---
title: The van der Waals Gas and Liquid-Gas Coexistence
module: Interacting Gases
moduleNumber: 10
lessonNumber: 2
order: 1002
summary: >
  Resumming the second virial coefficient $B_2=b-a/k_BT$ into an equation of
  state gives the van der Waals model $(P+a/v^2)(v-b)=k_BT$, the simplest theory
  of a fluid that condenses. Below the critical temperature its isotherms develop
  a mechanically unstable loop; the Maxwell equal-area construction replaces the
  loop with a coexistence tie line. The critical point sits at $v_c=3b$,
  $k_BT_c=8a/27b$, $P_c=a/27b^2$, and the model predicts universal but incorrect
  critical exponents because it ignores fluctuations.
topics: [Interacting Gases]
sources:
  - book: Schroeder
    ref: "Ch. 5 — Free Energy and Chemical Thermodynamics; §5.3 The van der Waals Model"
  - book: Kardar (Statistical Physics of Particles)
    ref: "Ch. 5 — Interacting Particles; §5.3"
  - book: Pathria & Beale
    ref: "Ch. 12 — Phase Transitions: Criticality, Universality, and Scaling; §12.2"
  - book: Reif
    ref: "Ch. 10 — Systems of Interacting Particles; §10.5"
draft: false
---

The second virial coefficient of the [previous lesson](/statistical-mechanics/interactions/the-cluster-expansion-and-virial-coefficients)
is a small correction valid only when the gas is dilute. Condensation — the
abrupt appearance of a dense liquid phase — lies far outside a low-density
expansion. van der Waals's achievement in 1873 was to keep the two physical
effects behind $B_2$, the excluded volume of the molecular cores and the mutual
attraction, and resum them into a single equation of state that describes both
the gas and the liquid and the transition between them. The model is
quantitatively wrong near the critical point, but it is the prototype of a
**mean-field theory**, and every later, more accurate treatment is measured
against it.

## From the virial coefficient to the van der Waals equation

For a potential with a hard core of diameter $\sigma$ and a weak long-range
attraction, the second virial coefficient splits into a temperature-independent
repulsive part and a temperature-dependent attractive part. Inside the core the
Mayer function is $f=-1$; outside it, where $|u|\ll k_BT$, the linearization
$f\approx -\beta u$ holds, and

$$
B_2(T) = -\frac12\int f\,\d^3 r
= \underbrace{\frac{2\pi}{3}\sigma^3}_{b}
\;-\;\frac{1}{k_BT}\underbrace{\Bigl(2\pi\!\int_\sigma^\infty |u(r)|\,r^2\,\d r\Bigr)}_{a}
= b - \frac{a}{k_BT},
$$

with $b$ the excluded volume (four times the molecular volume) and $a>0$ the
integrated strength of the attraction. Substituting into the virial expansion
gives the pressure to first order in density,

$$
\frac{P}{k_BT} = n + B_2\,n^2 + \cdots = n + \Bigl(b - \frac{a}{k_BT}\Bigr)n^2 + \cdots.
$$

van der Waals resummed this. The excluded volume means each molecule moves not in
the full volume $V$ but in the reduced volume $V-Nb$, replacing $n=N/V$ by the
denser $N/(V-Nb)$ in the ideal term; the attraction lowers the pressure by a term
proportional to the density squared, $a n^2 = a/v^2$ per particle. With the
per-particle volume $v = V/N$,

$$
P = \frac{k_BT}{v-b} - \frac{a}{v^2},
\qquad\text{equivalently}\qquad
\Bigl(P + \frac{a}{v^2}\Bigr)(v-b) = k_BT.
$$

Expanding $k_BT/(v-b) = (k_BT/v)(1 + b/v + \cdots)$ recovers the virial form with
exactly $B_2 = b - a/k_BT$, so the two constants are fixed by the pair potential
and the model interpolates between the dilute gas and a dense fluid bounded by the
close-packing volume $b$.

> **The van der Waals equation of state.** $\displaystyle P = \frac{k_BT}{v-b} -
> \frac{a}{v^2}$, with $b=\tfrac{2\pi}{3}\sigma^3$ the excluded volume per particle
> and $a$ the integrated attraction. The first term is a hard-core-corrected ideal
> pressure that diverges as $v\to b$; the second is a uniform attractive
> background that pulls the pressure down.

