A container with insulating walls is divided into two equal parts by a partition fitted with a valve. One part is filled with an ideal gas at a pressure P and temperature T, whereas the other part is completely evacuated . If the valve is suddenly opened, the pressure and temperature of the gas will be

Options

A. `(P)/(2), T`
B. `(P)/(2), (T)/(2)`
C. P, T
D. `P, (T)/(2)`

Explanation

To solve the problem of what happens when the valve is suddenly opened in a container divided into two equal parts, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the System**:
- We have a container divided into two equal parts by a partition.
- One part contains an ideal gas at pressure \( P \) and temperature \( T \).
- The other part is a vacuum (completely evacuated).

2. **Opening the Valve**:
- When the valve is opened, the gas will expand into the vacuum. This is known as free expansion.

3. **Characteristics of Free Expansion**:
- In free expansion, the gas does not perform work on the surroundings, and since the walls are insulating, there is no heat exchange with the environment.
- Therefore, the process is adiabatic, and the internal energy of the gas remains constant.

4. **Temperature Change**:
- For an ideal gas undergoing free expansion, the temperature remains constant. Hence, the final temperature \( T' \) of the gas after expansion will still be \( T \).

5. **Pressure Change**:
- Initially, the volume of the gas is \( V/2 \) (where \( V \) is the total volume of the container).
- After the gas expands to fill the entire volume \( V \), we can use the ideal gas law, which states that \( PV = nRT \).
- Initially, we have:
\[ P \cdot \frac{V}{2} = nRT \]
- After expansion, the new pressure \( P' \) can be calculated as:
\[ P' \cdot V = nRT \]
- By equating the two expressions for \( nRT \):
\[ P \cdot \frac{V}{2} = P' \cdot V \]
- Simplifying this gives:
\[ P' = \frac{P}{2} \]

6. **Final Results**:
- After the valve is opened, the pressure of the gas will be \( \frac{P}{2} \) and the temperature will remain \( T \).

### Conclusion:
- The final pressure of the gas is \( \frac{P}{2} \) and the final temperature is \( T \).