Question

60 cm\(^3\) of SO\(_2\) gas diffused through a porous membrane in \( x \) minutes. Under similar conditions, 360 cm\(^3\) of another gas (molar mass 4 g mol\(^{-1}\)) diffused in \( y \) minutes. The ratio of \( x \) and \( y \) is:

• A. \( 3:2 \)
• B. \( 2:3 \)
• C. \( 1:3 \)
• D. \( 3:1 \)

Hint:

Graham’s law states that the rate of diffusion of a gas is inversely proportional to the square root of its molar mass.

Answer 1

The Correct Option is B

To solve this problem, we will use Graham's Law of Diffusion, which states that the rate of diffusion of a gas is inversely proportional to the square root of its molar mass. Mathematically, Graham's Law is expressed as:

\[ \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}, \]

where:

• \( r_1 \) and \( r_2 \) are the rates of diffusion of gases 1 and 2, respectively,
• \( M_1 \) and \( M_2 \) are the molar masses of gases 1 and 2, respectively.

Step 1: Identify the gases and their molar masses

• Gas 1: \( \text{SO}_2 \) (sulfur dioxide), with molar mass \( M_1 = 64 \, \text{g mol}^{-1} \).
• Gas 2: Another gas, with molar mass \( M_2 = 4 \, \text{g mol}^{-1} \).

Step 2: Express the rates of diffusion The rate of diffusion \( r \) is given by the volume of gas diffused per unit time:

\[ r = \frac{V}{t}, \]

where:

• \( V \) is the volume of gas diffused,
• \( t \) is the time taken for diffusion.

For \( \text{SO}_2 \):

\[ r_1 = \frac{60 \, \text{cm}^3}{x \, \text{minutes}}. \]

For the other gas:

\[ r_2 = \frac{360 \, \text{cm}^3}{y \, \text{minutes}}. \]

Step 3: Apply Graham's Law Using Graham's Law:

\[ \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}. \]

Substitute the expressions for \( r_1 \) and \( r_2 \):

\[ \frac{\frac{60}{x}}{\frac{360}{y}} = \sqrt{\frac{4}{64}}. \]

Simplify the left-hand side:

\[ \frac{60}{x} \cdot \frac{y}{360} = \frac{y}{6x}. \]

Simplify the right-hand side:

\[ \sqrt{\frac{4}{64}} = \sqrt{\frac{1}{16}} = \frac{1}{4}. \]

Thus:

\[ \frac{y}{6x} = \frac{1}{4}. \]

Step 4: Solve for the ratio \( \frac{x}{y} \) Cross-multiply:

\[ 4y = 6x. \]

Divide both sides by 2:

\[ 2y = 3x. \]

Rearrange to find the ratio \( \frac{x}{y} \):

\[ \frac{x}{y} = \frac{2}{3}. \]

Final Answer: The ratio of \( x \) to \( y \) is:

\[ \boxed{2:3} \]