## Isotherms and the unstable loop

At high temperature the attractive term is negligible and the isotherms $P(v)$
fall monotonically, close to ideal-gas hyperbolas shifted by the excluded volume.
As $T$ drops, the $-a/v^2$ term deepens and the isotherm develops a wiggle: below
a critical temperature $T_c$ the curve $P(v)$ is no longer monotonic but has a
local maximum and a local minimum, so a single pressure can correspond to three
volumes.

$$
% caption: van der Waals isotherms in reduced units $P_r(v_r)$ for $T_r=1.15$ (monotonic, gas-like), $T_r=1$ (critical, with a horizontal inflection at the critical point), and $T_r=0.85$ (a non-monotonic loop). The middle branch of the loop has $\partial P/\partial v>0$ and is mechanically unstable.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(6.6,0) node[right,black]{$v_r$};
\draw[->,black] (0,0)--(0,4.6) node[above,black]{$P_r$};
\begin{scope}
\clip (0.6,0) rectangle (6.5,4.5);
\draw[black,very thick,densely dashed] plot[domain=0.36:3.1,samples=140]
  ({\x*2.1},{(8*1.15/(3*\x-1)-3/(\x*\x))*2.2});
\draw[black!72,very thick] plot[domain=0.36:3.1,samples=140]
  ({\x*2.1},{(8*1.0/(3*\x-1)-3/(\x*\x))*2.2});
\draw[acc,very thick] plot[domain=0.36:3.1,samples=160]
  ({\x*2.1},{(8*0.85/(3*\x-1)-3/(\x*\x))*2.2});
\end{scope}
\filldraw[black] (2.1,2.2) circle (2.2pt);
\node[black,above right] at (2.1,2.25) {critical point};
\node[black] at (5.6,3.6) {$T_r>1$};
\node[acc] at (5.6,1.15) {$T_r<1$};
\node[acc] at (2.9,0.62) {unstable loop};
\end{tikzpicture}
$$

The middle branch, where $\partial P/\partial v>0$, is unphysical. A fluid on it
would respond to a small compression by lowering its pressure, and any density
fluctuation would grow without bound: the isothermal compressibility
$\kappa_T = -v^{-1}(\partial v/\partial P)_T$ is negative, violating mechanical
stability. The real substance avoids the unstable branch by splitting into two
coexisting phases — a low-density gas and a high-density liquid — connected by a
horizontal segment of constant pressure. The van der Waals loop, taken literally,
is wrong; the equation must be supplemented by a rule that locates the flat
coexistence line.

## The Maxwell equal-area construction

The coexistence pressure follows from the requirement that the two phases share
not only temperature and pressure but chemical potential, so that no particle has
a thermodynamic incentive to move between them. At fixed $T$, the Gibbs free
energy per particle $g=\mu$ satisfies $\d g = v\,\d P$ along the isotherm, so the
condition $g_{\text{liq}} = g_{\text{gas}}$ integrated between the two coexistence
volumes reads

$$
\mu_{\text{gas}} - \mu_{\text{liq}} = \int_{\text{liq}}^{\text{gas}} v\,\d P = 0.
$$

Rewriting the integral around the loop as an area in the $P$–$v$ plane converts
this into the **Maxwell equal-area construction**: the horizontal coexistence
line is drawn at the pressure $P_0$ for which the two areas enclosed between the
line and the van der Waals loop are equal,

$$
\int_{v_{\text{liq}}}^{v_{\text{gas}}}\bigl[P(v) - P_0\bigr]\,\d v = 0.
$$

$$
% caption: The Maxwell construction. The coexistence pressure $P_0$ is chosen so the two shaded lobes between the horizontal tie line and the van der Waals loop have equal area, which enforces equal chemical potential $\mu_{\text{liq}}=\mu_{\text{gas}}$ of the coexisting liquid and gas at the endpoints.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(7.2,0) node[right,black]{$v$};
\draw[->,black] (0,0)--(0,4.2) node[above,black]{$P$};
% van der Waals loop as a hand-drawn S-curve
\draw[acc,very thick] (0.9,3.7)
  .. controls (1.4,2.3) and (1.7,2.05) .. (2.3,2.7)
  .. controls (2.9,3.3) and (3.6,3.2) .. (4.2,1.9)
  .. controls (4.9,0.6) and (5.8,1.05) .. (6.6,0.95);
% coexistence tie line
\draw[black!70,thick,dashed] (1.15,2.35)--(5.55,2.35);
\node[black!70,left] at (1.1,2.35) {$P_0$};
% endpoints
\filldraw[black!70] (1.15,2.35) circle (2pt) node[below left,black]{liquid};
\filldraw[black!70] (5.55,2.35) circle (2pt) node[below right,black]{gas};
% area labels
\node[acc] at (2.0,2.75) {$A_1$};
\node[acc] at (4.55,1.95) {$A_2$};
\node[black] at (3.6,3.55) {$A_1=A_2$};
\end{tikzpicture}
$$

Between the two endpoints the substance is a mixture whose overall volume slides
along the tie line as the proportion of liquid to gas changes, at fixed pressure
$P_0(T)$ and temperature. The locus of endpoints traced over all subcritical
temperatures is the **coexistence curve** (binodal), a dome in the $P$–$v$ plane
that closes at the critical point.

