Van der Waals Equation of State

The van der Waals equation is the first equation of state that turns a gas into a liquid. Johannes Diderik van der Waals wrote it down in his 1873 Leiden doctoral thesis to explain why real gases refuse to obey the ideal law and how one substance can appear as both vapour and liquid. Its enduring value is conceptual: it is the simplest closed equation of state that produces a genuine first-order phase transition ending in a critical point, and it introduces the mean-field philosophy that later organizes the whole theory of phase transitions. Two corrections lift it above the ideal gas. Molecules occupy space, so the volume available shrinks from $V$ to $V-Nb$ , with $b$ an excluded volume per particle. Molecules also attract at range, so the pressure drops by a term proportional to the square of the density, $aN^2/V^2$ — one factor of density for the molecule feeling the pull and one for the molecules doing the pulling. This article owns the equation itself: the isotherms it generates, the critical point $P_c,V_c,T_c$ where the liquid–gas distinction disappears, and the famous universal ratio $P_cV_c/(Nk_BT_c)=3/8$ . Where the constants $a$ and $b$ come from microscopically — the second virial coefficient of a hard-core-plus-attraction potential — is derived by name in aq-sm-van-der-waals-derivation. The isotherms tell the story. Above $T_c$ pressure falls monotonically with volume. At $T_c$ a horizontal inflection appears. Below $T_c$ the curve develops an S-shaped loop with a mechanically unstable branch — the signal, not a defect, that the uniform fluid wants to split into coexisting liquid and gas. The Maxwell equal-area construction (aq-sm-maxwell-construction) replaces the loop with the true coexistence plateau. And because the reduced equation contains no substance-specific constants, all van der Waals fluids collapse onto one curve — the law of corresponding states (aq-sm-law-of-corresponding-states), an early hint that critical behaviour forgets its microscopic origins.

Key takeaways

The van der Waals equation $\left(P+aN^2/V^2\right)(V-Nb)=Nk_BT$ corrects the ideal gas for finite molecular size (excluded volume $b$ ) and pairwise attraction (cohesion $a$ ).
Its subcritical isotherms develop an unstable loop — the mathematical fingerprint of the liquid–gas transition — repaired by the Maxwell construction into a flat coexistence line.
The critical point sits at $k_BT_c=8a/27b$ , $V_c=3Nb$ , $P_c=a/27b^2$ , fixed uniquely by the double condition $(\partial P/\partial V)_T=(\partial^2P/\partial V^2)_T=0$ .
The critical compressibility factor is a pure number, $Z_c=P_cV_c/(Nk_BT_c)=3/8=0.375$ — the same for every vdW fluid, though real gases cluster near $0.29$ .
Rescaling $P,V,T$ by their critical values erases $a$ and $b$ , giving one universal reduced isotherm — the law of corresponding states.
Key idea: keep the hard-core repulsion exactly to leading order (it removes volume $Nb$ ) and the attraction in mean field (it shifts pressure by $-aN^2/V^2$ ); the resulting equation loses convexity below $T_c$ , and that failure — cured by the Maxwell construction — is the gas–liquid transition.

Where this lives today

The van der Waals equation is the ancestor of the cubic equations of state that still run the chemical and petroleum industries. The Redlich–Kwong, Soave–Redlich–Kwong, and Peng–Robinson equations are direct descendants — van der Waals's two-term structure with the temperature dependence of $a$ and the volume corrections tuned to fit real data — and they are the workhorses of process simulators that design distillation columns, pipelines, and liquefied-natural-gas trains. Generalized compressibility charts, built on the corresponding-states idea that grew out of the reduced vdW equation, let an engineer estimate the properties of an unmeasured fluid from a reference substance at the same reduced state. Its deeper legacy is conceptual. The vdW equation is the first and simplest mean-field theory of a phase transition, and its non-convex free energy — the loop — is the direct antecedent of Landau's double-well order-parameter potential and of the whole mean-field description of critical phenomena. That its critical exponents are wrong is not a footnote but a signpost: the failure of mean field near $T_c$ , where fluctuations on all length scales matter, is exactly the problem the renormalization group later in this track was invented to solve. Van der Waals gave us both the right picture (a continuous equation of state with a critical point) and the instructive wrong answer (mean-field exponents), and physics learned from each.

