Consider a linear polyatomic gas having f active vibrational modes....

For a linear polyatomic gas, there are 3 translational degrees of freedom and 2 rotational degrees of freedom.

Let the number of active vibrational modes be $f$. Each vibrational mode contributes $R$ to the molar heat capacity at constant volume ($C_v$).

The total $C_v$ is given by:
$C_v = \dfrac{3}{2}R + \dfrac{2}{2}R + fR = \dfrac{5}{2}R + fR = \left(\dfrac{5 + 2f}{2}\right)R$

The molar heat capacity at constant pressure ($C_p$) is:
$C_p = C_v + R = \left(\dfrac{5 + 2f}{2}\right)R + R = \left(\dfrac{7 + 2f}{2}\right)R$

The ratio of specific heats $\gamma$ is:
$\gamma = \dfrac{C_p}{C_v} = \dfrac{7 + 2f}{5 + 2f}$

Rearranging to solve for $f$:
$\gamma(5 + 2f) = 7 + 2f$
$5\gamma + 2f\gamma = 7 + 2f$
$2f(\gamma - 1) = 7 - 5\gamma$
$f = \dfrac{7 - 5\gamma}{2(\gamma - 1)}$

Using the formula for a non-linear polyatomic gas would incorrectly yield $\dfrac{4 - 3\gamma}{\gamma - 1}$, and forgetting the factor of 2 in the denominator during algebraic isolation leads to $\dfrac{7 - 5\gamma}{\gamma - 1}$.

Answer: $\dfrac{7 - 5\gamma}{2(\gamma - 1)}$