Example: van der Waals gas

Mean-field model of an interacting fluid with excluded volume and attraction: equation of state, free energy, and critical point.

Example: van der Waals gas

A van der Waals gas is a simple interacting-fluid model that modifies the ideal gas by (i) an excluded volume per particle $b$ and (ii) a mean-field attraction strength $a$ .

Equation of state

With number density $n=N/V$ , the van der Waals equation is

\big(p + a n^2\big)\,(1-b n)=n k_B T,

equivalently

\bigg(p + a\Big(\frac{N}{V}\Big)^2\bigg)\,(V-Nb)=N k_B T,

where $p$ is the pressure and $T$ the temperature .

A free-energy representation (useful for coexistence)

A standard mean-field Helmholtz free energy model is

F(T,V,N)=F_{\mathrm{id}}(T,V-Nb,N)-a\frac{N^2}{V},

where $F_{\mathrm{id}}$ is the ideal-gas free energy evaluated at the reduced volume $V-Nb$ . Differentiating gives the equation of state via

p = -\left(\frac{\partial F}{\partial V}\right)_{T,N}.

This form is also a natural entry point to metastability and coexistence (see metastable states and interfaces and surface tension ).

Critical point (molar form)

Using molar volume $V_m$ and gas constant $R$ ,

p=\frac{RT}{V_m-b}-\frac{a}{V_m^2}.

The critical point occurs where $(\partial p/\partial V_m)_T=(\partial^2 p/\partial V_m^2)_T=0$ , yielding

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

and the critical compressibility factor is

Z_c=\frac{p_c V_{m,c}}{R T_c}=\frac{3}{8}.

Remarks

The van der Waals model captures a liquid–gas phase transition in a mean-field way, but does not reproduce non-mean-field critical behavior (compare mean-field approximation , critical exponents , and universality classes ).
Stability criteria can be phrased using thermodynamic stability (e.g., positivity of isothermal compressibility outside the spinodal region).