$$
% caption: The coexistence dome (binodal). Its left branch is the saturated liquid, its right branch the saturated gas; inside, the fluid is a two-phase mixture and the isotherms are flat tie lines. The dome closes at the critical point, where the two phases become identical.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(7.2,0) node[right,black]{$v$};
\draw[->,black] (0,0)--(0,4.4) node[above,black]{$P$};
% dome
\draw[very thick] (1.0,0.5)
  .. controls (1.6,2.6) and (2.6,3.5) .. (3.6,3.5)
  .. controls (4.6,3.5) and (5.8,1.7) .. (6.6,0.55);
\fill[black!5] (1.0,0.5) .. controls (1.6,2.6) and (2.6,3.5) .. (3.6,3.5)
  .. controls (4.6,3.5) and (5.8,1.7) .. (6.6,0.55) -- cycle;
\filldraw[acc] (3.6,3.5) circle (2.2pt) node[above,acc]{critical point};
\node[black] at (1.55,1.35) {liquid};
\node[black] at (5.75,1.1) {gas};
\node[black] at (3.6,1.75) {two-phase};
\node[black] at (3.6,1.25) {mixture};
% a subcritical coexistence tie line inside the dome
\draw[black,thick] (1.9,2.1)--(5.25,2.1);
\filldraw[black] (1.9,2.1) circle (1.6pt);
\filldraw[black] (5.25,2.1) circle (1.6pt);
\end{tikzpicture}
$$

## The critical point

The critical temperature is where the loop first appears. At exactly $T_c$ the
local maximum and minimum of $P(v)$ merge into a single horizontal inflection
point, so the critical point is defined by the simultaneous vanishing of the
first two volume derivatives,

$$
\left(\frac{\partial P}{\partial v}\right)_{T_c} = 0,
\qquad
\left(\frac{\partial^2 P}{\partial v^2}\right)_{T_c} = 0.
$$

Applying these to $P = k_BT/(v-b) - a/v^2$ gives two equations,
$k_BT/(v-b)^2 = 2a/v^3$ and $k_BT/(v-b)^3 = 3a/v^4$. Their ratio fixes $v-b =
\tfrac23 v$, hence the critical volume, and back-substitution gives the critical
temperature and pressure,

$$
v_c = 3b,
\qquad
k_BT_c = \frac{8a}{27b},
\qquad
P_c = \frac{a}{27b^2}.
$$

These three combine into a dimensionless ratio independent of $a$ and $b$,

$$
Z_c \equiv \frac{P_c v_c}{k_BT_c} = \frac{3}{8} = 0.375,
$$

the **critical compressibility factor**, the same number for every van der Waals
substance. Real fluids cluster near $Z_c\approx 0.29$, below the mean-field
prediction — the first quantitative sign that the model is only approximate.

## The law of corresponding states

Measuring pressure, volume, and temperature in units of their critical values,
$P_r = P/P_c$, $v_r = v/v_c$, $T_r = T/T_c$, eliminates $a$ and $b$ entirely. The
van der Waals equation becomes

$$
\Bigl(P_r + \frac{3}{v_r^2}\Bigr)\bigl(3v_r - 1\bigr) = 8\,T_r,
$$

a single universal relation with no substance-dependent parameters. This is the
**law of corresponding states**: all van der Waals fluids, plotted in reduced
variables, fall on one surface, and two fluids at the same $(P_r,T_r)$ occupy
corresponding states with the same $v_r$. The law holds approximately for real
simple fluids whose molecules interact through similar potentials differing only
in the scales $\sigma$ and $\varepsilon$ — the same scaling that collapsed the
Lennard-Jones $B_2(T)$ data in the previous lesson.

$$
% caption: Corresponding states: reduced isotherms of different substances collapse onto one curve when pressure, volume, and temperature are scaled by their critical values. Data points from distinct gases (circles, squares, triangles) fall on the common reduced isotherm.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(6.8,0) node[right,black]{$v_r$};
\draw[->,black] (0,0)--(0,4.6) node[above,black]{$P_r$};
\begin{scope}
\clip (0.9,0) rectangle (6.6,4.5);
\draw[acc,very thick] plot[domain=0.5:3.0,samples=160]
  ({\x*2.1},{(8*1.3/(3*\x-1)-3/(\x*\x))*1.0});
\end{scope}
\filldraw[black!75] (1.68,2.74) circle (2.4pt);
\draw[black!75,thick] (2.66,1.74) rectangle ++(0.14,0.14);
\draw[black!75,thick] (3.92,1.30)--++(0.18,0)--++(-0.09,0.16)--cycle;
\node[black!75,above right] at (1.7,2.8) {gas 1};
\node[black!75,above right] at (2.85,1.78) {gas 2};
\node[black!75,above right] at (4.15,1.3) {gas 3};
\node[acc] at (5.4,0.7) {reduced isotherm};
\end{tikzpicture}
$$

## Critical exponents and the failure of mean field

Near the critical point the van der Waals equation predicts power-law behavior
governed by **critical exponents**. Expanding the reduced equation about the
critical point with $\omega = v_r - 1$ and $t = T_r - 1$ gives, to the leading
orders,

$$
P_r = 1 + 4t - 6t\,\omega - \frac{3}{2}\,\omega^3 + \cdots.
$$

Three exponents follow directly.

