When Molecules Bump
Today, a historical injustice. The University of Houston presents this series about the machines that make our civilization run, and the people whose ingenuity created them.
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The air around us is a cacophony of molecules flying about with average speeds over a thousand miles an hour. And they often bump into each other. Well, they don’t actually bump. Here’s how what we call a collision really works: When they get close, they first attract each other – and get pulled out of their flight paths. Then, when they get very close, they repel each other, and fly off. (They never actually touch.)
We have an equation, the Ideal Gas Law. It just says that the air pressure times its volume is proportional to its temperature. And it’s true as long as molecules, like those around us, spend far more time flying than colliding. They’ll typically move a couple hundred times their own size before they collide. So our Gas law works just fine in the air we breathe.
But what happens when air is highly pressurized or very cold? Now molecules crowd in and suffer more collisions. The ideal gas law is no longer accurate. Enter now the van der Waals equation. It takes account of the fact that, while its molecules attract, their size takes up a noticeable fraction of the volume in which they move.
It turns out that van der Waals’ equation is perfectly accurate for molecules that collide like billiard balls (well – if billiard balls slightly attracted one another). So: What sort of molecules behave like that?
One answer is: Big ones – like metals that’ve first melted, then boiled into a gas. The idea of boiling metals might sound odd. But: Any metal will boil at a high enough temperature. And that’s no big deal with, say, mercury.
But we have limited data for gaseous metals. And, since people have largely applied van der Waals’ equation to materials with far lighter atoms – like common gases – it has appeared to be inaccurate. The engineering textbooks I’ve used all treated it as an interesting approximation. But it’s actually accurate for the molecules it describes. (I’ll post links to more detailed work that coworkers and I have done.)
The creator of the equation, Johannes van der Waals, is justly famous for giving us a better equation than the ideal gas law – and for doing so back when people were just beginning to accept the existence of atoms and molecules. He went even further. Toward the end of his life he even showed how his equation gave a basis for predicting other properties – like surface tension.
Johannes Diderik van der Waals (Image courtesy of Wikimedia commons)
But pick up any textbook today, and it’ll still tell us that his famous equation is just an approximation ... overlooking the fact that it is accurate when we use it to predict properties of the materials which it actually describes. Accuracy that stands to improve our very understanding of physical properties.
I’m John Lienhard, at the University of Houston, where we’re interested in the way inventive minds work.
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Some Sources
Johannes Diderik van der Waals - Wikipedia
Van der Waals equation - Wikipedia
van der Waals‘ second important paper about his equation was: van der Waals, J. D., “Thermodynamische Theorie der Kapillarität unter Voraussetzung stetiger Dichteänderung,” Zeitschrift für Physikalische Chemie, 13 (1894), 657–725. This is where he uses his equation to represent surface tension.
Here we see the comparison of van der Waals equation with data for materials which it accurately describes. See especially Fig. 2 correspondingstatesmetastable.pdf
Here, van der Waals equation actually predicts surface tension: cubicequationsurfacetension.pdf
And this paper summarizes many uses of van der Waals equation. SuperheatedLiquids.pdf
Here we find a conventional dismissal of the van der Waals equation under non-ideal conditions.
The Sutherland potential describes a van der Waals molecular collision. See: Sutherland potential page on SklogWiki - a wiki for statistical mechanics and thermodynamics
This episode first aired on September 8, 2026.