Two gases A and B are filled at the same pressure in separate cylinders with movable pistons of radi

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Question

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

Accepted Answer

Correct Answer: C — rA/rB = 3/4Step-by-step Solution using First Law of Thermodynamics:First Law: ΔQ = ΔU + W = ΔU + PΔVGiven: ΔQ is same for both; ΔU is same for both.Therefore: WA = WBWork done at constant pressure: W = PΔV = P × A × d (where A = piston area, d = displacement)Since pressure P is the same for both:AA × dA = AB × dBπrA² × 16 = πrB² × 9rA²/rB² = 9/16rA/rB = 3/4Why Option A (3/2) is wrong: This is not consistent with rA²/rB² = 9/16; taking the square root gives 3/4, not 3/2.Why Option B (4/3) is wrong: This is the exact reciprocal of the correct answer (rB/rA). Arises from swapping the displacement values.Why Option D (2/3) is wrong: Arises from incorrectly taking W ∝ r × d (linear in r) rather than W ∝ r² × d (area-dependent).Key Concepts from Thermodynamics (NCERT Class 11, Chapter 12):First Law of Thermodynamics: ΔQ = ΔU + W. It is a statement of conservation of energy.Work done at constant pressure: W = PΔV = P × A × d. For a piston of area A = πr² displaced by d: W = P × πr² × d.In an isobaric process: ΔQ = nCPΔT; ΔU = nCVΔT; W = nRΔT.Ratio W/ΔQ = R/CP = (γ−1)/γ for an ideal gas at constant pressure.

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