The van der Waals Gas and Liquid-Gas Coexistence

The second virial coefficient of the previous lesson 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 $|u|\ll k_{B}T$, the linearization $f\approx -\beta u$ holds, and

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,

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 $v=V/N$,

Expanding $k_{B}T/(v-b)=(k_{B}T/v)(1+b/v+\cdots )$ recovers the virial form with exactly $B_2=b-a/k_{B}T$, 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 $b$.

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.

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 $\mathbf{d}g=v\mathbf{d}P$ along the isotherm, so the condition $g_{\text{liq}}=g_{\text{gas}}$ integrated between the two coexistence volumes reads

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,

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.

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,

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

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

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

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 $\epsilon$ — the same scaling that collapsed the Lennard-Jones $B_2(T)$ data in the previous lesson.

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,

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 $\beta =\frac{1}{2}$.
• Critical isotherm, $\delta$. At $t=0$ the expansion reads $P_r-1=-\frac{3}{2}{\omega }^3$, 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.

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.¹

Summary

• van der Waals resums the two effects behind $B_2=b-a/k_{B}T$: the excluded volume $b$ replaces $V$ by $V-Nb$, and the attraction $a$ subtracts $a/v^2$ from the pressure, giving $P=k_{B}T/(v-b)-a/v^2$.
• 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, ${\int }_{\text{liq}}^{\text{gas}}(P-P_0)\mathbf{d}v=0$; the endpoints trace the coexistence dome closing at the critical point.
• The critical point is $v_c=3b$, $k_B{T}_c=8a/27b$, $P_c=a/27b^2$, with universal $Z_c=P_c{v}_c/k_B{T}_c=3/8$. 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 $\alpha =0$, $\beta =\frac{1}{2}$, $\gamma =1$, $\delta =3$, universal but wrong, because it discards the density fluctuations that dominate near $T_c$ in three dimensions.