A mixture of two gases A and B in the mole ratio `2:3` is kept in a 2 litre vessel. A second 3 litre vessel has the same two gases in the mole ratio `3:5`. Both gas mixtures have the same temperature and same pressure. They are allowed to intermix and the final temperature and pressure are the same as the initial values, the final volume beingh 5 litres. Given theat the molar masses are `M_(A)` and `M_(B)` . what is the mean molar mass of the final mixture `:`

Options

A. `(77M_(A)+123M_(B))/(200)`
B. `(123M_(A)+77M_(B))/(200)`
C. `(77M_(A)+123M_(B))/(250)`
D. `(123M_(A)+77M_(B))/(250)`

Answer

To find the mean molar mass of the final mixture of gases A and B, we will follow these steps:

### Step 1: Calculate the number of moles of gases A and B in the first vessel.
Given the mole ratio of A to B is 2:3 in a 2-liter vessel:

- Total moles = 2 + 3 = 5
- Moles of A = (2/5) * (P * V) / (R * T) = (2/5) * (P * 2) / (R * T)
- Moles of A = (4/5) * (P / (R * T))

- Moles of B = (3/5) * (P * V) / (R * T) = (3/5) * (P * 2) / (R * T)
- Moles of B = (6/5) * (P / (R * T))

### Step 2: Calculate the number of moles of gases A and B in the second vessel.
Given the mole ratio of A to B is 3:5 in a 3-liter vessel:

- Total moles = 3 + 5 = 8
- Moles of A = (3/8) * (P * V) / (R * T) = (3/8) * (P * 3) / (R * T)
- Moles of A = (9/8) * (P / (R * T))

- Moles of B = (5/8) * (P * V) / (R * T) = (5/8) * (P * 3) / (R * T)
- Moles of B = (15/8) * (P / (R * T))

### Step 3: Combine the moles of gases A and B from both vessels.
Total moles of A from both vessels:
- Moles of A (total) = (4/5) + (9/8)

Total moles of B from both vessels:
- Moles of B (total) = (6/5) + (15/8)

### Step 4: Calculate the total molar mass of the mixture.
The total molar mass of the mixture can be calculated using the formula:
\[ \text{Total Molar Mass} = n_A \cdot M_A + n_B \cdot M_B \]

Where:
- \( n_A \) and \( n_B \) are the total moles of gases A and B respectively.
- \( M_A \) and \( M_B \) are the molar masses of gases A and B respectively.

### Step 5: Calculate the mean molar mass of the final mixture.
The mean molar mass is given by:
\[ \text{Mean Molar Mass} = \frac{\text{Total Molar Mass}}{\text{Total Moles}} \]

Where:
- Total Moles = Moles of A (total) + Moles of B (total)

### Final Calculation:
1. Calculate the total moles of A and B.
2. Substitute into the total molar mass equation.
3. Divide by the total moles to find the mean molar mass.