Question:
The mean free path of a gas molecule is inversely proportional to
The mean free path of a gas molecule is inversely proportional to
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Remember that in kinetic theory, the mean free path depends on the size of molecules and their density, not directly on temperature.
Updated On: May 31, 2026
- Square of the molecular diameter
- Molecular diameter
- Temperature
- Square root of temperature
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The Correct Option is A
Solution and Explanation
Step 1: Concept
The mean free path (\(\lambda\)) of a gas molecule is defined as the average distance traveled by a molecule between two successive collisions. It depends on the size of the molecules and the density of the gas.
Step 2: Meaning
Inversely proportional means that if one quantity increases, the other decreases in such a way that their product remains constant.
Step 3: Analysis
The mean free path \(\lambda\) is given by: \[\lambda = \frac{1}{\sqrt{2} \pi d^2 n}\] where \(d\) is the molecular diameter and \(n\) is the number density of molecules. Notice that \(\lambda\) is inversely proportional to \(d^2\). This relationship can be understood as follows: As the molecular diameter \(d\) increases, the area over which a molecule can collide with another decreases, leading to a longer mean free path. Conversely, if \(d\) decreases (molecules become smaller), the area for collisions increases, reducing the mean free path. Let's examine each option: A) Square of the molecular diameter: \(\lambda \propto \frac{1}{d^2}\). This is correct because an increase in \(d^2\) leads to a decrease in \(\lambda\). B) Molecular diameter: \(\lambda \propto \frac{1}{d}\). This would imply that doubling the molecular diameter halves the mean free path, which does not match the given relationship. C) Temperature: The mean free path is independent of temperature. While higher temperatures increase the speed of molecules, they also increase the number density \(n\), balancing out to keep \(\lambda\) constant at a fixed pressure and volume. D) Square root of temperature: This would imply that \(\lambda \propto \frac{1}{\sqrt{T}}\). However, since the mean free path is independent of temperature under ideal gas assumptions, this option is incorrect.
Step 4: Conclusion
The correct relationship between the mean free path and the molecular diameter is that they are inversely proportional to the square of the molecular diameter. Final Answer: (A)
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