For a monotomic gas at temperature T, match the following columns.
`{:(,"ColumnI",, "ColumnII"),((A),"Speed of sound", (p),sqrt(2RT//M)),((B),"RMS speed of gas molecules",(q),sqrt(8RT//piM)),((C),"Average speed of gas molecules",(r),sqrt(3RT//M)),((D),"Most probable speed of gas molecules",(s),sqrt(5RT//3M)):}`

Explanation

To solve the problem of matching the properties of a monatomic gas at temperature T with their respective formulas, we will analyze each option in Column I and find the correct match in Column II.

### Step-by-Step Solution:

1. **Speed of Sound (A)**:
The speed of sound in a gas is given by the formula:
\[ v_s = \sqrt{\frac{\gamma RT}{M}} \]
For a monatomic gas, \(\gamma = \frac{5}{3}\). Therefore, the speed of sound becomes:
\[ v_s = \sqrt{\frac{5RT}{3M}} \]
This corresponds to option (p) in Column II.

2. **RMS Speed of Gas Molecules (B)**:
The root mean square (RMS) speed of gas molecules is given by:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
This corresponds to option (r) in Column II.

3. **Average Speed of Gas Molecules (C)**:
The average speed of gas molecules can be derived from statistical mechanics and is given by:
\[ v_{avg} = \sqrt{\frac{8RT}{\pi M}} \]
This corresponds to option (q) in Column II.

4. **Most Probable Speed of Gas Molecules (D)**:
The most probable speed of gas molecules is given by:
\[ v_{mp} = \sqrt{\frac{2RT}{M}} \]
This corresponds to option (s) in Column II.

### Final Matching:
- A (Speed of sound) → (p) \(\sqrt{\frac{5RT}{3M}}\)
- B (RMS speed of gas molecules) → (r) \(\sqrt{\frac{3RT}{M}}\)
- C (Average speed of gas molecules) → (q) \(\sqrt{\frac{8RT}{\pi M}}\)
- D (Most probable speed of gas molecules) → (s) \(\sqrt{\frac{2RT}{M}}\)

### Summary of Matches:
- A → p
- B → r
- C → q
- D → s