Displacement work: constant P, constant T, polytropic

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).