Equal volumes of two gases which do not react together are enclosed in separate vessel. Their pressure at 100 mm and 400 mm respectively. If the two vessel are joined together, then what will be the pressure of the resulting mixture (temperature remaining constant)?

Options

A. 125 mm
B. 500 mm
C. 1000 mm
D. 250 mm

Explanation

To solve the problem, we will use the ideal gas law and the concept of partial pressures. Here’s a step-by-step solution:

### Step 1: Understand the Given Information
We have two gases in separate vessels:
- Gas 1 has a pressure of \( P_1 = 100 \, \text{mm} \)
- Gas 2 has a pressure of \( P_2 = 400 \, \text{mm} \)
- Both gases occupy equal volumes \( V \) and are at the same temperature \( T \).

### Step 2: Calculate the Number of Moles of Each Gas
Using the ideal gas equation \( PV = nRT \), we can express the number of moles for each gas.

For Gas 1:
\[ n_1 = \frac{P_1 V}{RT} = \frac{100 \, \text{mm} \cdot V}{RT} \]

For Gas 2:
\[ n_2 = \frac{P_2 V}{RT} = \frac{400 \, \text{mm} \cdot V}{RT} \]

### Step 3: Find the Total Number of Moles
When the two vessels are joined, the total number of moles \( n_{total} \) is the sum of the moles of both gases:
\[ n_{total} = n_1 + n_2 = \frac{100 \, \text{mm} \cdot V}{RT} + \frac{400 \, \text{mm} \cdot V}{RT} \]
\[ n_{total} = \frac{(100 + 400) \, \text{mm} \cdot V}{RT} = \frac{500 \, \text{mm} \cdot V}{RT} \]

### Step 4: Calculate the Volume of the Resulting Mixture
When the two gases are mixed, the total volume \( V_{total} \) becomes:
\[ V_{total} = V + V = 2V \]

### Step 5: Use the Ideal Gas Law to Find the Pressure of the Mixture
We can now use the ideal gas equation again to find the pressure \( P \) of the resulting mixture:
\[ PV_{total} = n_{total} RT \]
Substituting the values we found:
\[ P \cdot (2V) = \frac{500 \, \text{mm} \cdot V}{RT} \cdot RT \]
\[ P \cdot (2V) = 500 \, \text{mm} \cdot V \]

### Step 6: Solve for Pressure \( P \)
Dividing both sides by \( 2V \):
\[ P = \frac{500 \, \text{mm} \cdot V}{2V} = \frac{500 \, \text{mm}}{2} = 250 \, \text{mm} \]

### Final Answer
The pressure of the resulting mixture is \( 250 \, \text{mm} \).

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