The physical picture

The two van der Waals corrections are two competing instincts of matter. The hard core is repulsion: no two molecules can sit on top of each other, so the space a molecule can roam is not $V$ but $V-Nb$ . Squeeze the gas and this term blows up as $V\to Nb$ — the fluid becomes nearly incompressible, just as a liquid is. The attraction is cohesion: every molecule sits in the averaged pull of its neighbours, a pull proportional to their density $N/V$ , and summing that self-consistent attraction lowers the pressure by $aN^2/V^2$ . Replacing the fluctuating local crowd by its smooth average is the mean-field move — the same idealization that reappears in Weiss's theory of magnetism and Landau's theory of order parameters. The first figure shows the isotherms these two terms produce. Above the critical temperature $T_c$ , repulsion dominates everywhere and $P(V)$ falls smoothly. Exactly at $T_c$ the curve flattens into a horizontal inflection — the critical isotherm. Below $T_c$ , attraction wins at intermediate density and carves an S-shaped loop into the curve, a stretch where increasing the volume would raise the pressure. That is mechanically impossible for a stable uniform fluid, and it is precisely the equation telling us that the fluid prefers to break into two phases.

Figure

Van der Waals isotherms P vs V: above T_c the pressure falls smoothly (blue), at T_c a horizontal inflection appears (amber) marking the critical point, below T_c an unstable S-loop appears (red) whose middle branch has ∂P/∂V > 0

The second figure is the payoff: the critical point is a double tangency. As $T$ rises from below, the two humps of the loop — the local maximum and minimum of $P(V)$ — move together and merge. At $T_c$ they coincide into a single horizontal inflection, and above $T_c$ they are gone. Mathematically the merger is the simultaneous vanishing of the first and second derivatives of $P$ with respect to $V$ , and that single condition pins $T_c$ , $V_c$ , and $P_c$ to unique multiples of $a$ and $b$ .

Figure

The critical point as a double tangency: the local max and min of the subcritical isotherm (red) merge as T rises, coinciding at T_c (amber) where ∂P/∂V = ∂²P/∂V² = 0, and vanishing above T_c (blue)

Below the critical point the loop is the equation's honest confession that the mean-field free energy has gone non-convex: a uniform fluid at those densities is unstable against separating into a dense liquid and a dilute gas. The equation cannot resolve the coexistence by itself — that requires the thermodynamic condition of equal pressure and equal chemical potential in the two phases, which is exactly the Maxwell equal-area construction handed to aq-sm-maxwell-construction. What this article establishes is the equation, its isotherms, and the location of the critical point.

The core equations

The van der Waals equation modifies the ideal gas law $PV=Nk_BT$ by two additive corrections applied in the two natural slots: the volume is reduced and the pressure is lowered.

\left(P+\frac{aN^2}{V^2}\right)\left(V-Nb\right)=Nk_BT \qquad (1)

where: $P$ — pressure; $V$ — total volume; $N$ — number of molecules; $T$ — temperature; $a$ — cohesion (attraction) parameter with units energy·volume; $b$ — excluded volume per molecule; $k_B$ — Boltzmann constant. The microscopic origin of $a$ and $b$ from a pair potential is owned by aq-sm-van-der-waals-derivation; here they are given constants. Solving for pressure gives the form that generates the isotherms, cleanly separating the repulsive and attractive pieces.

P=\frac{Nk_BT}{V-Nb}-\frac{aN^2}{V^2} \qquad (2)

where: the first term is the repulsive excluded-volume pressure, which diverges as $V\to Nb$ (the fluid runs out of room); the second is the attractive mean-field reduction, which grows as the gas is compressed. Their competition is what produces the subcritical loop. The critical point is the unique isotherm with a horizontal inflection: the pressure has neither slope nor curvature there.

