★★ Intermediate

🧪 Van der Waals Gas

Real gases differ from ideal ones: molecules attract each other at moderate distances and repel at close range. Van der Waals corrected the ideal gas law with two parameters — a (intermolecular attraction) and b (excluded volume) — producing P-V isotherms that capture gas, liquid-vapour coexistence, and the critical point.

a = 0.364 Pa·m⁶/mol² b = 4.27 ×10⁻⁵ m³/mol T = 310 K

T_c = — K P_c = — bar V_c = — L/mol T/T_c = — Z_c = 3/8 = 0.375

(P + a/V_m²)(V_m − b) = RT | T_c = 8a/(27Rb) | P_c = a/(27b²) | V_c = 3b

Van der Waals Equation of State (1873)

Johannes Diderik van der Waals proposed two corrections to the ideal gas law PV = nRT:

Pressure correction (+a/V_m²) — molecules attract each other, reducing the pressure that would be exerted on the walls.
Volume correction (−b) — molecules have a finite size, so the free volume available is less than the total volume.

Below the critical temperature T_c, the isotherm develops an S-shaped loop (the van der Waals loop). The physical equilibrium line — the horizontal coexistence plateau — is determined by Maxwell's equal-area rule: the areas above and below the horizontal tie-line are equal.

At the critical point, the three roots of the cubic in V merge: T_c = 8a/(27Rb), P_c = a/(27b²), V_c = 3b, giving a universal compressibility factor Z_c = P_cV_c/(RT_c) = 3/8.