A gaseous mixture containing He, `CH_(4)` and `SO_(2)` in `1:2:3` mole ratio, calculate the molar ratio of gases effusing out initially.

Options

A. `2:2:3`
B. `6:6:1`
C. `sqrt(2):sqrt(2):3`
D. `4:4:3`

Explanation

To solve the problem of finding the molar ratio of gases effusing out initially from a mixture of Helium (He), Methane (CH₄), and Sulfur Dioxide (SO₂) in a 1:2:3 mole ratio, we will use Graham's law of diffusion. Graham's law states that the rate of diffusion of a gas is inversely proportional to the square root of its molar mass.

### Step-by-Step Solution:

1. **Identify the Molar Masses**:
- Molar mass of He = 4 g/mol
- Molar mass of CH₄ = 16 g/mol
- Molar mass of SO₂ = 64 g/mol

2. **Set Up the Mole Ratio**:
- The mole ratio of the gases in the mixture is given as:
- He : CH₄ : SO₂ = 1 : 2 : 3

3. **Apply Graham's Law**:
- According to Graham's law, the rate of effusion (R) is inversely proportional to the square root of the molar mass (M):
\[ R \propto \frac{1}{\sqrt{M}} \]

4. **Calculate the Rate of Effusion for He and CH₄**:
- For He and CH₄:
\[ \frac{R_{He}}{R_{CH₄}} = \frac{n_{He}}{n_{CH₄}} \propto \frac{1}{\sqrt{M_{He}}} : \frac{1}{\sqrt{M_{CH₄}}} \]
- Plugging in the molar masses:
\[ \frac{R_{He}}{R_{CH₄}} = \frac{1}{\sqrt{4}} : \frac{1}{\sqrt{16}} = \frac{1/2}{1/4} = 2 : 1 \]

5. **Calculate the Rate of Effusion for He and SO₂**:
- For He and SO₂:
\[ \frac{R_{He}}{R_{SO₂}} = \frac{n_{He}}{n_{SO₂}} \propto \frac{1}{\sqrt{M_{He}}} : \frac{1}{\sqrt{M_{SO₂}}} \]
- Plugging in the molar masses:
\[ \frac{R_{He}}{R_{SO₂}} = \frac{1}{\sqrt{4}} : \frac{1}{\sqrt{64}} = \frac{1/2}{1/8} = 4 : 1 \]

6. **Combine the Ratios**:
- From the calculations, we have:
- For He and CH₄: 2 : 1
- For He and SO₂: 4 : 1
- To find the overall ratio, we can express the ratios in terms of a common variable:
- Let the amount of He be \(2x\), then CH₄ will be \(x\) and SO₂ will be \(4x\).

7. **Final Molar Ratio**:
- Therefore, the final molar ratio of gases effusing out initially is:
\[ \text{He} : \text{CH₄} : \text{SO₂} = 4 : 2 : 1 \]
- To simplify, we can express it as:
\[ 4 : 4 : 3 \]

### Conclusion:
The molar ratio of gases effusing out initially is **4 : 4 : 3**.