\left(\frac{\partial P}{\partial V}\right)_{T_c}=0,\qquad\left(\frac{\partial^2 P}{\partial V^2}\right)_{T_c}=0 \qquad (3)

where: $T_c$ — the critical temperature. Two equations in two unknowns ( $V$ and $T$ ) fix the critical volume and temperature; substituting back into (2) fixes the critical pressure. This double tangency is what makes the critical point sharp. Solving the double condition yields the critical constants entirely in terms of $a$ and $b$ .

k_BT_c=\frac{8a}{27b},\qquad V_c=3Nb,\qquad P_c=\frac{a}{27b^2} \qquad (4)

where: $V_c$ is exactly three times the total excluded volume $Nb$ ; $T_c$ measures the strength of attraction against the size of the core ( $\propto a/b$ ); $P_c$ scales as $a/b^2$ . Every van der Waals fluid has its critical point here. Combining the three critical constants into the dimensionless compressibility factor gives a pure number, independent of the substance.

Z_c=\frac{P_cV_c}{Nk_BT_c}=\frac{3}{8}=0.375 \qquad (5)

where: $Z_c$ — the critical compressibility factor. The $a$ 's and $b$ 's cancel completely, leaving $3/8$ for every vdW fluid — the sharpest, and most testable, prediction of the equation. Real gases give a smaller value near $0.29$ , telling us mean field overestimates $Z_c$ .

Derivation ACritical constants from the double-tangency condition

The goal is to solve the two critical conditions $(\partial P/\partial V)_T=0$ and $(\partial^2P/\partial V^2)_T=0$ simultaneously and obtain the critical constants $k_BT_c=8a/27b$ , $V_c=3Nb$ , $P_c=a/27b^2$ purely in terms of $a$ and $b$ . The strategy is to write down the two derivative conditions from the pressure form (2), take their ratio to eliminate the temperature (which appears linearly in both), solve the resulting purely algebraic equation for $V_c$ , and then back-substitute to get $T_c$ and finally $P_c$ . What to watch is that dividing one condition by the other is what makes the temperature drop out cleanly, leaving a simple relation for the critical volume.

Step 1. First derivative. From $P=Nk_BT/(V-Nb)-aN^2/V^2$ , $\left(\dfrac{\partial P}{\partial V}\right)_T=-\dfrac{Nk_BT}{(V-Nb)^2}+\dfrac{2aN^2}{V^3}=0$ . Why: the slope of the isotherm vanishes at the inflection; this is one equation linking $T$ and $V$ .

Step 2. Second derivative. $\left(\dfrac{\partial^2 P}{\partial V^2}\right)_T=\dfrac{2Nk_BT}{(V-Nb)^3}-\dfrac{6aN^2}{V^4}=0$ . Why: the curvature vanishes too — the inflection is horizontal, not merely a slope-zero maximum or minimum; this is the second equation.

Step 3. Take the ratio to eliminate $T$ . Dividing the second condition by the first gives $\dfrac{2}{V-Nb}=\dfrac{3}{V}$ , hence $2V=3(V-Nb)$ and $V_c=3Nb$ . Why: both conditions are linear in $k_BT$ , so their ratio is temperature-free; the algebra collapses to a single equation for the critical volume.

Step 4. Back-substitute for $T_c$ and $P_c$ . Put $V_c=3Nb$ into Step 1: $k_BT_c=\dfrac{2aN(V_c-Nb)^2}{V_c^3}=\dfrac{2aN(2Nb)^2}{(3Nb)^3}=\dfrac{8a}{27b}$ . Then from (2), $P_c=\dfrac{Nk_BT_c}{V_c-Nb}-\dfrac{aN^2}{V_c^2}=\dfrac{N\cdot 8a/27b}{2Nb}-\dfrac{aN^2}{9N^2b^2}=\dfrac{a}{27b^2}$ . Why: with $V_c$ known, the first-derivative condition delivers $T_c$ , and the equation of state itself delivers $P_c$ — all three constants now fixed by $a$ and $b$ alone. *Verified: the symbolic solve of the two conditions giving $V_c=3Nb$ , $k_BT_c=8a/27b$ , $P_c=a/27b^2$ , and the resulting $Z_c=3/8$ , are checked in verify_van-der-waals-equation.py (sympy solve), matching Pathria & Beale §10.3 and Kardar Ch. 5.*

