1. For a monoatomic ideal gas

A monoatomic gas has only translational degrees of freedom: $f_1=3$

So, $C_{V1} = \frac{f_1}{2}R = \frac{3}{2}R$ $C_{P1} = C_{V1}+R = \frac{5}{2}R$

Hence, $\gamma_1 = \frac{C_{P1}}{C_{V1}} = \frac{\frac{5}{2}R}{\frac{3}{2}R} = \frac{5}{3}$

2. For a diatomic ideal gas treated as a rigid rotator

A rigid rotator diatomic molecule has:

• 3 translational degrees of freedom
• 2 rotational degrees of freedom

So total degrees of freedom: $f_2=5$

Thus, $C_{V2} = \frac{f_2}{2}R = \frac{5}{2}R$ $C_{P2} = C_{V2}+R = \frac{7}{2}R$

Hence, $\gamma_2 = \frac{C_{P2}}{C_{V2}} = \frac{\frac{7}{2}R}{\frac{5}{2}R} = \frac{7}{5}$

3. Required ratio

$\frac{\gamma_1}{\gamma_2} = \frac{\frac{5}{3}}{\frac{7}{5}} = \frac{5}{3}\cdot\frac{5}{7} = \frac{25}{21}$

4. Match with options

$\frac{\gamma_1}{\gamma_2} = \frac{25}{21}$

So the correct option is B.

5. Comparison with stored answer

Stored correct answer: B

Our derived answer: B

They match.