---
title: "Displacement work: constant P, constant T, polytropic"
subject: "Fundamental of Thermodynamics & Heat Transfer"
chapter: 2
key: "semester_2/fundamental_of_thermodynamics_and_heat_transfer:displacement-work-three-processes"
source: https://www.bctengineer.com/q/semester_2/fundamental_of_thermodynamics_and_heat_transfer/displacement-work-three-processes
site: BCT Engineering
---

# Displacement work: constant P, constant T, polytropic

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

- Asked in: 2072 Asoj, 4 marks
- Video: [Boundary work | Constant pressure process | Constant temperature process | Adiabatic process](https://www.youtube.com/watch?v=eiOzWu5HVDY) — Adriaan Van Niekerk
- 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-three-processes
- Whole chapter: https://www.bctengineer.com/md/semester_2/fundamental_of_thermodynamics_and_heat_transfer/chapter-2

## Question

Derive expressions for displacement work transfer for the following process:
a) Constant-Pressure process
b) Constant temperature process and
c) Polytropic process

## Short answer

Displacement work for a reversible process is W = ∫ P dV. Constant pressure: W = P(V₂ - V₁). Isothermal (ideal gas): W = P₁V₁ ln(V₂/V₁). Polytropic (PVⁿ = C, n≠1): W = (P₁V₁ - P₂V₂)/(n-1).

## Answer

Displacement work for a reversible process 1 → 2:  **W₁₂ = ∫₁² P dV**

#### a) Constant-pressure process (P = C)
W₁₂ = P ∫₁² dV = **P (V₂ − V₁)**  
For an ideal gas, W₁₂ = mR (T₂ − T₁).

#### b) Constant-temperature process (isothermal, ideal gas)
PV = mRT = C, so P = mRT/V.

W₁₂ = ∫₁² (mRT/V) dV = mRT ln(V₂/V₁)

**W₁₂ = P₁V₁ ln(V₂/V₁) = P₁V₁ ln(P₁/P₂)**

#### c) Polytropic process (PVⁿ = C)
P = C/Vⁿ, so

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

With C = P₁V₁ⁿ = P₂V₂ⁿ:

**W₁₂ = (P₁V₁ − P₂V₂)/(n − 1) = mR (T₁ − T₂)/(n − 1)**  (n ≠ 1)

For n = 1 this becomes the isothermal result in (b).

## Check yourself

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

1. Work for a constant-pressure process is
   A. P ln(V2/V1)
   B. P (V2 - V1)
   C. (V2 - V1)/P
   D. P V1
   Answer: B. P comes out of the integral, leaving P(V2 - V1). (Key point: Constant pressure)

2. Isothermal work can also be written as
   A. P1V1 ln(P2/P1)
   B. P1V1 ln(P1/P2)
   C. P1V1 (P1 - P2)
   D. P1V1 ln(V1/V2)
   Answer: B. Since P1V1 = P2V2, V2/V1 = P1/P2, so work = P1V1 ln(P1/P2). (Key point: Pressure ratio form)

3. Which step turns C(V2^(1-n) - V1^(1-n))/(1 - n) into the final polytropic form?
   A. Setting n = 1
   B. Setting T = constant
   C. Setting P = constant
   D. Using C = P1V1^n = P2V2^n
   Answer: D. Substituting C gives P2V2 and P1V1 terms, so W = (P1V1 - P2V2)/(n - 1). (Key point: Polytropic substitution)

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