Question:

60 cm3^3 of SO2_2 gas diffused through a porous membrane in x x minutes. Under similar conditions, 360 cm3^3 of another gas (molar mass 4 g mol1^{-1}) diffused in y y minutes. The ratio of x x and y y is:

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Graham’s law states that the rate of diffusion of a gas is inversely proportional to the square root of its molar mass.
Updated On: Mar 13, 2025
  • 3:2 3:2
  • 2:3 2:3
  • 1:3 1:3
  • 3:1 3:1
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The Correct Option is B

Solution and Explanation

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:

r1r2=M2M1, \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}},

where:
  • r1 r_1 and r2 r_2 are the rates of diffusion of gases 1 and 2, respectively,
  • M1 M_1 and M2 M_2 are the molar masses of gases 1 and 2, respectively.

Step 1: Identify the gases and their molar masses
  • Gas 1: SO2 \text{SO}_2 (sulfur dioxide), with molar mass M1=64g mol1 M_1 = 64 \, \text{g mol}^{-1} .
  • Gas 2: Another gas, with molar mass M2=4g mol1 M_2 = 4 \, \text{g mol}^{-1} .

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

r=Vt, r = \frac{V}{t},

where:
  • V V is the volume of gas diffused,
  • t t is the time taken for diffusion.
For SO2 \text{SO}_2 :

r1=60cm3xminutes. r_1 = \frac{60 \, \text{cm}^3}{x \, \text{minutes}}.

For the other gas:

r2=360cm3yminutes. r_2 = \frac{360 \, \text{cm}^3}{y \, \text{minutes}}.


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

r1r2=M2M1. \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}.

Substitute the expressions for r1 r_1 and r2 r_2 :

60x360y=464. \frac{\frac{60}{x}}{\frac{360}{y}} = \sqrt{\frac{4}{64}}.

Simplify the left-hand side:

60xy360=y6x. \frac{60}{x} \cdot \frac{y}{360} = \frac{y}{6x}.

Simplify the right-hand side:

464=116=14. \sqrt{\frac{4}{64}} = \sqrt{\frac{1}{16}} = \frac{1}{4}.

Thus:

y6x=14. \frac{y}{6x} = \frac{1}{4}.


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

4y=6x. 4y = 6x.

Divide both sides by 2:

2y=3x. 2y = 3x.

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

xy=23. \frac{x}{y} = \frac{2}{3}.


Final Answer: The ratio of x x to y y is:

2:3 \boxed{2:3}

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