Q.2 Physics ⭐ 360° Option Elimination Active

NEET 2026 Physics Question 2 Solution

Topic: Kinetic Theory of Gases · Official NTA Booklet Code 50

Question

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, then the correct option is:

✓

Verified Against NTA Final Answer Key

Option (4) : n_A = (1/2) * n_B

📚 Kinetic Theory of Gases Medium

📋 Given Parameters & Problem Setup

Mean Free Path of Gas A (lambda_A) (1/2) * lambda_B

Molecular Diameter of Gas A (d_A) 2 * d_B

Target Variable Ratio of number densities n_A / n_B

⚡

The 30-Second Exam Hall Shortcut: ⚡ Inverse Power Law Scaling

⏱️ Solves in 15 seconds without writing full constants.

Isolate number density n in terms of lambda and d

1 Recall fundamental relation: lambda is inversely proportional to (n * d^2).

2 Rearrange for number density: n is proportional to 1 / (lambda * d^2).

3 Form the scaling ratio: n_A / n_B = (lambda_B / lambda_A) * (d_B / d_A)^2.

4 Substitute given values: (2) * (1/2)^2 = 2 * (1/4) = 1/2 ==> n_A = (1/2) * n_B.

🛡️

Why You Can 100% Trust This Shortcut: Collision cross-section area sigma scales as the square of the diameter (pi * d^2). Doubling diameter quadruples collision probability, requiring density to halve to balance the halved path length.

📐

Step-by-Step Rigorous Workflow & Mathematical Proof

Step 1

State the Clausius-Maxwell Mean Free Path Formula

For an ideal gas with spherical molecules of diameter d and number density n, the mean free path lambda is given by:

\lambda = \frac{1}{\sqrt{2} \pi n d^2}

• The sqrt(2) factor accounts for the relative motion of colliding molecules.
• pi * d^2 represents the effective spherical collision cross-section.

Step 2

Express Number Density Explicitly

Rearranging the formula to isolate the number density n:

n = \frac{1}{\sqrt{2} \pi \lambda d^2} \implies n \propto \frac{1}{\lambda d^2}

• Number density is inversely proportional to the mean free path.
• Number density is inversely proportional to the square of molecular diameter.

Step 3

Substitute Ratios and Solve

Calculate the ratio of number densities for Gas A relative to Gas B:

\frac{n_A}{n_B} = \left( \frac{\lambda_B}{\lambda_A} \right) \times \left( \frac{d_B}{d_A} \right)^2 = (2) \times \left( \frac{1}{2} \right)^2 = 2 \times \frac{1}{4} = \frac{1}{2}

• lambda_B / lambda_A = 2 (since lambda_A is half of lambda_B).
• (d_B / d_A)^2 = (1/2)^2 = 1/4 (since d_A is twice d_B).
• Multiplying gives n_A / n_B = 1/2 ==> n_A = (1/2) * n_B.

⭐

360° Option Elimination Engine Key Feature

Diagnose every choice: Know exactly why the right option works and why distractors fail.

4-Option Deep Scan

1 Option (1)

✗ Incorrect

Verdict & Scientific Reason:

Ignores the quadratic effect of molecular diameter on collision rate.

⚠️

Student Fallacy (Why candidates pick this trap): Assumes mean free path depends linearly on diameter (lambda ~ 1/(n*d)), causing a false 2 * (1/2) = 1 cancellation.

Note: Collision cross-section is an AREA (d^2), not a linear length (d).

2 Option (2)

✗ Incorrect

Verdict & Scientific Reason:

Inverts the inverse relationship.

⚠️

Student Fallacy (Why candidates pick this trap): Multiplies diameter ratio directly without taking the reciprocal.

Note: Larger molecules collide much more frequently; their density must be lower, not higher, for a given path.

3 Option (3)

✗ Incorrect

Verdict & Scientific Reason:

Accounts for diameter squared but completely forgets the mean free path ratio.

⚠️

Student Fallacy (Why candidates pick this trap): Only computes (d_B / d_A)^2 = 1/4 and omits the (lambda_B / lambda_A) = 2 multiplier.

Note: Both parameters (lambda and d) were altered in the problem statement.

4 Option (4)

✓ 100% Correct

Verdict & Scientific Reason:

100% scientifically accurate and confirmed by Official NTA Final Answer Key.

💡

Why this is airtight: Correctly accounts for both the inverse linear path ratio and inverse quadratic diameter ratio.

Note: Exact match to Clausius-Maxwell collision dynamics.

⚠️

NTA Trap Alert: The Linear Diameter Trap

Many candidates mistakenly remember lambda ~ 1/(n*d) instead of lambda ~ 1/(n*d^2), leading them to mark Option (1).

🛡️ How to avoid the -1 mark: Remember dimensional consistency: n is per unit volume (m^-3) and lambda is length (m). To make the right-hand side meters, the denominator must have m^-3 * m^2 = m^-1, which requires diameter squared (d^2)!

📖

Core NEET Syllabus Standard

NEET UG Physics Core Standards · Kinetic Theory of Gases Standard Benchmark

Core Scientific Principle Tested:

"Mean free path is the average distance traversed between successive collisions: lambda = 1 / (sqrt(2) * pi * n * d^2)."

Syllabus Focus: Mean Free Path & Molecular Dimensions

✓ Core NEET UG Syllabus Anchor Syllabus & Academic Standards ↗

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NEET 2026 Physics Question 2 Solution

Question

The 30-Second Exam Hall Shortcut: ⚡ Inverse Power Law Scaling

Step-by-Step Rigorous Workflow & Mathematical Proof

State the Clausius-Maxwell Mean Free Path Formula

Express Number Density Explicitly

Substitute Ratios and Solve

360° Option Elimination Engine Key Feature

NTA Trap Alert: The Linear Diameter Trap

Core NEET Syllabus Standard

Ready to attempt under real exam pressure?

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