Van Der Waals Equation Of State

You’ve memorized the ideal gas law. Now, you can recite PV = nRT in your sleep. And for a lot of introductory chemistry problems, it works perfectly fine.

But then you hit a problem involving high pressure. Still, or low temperature. Or a gas that just refuses to behave — carbon dioxide, ammonia, water vapor. Suddenly the numbers don’t match the lab data. In real terms, the volume is too low. In practice, the pressure is too high. The ideal gas law shrugs and says, "Well, gases are point particles with no feelings, so deal with it.

Real gases have volume. Real gases attract each other. And in 1873, a Dutch physicist named Johannes Diderik van der Waals decided to do something about it. His equation of state didn’t just patch the ideal gas law — it introduced the very concept of a critical point and won him a Nobel Prize. It’s the grandfather of every modern cubic equation of state used in chemical engineering today.

Here’s what it is, why it matters, and where it still trips people up.

What Is the van der Waals Equation

At its core, the van der Waals equation is a corrected version of the ideal gas law. It adds two parameters — a and b — to account for the two big lies the ideal gas law tells.

The equation looks like this:

(P + a(n/V)²)(V - nb) = nRT

Or, in molar form (using molar volume Vₘ = V/n):

(P + a/Vₘ²)(Vₘ - b) = RT

Let’s break the corrections down.

The pressure correction: a

Ideal gas molecules don’t interact. Real ones do. When molecules attract each other — London dispersion forces, dipole-dipole, hydrogen bonding — they pull inward slightly. That reduces the force with which they hit the container walls. The measured pressure P is lower than the "ideal" pressure would be.

Van der Waals argued the reduction is proportional to the square of the molar density (n/V)². That's why why squared? Because the attractive force depends on pairs of molecules interacting. Double the density, you quadruple the pairwise interactions. Here's the thing — the constant a quantifies how strong that attraction is for a specific gas. Units? Usually L²·bar·mol⁻² or Pa·m⁶·mol⁻².

The volume correction: b

Ideal gas molecules are points. In real terms, real molecules have size. They occupy space. The volume available for molecules to move around in isn’t the container volume V — it’s V minus the volume the molecules themselves take up.

That excluded volume per mole is b. It’s not exactly the physical volume of one mole of molecules (that would be Nₐ times the volume of one molecule). Because of the way spheres pack and exclude each other, b is roughly four times the actual molecular volume. Units: L·mol⁻¹ or m³·mol⁻¹.

The constants are specific

Every gas gets its own a and b. That's why helium has a tiny a (weak forces) and a tiny b (small atoms). Something like n-butane has a large a (lots of electrons, strong dispersion) and a larger b. You look these up in a table. You don’t calculate them from first principles — not usually.

Why It Matters

You might ask: if we have supercomputers and molecular dynamics simulations, why does a 150-year-old cubic equation still show up in textbooks and process simulators?

It introduced the critical point

This is the big one. Which means above T_c, you cannot liquefy the gas no matter how much pressure you apply. The van der Waals equation predicts a critical temperature T_c, critical pressure P_c, and critical volume V_c. On top of that, the meniscus disappears. The distinction between gas and liquid vanishes.

The ideal gas law has no critical point. That said, it predicts gases just compress indefinitely. Now, van der Waals showed that a simple mean-field model — treating the attraction as a uniform background field — naturally produces a critical isotherm with an inflection point. That was revolutionary.

It connects to measurable constants

Here’s the practical magic. You can express a and b in terms of the critical constants:

a = 27(R²T_c²) / (64P_c) b = RT_c / (8P_c)

Want to learn more? We recommend what happens to atoms during a chemical reaction and multi-objective optimization of industrial ammonia synthesis pdf for further reading.

Or invert them to find critical constants from a and b:

T_c = 8a / (27Rb) P_c = a / (27b²) V_c = 3b

This means if you know T_c and P_c for a substance (easy to measure), you have the equation of state. No fitting required. That’s why it became the workhorse of early chemical engineering.

It’s the ancestor of modern EOS

Redlich-Kwong. Soave-Redlich-Kwong. Day to day, peng-Robinson. These are all modifications of the van der Waals structure. They tweak the attraction term to be temperature-dependent, they adjust the volume correction, they add a third parameter. But the skeleton — cubic in volume, pressure correction, volume correction — is pure van der Waals. If you understand this one, the others make sense.

How It Works (and How to Use It)

The equation is cubic in volume. That means for a given P and T, you can get up to three real roots for V. This is a feature, not a bug.

The cubic nature

Expand the molar form:

Vₘ³ - (b + RT/P)Vₘ² + (a/P)Vₘ - ab/P = 0*

Three roots. What do they mean?

• One real root (supercritical or high T): Only one phase exists. That root is the molar volume.
• Three real roots (subcritical): The largest root is the vapor molar volume. The smallest root is the

The smallest root is the liquid molar volume. The middle root, while mathematically real, corresponds to a metastable or unstable state that lies on the van der Waals loop; it does not represent any observable phase and is discarded when constructing phase‑equilibrium data.

Using the cubic form in practice
When pressure and temperature are specified below the critical point, solving the cubic yields three real volumes. The vapor and liquid volumes are taken as the largest and smallest roots, respectively. To obtain the saturation pressure at a given temperature, one imposes the Maxwell equal‑area rule: the areas above and below the isotherm between the liquid and vapor roots must be equal. This condition adjusts the pressure until the chemical potentials of the two phases match, producing the coexistence curve that the van der Waals model predicts.

Because the equation is analytic, the roots can be found with a closed‑form Cardano solution or, more conveniently, with a numerical root‑finder (Newton‑Raphson, Durand‑Kerner, etc.So ). Many early process simulators implemented a direct cubic solver and then applied the Maxwell construction to generate vapor‑pressure tables, enthalpy departures, and fugacity coefficients.

What the model gets right – and where it falls short
The van der Waals EOS captures the qualitative topology of fluid phase behavior: a critical point, a liquid‑vapor dome, and the correct trend that attractive forces lower pressure while repulsive volume raises it. Quantitatively, however, it predicts a critical compressibility factor

[ Z_c = \frac{P_c V_c}{RT_c} = \frac{3}{8}=0.375, ]

whereas most real fluids have (Z_c) in the range 0.On top of that, 23–0. So naturally, the model overestimates the liquid density and underestimates the vapor pressure near the critical region. But 29. These systematic errors motivated the succession of refined cubic EOS — Redlich‑Kwong, Soave‑Redlich‑Kwong, Peng‑Robinson — each preserving the van der Waals skeleton while introducing temperature‑dependent attraction terms, acentric‑factor corrections, or volume‑shift parameters to improve accuracy for hydrocarbons, polar substances, and high‑pressure applications.

Legacy and continued relevance
Despite its shortcomings, the van der Waals equation remains a cornerstone of thermodynamics education because it demonstrates, in a single compact formula, how intermolecular forces and finite molecular size conspire to produce non‑ideal behavior and a critical point. Its algebraic simplicity makes it an ideal testbed for illustrating concepts such as stability criteria, phase‑split calculations, and the meaning of equations of state. In modern engineering workflows, the van der Waals form is rarely used directly for design calculations, but its conceptual framework underpins every cubic EOS that powers process simulators, reflux‑drum calculators, and equation‑of‑state‑based optimization tools.

In short, the van der Waals equation taught us that a modest correction to the ideal gas law could reproduce the rich phenomenology of real fluids — a lesson that continues to echo in the sophisticated models of today.