The van der Waals equation is a modified version of the ideal gas law that accounts for two things real gas molecules actually do: take up space and attract each other. Proposed by Johannes Diderik van der Waals in his 1873 doctoral thesis, it was the first equation of state to explain why gases can be squeezed into liquids and why the boundary between those two phases can vanish altogether at high enough temperatures and pressures. The equation is far from perfect for precision engineering, but it remains one of the most important conceptual tools in physical science because it captures, with just two extra parameters, the essential physics that the ideal gas law ignores.
Why the Ideal Gas Law Needed Fixing
The ideal gas law treats gas molecules as infinitely tiny points that never interact with each other. That works surprisingly well at everyday temperatures and low pressures, where molecules are far apart and moving fast enough that their mutual attraction barely matters. But push a gas to higher pressures or cool it down, and the approximation starts to fail badly. Molecules get crowded together, their finite size matters, and the attractive forces between them become significant enough to change how the gas behaves. At extreme conditions the ideal gas law cannot even hint at the existence of a liquid phase, let alone predict when a gas will condense into one.
Van der Waals recognized both problems and addressed them in a single stroke. His 1873 thesis, titled “On the continuity of the gas and liquid state,” argued that gases and liquids are not fundamentally different kinds of matter but rather two manifestations of the same substance under different conditions. As one contemporary account noted, the thesis demonstrated that gaseous and liquid states “not only merge into each other in a continuous manner but also that they are in fact the same nature.”1The Journal of Supercritical Fluids. Gas solubilities in ionic liquids using a generic van der Waals equation of state Maxwell himself praised the work in Nature, and van der Waals eventually received the Nobel Prize in Physics in 1910 for it.
The Two Corrections and What They Mean
The ideal gas law says that pressure times volume equals the number of molecules times temperature (scaled by a constant). Van der Waals kept that skeleton but added a correction for each of the two things ideal gases get wrong.
The first correction deals with molecular size. Real molecules are not dimensionless points; they have a physical volume you cannot compress away no matter how hard you squeeze. Van der Waals introduced a parameter, usually called b, that represents the effective volume excluded by the molecules themselves. In the equation, this shows up as a reduction in the available volume: instead of the full container volume, you subtract off the space the molecules physically occupy. Without attractive forces in the picture, the equation simplifies to a form where pressure depends on the density divided by one minus the density times b, which captures the idea that as you pack molecules tighter, the remaining free space shrinks faster than you might expect.2Journal of Molecular Liquids. Excluded volume of the system of hard-core spheres revisited: New insights from computer simulations
The second correction deals with intermolecular attraction. Gas molecules are not indifferent to each other; they pull on their neighbors through weak forces (the same forces, confusingly also called van der Waals forces, that make geckos stick to walls and allow liquids to form in the first place). This mutual tug reduces the pressure a gas exerts on its container walls, because molecules near the wall are being pulled inward by the molecules behind them. Van der Waals captured this with a second parameter, a, which appears in the equation as a pressure-reducing term that grows stronger as molecules get more crowded. Together, the two parameters transform the ideal gas law into something that can describe real gas behavior across a far wider range of conditions.
What the Constants Tell You About a Molecule
The parameters a and b are not arbitrary fitting numbers. They reflect genuine physical properties of the molecules in question. Research has shown that both constants are linearly proportional to the polarizability volume of the molecules in a gas or liquid.3The Journal of Physical Chemistry B. van der Waals Equation of State Revisited: Importance of the Dispersion Correction Polarizability is a measure of how easily the electron cloud around a molecule can be distorted. A highly polarizable molecule has a large, loosely held electron cloud, which means it generates stronger fleeting electrical attractions with its neighbors and also tends to be physically bigger. That is why a (the attraction parameter) and b (the size parameter) both scale with polarizability: a fatter, squishier electron cloud means both stronger stickiness and a larger excluded volume.
This connection gives the equation an appealing physical transparency. If you know something about a molecule’s electronic structure, you can make a reasonable guess at its van der Waals constants, and vice versa. Small, tightly held molecules like helium have tiny values of both a and b; large, electron-rich molecules like heavy hydrocarbons have much larger ones. The practical upshot is that the equation is not just a mathematical trick for fitting pressure-volume data. It encodes something real about how molecules interact at the atomic scale.
Phase Transitions and the Critical Point
One of the most striking things the van der Waals equation does, and the reason van der Waals wrote it in the first place, is predict that gases can condense into liquids. If you plot the pressure predicted by the equation against volume at a fixed temperature, the resulting curve (called an isotherm) changes shape depending on how hot the gas is. At high temperatures, the curve looks smooth and monotonically decreasing, much like what the ideal gas law would give you. But below a certain temperature, the curve develops a wiggle: it dips down, rises, and then dips again. That wiggle is the equation’s way of signaling a phase transition. Part of the curve represents gas, part represents liquid, and the unstable middle part is physically unreachable.
