Problem 12 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

Sulphur dioxide and oxygen were allowed to diffuse through a porous partition. \(20 \mathrm{dm}^{3}\) of \(\mathrm{SO}_{2}\) diffuses through the porous partition in 60 seconds. The volume of \(\mathrm{O}_{2}\) in \(\mathrm{dm}^{3}\) which diffuses under the similar condition in 30 seconds will be (atomic mass of sulphur \(=32 \mathrm{u}\) ): (a) \(7.09\) (b) \(14.1\) (c) \(10.0\) (d) \(28.2\)

Step-by-Step Solution

Answer

The volume of \( \text{O}_2 \) that diffuses is 14.1 dm³ (option b).

Step 1: Use Graham's Law of Diffusion

According to Graham's Law of Diffusion, the rate of diffusion of a gas is inversely proportional to the square root of its molar mass. For two gases, this 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 diffusion of \( \text{SO}_2 \) and \( \text{O}_2 \), respectively, and \( M_1 \) and \( M_2 \) are their molar masses.

Step 2: Calculate Molar Masses

Calculate the molar mass of \( \text{SO}_2 \) and \( \text{O}_2 \). \( \text{SO}_2 \) consists of 1 sulphur atom (32 u) and 2 oxygen atoms (16 u each): \( 32 + 2 \times 16 = 64 \) u. \( \text{O}_2 \) consists of 2 oxygen atoms: \( 2 \times 16 = 32 \) u.

Step 3: Calculate Rate of Diffusion of SO2

Given, 20 dm³ of \( \text{SO}_2 \) diffuses in 60 seconds. So, the rate of diffusion \( r_1 \) for \( \text{SO}_2 \) is \( \frac{20 \text{ dm}^3}{60 \text{ s}} = \frac{1}{3} \text{ dm}^3/\text{s} \).

Step 4: Apply Graham's Law

Using Graham's law formula: \[ \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} \] Substitute: \( M_1 = 64 \), \( M_2 = 32 \), and \( r_1 = \frac{1}{3} \): \[ \frac{\frac{1}{3}}{r_2} = \sqrt{\frac{32}{64}} \] Simplifying, \[ \frac{1}{3r_2} = \frac{1}{\sqrt{2}} \].

Step 5: Solve for \( r_2 \)

Rearrange the equation to solve for \( r_2 \): \[ 3r_2 = \sqrt{2} \] \[ r_2 = \frac{\sqrt{2}}{3} \] Calculate \( \sqrt{2} \approx 1.414 \), so \( r_2 \approx \frac{1.414}{3} \approx 0.471 \text{ dm}^3/\text{s} \).

Step 6: Calculate Volume of O2 Diffused in 30 seconds

Using the calculated rate of diffusion \( r_2 \), the volume of \( \text{O}_2 \) diffused in 30 seconds is: \[ \text{Volume} = r_2 \times 30 = 0.471 \times 30 = 14.13 \text{ dm}^3 \].

Step 7: Choose Nearest Option

Match the calculated volume to the closest option, 14.1 dm³ corresponds to option (b).

Key Concepts

Diffusion Rate Calculation

Understanding how gases spread out or diffuse can be quite fascinating. Diffusion happens when gas particles move from an area of high concentration to an area of low concentration. To calculate this diffusion rate, chemists often rely on Graham's Law of Diffusion. This law tells us that the diffusion rate of a gas is inversely proportional to the square root of its molar mass. This means lighter gases diffuse faster than heavier ones.
For example, if you know how quickly one gas is diffusing, you can determine the rate of a second gas by comparing their molar masses using the formula from Graham's Law. In practice, this equation looks like this: \[\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}\]where \(r_1\) and \(r_2\) are the rates of diffusion, and \(M_1\) and \(M_2\) are the molar masses of two different gases.
By applying this equation, you can solve real-world problems, like finding out how much oxygen diffuses compared to sulfur dioxide in a given time period.

Molar Mass Calculation

Calculating molar mass is another crucial step in chemistry problem-solving, especially in diffusion calculations. The molar mass of a substance is the mass of one mole of its particles. It is usually expressed in grams per mole (g/mol) and is crucial for understanding how quickly different gases will diffuse.
For example, the molar mass of sulfur dioxide, \(\text{SO}_2\), can be computed by adding the atomic masses of one sulfur atom (32 u) and two oxygen atoms (16 u each). This gives you:\[32 + 2 \times 16 = 64 \text{ u}\]Similarly, for oxygen gas, \(\text{O}_2\), which consists of two oxygen atoms, the calculation is:\[2 \times 16 = 32 \text{ u}\]These molar masses are essential when using Graham's Law of Diffusion to find how different gases will act under similar conditions.
Knowing how to calculate and use molar mass helps you to make predictions and solve chemistry puzzles involving different substances.

Problem-Solving in Chemistry

Problem-solving in chemistry often involves a sequence of logical steps to find a solution. Let's consider a scenario involving diffusion. First, identify what's given and what you need to find. In our exercise, we are given the volume and time for \(\text{SO}_2\) to diffuse, and need to find out how much \(\text{O}_2\) will diffuse in a shorter period.
Follow these simple steps:

• Calculate the rate of diffusion for the known gas using the formula \(\text{Rate} = \frac{\text{Volume}}{\text{Time}}\).
• Use Graham's Law to relate the diffusion rates of the two gases.
• Substitute the molar masses you've calculated into Graham's Law equation.
• Solve for the unknown rate of the second gas.
• Finally, use this rate to calculate the new volume for the given time period.

This approach shows you how problem-solving in chemistry combines mathematical skills with chemical knowledge to find solutions.
With practice, these steps can become instinctive, making complex problems much easier to tackle.