An ideal gas is made of polyatomic molecules. Each of the molecules...

The internal energy of one mole of the polyatomic gas is given by the equipartition theorem. Each translational and rotational degree of freedom contributes $\dfrac{1}{2}RT$ to the internal energy, while each vibrational mode contributes $RT$ (since it has both kinetic and potential energy components).

$U = 3 \left(\dfrac{1}{2}RT\right) + 3 \left(\dfrac{1}{2}RT\right) + f(RT) = (3+f)RT$

The molar heat capacity at constant volume is:

$C_V = \dfrac{dU}{dT} = (3+f)R$

The molar heat capacity at constant pressure is:

$C_P = C_V + R = (4+f)R$

The ratio of heat capacities is given as $\dfrac{8}{7}$:

$\dfrac{C_P}{C_V} = \dfrac{4+f}{3+f} = \dfrac{8}{7}$

$28 + 7f = 24 + 8f$

$f = 4$

Answer: $4$