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, which states that \( PV = nRT \), where:

- \( P \) is the pressure,
- \( V \) is the volume,
- \( n \) is the number of moles,
- \( R \) is the universal gas constant,
- \( T \) is the temperature.

### Step-by-Step Solution:

1. **Determine Moles of Gases A and B:**
- For gas A:
\[ n_A = \frac{PV}{RT} \]
- For gas B:
\[ n_B = \frac{PV}{RT} \]

2. **Total Moles in the Mixture:**
- When gases A and B are mixed, the total number of moles \( n_{total} \) is:
\[ n_{total} = n_A + n_B = \frac{PV}{RT} + \frac{PV}{RT} = \frac{2PV}{RT} \]

3. **Final Conditions of the Mixture:**
- The volume of the mixture is given as \( V/2 \).
- The temperature remains unchanged at \( T \).

4. **Using the Ideal Gas Law for the Mixture:**
- The ideal gas law for the final state of the mixture can be expressed as:
\[ P_{final} \cdot \left(\frac{V}{2}\right) = n_{total} \cdot RT \]
- Substituting \( n_{total} \):
\[ P_{final} \cdot \left(\frac{V}{2}\right) = \left(\frac{2PV}{RT}\right) \cdot RT \]
- Simplifying the right side:
\[ P_{final} \cdot \left(\frac{V}{2}\right) = 2PV \]

5. **Solving for \( P_{final} \):**
- Rearranging the equation to solve for \( P_{final} \):
\[ P_{final} = \frac{2PV}{\frac{V}{2}} = \frac{2PV \cdot 2}{V} = 4P \]

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

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