For a polyatomic molecule, the total degrees of freedom ($f_{total}$) is the sum of translational, rotational, and vibrational modes. Given: translational = 3, rotational = 3, and vibrational = $f$. Each vibrational mode contributes 2 degrees of freedom (kinetic and potential energy). Thus, $f_{total} = 3 + 3 + 2f = 6 + 2f$.
The molar heat capacity at constant volume is given by $C_V = \frac{f_{total}}{2}R$. The molar heat capacity at constant pressure is $C_P = C_V + R = (\frac{f_{total}}{2} + 1)R$. The ratio $\gamma = \frac{C_P}{C_V} = 1 + \frac{2}{f_{total}}$.
Given $\gamma = \frac{8}{7}$, we set up the equation: $$\frac{8}{7} = 1 + \frac{2}{6 + 2f}$$ $$\frac{1}{7} = \frac{2}{6 + 2f}$$ $$6 + 2f = 14$$ $$2f = 8$$ $$f = 4$$
Final Answer: 4
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