For a diatomic gas, if ${\gamma }_1=\left(\frac{C_P}{C_V}\right)$ for rigid molecules and ${\gamma }_2=\left(\frac{C_P}{C_V}\right)$ for another diatomic molecules, but also having vibrational modes. Then, which one of the following options is correct ?
(C_P and C_V are specific heats of the gas at constant pressure and volume)

Step 1: Understanding the Heat Capacity Ratio ($\gamma$)

The adiabatic index, $\gamma$, is defined as:

$$\gamma = \frac{C_P}{C_V}$$

For a diatomic gas, the specific heat capacities depend on the degrees of freedom ($f$):

$$C_V = \frac{f}{2} R, \quad C_P = C_V + R = \frac{f}{2} R + R$$

 $$\gamma = \frac{C_P}{C_V} = \frac{\frac{f}{2} R + R}{\frac{f}{2} R} = \frac{f+2}{f}$$ 

Step 2: Case for Rigid Diatomic Molecule ($\gamma_1$)

A rigid diatomic gas has only translational (3) and rotational (2) degrees of freedom.

So, total degrees of freedom:

• $$f_1 = 3 + 2 = 5$$

Using the formula for $\gamma$:

• $$\gamma_1 = \frac{f_1+2}{f_1} = \frac{5+2}{5} = \frac{7}{5}$$

Step 3: Case for Diatomic Molecule with Vibrational Modes ($\gamma_2$)

When vibrational modes are considered, each vibrational mode contributes 2 extra degrees of freedom (one for kinetic energy, one for potential energy).

A diatomic molecule has one vibrational mode, adding 2 more degrees of freedom.

So, the total degrees of freedom now:

• $$f_2 = 3 + 2 + 2 = 7$$

Using the formula for $\gamma$:

• $$\gamma_2 = \frac{f_2+2}{f_2} = \frac{7+2}{7} = \frac{9}{7}$$

Step 4: Comparing $\gamma_1$ and $\gamma_2$

$$\gamma_1 = \frac{7}{5}, \quad \gamma_2 = \frac{9}{7}$$

Since:

$$\frac{7}{5} > \frac{9}{7}$$

We conclude:

$$\gamma_1 > \gamma_2$$