Entropy relations: ideal gases and incompressible substance

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

Basic relations (per unit mass)

(1) T ds = du + P dv (2) T ds = dh − v dP

Ideal gas

Equation of state Pv = RT, du = cv dT, dh = cp dT, R = cp − cv.

From (1): ds = cv dT/T + R dv/v From (2): ds = cp dT/T − R dP/P

For constant specific heats, integrate between states 1 and 2:

s₂ − s₁ = cv ln(T₂/T₁) + R ln(v₂/v₁) s₂ − s₁ = cp ln(T₂/T₁) − R ln(P₂/P₁) s₂ − s₁ = cv ln(P₂/P₁) + cp ln(v₂/v₁)

If cp varies with temperature, use the tabulated absolute entropy s°(T):

s₂ − s₁ = s₂° − s₁° − R ln(P₂/P₁)

Isentropic process: T₂/T₁ = (v₁/v₂)^(k−1) = (P₂/P₁)^((k−1)/k).

Incompressible substance (solids, liquids)

The specific volume is constant, so dv = 0, and cp ≈ cv = c. From (1):

ds = du/T = c dT/T

s₂ − s₁ = ∫ c(T) dT/T, and for constant c, s₂ − s₁ = c ln(T₂/T₁)

Entropy depends only on temperature, not on pressure. For an isentropic process, T₂ = T₁.

Example: 2 kg of water (c = 4.18 kJ/kg·K) heated from 20 °C to 80 °C: ΔS = 2 × 4.18 × ln(353.15/293.15) = 1.56 kJ/K.