Problem 24 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

Rate of diffusion of a gas is: (a) directly proportional to its density. (b) directly proportional to its molecular weight. (c) directly proportional to the square root of its molecular weight. (d) inversely proportional to the square root of its molecular weight.

Step-by-Step Solution

Answer

(d) inversely proportional to the square root of its molecular weight.

Step 1: Understanding Graham's Law

Graham's Law states that the rate of diffusion of a gas is inversely proportional to the square root of its molar mass or molecular weight. This is because lighter gases diffuse more quickly than heavier ones.

Step 2: Identifying Key Terms

In the given options, we need to focus on two key terms: "directly proportional" and "inversely proportional." Directly proportional means that as one quantity increases, the other also increases, and inversely proportional means that as one increases, the other decreases.

Step 3: Analyzing the Options

We need to match the description from Graham's Law to the provided options. According to Graham's Law, the rate of diffusion is inversely proportional to the square root of the molecular weight. This matches option (d).

Step 4: Conclusion

Based on Graham's Law and the option analysis, the correct answer is that the rate of diffusion of a gas is inversely proportional to the square root of its molecular weight, which corresponds to option (d).

Key Concepts

Rate of Diffusion

The rate of diffusion is a measure of how quickly a gas spreads out or mixes with another gas. This process is crucial in many natural and industrial processes, such as the exchange of oxygen and carbon dioxide in human lungs or the mixing of gases in chemical engineering. The rate at which this happens depends on several factors, including temperature, pressure, and particularly the molecular weight of the gas.

Diffusion rate is affected by the speed at which gas particles move. When considering Graham's Law, lighter gases, which have lower molecular weights, tend to move faster than heavier gases. This increased speed allows them to diffuse more rapidly.
To put it simply, imagine you have two differently sized balls. If you roll both with the same force, the lighter ball will go farther and faster than the heavier one. Similarly, in the world of gases, lighter particles diffuse faster than heavier ones.

Molecular Weight

Molecular weight is an important concept when discussing the diffusion of gases. It refers to the mass of one molecule of a chemical compound and is usually measured in atomic mass units (amu). A gas's molecular weight affects how quickly it will diffuse in a given environment.

To illustrate, consider comparing helium and neon: helium has a molecular weight of about 4 amu, while neon has a molecular weight of about 20 amu. Because helium is significantly lighter than neon, it diffuses more rapidly according to Graham's Law.
Understanding molecular weight helps predict how gases will behave when mixed. It's a key factor in industries that deal with gas separations, chemical reactions, and environmental sciences.

Inverse Proportionality

Inverse proportionality is a fundamental principle that describes a specific relationship between two variables. Specifically, in terms of Graham's Law, as the molecular weight of a gas increases, the rate of diffusion decreases. This is a clear example of inverse proportionality.

Mathematically, this relationship is expressed as:

• \( ext{Rate of diffusion} \propto \frac{1}{\sqrt{\text{molecular weight}}} \)

This means that if you increase the molecular weight of the gas, the rate at which it diffuses will decrease. Conversely, if the molecular weight is decreased, the rate of diffusion will increase.
This concept is crucial in applications such as respiratory therapy and environmental modeling, where understanding how different gases move can affect human health and ecological balance.