In the proscess T`prop` W, pressure of the gas increase from `p_(o)` to `t_(o)` Match the columns.
`{:(,"ColumnI",, "ColumnII"),((A),"Temperature of teh gas",(p),"Pssitive"),((B),"Volume of the gas",(q),"Nagative"),((C),"work done by the gas",(r),"Two times"),((D),"Heat supplies to the gas",(s),"Cannot say anything"):}`

Explanation

To solve the problem and match the columns correctly, we will analyze each of the options in Column I and determine their corresponding values in Column II based on the given process where the pressure of the gas increases from \( p_0 \) to \( T_0 \).

### Step-by-step Solution:

1. **Temperature of the gas (A)**:
- According to the ideal gas law, if pressure increases while the volume remains constant, the temperature must also increase.
- From the video transcript, it is mentioned that temperature is proportional to the square root of pressure. Therefore, if pressure increases, temperature also increases.
- If pressure becomes four times, temperature becomes two times.
- **Conclusion**: A matches with (r) "Two times".

2. **Volume of the gas (B)**:
- The relationship between pressure and volume is given by Boyle's Law, which states that pressure is inversely proportional to volume when temperature is constant.
- As pressure increases, volume must decrease. Therefore, the volume of the gas decreases.
- **Conclusion**: B matches with (p) "Positive".

3. **Work done by the gas (C)**:
- The work done by a gas is given by the formula \( W = P \Delta V \).
- If the volume decreases (as established in the previous step), the work done by the gas will be negative because the gas is doing work on the surroundings.
- **Conclusion**: C matches with (q) "Negative".

4. **Heat supplied to the gas (D)**:
- The heat supplied to the gas cannot be determined solely based on the changes in pressure and volume without additional information about the system.
- As stated in the video, we cannot conclude anything about the heat supplied to the gas.
- **Conclusion**: D matches with (s) "Cannot say anything".

### Final Matching:
- A → r (Two times)
- B → p (Positive)
- C → q (Negative)
- D → s (Cannot say anything)