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 step-by-step, we can follow these logical steps:

### 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 \( T \), while the other part is a vacuum (pressure = 0).

### Step 2: Analyze the Process When the Valve is Opened
When the valve is suddenly opened, the gas will expand to fill the entire volume of the container. Since the walls are insulating, no heat can enter or leave the system, indicating that the process is adiabatic.

### Step 3: Apply the First Law of Thermodynamics
For an adiabatic process involving an ideal gas, the internal energy remains constant if there is no heat transfer and no work done on or by the system. Therefore, the internal energy before and after the expansion must be the same:
\[ U_1 = U_2 \]
Since the internal energy of an ideal gas is a function of temperature, this implies that:
\[ T_1 = T_2 \]
Thus, the temperature of the gas after the expansion remains \( T \).

### Step 4: Use Boyle's Law to Find the New Pressure
Since the process is isothermal (temperature remains constant), we can apply Boyle's Law, which states:
\[ P_1 V_1 = P_2 V_2 \]
Where:
- \( P_1 = P \) (initial pressure)
- \( V_1 = \frac{V}{2} \) (initial volume of gas)
- \( V_2 = V \) (final volume of gas after expansion)

Substituting these values into Boyle's Law gives:
\[ P \left(\frac{V}{2}\right) = P_2 (V) \]

### Step 5: Solve for the Final Pressure \( P_2 \)
Rearranging the equation to solve for \( P_2 \):
\[ P_2 = \frac{P \left(\frac{V}{2}\right)}{V} = \frac{P}{2} \]

### Conclusion
After the valve is opened, the pressure of the gas will be \( \frac{P}{2} \) and the temperature will remain \( T \).

### Final Answer
- Pressure \( P_2 = \frac{P}{2} \)
- Temperature \( T_2 = T \)