Question:
The mean free path of a gas molecule of diameter d and number density n (number of molecules per unit volume) is inversely proportional to:
The mean free path of a gas molecule of diameter d and number density n (number of molecules per unit volume) is inversely proportional to:
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Think of the collision space geometrically: a molecule sweeps out a target collision cylinder of cross-sectional area = d^2. The higher the density (n) or the wider this target area ( d^2), the more frequently collisions occur, meaning the mean free path () shrinks: 1Area Density = 1 d^2 n.
Updated On: Jun 10, 2026
- n d
- n^2 d
- n d^2
- n d
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The Correct Option is C
Solution and Explanation
Concept:
In a real gas, molecules are continuously moving in random directions and colliding with one another. The path traversed by an individual molecule between two successive elastic collisions is a straight line covered at a constant speed. The length of these segments varies randomly.
The statistical average of the distances covered between successive collisions is defined as the Mean Free Path (). According to the kinetic theory of gases, the expression for the mean free path taking into account the relative velocity distribution of molecules is given by:
where:
• n is the number density of the gas molecules (total number of molecules per unit volume, n = N/V).
• d is the effective collision diameter of a gas molecule.
Step 1: Analyzing the Functional Dependencies
From the mathematical definition of the mean free path, we can examine how changing the physical parameters of the gas grid dynamically impacts the straight-line travel budget of a molecule: Combining these individual inverse proportionalities yields:
Step 2: Matching with the Question Requirement
The question asks to find the specific expression to which the mean free path is inversely proportional. Since we can write = constantn d^2, it implies that is inversely proportional to the compound term n d^2. Let us evaluate the options:
• n d: Incorrect, misses the quadratic dependence on the molecular cross-sectional collision sweep area.
• n^2 d: Incorrect, over-represents density scaling.
• n d^2: Correct, perfectly matches the denominator group of the standard Maxwellian derivation.
• n d: Incorrect.
Therefore, the mean free path is inversely proportional to n d^2, which corresponds to Option (C).
• n is the number density of the gas molecules (total number of molecules per unit volume, n = N/V).
• d is the effective collision diameter of a gas molecule.
Step 1: Analyzing the Functional Dependencies
From the mathematical definition of the mean free path, we can examine how changing the physical parameters of the gas grid dynamically impacts the straight-line travel budget of a molecule: Combining these individual inverse proportionalities yields:
Step 2: Matching with the Question Requirement
The question asks to find the specific expression to which the mean free path is inversely proportional. Since we can write = constantn d^2, it implies that is inversely proportional to the compound term n d^2. Let us evaluate the options:
• n d: Incorrect, misses the quadratic dependence on the molecular cross-sectional collision sweep area.
• n^2 d: Incorrect, over-represents density scaling.
• n d^2: Correct, perfectly matches the denominator group of the standard Maxwellian derivation.
• n d: Incorrect.
Therefore, the mean free path is inversely proportional to n d^2, which corresponds to Option (C).
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