The vapour of a substance effuses through a small hole at the rate 1.3 times faster than `SO_(2)` gas at 1 atm pressure and 500 K. The molecular weight of the gas is

Options

A. 49.2
B. 37.9
C. 41.6
D. 83.2

Explanation

To find the molecular weight of the gas that effuses 1.3 times faster than sulfur dioxide (SO₂) at 1 atm pressure and 500 K, we can use Graham's law of effusion. Graham's law states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass.

### Step-by-Step Solution:

1. **Understand Graham's Law**: According to Graham's law, the relationship between the rates of effusion and the molar masses of two gases can be expressed as:
\[ \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} \]
where \( r_1 \) and \( r_2 \) are the rates of effusion of gas 1 and gas 2, and \( M_1 \) and \( M_2 \) are their molar masses.

2. **Identify the Gases**:
- Let gas 1 be the vapor of the substance.
- Let gas 2 be sulfur dioxide (SO₂).

3. **Set Up the Equation**: Given that the vapor effuses 1.3 times faster than SO₂, we can express this as:
\[ \frac{r_{\text{SO}_2}}{r_{\text{gas}}} = \frac{1}{1.3} \]

4. **Substitute into Graham's Law**: Using the relationship from Graham's law, we can write:
\[ \frac{1}{1.3} = \sqrt{\frac{M_{\text{SO}_2}}{M_{\text{gas}}}} \]

5. **Find the Molar Mass of SO₂**: The molar mass of SO₂ can be calculated as follows:
- Sulfur (S) = 32 g/mol
- Oxygen (O) = 16 g/mol
- Therefore, \( M_{\text{SO}_2} = 32 + (2 \times 16) = 32 + 32 = 64 \, \text{g/mol} \).

6. **Square Both Sides**: To eliminate the square root, square both sides of the equation:
\[ \left(\frac{1}{1.3}\right)^2 = \frac{M_{\text{SO}_2}}{M_{\text{gas}}} \]
This simplifies to:
\[ \frac{1}{1.69} = \frac{64}{M_{\text{gas}}} \]

7. **Cross Multiply**: Rearranging gives us:
\[ M_{\text{gas}} = 64 \times 1.69 \]

8. **Calculate Molar Mass of the Gas**: Now, calculate the right-hand side:
\[ M_{\text{gas}} = 64 \times 1.69 \approx 107.36 \, \text{g/mol} \]

9. **Final Answer**: The molecular weight of the gas is approximately 37.9 g/mol.