Problem Analysis:

The question asks for the number of vibrational modes ($f$) for an ideal polyatomic gas, given its heat capacities ratio ($C_P/C_V$) is $8/7$. The gas molecules have 3 translational, 3 rotational, and $f$ vibrational modes.

Degrees of Freedom and Heat Capacity

For an ideal gas molecule:

• Translational degrees of freedom = 3. Contribution to $C_V$ = $\frac{3}{2}R$.
• Rotational degrees of freedom = 3. Contribution to $C_V$ = $\frac{3}{2}R$.
• Vibrational modes = $f$. Each vibrational mode contributes $R$ to $C_V$ (assuming high temperature where vibrations are active). Contribution to $C_V$ = $fR$.

The specific heat at constant volume ($C_V$) per mole is the sum of contributions from all degrees of freedom:

$ C_V = \left(\frac{3}{2} + \frac{3}{2} + f\right) R = (3 + f) R $

Calculating Heat Capacity Ratio ($\gamma$)

Using Mayer's relation, the specific heat at constant pressure ($C_P$) is:

$ C_P = C_V + R $

$ C_P = (3 + f) R + R = (4 + f) R $

The ratio of heat capacities ($\gamma$) is:

$ \gamma = \frac{C_P}{C_V} = \frac{(4 + f) R}{(3 + f) R} = \frac{4 + f}{3 + f} $

Solving for Vibrational Modes ($f$)

We are given that $\gamma = C_P/C_V = 8/7$. Setting the derived formula equal to the given value:

$ \frac{4 + f}{3 + f} = \frac{8}{7} $

Cross-multiplying to solve for $f$:

$ 7(4 + f) = 8(3 + f) $

$ 28 + 7f = 24 + 8f $

Rearranging the terms:

$ 8f - 7f = 28 - 24 $

$ f = 4 $

Conclusion

The value of $f$, representing the number of vibrational modes, is 4. This corresponds to Option D.