number of translational degrees of freedom is $3$ and the number of rotational degrees of freedom is $3$.

Each vibrational mode contributes $2$ degrees of freedom, one associated with kinetic energy and the other with potential energy. Therefore, $f$ vibrational modes contribute $2f$ degrees of freedom.

The total number of degrees of freedom $N$ is:

$N = 3 + 3 + 2f = 6 + 2f$

The molar specific heat at constant volume $C_v$ is given by $\dfrac{N}{2}R$.

$C_v = \dfrac{6 + 2f}{2}R = (3 + f)R$

Using Mayer's relation, the molar specific heat at constant pressure $C_p$ is:

$C_p = C_v + R$

$C_p = (3 + f)R + R = (4 + f)R$

The ratio of specific heats $\gamma$ is given by $\dfrac{C_p}{C_v}$.

$\gamma = \dfrac{(4 + f)R}{(3 + f)R}$

$\gamma = \dfrac{4 + f}{3 + f}$