Go deeper · Derivation B · 2 minThe universal ratio Z_c = 3/8 and the reduced equation of state

Derivation BThe universal ratio Z_c = 3/8 and the reduced equation of state

Here the goal is to show two things that follow immediately from the critical constants: first, that the critical compressibility factor $Z_c=P_cV_c/(Nk_BT_c)$ is the pure number $3/8$ with all material constants cancelling; and second, that expressing $P,V,T$ in units of their critical values collapses the van der Waals equation to a single universal reduced form containing no $a$ or $b$ . The route is a direct substitution: assemble $Z_c$ from the boxed constants, then insert $P=pP_c$ , $V=vV_c$ , $T=tT_c$ into the original equation and watch every substance-specific constant divide out. The key point to watch is that this cancellation is what the law of corresponding states means — one equation for all fluids.

Step 1. Assemble $Z_c$ . $Z_c=\dfrac{P_cV_c}{Nk_BT_c}=\dfrac{(a/27b^2)(3Nb)}{N(8a/27b)}=\dfrac{3a/9b}{8a/27b}=\dfrac{3}{8}$ . Why: the $a$ 's and $b$ 's and $N$ 's all cancel; only pure numbers survive, so $Z_c=3/8$ for every vdW fluid regardless of the substance.

Step 2. Introduce reduced variables. Define $p=P/P_c$ , $v=V/V_c$ , $t=T/T_c$ . Substitute $P=pP_c$ , $V=vV_c=3Nbv$ , $T=tT_c$ into $\left(P+aN^2/V^2\right)(V-Nb)=Nk_BT$ . Why: measuring each quantity as a multiple of its critical value is the natural non-dimensionalization; if the equation is truly universal, the constants must vanish.

Step 3. Simplify. Using $P_c=a/27b^2$ and $V_c=3Nb$ , the equation reduces (all $a,b,N,k_B$ dividing out) to $\left(p+\dfrac{3}{v^2}\right)\left(3v-1\right)=8t$ . Why: this is the reduced van der Waals equation — no $a$ , no $b$ , no substance identity remains, so all vdW fluids obey the same isotherm in reduced coordinates.

Step 4. Read the consequence. Two fluids at the same reduced $t$ and $v$ have the same reduced $p$ : they are in corresponding states. The prediction $Z_c=3/8$ is the reduced equation evaluated at the critical point $p=v=t=1$ . Why: the universality is exactly the cancellation of $a,b$ ; the quantitative gap between $3/8$ and the measured $\approx0.29$ (owned by aq-sm-law-of-corresponding-states) measures how far real fluids depart from mean field. *Verified: $Z_c=3/8$ and the reduction to $(p+3/v^2)(3v-1)=8t$ with all of $a,b,N,k_B$ cancelling from the shape are checked symbolically in verify_van-der-waals-equation.py; the tabulated real-gas $Z_c$ (Ar 0.290, N₂ 0.289, CO₂ 0.274, H₂O 0.229) are all confirmed below $0.375$ .*

Limiting cases

The equation interpolates between the ideal gas and its own breakdown, and exposes each regime cleanly.

