Two gases A and B having the same temperature T, same pressure P and same volume V are mixed. If the mixture is at the same temperature and occupies a volume V. The pressure of the mixture is

Options

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

Answer

To solve the problem of finding the pressure of the mixture of two gases A and B, we can use the ideal gas law, which states that for a given amount of gas at constant temperature, the pressure and volume are related. Here’s the step-by-step solution:

### Step 1: Understand the Initial Conditions
We have two gases A and B, both at the same temperature (T), pressure (P), and volume (V).

### Step 2: Determine the Total Initial Volume
Since there are two gases, the total initial volume (V1) when they are mixed is:
\[ V1 = V_A + V_B = V + V = 2V \]

### Step 3: Identify the Final Conditions
After mixing, the gases occupy a volume (V2) which is given as:
\[ V2 = V \]

### Step 4: Apply the Ideal Gas Law
According to the ideal gas law, the relationship between pressure and volume at constant temperature can be expressed as:
\[ P_1 V_1 = P_2 V_2 \]
Where:
- \( P_1 \) is the initial pressure (P)
- \( V_1 \) is the initial volume (2V)
- \( P_2 \) is the final pressure (which we need to find)
- \( V_2 \) is the final volume (V)

### Step 5: Substitute the Known Values
Substituting the known values into the equation:
\[ P \cdot 2V = P_2 \cdot V \]

### Step 6: Solve for the Final Pressure (P2)
To find \( P_2 \), we can rearrange the equation:
\[ P_2 = \frac{P \cdot 2V}{V} \]
This simplifies to:
\[ P_2 = 2P \]

### Conclusion
Thus, the pressure of the mixture is:
\[ \text{Pressure of the mixture} = 2P \]

### Final Answer
The correct option is \( 2P \).

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