- **Coexistence, $\beta$.** For $t<0$ the equal-area construction pairs a liquid
  and gas with $\omega_{\text{liq}} = -\omega_{\text{gas}}$; equal pressure forces
  $\omega \propto (-t)^{1/2}$, so the density difference vanishes as
  $\rho_{\text{liq}}-\rho_{\text{gas}} \propto (T_c-T)^{1/2}$ and $\beta = \tfrac12$.
- **Critical isotherm, $\delta$.** At $t=0$ the expansion reads
  $P_r - 1 = -\tfrac32\omega^3$, so $P-P_c \propto (v-v_c)^3$ and $\delta = 3$.
- **Compressibility, $\gamma$.** On the critical isochore $\omega=0$,
  $\partial P_r/\partial\omega = -6t$, so
  $\kappa_T \propto (\partial P/\partial v)^{-1} \propto |t|^{-1}$ and $\gamma = 1$.
  The heat capacity $C_V$ jumps discontinuously but does not diverge, giving
  $\alpha = 0$.

These values are universal within the model, but they disagree with experiment
and with the exact two-dimensional results. Real three-dimensional fluids belong
to the Ising universality class, with measurably different exponents.

| Exponent | Definition | van der Waals (mean field) | 3D Ising / experiment |
|---|---|---|---|
| $\alpha$ | $C_V \sim \lvert t\rvert^{-\alpha}$ | $0$ (jump) | $\approx 0.11$ |
| $\beta$ | $\Delta\rho \sim (-t)^{\beta}$ | $\tfrac12$ | $\approx 0.326$ |
| $\gamma$ | $\kappa_T \sim \lvert t\rvert^{-\gamma}$ | $1$ | $\approx 1.24$ |
| $\delta$ | $P-P_c \sim \lvert\Delta v\rvert^{\delta}$ | $3$ | $\approx 4.8$ |

The discrepancy has a definite cause. The van der Waals derivation replaced the
fluctuating local environment of each molecule by a uniform average — every
particle feels the same mean attraction $a$, regardless of the instantaneous
positions of its neighbors. That mean-field assumption discards density
correlations, which near $T_c$ grow to macroscopic scale as the correlation length
diverges. When fluctuations on all length scales dominate, an analytic expansion
of the free energy in the order parameter cannot reproduce the true singularities.
The mean-field description is exact only above four spatial dimensions; in three
dimensions the fluctuation corrections change every exponent, a failure repaired
by the renormalization group in the [phase-transitions module](/statistical-mechanics/phase-transitions/critical-exponents-and-landau-theory).[^ginzburg]

[^ginzburg]: The condition for mean-field validity is the **Ginzburg criterion**,
and the upper critical dimension for the liquid-gas and Ising transitions is
$d_c=4$. **Kardar**, _Statistical Physics of Particles_, §5.3, and **Pathria &
Beale**, _Statistical Mechanics_ (4th ed.), §12.2, §13.4; MIT OCW 8.334,
<https://ocw.mit.edu/courses/8-334-statistical-mechanics-ii-statistical-mechanics-of-fields-spring-2014/>.

## Summary

- van der Waals resums the two effects behind $B_2 = b - a/k_BT$: the excluded
  volume $b$ replaces $V$ by $V-Nb$, and the attraction $a$ subtracts $a/v^2$ from
  the pressure, giving $P = k_BT/(v-b) - a/v^2$.
- Below $T_c$ the isotherms have a loop with an unstable middle branch
  ($\partial P/\partial v>0$, $\kappa_T<0$); the physical isotherm is flat across
  the two-phase region.
- The Maxwell equal-area construction fixes the coexistence pressure by equating
  the chemical potentials of the two phases,
  $\int_{\text{liq}}^{\text{gas}}(P-P_0)\,\d v = 0$; the endpoints trace the
  coexistence dome closing at the critical point.
- The critical point is $v_c=3b$, $k_BT_c=8a/27b$, $P_c=a/27b^2$, with universal
  $Z_c = P_cv_c/k_BT_c = 3/8$. Reduced variables give the parameter-free law of
  corresponding states $(P_r+3/v_r^2)(3v_r-1)=8T_r$.
- The model predicts mean-field exponents $\alpha=0$, $\beta=\tfrac12$, $\gamma=1$,
  $\delta=3$, universal but wrong, because it discards the density fluctuations
  that dominate near $T_c$ in three dimensions.