Case	Result	Meaning
Dilute / high- $T$ ( $N/V\to0$ )	$P\to Nk_BT/V$	Both corrections vanish; the ideal gas law is recovered.
Leading density correction	$\dfrac{PV}{Nk_BT}=1+\left(b-\dfrac{a}{k_BT}\right)\dfrac{N}{V}+\cdots$	The second virial coefficient is $B_2(T)=b-a/(k_BT)$ — the microscopic bridge, owned by aq-sm-van-der-waals-derivation.
Boyle temperature	$k_BT_B=a/b$ , so $B_2(T_B)=0$	Attraction and excluded volume cancel to leading order; here $T_B/T_c=27/8\approx3.375$ .
Critical point	$k_BT_c=8a/27b,\ V_c=3Nb,\ P_c=a/27b^2$	Double tangency; $Z_c=P_cV_c/(Nk_BT_c)=3/8$ for all vdW fluids.
Incompressible limit ( $V\to Nb$ )	$P\to+\infty$	The excluded volume acts as a hard floor — the liquid-like near-incompressibility.

Common trapThe subcritical loop is a real pressure the gas can reach

No: the S-shaped loop below $T_c$ is not a physical isotherm. Its central branch has $(\partial P/\partial V)_T>0$ , i.e. negative isothermal compressibility — squeeze it and it pushes back less, which is mechanically unstable and cannot occur in equilibrium. The true isotherm replaces the loop with a flat coexistence line at the pressure set by the Maxwell equal-area construction (aq-sm-maxwell-construction); only short metastable stretches (superheated liquid, supercooled vapour) survive experimentally, up to the spinodal. A second, independent trap: quoting the vdW critical exponents ( $\beta=1/2,\ \delta=3,\ \gamma=1$ ) as if they matched experiment. They are mean-field values; the measured 3D-Ising exponents ( $\beta\approx0.326$ ) differ because vdW ignores critical fluctuations.

Worked examples

1 · The universal critical ratio, and why real gases fall short (confidence builder)

We compute $Z_c$ for the van der Waals equation from the critical constants, then compare to measured values — a real number that tests the theory.

Step 1. From (4), $P_c=a/27b^2$ , $V_c=3Nb$ , $k_BT_c=8a/27b$ . Form the ratio $Z_c=\dfrac{P_cV_c}{Nk_BT_c}=\dfrac{(a/27b^2)(3Nb)}{N(8a/27b)}$ .

Step 2. Cancel: numerator $=3aN/(27b)=aN/9b$ ; denominator $=8aN/(27b)$ . So $Z_c=\dfrac{aN/9b}{8aN/27b}=\dfrac{27}{9\cdot 8}=\dfrac{3}{8}=\mathbf{0.375}$ . Every substance-specific constant has vanished.

Step 3. Compare to experiment: argon $Z_c=0.290$ , nitrogen $0.289$ , carbon dioxide $0.274$ , water $0.229$ . All lie below $0.375$ .

Step 4. The verdict: van der Waals gets the existence of a universal $Z_c$ right — a genuine, correct qualitative prediction — but overestimates its value because the mean-field treatment ignores the density correlations that make real fluids more compressible near the critical point. A single dimensionless number both vindicates and indicts the theory.

2 · Argon: extracting molecular constants from critical data (exam level)

We invert measured critical data for argon to recover its van der Waals constants and an effective molecular size.

Step 1. Argon has $T_c=150.7$ K and $P_c=4.86\times10^6$ Pa. Invert (4): from $k_BT_c=8a/27b$ and $P_c=a/27b^2$ , dividing gives $k_BT_c/P_c=8b$ , so the molar $b=RT_c/(8P_c)$ .

Step 2. Numerically, $b=\dfrac{(8.314)(150.7)}{8(4.86\times10^6)}=\mathbf{3.22\times10^{-5}\ m^3/mol}$ . Then $a=27(RT_c)^2/(64P_c)=\mathbf{0.136\ Pa\,m^6/mol^2}$ .

Step 3. From $b=\tfrac23\pi N_A\sigma^3$ (derivation owned by aq-sm-van-der-waals-derivation), the effective hard-core diameter is $\sigma=\left(\dfrac{3b}{2\pi N_A}\right)^{1/3}=\mathbf{0.29\ nm}$ , comparing well with the Lennard-Jones value $\sigma\approx0.34$ nm.

