Q.41 Physics ⭐ 360° Option Elimination Active

NEET 2025 Physics Question 41 Solution

Topic: Thermodynamics · Official NTA Booklet Code 45

Question

Two gases A and B are filled at the same pressure in separate cylinders with movable pistons of radius r_A and r_B, respectively. On supplying an equal amount of heat to both the systems reversibly under constant pressure, the pistons of gas A and B are displaced by 16 cm and 9 cm, respectively. If the change in their internal energy is the same, then the ratio r_A/r_B is equal to

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Verified Against NTA Final Answer Key

Option (2) : \frac{3}{4}

📚 Thermodynamics Easy

📋 Given Parameters & Problem Setup

Initial Condition Both gases filled at identical constant pressure P

Heat Supplied Equal heat supplied reversibly: ΔQ_A = ΔQ_B

Internal Energy Change Identical: ΔU_A = ΔU_B

Piston Displacements Δx_A = 16 cm, Δx_B = 9 cm

Target Variable Ratio of piston radii r_A / r_B

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The 30-Second Exam Hall Shortcut: ⚡ First Law Equal Work Shortcut

⏱️ Save ~35 seconds

ΔQ = ΔU + W ⟹ Since ΔQ and ΔU are equal, W_A = W_B! W = P ΔV = P (π r² Δx) ⟹ r_A² Δx_A = r_B² Δx_B ⟹ r_A / r_B = √(9 / 16) = 3/4!

1 Apply First Law of Thermodynamics: ΔQ = ΔU + W ⟹ W = ΔQ - ΔU.

2 Since both systems receive the same heat (ΔQ_A = ΔQ_B) and have the same internal energy change (ΔU_A = ΔU_B), work done is identical: W_A = W_B.

3 Isobaric work done on piston: W = P ΔV = P · (Area × displacement) = P (π r² Δx).

4 Both cylinders have equal pressure P, so: P π r_A² Δx_A = P π r_B² Δx_B ⟹ r_A² Δx_A = r_B² Δx_B.

5 Solve for radius ratio: r_A / r_B = √(Δx_B / Δx_A) = √(9 / 16) = 3 / 4.

6 Match Option (2) in under 15 seconds!

🛡️

Why You Can 100% Trust This Shortcut: First Law of Thermodynamics (NCERT Physics · Thermodynamics: ΔQ = ΔU + W).

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Step-by-Step Rigorous Workflow & Mathematical Proof

Step 1

Apply First Law of Thermodynamics

For any thermodynamic system, heat added relates to internal energy change and work done by:

\Delta Q = \Delta U + W \implies W = \Delta Q - \Delta U

• Given ΔQ_A = ΔQ_B and ΔU_A = ΔU_B, it follows that the work done by both gases must be strictly identical: W_A = W_B.

Step 2

Express Isobaric Work Done in Terms of Piston Geometry

The cylinders are maintained under constant pressure P. The work done in displacing a cylindrical piston of radius r by a distance Δx is:

W = P \Delta V = P (A \cdot \Delta x) = P (\pi r^2 \Delta x)

• A = π r² is the cross-sectional area of the piston.

Step 3

Equate Work Done and Solve for Radii Ratio

Equate the work expressions for gas A and gas B:

P \pi r_A^2 \Delta x_A = P \pi r_B^2 \Delta x_B \implies r_A^2 \Delta x_A = r_B^2 \Delta x_B

• Rearranging for the ratio of radii:
• \frac{r_A}{r_B} = \sqrt{\frac{\Delta x_B}{\Delta x_A}} = \sqrt{\frac{9\text{ cm}}{16\text{ cm}}} = \frac{3}{4}.
• Corresponds to Option (2).

⭐

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 A

✗ Incorrect

Verdict & Scientific Reason:

Incorrect. Inverted ratio r_B / r_A = √(16/9) = 4/3.

⚠️

Student Fallacy (Why candidates pick this trap): Inverting the ratio of displacements.

2 B

✓ 100% Correct

Verdict & Scientific Reason:

100% correct. r_A / r_B = √(9/16) = 3/4.

💡

Why this is airtight: None; flawless thermodynamic deduction.

3 C

✗ Incorrect

Verdict & Scientific Reason:

Incorrect. Taking fourth roots or arbitrary arithmetic.

⚠️

Student Fallacy (Why candidates pick this trap): Taking fourth root of 16/9.

4 D

✗ Incorrect

Verdict & Scientific Reason:

Incorrect. Taking square root of 3/4 instead of 9/16.

⚠️

Student Fallacy (Why candidates pick this trap): Double square-rooting.

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NTA Trap Alert: Inverting the Ratio (4/3 instead of 3/4)

Because Δx_A = 16 and Δx_B = 9, students hastily write r_A / r_B = √(16/9) = 4/3. But larger displacement requires a smaller piston area to keep volume change equal: r_A² ∝ 1 / Δx_A!

🛡️ How to avoid the -1 mark: Physical check: Gas A moves farther (16 cm > 9 cm), so its piston MUST be narrower (r_A < r_B) ⟹ ratio < 1!

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Core NEET Syllabus Standard

NEET Physics Core Syllabus · Thermodynamics Standard Benchmark

Core Scientific Principle Tested:

"W = P ΔV = P π r² Δx; equal work implies r² Δx = constant."

Syllabus Focus: First Law of Thermodynamics and Work Done by Gas

✓ Core NEET UG Syllabus Anchor Syllabus & Academic Standards ↗

Bigger move = narrower piston: r_A / r_B = √(9/16) = 3/4!

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NEET 2025 Physics Question 41 Solution

Question

The 30-Second Exam Hall Shortcut: ⚡ First Law Equal Work Shortcut

Step-by-Step Rigorous Workflow & Mathematical Proof

Apply First Law of Thermodynamics

Express Isobaric Work Done in Terms of Piston Geometry

Equate Work Done and Solve for Radii Ratio

360° Option Elimination Engine Key Feature

NTA Trap Alert: Inverting the Ratio (4/3 instead of 3/4)

Core NEET Syllabus Standard

Ready to attempt under real exam pressure?

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Entropy Changes in Isothermal Free Expansion of an Ideal Gas

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