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.
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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