Thermodynamic consistency in multicomponent systems
—μ₁ at x₁=0.5
—μ₂ at x₂=0.5
—G_mix at x=0.5
—GD check Σxᵢdμᵢ
—Phase state
Gibbs-Duhem equation: SdT − VdP + Σnᵢdμᵢ = 0. At constant T,P: Σxᵢdμᵢ = 0 — chemical potentials are not independent; knowledge of μ₁(x) determines μ₂(x). Regular solution: G_mix = RT[x ln x + (1-x)ln(1-x)] + χRT·x(1-x). Spinodal decomposition occurs when ∂²G/∂x² < 0 (shown as red region). The Gibbs-Duhem relation is the thermodynamic consistency constraint that prevents unphysical models.