Displacement work and polytropic process

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).