--- title: "General displacement work; constant temperature" subject: "Fundamental of Thermodynamics & Heat Transfer" chapter: 2 key: "semester_2/fundamental_of_thermodynamics_and_heat_transfer:displacement-work-isothermal-piston" source: https://www.bctengineer.com/q/semester_2/fundamental_of_thermodynamics_and_heat_transfer/displacement-work-isothermal-piston site: BCT Engineering --- # General displacement work; constant temperature Fundamental of Thermodynamics & Heat Transfer · Chapter 2: Energy and Energy Transfer - Asked in: 2068 Magh, 4 marks - Video: [Work equation for various process||Displacement work](https://www.youtube.com/watch?v=zUqkyMiN9cY) — Aero Beast - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_2/fundamental_of_thermodynamics_and_heat_transfer/displacement-work-isothermal-piston - Whole chapter: https://www.bctengineer.com/md/semester_2/fundamental_of_thermodynamics_and_heat_transfer/chapter-2 ## Question Derive a general expression for the displacement work transfer for a piston cylinder device. Also reduce it for an ideal gas undergoing constant temperature process. ## Short answer For a reversible piston-cylinder device, displacement work is W = ∫ P dV. For an ideal gas undergoing constant temperature process, work is given by mRT ln(V2/V1) = P1V1 ln(P1/P2). ## Answer #### General expression Let a gas at pressure P be in a frictionless piston-cylinder of area A. The force on the piston is F = PA. For a small displacement dx: δW = F dx = P A dx = P dV For a reversible (quasi-static) process from state 1 to 2: **W₁₂ = ∫₁² P dV** This equals the area under the curve on the P-V diagram; it is positive in expansion and negative in compression, and depends on the path. #### Isothermal process of an ideal gas PV = mRT = constant, so P = mRT/V. W₁₂ = mRT ∫₁² dV/V = mRT ln(V₂/V₁) Since P₁V₁ = P₂V₂ = mRT: **W₁₂ = P₁V₁ ln(V₂/V₁) = P₁V₁ ln(P₁/P₂) = mRT ln(V₂/V₁)** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How does the general displacement work expression arise for a piston? A. Force F = PA times displacement dx gives dW = P dV B. Work = mass times acceleration times dx C. Work = V dP, since pressure changes during motion D. Work = temperature change times pressure Answer: A. With F = PA and dV = A dx, dW = F dx = P A dx = P dV. (Key point: Origin of P dV) 2. What does W12 = integral of P dV represent, and what does it depend on? A. A state property that depends only on end states B. Area under the P-V curve; it depends on the path C. Area under the T-S curve; it is path independent D. Area to the left of the curve, independent of path Answer: B. Work is an area under the P-V curve and is a path function; it is positive for expansion and negative for compression. (Key point: Meaning of integral) 3. For an ideal gas at constant T, why does the integral give a logarithm? A. P is constant so W = P(V2 - V1) B. PV^n = C with n = 1.4 gives a power law C. P = mRT/V so the integral of dV/V is ln(V2/V1) D. Because dV = 0 for constant temperature Answer: C. Substituting P = mRT/V gives W = mRT integral dV/V = mRT ln(V2/V1). (Key point: Isothermal integration) 4. Which expression is equal to mRT ln(V2/V1) for the isothermal process? A. P1V1 ln(P2/P1) B. P1V1 (V2 - V1) C. (P2V2 - P1V1)/(1 - n) D. P1V1 ln(P1/P2) Answer: D. Because P1V1 = mRT and V2/V1 = P1/P2, W = P1V1 ln(V2/V1) = P1V1 ln(P1/P2). 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