To figure out exactly where the gas-to-liquid jump happens along that wiggly isotherm, physicists use what is known as the Maxwell construction, or the equal-area rule. The idea is to draw a horizontal line across the wiggle at just the right pressure so that the area of the curve above the line equals the area below it. That horizontal line represents the actual behavior of the substance: instead of following the mathematical wiggle, the real material simply transitions from gas to liquid (or back) at that constant pressure. The equal-area rule was devised specifically for vapor-liquid equilibrium calculations with the van der Waals equation.4arXiv. The Maxwell crossover and the van der Waals equation of state
At one special temperature, the wiggle just barely disappears, collapsing into a single inflection point. That is the critical point: the temperature and pressure above which there is no distinction between liquid and gas. The substance becomes a supercritical fluid, a state that has properties of both phases simultaneously. The van der Waals equation predicts the existence of the critical point directly from the values of a and b, which was a remarkable achievement for a nineteenth-century model. It gets the qualitative story exactly right, though its quantitative predictions for critical-point properties are only approximate.
Where the Equation Falls Short
For all its conceptual elegance, the van der Waals equation is not accurate enough for most engineering or industrial calculations. Its biggest weakness is that it systematically mispredicts what is called the compressibility factor, which is a measure of how much a real gas deviates from ideal behavior. The deviation between an actual gas and an ideal gas comes from exactly the two effects the equation tries to correct, molecular volume and intermolecular forces, but the specific mathematical forms van der Waals chose for those corrections are too simple to capture the full picture.5International Journal of Hydrogen Energy. Study on real-gas equations of high pressure hydrogen
The equation also struggles with certain categories of substance. Actual gases can be divided roughly into nonpolar gases (like nitrogen or methane), polar gases (like water vapor or ammonia), and so-called quantum gases (like hydrogen and helium at very low temperatures). The van der Waals equation handles nonpolar gases reasonably well, but its two-parameter approach does not capture the directional, orientation-dependent forces that polar molecules exert on each other, and it completely ignores quantum effects that become important for the lightest gases near absolute zero.
Even for the phase-transition predictions that are the equation’s strong suit, the numbers are only roughly correct. The critical compressibility factor it predicts is the same for all substances, a value of 3/8. Real substances have critical compressibility factors that range from about 0.23 to 0.29, with most common gases clustering around 0.27. The equation overshoots by about 40 percent. That kind of systematic error matters if you are designing a chemical plant but matters less if you are trying to understand why gases condense in the first place.
Modern Equations That Grew Out of It
Almost every widely used equation of state in chemical engineering today is a direct descendant of the van der Waals equation. The family tree starts with the Redlich-Kwong equation from 1949, which kept the same basic structure but used a more sophisticated expression for the attraction term, making it temperature-dependent. The Soave modification of Redlich-Kwong (SRK) came in 1972 and improved predictions for vapor-liquid equilibrium. Then in 1976, Peng and Robinson introduced their equation, which further refined the attraction term and gave better liquid-density predictions. All of these are sometimes called “cubic” equations of state because, like the original van der Waals equation, they can be rearranged into a cubic polynomial in volume.
One area where these descendants have been particularly refined is in handling mixtures. When you have a blend of different gases or liquids, you need rules for combining the individual molecules’ parameters into effective values for the mixture. The original approach, now called the van der Waals mixing rules, uses simple averages. More advanced mixing rules based on statistical-mechanical theory have been developed that improve predictions substantially. Testing these newer rules with the Redlich-Kwong and Peng-Robinson equations has shown they can predict things like the solubility of heavy solids in supercritical fluids more accurately than the original mixing rules.6Chemical Engineering Science. Van der waals mixing rules for cubic equations of state. Applications for supercritical fluid extraction modelling Supercritical fluid extraction is used commercially for things like decaffeinating coffee and extracting essential oils, so getting these mixture calculations right has real economic stakes.
Despite their improvements, all of these modern equations retain the conceptual architecture van der Waals established: a repulsive term that accounts for molecular size and an attractive term that accounts for intermolecular pull. The refinements are in the details of those terms, not in the overall framework. That is why the van der Waals equation is still taught first: understanding it gives you the mental scaffolding for every equation of state that followed.
Predicting Joule-Thomson Behavior
One practical test of any equation of state is whether it can predict the Joule-Thomson effect, the temperature change a gas undergoes when it expands through a valve or porous plug without gaining or losing heat to its surroundings. This effect is the basis of most industrial refrigeration and gas liquefaction. Every real gas has an inversion temperature: above it, the gas warms when it expands; below it, the gas cools. The inversion curve, which plots this boundary across a range of pressures, is notoriously hard for equations of state to get right.
