Johannes Diderik van der Waals

How to use this course

Johannes Diderik van der Waals wrote down, in his doctoral thesis of 1873, an equation that every chemistry and physics student still learns: \((p+a/V_m^2)(V_m-b)=RT\). Two small corrections to the ideal gas law, one for the size of molecules and one for their attraction, were enough to produce condensation, a critical point and a single theory of gas and liquid. He then derived a law of corresponding states that guided the liquefaction of hydrogen and helium, a theory of mixtures and a theory of the liquid surface that anticipated modern field theories of interfaces. The short-range attractions between neutral molecules now bear his name.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Throughout, \(V_m\) is the molar volume, \(R=8.314\ \mathrm{J\,mol^{-1}\,K^{-1}}=0.083145\ \mathrm{L\,bar\,mol^{-1}\,K^{-1}}\), and reduced variables are \(p_r=p/p_c\), \(v_r=V_m/V_c\), \(t=T/T_c\).

Timeline

Leiden, 1837–1862

Johannes Diderik van der Waals was born in Leiden on 23 November 1837, the eldest of ten children of Jacobus van der Waals, a carpenter, and Elisabeth van den Berg. Leiden had the oldest university in the Netherlands, but children of working families did not go there. Entry required the classical secondary school, the gymnasium, with its Latin and Greek, and van der Waals did not attend one.

He went instead to a school of advanced primary education, which he finished at fifteen. He then became a pupil-teacher, the usual path for a capable boy without means: he taught younger children while continuing his own studies. Between 1856 and 1861 he passed the examinations that qualified him first as a primary-school teacher and then as a head teacher. He worked in primary schools in these years.

The long apprenticeship in teaching left a mark. Colleagues and students later remembered him as a clear and impressive lecturer, and much of his scientific writing has the step-by-step character of a teacher explaining an argument to himself.

Teacher and outside student, 1862–1873

In 1862 van der Waals began to attend lectures in mathematics, physics and astronomy at Leiden University. Without a classical education he could not matriculate, so he came as an outside student, permitted to follow a limited number of courses each year. From 1863 he studied in his spare time for the examinations that licensed teachers of mathematics and physics in the new secondary schools, the hogere burgerscholen (HBS), created by a law of 1863.

He passed, and in 1865 became physics teacher at the HBS in Deventer. He married Anna Magdalena Smit around the same time; they had three daughters and a son. In 1866 he moved to an HBS in The Hague, close enough to Leiden to resume university study, and later became the school’s director.

The exemption

A doctorate still required Latin and Greek. When the law changed to allow the minister of education to grant dispensation from the classical languages for students of science, van der Waals obtained one. He passed the qualifying examinations in physics and mathematics and began a thesis under Pieter Rijke, professor of physics at Leiden.

The subject came from his reading. Rudolf Clausius’s 1857 paper on the kind of motion we call heat derived the gas laws from moving molecules. Pierre-Simon Laplace’s theory of capillarity, from 1806, had attributed the cohesion of liquids to short-range attractions producing a large internal pressure. In 1869 Thomas Andrews in Belfast had shown that carbon dioxide above about 31 °C cannot be liquefied by any pressure and that gas and liquid can be transformed into each other continuously. Van der Waals set out to explain Andrews’s observations with Clausius’s molecules and Laplace’s attractions.

The thesis and its reception, 1873–1877

On 14 June 1873 van der Waals defended Over de Continuïteit van den Gas- en Vloeistoftoestand (On the Continuity of the Gaseous and Liquid State) at Leiden. He was 35. The thesis introduced two molecular quantities, an effective volume of the molecules and the strength of their mutual attraction, and from them derived the equation of state that bears his name. Its central claim was in the title: the gas and the liquid are one fluid at different densities, governed by one equation.

The thesis was in Dutch, which few physicists abroad could read. James Clerk Maxwell read it. His review in Nature in 1874 was strongly favourable; he predicted that van der Waals would soon rank among the leaders of molecular science, while criticizing parts of the derivation. A German translation followed in 1881, and an English one in 1890.

