The explanation uses the equation for the heat capacity ratio, $ \gamma = 1 + \frac{2}{\text{f}} $, where $ \text{f} $ is the degrees of freedom of the gas. Here's how it applies to different types of gases:

Triatomic Rigid Gas:

Degrees of freedom ($ \text{f} $) = 6

$ \gamma = 1 + \frac{2}{6} = \frac{4}{3} $

Diatomic Non-Rigid Gas:

Degrees of freedom ($ \text{f} $) = 7

$ \gamma = 1 + \frac{2}{7} = \frac{9}{7} $

Diatomic Rigid Gas:

Degrees of freedom ($ \text{f} $) = 5

$ \gamma = 1 + \frac{2}{5} = \frac{7}{5} $

Monoatomic Gas:

Degrees of freedom ($ \text{f} $) = 3

$ \gamma = 1 + \frac{2}{3} = \frac{5}{3} $

Therefore, when matching List I with List II, the correct pairing is:

A (Triatomic rigid gas) - III $(\frac{4}{3})$

B (Diatomic non-rigid gas) - IV $(\frac{9}{7})$

C (Monoatomic gas) - I $(\frac{5}{3})$

D (Diatomic rigid gas) - II $(\frac{7}{5})$

Thus, the correct answer from the options given is:

A-III, B-IV, C-I, D-II