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

Explanation

To solve the problem step by step, we will analyze the situation before and after the valve is opened.

### Step 1: Understand the Initial Conditions
- We have a container divided into two equal parts.
- One part contains an ideal gas at pressure \( P \) and temperature \( T \).
- The other part is evacuated (no gas, vacuum).

### Step 2: Identify the Volume
- Let the total volume of the container be \( V \).
- Each part of the container has a volume of \( \frac{V}{2} \).

### Step 3: Initial State of the Gas
- Before the valve is opened:
- Volume of gas, \( V_1 = \frac{V}{2} \)
- Pressure of gas, \( P_1 = P \)
- Temperature of gas, \( T_1 = T \)

### Step 4: Opening the Valve
- When the valve is suddenly opened, the gas expands into the entire volume of the container.
- The gas now occupies the total volume \( V \).

### Step 5: Final State of the Gas
- After the valve is opened:
- Volume of gas, \( V_2 = V \)
- The process is an expansion, and since it is sudden, we assume it is an isothermal process (temperature remains constant).
- Therefore, \( T_2 = T_1 = T \).

### Step 6: Applying the Ideal Gas Law
- According to the ideal gas law, we can write:
\[ P_1 V_1 = nRT_1 \quad \text{(initial state)} \]
\[ P_2 V_2 = nRT_2 \quad \text{(final state)} \]
- Since \( T_1 = T_2 \), we can equate the two equations:
\[ P_1 V_1 = P_2 V_2 \]

### Step 7: Substitute the Known Values
- Substitute \( P_1 = P \), \( V_1 = \frac{V}{2} \), and \( V_2 = V \):
\[ P \left(\frac{V}{2}\right) = P_2 (V) \]

### Step 8: Solve for Final Pressure \( P_2 \)
- Rearranging gives:
\[ P_2 = \frac{P \left(\frac{V}{2}\right)}{V} = \frac{P}{2} \]

### Conclusion
- After the valve is opened, the pressure of the gas becomes \( \frac{P}{2} \) and the temperature remains \( T \).

### Final Answer
- The pressure of the gas after the valve is opened is \( \frac{P}{2} \) and the temperature remains \( T \).

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