Equal-area rule resolves the unphysical van der Waals isotherm — revealing liquid-vapor coexistence
van der Waals Equation
(P + a/V²)(V − b) = RT
In reduced variables (T_r=T/T_c, P_r=P/P_c, V_r=V/V_c):
(P_r + 3/V_r²)(3V_r − 1) = 8T_r
Critical point (law of corresponding states):
T_c = 8a/(27bR)
P_c = a/(27b²)
V_c = 3b
For T < T_c, the isotherm has an unphysical S-shaped wiggle (spinodal region with ∂P/∂V > 0).
Maxwell Equal-Area Construction
The Maxwell construction replaces the unphysical wiggle with a flat line at the coexistence pressure P*, chosen so that the two shaded areas are equal:
∫_{V_liq}^{V_gas} (P_vdW − P*) dV = 0
This minimizes the Gibbs free energy — the system undergoes a first-order phase transition with discontinuous V (latent heat).
The coexistence curve (liquid-vapor boundary) ends at the critical point T_c. Above T_c, there is no distinction between liquid and gas — a supercritical fluid.