A `4 : 1` mixture of helium and methane is confined in a vessel at `10` bar pressure . Due to a hole in the vessel, the gas mixture effusing out initially is

Options

A. ` 8 : 1`
B. ` 8 : 3`
C. `4 : 1`
D. ` 1:1`

Explanation

To solve the problem regarding the effusion of a gas mixture of helium and methane, we can follow these steps:

### Step 1: Understand the Mixture
We have a mixture of helium (He) and methane (CH₄) in a ratio of 4:1. This means for every 4 moles of helium, there is 1 mole of methane.

### Step 2: Determine Molar Masses
Next, we need to find the molar masses of both gases:
- Molar mass of helium (He) = 4 g/mol
- Molar mass of methane (CH₄) = 12 (C) + 4 (H) = 16 g/mol

### Step 3: Apply Graham's Law of Effusion
According to Graham's Law of Effusion, the rate of effusion of a gas is inversely proportional to the square root of its molar mass. The formula is given by:

\[ \frac{\text{Rate of effusion of He}}{\text{Rate of effusion of CH₄}} = \sqrt{\frac{\text{Molar mass of CH₄}}{\text{Molar mass of He}}} \]

### Step 4: Calculate the Rate of Effusion
Substituting the molar masses into the equation:

\[ \frac{\text{Rate of effusion of He}}{\text{Rate of effusion of CH₄}} = \sqrt{\frac{16}{4}} = \sqrt{4} = 2 \]

This means the rate of effusion of helium is twice that of methane.

### Step 5: Determine the Overall Mole Ratio
Since the initial mole ratio of the gases is 4:1 (He:CH₄), we can calculate the effective rate of effusion for the mixture.

Let the rate of effusion of methane be \( x \). Therefore, the rate of effusion of helium will be \( 2x \).

Now, considering the mole ratio:
- Moles of He = 4
- Moles of CH₄ = 1

The total effective rate of effusion can be calculated as follows:

\[ \text{Total rate} = 4 \times (2x) + 1 \times x = 8x + x = 9x \]

### Step 6: Calculate the Final Ratio
The ratio of the moles of gases effusing out can be expressed as:

\[ \frac{\text{Moles of He}}{\text{Moles of CH₄}} = \frac{8}{1} \]

Thus, the overall mole ratio of the gases effusing out is 8:1.

### Conclusion
The gas mixture effusing out initially is in the ratio of 8:1 (He:CH₄).

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