---
title: "Displacement work and polytropic process"
subject: "Fundamental of Thermodynamics & Heat Transfer"
chapter: 2
key: "semester_2/fundamental_of_thermodynamics_and_heat_transfer:displacement-work-polytropic"
source: https://www.bctengineer.com/q/semester_2/fundamental_of_thermodynamics_and_heat_transfer/displacement-work-polytropic
site: BCT Engineering
---

# Displacement work and polytropic process

Fundamental of Thermodynamics & Heat Transfer · Chapter 2: Energy and Energy Transfer

- Asked in: 2075 Baisakh, 4 marks; 2071 Magh, 4 marks (asked 2 times)
- Video: [Work Done Equation by Polytropic Process | Derivation | BME](https://www.youtube.com/watch?v=8kKZkkkWdm0) — Gautam Varde
- 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-polytropic
- Whole chapter: https://www.bctengineer.com/md/semester_2/fundamental_of_thermodynamics_and_heat_transfer/chapter-2

## Question

Derive the mathematical expression of displacement work and simplify it for a polytropic process.

## Short answer

Displacement work is W = ∫ P dV. For a polytropic process PV^n = C, W = (P₂V₂ - P₁V₁)/(1-n) = (P₁V₁ - P₂V₂)/(n-1); for ideal gas, W = mR(T₁-T₂)/(n-1).

## Answer

#### Displacement work
Consider a gas of pressure P in a frictionless piston-cylinder, piston area A. When the piston moves a small distance dx, the force on it is PA, so

δW = F dx = (P A) dx = P dV   (since A dx = dV)

For a process 1 → 2, the displacement (boundary) work is  
**W₁₂ = ∫₁² P dV**

It equals the area under the process curve on the P-V diagram. It is positive for expansion (dV > 0) and negative for compression.

#### Polytropic process (PVⁿ = C)
P = C / Vⁿ, with C = P₁V₁ⁿ = P₂V₂ⁿ.

W₁₂ = ∫₁² C V⁻ⁿ dV = C [V¹⁻ⁿ/(1 − n)]₁²  
 = (C V₂¹⁻ⁿ − C V₁¹⁻ⁿ)/(1 − n)

Using C V₂¹⁻ⁿ = P₂V₂ and C V₁¹⁻ⁿ = P₁V₁:

**W₁₂ = (P₂V₂ − P₁V₁)/(1 − n) = (P₁V₁ − P₂V₂)/(n − 1)**   (n ≠ 1)

For an ideal gas (PV = mRT):  
**W₁₂ = m R (T₁ − T₂)/(n − 1)**

For n = 1 (isothermal): W₁₂ = P₁V₁ ln(V₂/V₁).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How is δW = P dV obtained?
   A. From the force P A times distance dx, with A dx = dV
   B. From V dP
   C. From the temperature rise
   D. From the piston mass
   Answer: A. Force on the piston is PA; moving dx gives δW = PA dx = P dV. (Key point: Differential work)

2. Boundary work W = ∫P dV is
   A. positive for compression
   B. always zero
   C. negative for expansion
   D. positive for expansion and negative for compression
   Answer: D. Sign follows dV: positive when volume increases, negative when it decreases. (Key point: Sign of work)

3. For PV^n = C, the work for n not equal to 1 is
   A. (P1V1 - P2V2)/(n - 1)
   B. (P1V1 - P2V2)/n
   C. P1V1 ln(V2/V1)
   D. P(V2 - V1)
   Answer: A. Integrating C V^-n dV gives (P2V2 - P1V1)/(1 - n) = (P1V1 - P2V2)/(n - 1). (Key point: Polytropic result)

4. What does the polytropic work reduce to for n = 1?
   A. Zero
   B. P(V2 - V1)
   C. P1V1 ln(V2/V1)
   D. (P1V1 - P2V2)/0
   Answer: C. For n = 1, PV = C, and the integral gives a logarithm: P1V1 ln(V2/V1). (Key point: Isothermal case)

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