If one mole of the polyatomic gas is having two vibrational modes and $$\beta$$ is the ratio of molar specific heats for polyatomic gas $$\left(\beta = \frac{C_p}{C_v}\right)$$ then the value of $$\beta$$ is:

For a polyatomic gas, the degrees of freedom include 3 translational, 3 rotational, and the vibrational modes. Each vibrational mode contributes 2 degrees of freedom (one kinetic and one potential), so 2 vibrational modes contribute 4 degrees of freedom. The total degrees of freedom are $$f = 3 + 3 + 4 = 10$$.

The molar specific heat at constant volume is $$C_v = \frac{f}{2}R = \frac{10}{2}R = 5R$$, and at constant pressure $$C_p = C_v + R = 6R$$.

Therefore $$\beta = \frac{C_p}{C_v} = \frac{6R}{5R} = \frac{6}{5} = 1.2$$.