The Cluster's Van der Waals Equation

A page about the van der Waals equation — the correction to the ideal gas law for real gases.

The van der Waals equation

The van der Waals equation modifies the ideal gas law PV = N k_B T to account for finite molecular size and intermolecular attraction: (P + a n^2 / V^2) (V - N b) = N k_B T or equivalently: (P + a / v^2) (v - b) = k_B T where v = V / N is the volume per particle, a measures the strength of attraction, and b measures the excluded volume per particle (roughly the molecular volume).

In the cluster, the edit van der Waals equation modifies an edit ideal gas law.

The physical origin

• Finite size: Each molecule occupies volume ~ (4 pi / 3) r^3. The available volume is V - Nb, not V. The correction b ~ 4 x molecular volume.
• Attraction: Molecules near the wall are pulled inward by neighbors. The effective pressure is reduced by ~ a n^2. The factor of n^2 comes from: one factor for the molecule hitting the wall, one for the attracting neighbors.

In the cluster, the edit physical origin gives an edit correction.

The critical point

The van der Waals equation has a critical point where the liquid and gas phases become indistinguishable: P_c = a / (27 b^2), V_c = 3 Nb, T_c = 8a / (27 b k_B) In reduced variables (P_r = P / P_c, v_r = v / v_c, T_r = T / T_c): (P_r + 3 / v_r^2) (3 v_r - 1) = 8 T_r This is the law of corresponding states — all van der Waals fluids have the same behavior when scaled by their critical parameters.

In the cluster, the edit critical point gives an edit law.

The Maxwell construction

Below T_c, the van der Waals equation predicts an unphysical oscillation in P(v). Maxwell's equal-area construction replaces this with a horizontal line at P_sat such that: integral_{v_l}^{v_g} P_vdW(v) dv = P_sat (v_g - v_l) This determines the vapor pressure and the volumes of the coexisting liquid and gas phases.

In the cluster, the edit Maxwell construction gives an edit vapor pressure.

Applications

• Real gas behavior: Predicts P-V-T behavior of real gases
• Phase transitions: Qualitative description of liquid-gas transitions
• Critical phenomena: The critical exponents (beta = 1/2, delta = 3, gamma = 1, alpha = 0) are the mean-field values
• Equation of state: Foundation for more sophisticated equations of state
• Supercritical fluids: Above T_c, no phase transition — the fluid is a supercritical fluid

In the cluster, edit applications include:

• edit Real gas behavior
• edit Phase transitions
• edit Critical phenomena
• edit Equation of state
• edit Supercritical fluids

This equation

This page is about the van der Waals equation. (P + a/v^2)(v - b) = k_B T. Critical point: P_c = a/(27b^2), T_c = 8a/(27bk_B). The equation is real.