Chemistry 1873

On the Continuity of the Gaseous and Liquid States

Johannes Diderik van der Waals

Gas and liquid are not two substances but one, joined by a single equation.

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In depth · the introduction

Heat water and it boils into steam; cool the steam and it rains back down — but van der Waals found that gas and liquid are secretly the same thing, with no hard line between them at all.

One equation for two states

The simple gas law, PV = RT, pretends that gas molecules are tiny dots that take up no space and ignore each other. Van der Waals fixed both pretences. Molecules really do take up a little room, so there is less space inside the container than it seems. And molecules really do pull on each other a little, so the gas presses on the walls a touch more gently than it otherwise would.

Make just those two corrections and something remarkable happens: the very same equation now describes a thin gas and a dense liquid at once. Squeeze the gas hard enough and the formula itself shows it collapsing into a liquid. Gas and liquid stop being two different things and become two faces of one substance.

A schoolteacher in Leiden

Van der Waals came late and from the outside. He was a Dutch schoolteacher, largely self-taught in physics, who needed special dispensation even to sit his university exams because he had no Greek or Latin. He was thirty-six when he wrote up this idea as a doctoral thesis in 1873 — in Dutch, which almost guaranteed the world would overlook it.

It did not. James Clerk Maxwell, the towering physicist of the age, got hold of it, was so impressed that he is said to have taught himself Dutch to read it, and announced that this unknown teacher's name would soon stand among the foremost in molecular science. Thirty-seven years later, in 1910, van der Waals received the Nobel Prize in Physics for exactly this work.

Why it mattered

It explained a mystery the experimenters had just stumbled on: above a certain temperature — the “critical temperature” — no gas can be squeezed into a liquid, however hard you press. Van der Waals's equation showed why, and even predicted, from his theory, that every gas behaves the same way once you measure it against its own critical point. That single rule told experimenters precisely how cold and how compressed they had to get — and led, in 1908, to the first liquefaction of helium.

A crowded room

Picture people milling in a room. The ideal gas law treats them as ghosts: no body, never touching. Van der Waals adds two true things. Each person actually takes up space, so the room is more crowded than its floor area suggests. And people lightly hold hands with their neighbours, so the crowd pushes outward against the walls a little less. Pack them in tightly enough and the loose crowd suddenly condenses into a tight knot — a gas becoming a liquid, all from those two homely facts.

Its place in the story

Van der Waals built on the kinetic theory of Maxwell and Boltzmann (heat as molecules in motion) and on Thomas Andrews's discovery of the critical point in carbon dioxide. His equation became the model for every phase transition, and the weak attractions he proposed are now called van der Waals forces — the same forces that let geckos walk up walls and that hold the rungs of DNA together (see watson-crick-1953). When physicists finally understood the critical point exactly, a century later, they were completing a question van der Waals had opened.

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Original source text

J. D. van der Waals · doctoral thesis, Leiden · 1873 · structural map; equation and Nobel citation quoted verbatim

The puzzle of 1873

Four years earlier Thomas Andrews had shown that carbon dioxide, warmed past 31 °C, can no longer be turned to liquid by any pressure — there is a critical temperature above which the line between gas and liquid simply disappears. The kinetic theory of Clausius and Maxwell, meanwhile, pictured a gas as a swarm of point-like particles that neither take up room nor attract one another. Van der Waals asked the obvious, unasked question: what if the molecules do both?

Two corrections to the ideal gas

He kept the ideal gas law PV = RT and mended it twice. First, the molecules occupy space, so the room left for them to move in is not V but V − b, where b is roughly the volume of the molecules themselves. Second, they pull on one another, so a molecule near the wall is tugged back inward and presses a little less hard; this lost pressure grows as the gas is squeezed, as a/V². Putting both together gives a single equation of state.

(P + a/V²)(V − b) = RT

Read as a cubic in V, this one formula can have, at a fixed temperature and pressure, three solutions: a small volume (the dense liquid), a large volume (the dilute gas), and a third, unstable root between them. Below the critical temperature the isotherm carries a backward wiggle — the famous van der Waals loop — and the flat line of real condensation cuts across it where the two lobes have equal area (Maxwell's rule, 1875). Raise the temperature and the three roots draw together; at the critical point they merge into one. Above it, only a single volume remains for each pressure: gas and liquid have become one continuous fluid.

One law for every gas

Measuring each quantity against its value at the critical point — π = P/P_c, φ = V/V_c, τ = T/T_c — makes the constants a, b and R vanish, leaving (π + 3/φ²)(3φ − 1) = 8τ. The same dimensionless curve now describes oxygen, water, and carbon dioxide alike: the law of corresponding states. It is this prediction that guided Kamerlingh Onnes in Leiden to the liquefaction of hydrogen and, in 1908, of helium.

[ … ]

Maxwell, reviewing the work, is said to have learned enough Dutch to read it in the original, and told the British Association that van der Waals's name would soon be “among the foremost in molecular science.” The 1910 Nobel Prize in Physics was awarded to van der Waals

for his work on the equation of state for gases and liquids.

Leiden · 1873