For a diatomic gas, if $$\gamma_{1}=\left(\frac{Cp}{Cv}\right)$$ for rigid molecules and $$\gamma_{2}=\left(\frac{Cp}{Cv}\right)$$ for another diatomic molecules, but also having vibrational modes. Then, which one of the following options is correct ? (Cp and Cv are specific heats of the gas at constant pressure and volume)

To solve this problem, we need to compare the ratio of specific heats, $$\gamma = \frac{C_p}{C_v}$$, for two types of diatomic gases: one with rigid molecules (no vibrational modes) and another with vibrational modes. The specific heats depend on the degrees of freedom of the gas molecules.

First, recall the formula for $$\gamma$$ in terms of degrees of freedom $$f$$:

$$\gamma = 1 + \frac{2}{f}$$

This is derived from the relations $$C_v = \frac{f}{2} R$$ and $$C_p = C_v + R = \frac{f}{2} R + R$$, so:

$$\gamma = \frac{C_p}{C_v} = \frac{\frac{f}{2} R + R}{\frac{f}{2} R} = \frac{\frac{f}{2} + 1}{\frac{f}{2}} = 1 + \frac{2}{f}$$

Now, determine the degrees of freedom for each case:

Case 1: Rigid diatomic molecules (no vibration)

- Translational degrees of freedom: 3 (for motion along x, y, z axes)

- Rotational degrees of freedom: 2 (for diatomic molecules, rotation about axes perpendicular to the bond axis)

- Total degrees of freedom, $$f_1 = 3 + 2 = 5$$

Thus, $$\gamma_1 = 1 + \frac{2}{f_1} = 1 + \frac{2}{5} = 1.4$$

Case 2: Diatomic molecules with vibrational modes

- Translational degrees of freedom: 3

- Rotational degrees of freedom: 2

- Vibrational modes: 1 vibrational mode, but each vibrational mode contributes 2 degrees of freedom (kinetic and potential energy)

- Total degrees of freedom, $$f_2 = 3 + 2 + 2 = 7$$

Thus, $$\gamma_2 = 1 + \frac{2}{f_2} = 1 + \frac{2}{7} \approx 1.2857$$

Comparing $$\gamma_1$$ and $$\gamma_2$$:

$$\gamma_1 = 1.4$$ and $$\gamma_2 \approx 1.2857$$, so $$\gamma_2 < \gamma_1$$.

Now, evaluate the options:

A. $$\gamma_1 = \gamma_2$$ → False

B. $$2\gamma_2 = \gamma_1$$ → $$2 \times 1.2857 = 2.5714 \neq 1.4$$ → False

C. $$\gamma_2 < \gamma_1$$ → True

D. $$\gamma_2 > \gamma_1$$ → False

Therefore, the correct option is C.