# 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.