A non-linear polyatomic gas is heated to a temperature such that...

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.

According to the law of equipartition of energy, each translational and rotational degree of freedom contributes $\dfrac{1}{2}R$ to the molar heat capacity at constant volume ($C_v$). Each active vibrational mode contributes $R$ to $C_v$ (since it has both kinetic and potential energy components).

Given that there are 2 active vibrational modes:
$C_v = \dfrac{3}{2}R + \dfrac{3}{2}R + 2(R) = 3R + 2R = 5R$

The molar heat capacity at constant pressure ($C_p$) is:
$C_p = C_v + R = 5R + R = 6R$

The ratio of specific heats $\gamma$ is:
$\gamma = \dfrac{C_p}{C_v} = \dfrac{6R}{5R} = 1.20$

Assuming a vibrational mode contributes only $\dfrac{1}{2}R$ leads to the incorrect value of 1.25, while ignoring vibrational modes entirely gives 1.33.

Answer: $1.20$