NEET 2026 ALL Q #1933
COMPETENCY BASED
APPLY
4 Marks 2026 NTA-RE-NEET-2026 MCQ SINGLE
The mean free path of molecules in an ideal gas A is half that of another ideal gas B. The diameter of the spherical molecules of gas A is twice the diameter of the molecules of B. If number densities of the gases A and B are $n_{A}$ and $n_{B}$ respectively, the correct option is:
(A) $n_{A}=\frac{1}{2}n_{B}$
(B) $n_{A}=n_{B}$
(C) $n_{A}=2n_{B}$
(D) $n_{A}=\frac{1}{4}n_{B}$
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Correct Answer: A

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Detailed Solution

Step 1: Recall the formula for Mean Free Path

The mean free path ($ \lambda $) of gas molecules is given by the formula: $$ \lambda = \frac{1}{\sqrt{2} \pi d^2 n} $$ where $ d $ is the diameter of the molecule and $ n $ is the number density.

Step 2: Set up the ratio for gases A and B

Given $ \lambda_A = \frac{1}{2} \lambda_B $ and $ d_A = 2 d_B $. Using the formula: $$ \frac{\lambda_A}{\lambda_B} = \frac{d_B^2 n_B}{d_A^2 n_A} $$

Step 3: Substitute the given values

Substitute $ \frac{\lambda_A}{\lambda_B} = \frac{1}{2} $ and $ d_A = 2 d_B $: $$ \frac{1}{2} = \frac{d_B^2 n_B}{(2 d_B)^2 n_A} $$ $$ \frac{1}{2} = \frac{d_B^2 n_B}{4 d_B^2 n_A} $$ $$ \frac{1}{2} = \frac{n_B}{4 n_A} $$

Step 4: Solve for $ n_A $

Rearranging the equation: $$ 4 n_A = 2 n_B $$ $$ n_A = \frac{2}{4} n_B = \frac{1}{2} n_B $$

Final Answer: n_{A}=\frac{1}{2}n_{B}

AI Suggestion: Option A

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must utilize the theoretical formula for mean free path and perform algebraic manipulation to compare two distinct physical systems.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the application of a specific physical law (Kinetic Theory of Gases) through a multi-step mathematical derivation.
Syllabus Audit: In the context of NEET, this is classified as COMPETENCY. It tests the student's ability to relate microscopic parameters (diameter, number density) to macroscopic transport properties (mean free path), which is a core competency in Thermodynamics and Kinetic Theory.

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