Entropy relations for an ideal gas

Entropy is a property that measures the molecular disorder of a system; for a reversible process dS = (δQ/T)_rev (unit kJ/K).

Starting equations

For a pure compressible substance (per unit mass), from the first law δq = du + P dv and δq_rev = T ds:

(1) T ds = du + P dv (2) T ds = dh − v dP (since h = u + Pv, dh = du + P dv + v dP)

For an ideal gas: Pv = RT, du = cv dT, dh = cp dT, and R = cp − cv.

Relation 1: T and v

From (1): ds = du/T + (P/T) dv = cv dT/T + R dv/v

Integrating with constant cv:

s₂ − s₁ = cv ln(T₂/T₁) + R ln(v₂/v₁)

Relation 2: T and P

From (2): ds = dh/T − (v/T) dP = cp dT/T − R dP/P

s₂ − s₁ = cp ln(T₂/T₁) − R ln(P₂/P₁)

Relation 3: P and v

From T₂/T₁ = P₂v₂/(P₁v₁) in relation 1 (and cv + R = cp):

s₂ − s₁ = cv ln(P₂/P₁) + cp ln(v₂/v₁)

Variable specific heats

When cp changes with T, define the absolute entropy at 1 atm: s°(T) = ∫₀ᵀ cp dT/T. Then

s₂ − s₁ = s₂° − s₁° − R ln(P₂/P₁)

with s° read from gas tables (e.g. air tables).

Special cases

• Isothermal: s₂ − s₁ = R ln(v₂/v₁) = −R ln(P₂/P₁).
• Isentropic (s₂ = s₁): T₂/T₁ = (v₁/v₂)^(k−1) = (P₂/P₁)^((k−1)/k).

Example: air heated from 300 K to 600 K at constant pressure: Δs = 1.005 ln 2 = 0.697 kJ/kg·K.