Van der Waals Fluid, Critical Phenomena & Maxwell Construction

The van der Waals equation of state captures the essential physics of the gas-liquid transition through molecular volume ($b$) and intermolecular attraction ($a$). It exhibits a critical point and allows the full P-V-T surface to be explored analytically.

Van der Waals Equation

$$\left(p + \frac{a}{V_m^2}\right)(V_m - b) = RT$$

Critical Constants

Setting $\partial p/\partial V_m = 0$ and $\partial^2 p/\partial V_m^2 = 0$ simultaneously yields:

$$V_{m,c} = 3b, \qquad T_c = \frac{8a}{27Rb}, \qquad p_c = \frac{a}{27b^2}$$

The critical compressibility factor $Z_c = p_c V_{m,c}/(RT_c) = 3/8 = 0.375$ for all van der Waals fluids (experimental values for real fluids are 0.23–0.29, indicating the limitations of the vdW model).

Law of Corresponding States

In reduced variables $\Pi = p/p_c$, $\phi = V_m/V_{m,c}$, $\tau = T/T_c$:

$$\left(\Pi + \frac{3}{\phi^2}\right)(3\phi - 1) = 8\tau$$

All van der Waals fluids obey the same reduced equation — the Law of Corresponding States. Real fluids also approximately satisfy this law when plotted in reduced variables.

Maxwell Equal-Area Construction

Below $T_c$ the van der Waals isotherm has a physically unstable loop (region where $\partial p/\partial V_m > 0$). The actual phase transition occurs at the vapour pressure $p^*$ where the areas enclosed above and below the horizontal tie line are equal:

$$\int_{V_m^\ell}^{V_m^g} p\,dV_m = p^*(V_m^g - V_m^\ell)$$

This equal-area rule follows from requiring equal chemical potential of the two phases. It replaces the unphysical loop with a flat section (the two-phase region on the $p$-$V$ diagram).

Spinodal and Binodal

• Binodal (coexistence curve): locus of $(T, V_m)$ points where $\mu^\ell = \mu^g$ — determined by the Maxwell construction.
• Spinodal: locus where $\partial p/\partial V_m = 0$ — boundary of absolute mechanical instability. Between spinodal and binodal the system is metastable (superheated liquid or supersaturated vapour).