--- title: "Polytropic processes for different n; work" subject: "Fundamental of Thermodynamics & Heat Transfer" chapter: 2 key: "semester_2/fundamental_of_thermodynamics_and_heat_transfer:polytropic-process-graph-work" source: https://www.bctengineer.com/q/semester_2/fundamental_of_thermodynamics_and_heat_transfer/polytropic-process-graph-work site: BCT Engineering --- # Polytropic processes for different n; work Fundamental of Thermodynamics & Heat Transfer · Chapter 2: Energy and Energy Transfer - Asked in: 2073 Bhadra, 4 marks - Video: [Thermodynamics I - Polytropic Processes](https://www.youtube.com/watch?v=Zn5HHyDQ1IU) — Carl Wassgren - 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/polytropic-process-graph-work - Whole chapter: https://www.bctengineer.com/md/semester_2/fundamental_of_thermodynamics_and_heat_transfer/chapter-2 ## Question Define polytropic process. Sketch polytropic processes on a common graph for different values of n. Also derive an expression for work transfer for a polytropic process. ## Short answer A polytropic process obeys PVⁿ = constant. On a P-V diagram, higher n gives steeper curves: n=0 (constant pressure), n=1 (isothermal), n=k (adiabatic), n=∞ (constant volume). Work transfer is W = (P₁V₁ - P₂V₂)/(n-1) for n≠1, and W = P₁V₁ ln(V₂/V₁) for n=1. ## Answer #### Polytropic process A polytropic process is one in which the pressure and volume of a gas are related by **PVⁿ = constant**, where n is the polytropic index. It is a general process which includes many particular ones. #### Graphs for different n ```text P|* |*\ n = inf (V = const, vertical) |* \ \ n = k (adiabatic) |* \ \ \ n = 1 (isothermal) |*---\-\-\----- n = 0 (P = const) +----------------- V ``` The standard cases are: | n | Process | Relation | |---|---|---| | 0 | Constant pressure | P = C (horizontal line) | | 1 | Isothermal (ideal gas) | PV = C | | k = cp/cv | Reversible adiabatic | PVᵏ = C | | ∞ | Constant volume | V = C (vertical line) | Starting from the same point, the curve gets steeper as n increases: horizontal (n = 0), hyperbola (n = 1), steeper (n = k), vertical (n = ∞). #### Work transfer W₁₂ = ∫₁² P dV with P = C/Vⁿ: W₁₂ = C (V₂¹⁻ⁿ − V₁¹⁻ⁿ)/(1 − n) Since C = P₁V₁ⁿ = P₂V₂ⁿ: **W₁₂ = (P₁V₁ − P₂V₂)/(n − 1) = mR(T₁ − T₂)/(n − 1)** (n ≠ 1) For n = 1, W₁₂ = P₁V₁ ln(V₂/V₁). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. A polytropic process follows A. PV = constant only B. P V^n = constant C. P/V^n = constant D. V = constant only Answer: B. It is a general process PV^n = C with polytropic index n. (Key point: Definition) 2. Which pairing of n and process is correct? A. n = 0: constant volume B. n = 1: constant pressure C. n = infinity: isothermal D. n = 0: constant pressure Answer: D. n = 0 gives P = C; n = 1 isothermal; n = k adiabatic; n = infinity constant volume. (Key point: Special values of n) 3. From the same starting point on a P-V graph, as n increases the curve A. gets flatter B. stays the same C. gets steeper D. becomes horizontal Answer: C. Order of steepness: n = 0 (horizontal), 1 (hyperbola), k (steeper), infinity (vertical). (Key point: Steepness) 4. Work for a polytropic process (n not 1) is A. P(V2 - V1) B. (P1V1 - P2V2)/(n - 1) C. mRT ln(V2/V1) D. (P2V2 - P1V1)/(n - 1) Answer: B. Integrating P = C/V^n gives (P1V1 - P2V2)/(n - 1) = mR(T1 - T2)/(n - 1). 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