Van der Waals equation, Maxwell construction, spinodal decomposition, and miscibility gap
Van der Waals Isotherm — drag T slider
0.85 T_c
(P + a/V²)(V - b) = RT | Maxwell construction: equal areas above/below P*
Van der Waals: (P + a/V²)(V − b) = RT. Below T_c: S-shaped isotherm — mechanically unstable region (dP/dV > 0). Maxwell construction: horizontal line at P* with equal areas. Gives coexistence densities ρ_L and ρ_G.
Phase Diagram & Coexistence Curve
Binodal (coexistence curve): boundary of liquid-gas coexistence region. Spinodal (dP/dV = 0): limit of metastability. Between spinodal and binodal: metastable (nucleation needed). Inside spinodal: spinodal decomposition (spontaneous separation).
Grand Canonical: N(μ) at fixed T
-1.00
Grand canonical: N = -∂Ω/∂μ at constant T,V. Chemical potential μ controls particle number — at phase transition, μ is continuous but N jumps discontinuously.
Helmholtz Free Energy f(V) and Maxwell Construction
Helmholtz free energy F(V) at fixed T. Maxwell construction = common tangent construction on F(V). The coexistence line is the convex hull of F — regions where F is non-convex are unstable (spinodal). Thermodynamic stability requires d²F/dV² > 0.