Two gases A and B having the same temperature 'T' , Same pressure 'P' and same volume 'V' are mixed . If the temperature of mixture is unchanged and the volume occupied by it is `'V//2'` , then the pressure of the mixture will be

Options

A. P/2
B. P
C. 2P
D. 4P

Answer

To solve the problem, we will use the ideal gas law and the information provided about the gases A and B.

### Step-by-Step Solution:

1. **Understand the Ideal Gas Law**: The ideal gas law is given by the equation \( PV = nRT \), where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles
- \( R \) = ideal gas constant
- \( T \) = temperature

2. **Initial Conditions**:
- Both gases A and B are at the same temperature \( T \), pressure \( P \), and volume \( V \).
- Therefore, for gas A:
\[ P_A V = n_A R T \]
and for gas B:
\[ P_B V = n_B R T \]
- Since \( P_A = P_B = P \), we can say:
\[ PV = n_A R T \quad \text{and} \quad PV = n_B R T \]

3. **Total Moles of the Mixture**:
- Let the number of moles of gas A and gas B be \( n_A \) and \( n_B \) respectively.
- The total number of moles in the mixture is:
\[ n_{total} = n_A + n_B \]

4. **Volume of the Mixture**:
- The volume occupied by the mixture is given as \( \frac{V}{2} \).

5. **Pressure of the Mixture**:
- Using the ideal gas law for the mixture, we have:
\[ P_{mixture} \cdot \frac{V}{2} = n_{total} R T \]

6. **Relate Moles to Pressure**:
- Since both gases have the same pressure \( P \) and volume \( V \), we can express the total moles in terms of pressure:
\[ n_A = \frac{PV}{RT} \quad \text{and} \quad n_B = \frac{PV}{RT} \]
- Therefore, the total number of moles is:
\[ n_{total} = n_A + n_B = \frac{PV}{RT} + \frac{PV}{RT} = \frac{2PV}{RT} \]

7. **Substituting Back**:
- Substitute \( n_{total} \) back into the equation for the pressure of the mixture:
\[ P_{mixture} \cdot \frac{V}{2} = \left(\frac{2PV}{RT}\right) R T \]
- Simplifying this gives:
\[ P_{mixture} \cdot \frac{V}{2} = 2PV \]

8. **Solving for \( P_{mixture} \)**:
- Rearranging the equation:
\[ P_{mixture} = \frac{2PV}{\frac{V}{2}} = 4P \]

### Final Answer:
The pressure of the mixture will be \( 4P \).