Step 4. As a self-consistency check, the Boyle temperature is $k_BT_B=a/b$ , giving $T_B=27T_c/8=\mathbf{508\ K}=3.375\,T_c$ , and $Z_c=P_cV_c/(RT_c)$ recomputed from these $a,b$ returns exactly $3/8$ , confirming the inversion. A handful of critical measurements have delivered both van der Waals constants and a molecular size.

Origins · 1 minvan der Waals and how this was discovered

van der Waals

1837–1923, Dutch physicist, Leiden · the equation of state that condenses a gas (1873)

Johannes Diderik van der Waals began as a schoolteacher without a classical education, teaching himself physics and mathematics before earning his doctorate at Leiden in 1873 at the age of thirty-five. That thesis, Over de Continuïteit van den Gas- en Vloeistoftoestand ("On the Continuity of the Gaseous and Liquid States"), contained the equation that bears his name and, with it, the revolutionary claim that gas and liquid are not two different substances but two regimes of one continuous equation of state, joined through a critical point. James Clerk Maxwell was so impressed that he learned Dutch to read it. The equation's two constants $a$ and $b$ gave the first quantitative handle on molecular attraction and molecular size, and its reduced form led directly to the law of corresponding states, which van der Waals published in 1880 and which guided the race to liquefy the "permanent" gases. His work underpinned Kamerlingh Onnes's liquefaction of helium in 1908 in the very same Leiden laboratory. Van der Waals received the Nobel Prize in Physics in 1910 "for his work on the equation of state for gases and liquids." The weak intermolecular attractions he first quantified are named van der Waals forces in his honour, and his mean-field treatment of cohesion is the ancestor of every mean-field theory of phase transitions.

History

1873

— van der Waals's Leiden doctoral thesis introduces the equation of state and the continuity of the gas and liquid states.

1875

— Maxwell presents the equal-area construction that selects the true coexistence pressure from the vdW loop.

1880

— van der Waals formulates the law of corresponding states from the reduced equation.

1908

— Kamerlingh Onnes liquefies helium at Leiden, the culmination of the corresponding-states-guided quest.

1910

— van der Waals receives the Nobel Prize in Physics for the equation of state.

Going further

Texts. Pathria & Beale, Statistical Mechanics, 3rd ed., §§10.1–10.3 (van der Waals, critical point, corresponding states). Kardar, Statistical Physics of Particles, Ch. 5. Reif, Fundamentals of Statistical and Thermal Physics, §§8.6–8.7. Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity, Ch. 11. Landau & Lifshitz, Statistical Physics Part 1, §§74, 84.

Provenance. Base article: AllQFT track `sm`, "Van der Waals Equation of State" (STAT.INTER.VDWEQ.1250); source Pathria & Beale §§10.1–10.3, Kardar Ch. 5. All symbolic and numeric claims re-verified in verify_van-der-waals-equation.py: the critical-point solve ( $V_c=3Nb$ , $k_BT_c=8a/27b$ , $P_c=a/27b^2$ ), $Z_c=3/8$ , the reduction to $(p+3/v^2)(3v-1)=8t$ with all constants cancelling, $B_2=b-a/(k_BT)$ , $k_BT_B=a/b$ with $T_B/T_c=27/8$ , and the argon inversion ( $b\approx3.2\times10^{-5}$ m³/mol, recovered $Z_c=3/8$ ). No base physics error found; the tabulated real-gas $Z_c$ values (Ar 0.290, N₂ 0.289, CO₂ 0.274, H₂O 0.229) all confirm the wrong-but-famous point that vdW's $0.375$ overestimates.

Prerequisites. Configuration Integral and the Interacting Partition Function · Second Virial Coefficient · Virial Expansion of the Equation of State.

Leads to. Maxwell Construction and Coexistence · Law of Corresponding States · Van der Waals from the Virial Expansion.

Why is the accessible volume V − Nb and not V minus the actual volume of the molecules?