Studies comparing several van der Waals-type equations found that they generally do a good job predicting the low-temperature branch of the inversion curve but struggle with the high-temperature branch and the peak of the curve.7J-STAGE. Prediction of Joule–Thomson Inversion Curves from van der Waals Type Equations of State The high-temperature region turns out to be very sensitive to the exact form of the equation used. This is one of the benchmarks engineers use when evaluating whether a newer equation of state is worth adopting for a particular application. The original van der Waals equation can sketch the inversion curve qualitatively, but you would not want to use it to design a cryogenic cooling system.
Applications in Planetary Science
The van der Waals equation and its descendants have found a surprising second life in planetary science, far from the chemical-engineering context where they are most commonly used. When scientists model the atmospheres of other worlds, they need to know how gases behave at temperatures and pressures very different from those on Earth’s surface. The ideal gas law is a poor guide when you are dealing with the crushing atmosphere of Venus or the frigid, methane-rich air of Saturn’s moon Titan.
Recent work has used a family of temperature-dependent van der Waals-type equations to estimate the adiabatic lapse rate (the rate at which temperature drops with altitude) in the tropospheres of Titan and Venus.8Brazilian Journal of Physics. Adiabatic Lapse Rate Estimation Using a Van Der Waals-type Equation of State On Earth, the lapse rate is well-characterized by measurements, but for other planets, theoretical models are often the only tool available. The van der Waals framework is useful here because it captures the real-gas effects that matter most: molecular size and intermolecular attraction, both of which become significant in dense, cold atmospheres. Venus, with its thick carbon dioxide atmosphere and surface pressures around 90 times Earth’s, is a case where the ideal gas approximation breaks down badly, and van der Waals-type corrections bring the calculated lapse rate much closer to what spacecraft have measured.
Why It Persists in the Classroom
Given that more accurate equations exist, you might wonder why the van der Waals equation still dominates introductory chemistry and physics courses. The reason is that no other equation manages the same balance of physical insight and mathematical simplicity. It introduces exactly two ideas, molecular size and molecular attraction, and shows how each one modifies the ideal gas law in an intuitive direction. It predicts the existence of phase transitions, critical points, and the continuity between gas and liquid states, all from a formula you can write on a napkin.
The equation has also become a standard vehicle for teaching computational methods. In modern science education, students use tools like Python to plot van der Waals isotherms alongside those from more advanced equations such as Redlich-Kwong, visualizing how different models handle the same physical situation.9PubMed Central. Google Colab and Virtual Simulations: Practical e-Learning Tools to Support the Teaching of Thermodynamics and to Introduce Coding to Students Plotting the wiggly isotherms, applying the Maxwell construction, and watching the critical point emerge from the math gives students a visceral sense of how phase transitions work that no amount of reading a textbook can match. The equation is simple enough that you can write the code in an afternoon but rich enough that the output teaches real physics.
There is also something historically satisfying about it. The ideal gas law is one of the first quantitative relationships students learn in science. The van der Waals equation is the first time many of them see a model improved by asking “what did we leave out?” and then adding corrections based on physical reasoning rather than just fitting parameters to data. That process of identifying the assumptions in a model, figuring out which ones matter, and building something better is arguably the core skill of applied science. The van der Waals equation teaches it with unusual clarity.
The Equal-Area Rule as a Window Into First-Order Transitions
The Maxwell construction mentioned earlier deserves a closer look, because it connects the van der Waals equation to a much broader idea in physics: first-order phase transitions. A first-order transition is one where a substance jumps discontinuously from one state to another, like water turning to steam at 100°C under normal pressure. The hallmark is a sudden change in density (or volume, or entropy) at a specific temperature and pressure, with the two phases coexisting during the transition.
In the mathematical language of the van der Waals equation, this discontinuous jump shows up as a shock in the solution. Analysis of the equation’s behavior near the critical region has shown that above a certain threshold, the physical solution develops a shock discontinuity corresponding to a first-order phase transition, with the position of the shock determined by the equal-area rule.10Annals of Physics. Exact solution of the van der Waals model in the critical region The mathematics here is closely related to shock waves in fluid dynamics, which is a connection that has fascinated physicists for over a century. The van der Waals equation, simple as it is, turns out to be a gateway into some of the deepest ideas in thermodynamics and mathematical physics.
Near the critical point, however, the equal-area rule and the van der Waals equation begin to disagree with experiment in a fundamental way. Real substances near their critical points exhibit what are called critical exponents, power-law behaviors that describe how properties like density difference between liquid and gas vanish as you approach the critical temperature. The van der Waals equation predicts these exponents, but it gets them wrong: it gives so-called “mean-field” values that differ from the experimentally measured ones. This failure is not a quirk of the equation’s simplicity. It reflects a deep limitation of any model that treats molecular interactions as smooth averages rather than accounting for the wild fluctuations that dominate near a critical point. Understanding why the van der Waals equation fails here is what led to the development of the renormalization group theory in the 1970s, work that earned Kenneth Wilson the Nobel Prize in Physics in 1982. So even the equation’s failures have been scientifically productive.