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:

$$\boxed{2:3}$$