Because the relevant exclusion is a pairwise geometric effect, not a simple subtraction of molecular volumes. Two hard spheres of diameter $\sigma$ cannot have their centres closer than $\sigma$ , so each pair excludes a ball of radius $\sigma$ — volume $\tfrac43\pi\sigma^3$ , which is eight times one molecule's volume. Sharing that between the two partners gives an excluded volume per molecule of $b=\tfrac23\pi\sigma^3=4\times$ the molecular volume. So $b$ is four times the geometric molecular volume, and $V-Nb$ is the leading-order accessible volume. The full geometric derivation of this factor of four is owned by aq-sm-van-der-waals-derivation.

Where do the specific numbers 8/27, 3, and 1/27 in the critical constants come from?

They fall straight out of solving the double-tangency condition. Setting $(\partial P/\partial V)_T=0$ and $(\partial^2P/\partial V^2)_T=0$ and taking their ratio eliminates the temperature and gives $V_c=3Nb$ — that is the origin of the $3$ . Back-substituting into the first-derivative condition gives $k_BT_c=8a/27b$ , and plugging $T_c$ and $V_c$ into the equation of state gives $P_c=a/27b^2$ . The numbers are not fitted; they are exact rational consequences of the cubic structure of the vdW equation. Their most important combination is $Z_c=P_cV_c/(Nk_BT_c)=3/8$ , in which the $a$ and $b$ cancel entirely.

If the vdW critical exponents are wrong, why is the equation still taught and used?

Because it is right about everything qualitative and wrong only about the fine quantitative behaviour in a narrow window around $T_c$ . It correctly predicts the existence of a liquid–gas critical point, the continuity of the two phases, the shape of the phase diagram, coexistence via the Maxwell construction, and the universality captured by corresponding states. Its exponents ( $\beta=1/2$ , etc.) are mean-field and disagree with the measured Ising values because it ignores critical fluctuations — but that failure is itself pedagogically valuable, since it motivates the renormalization group. For engineering, its descendants (Peng–Robinson and the like) are tuned to fit data and are accurate enough to design real plants. It is the rare theory whose errors are as instructive as its successes.

What exactly does this article own versus the derivation article?

This article owns the equation as an object: the form $(P+aN^2/V^2)(V-Nb)=Nk_BT$ , the isotherms it generates, the critical-point conditions and the resulting constants $T_c,V_c,P_c$ , the universal ratio $Z_c=3/8$ , and the reduced equation of state. It treats $a$ and $b$ as given constants. The companion article aq-sm-van-der-waals-derivation owns their microscopic origin: computing the second virial coefficient $B_2(T)=b-a/(k_BT)$ from a hard-core-plus-attraction pair potential, showing $b=\tfrac23\pi N_A\sigma^3$ and $a$ as an integral over the attractive tail, and clarifying in what precise sense vdW is a resummation of the virial series. The two together form the full picture: what the equation says, and why.

Does the van der Waals equation predict the coexistence pressure by itself?

Not directly. The bare equation produces the S-shaped loop below $T_c$ , whose central branch is mechanically unstable, but it does not tell you at what pressure the two phases coexist. Fixing the coexistence pressure requires an extra thermodynamic input: the two phases must have equal pressure and equal chemical potential (equivalently, equal Gibbs free energy). That condition translates into the Maxwell equal-area rule — the horizontal line drawn so that the two lobes it cuts from the loop have equal area — and it is owned by aq-sm-maxwell-construction. So the equation supplies the isotherm and the instability; the Maxwell construction supplies the coexistence plateau.

Why does the attractive term scale as the square of the density?

Because it takes two densities to make an attraction contribute to the pressure. In the mean-field picture each molecule sits in the averaged attractive field of all the others, and the strength of that field is proportional to the local density $N/V$ — that is one factor of density, for the molecules doing the pulling. The pressure reduction then counts how many molecules feel that field, which brings a second factor of $N/V$ — for the molecules being pulled. Multiplying the two gives the $-aN^2/V^2=-a n^2$ dependence. The single constant $a$ packages the strength and range of the pair attraction; its value as an integral over the attractive tail is derived in aq-sm-van-der-waals-derivation.