Maxwell also improved the theory. In a lecture published in Nature in 1875 he showed how to choose the pressure at which liquid and vapour coexist on a van der Waals isotherm, the equal-area rule of Lesson 06. In 1875 van der Waals was elected to the Royal Netherlands Academy of Arts and Sciences.

In 1876 a new law on higher education raised Amsterdam’s Athenaeum Illustre to a municipal university. In September 1877 van der Waals became its first professor of physics. Among his new colleagues were the chemist Jacobus Henricus van ’t Hoff and the botanist Hugo de Vries. The schoolteacher without Latin had become a professor in one of the most productive faculties in Europe.

The equation of state

For the general reader. The ideal gas law treats molecules as points that never touch or attract. Real molecules take up room, and they pull on each other when close. Van der Waals added one correction for each effect. The result still fits on one line, yet it predicts that a gas, squeezed at a low enough temperature, turns into a liquid.

Excluded volume

Model molecules as hard spheres of diameter \(\sigma\). The centre of one molecule cannot come within \(\sigma\) of another’s centre, so each pair excludes a sphere of volume \(\tfrac43\pi\sigma^3\). Assigning half to each molecule, the excluded volume per mole is

four times the actual volume of the molecules, the factor van der Waals gave. The argument is valid at low density, where only pairs matter. He knew that \(b\) should decrease at liquid densities, and his Nobel Lecture of 1910 calls the search for its correct dependence on volume the hardest part of the theory.

Internal pressure

A molecule deep in the fluid is pulled equally in all directions. One near the wall is pulled back into the fluid, so it strikes the wall with less momentum. The reduction is proportional to the number of molecules near the wall and to the number pulling on each, hence to the square of the density \(1/V_m\). This gives the term \(a/V_m^2\), Laplace’s internal pressure in molecular form. In terms of a pair potential \(u(r)\) that is hard-core for \(r<\sigma\) and weakly attractive outside, the mean-field estimate is

The critical point

For the general reader. Andrews found that carbon dioxide has a temperature, about 31 °C, above which no pressure will turn it into a liquid. At exactly that temperature and a definite pressure, the liquid and its vapour become identical. The van der Waals equation produces such a point automatically and predicts where it lies from the two constants \(a\) and \(b\).

Critical constants

Written as a cubic in \(V_m\), \(pV_m^3-(pb+RT)V_m^2+aV_m-ab=0\). Below a certain temperature an isotherm has three real roots for a range of pressures; above it, one. At the critical point the three roots merge, so the isotherm has a horizontal inflection:

Derivation. The cubic must equal \(p_c(V_m-V_c)^3\). Comparing coefficients of \(V_m^2\), \(V_m\) and \(1\): \(3V_c=b+RT_c/p_c\), \(3V_c^2=a/p_c\), \(V_c^3=ab/p_c\). Dividing the third by the second gives \(V_c=3b\); then \(p_c=a/(3V_c^2)=a/(27b^2)\), and the first gives \(RT_c=p_c(9b-b)=8a/(27b)\).

The prediction \(Z_c=0.375\) is universal for all van der Waals fluids. Real fluids have \(Z_c\approx0.27\)–0.30 (argon 0.29, carbon dioxide 0.27, water 0.23). In his Nobel Lecture van der Waals quoted an experimental ratio near \(1/3.77\) and attributed most of the difference to the volume dependence of \(b\).

Near the critical point

In reduced variables, with \(t=1+\tau\) and \(v_r=1+\omega\), the equation becomes \(p_r=8t/(3v_r-1)-3/v_r^2\), and Taylor expansion gives

The absence of an \(\omega^2\) term and the cubic \(\omega^3\) are the algebraic signature of the critical point. The isothermal compressibility \(\kappa_T=-\frac1V(\partial V/\partial p)_T\) behaves as \(\kappa_T\propto1/(6\tau)\) on the critical isochore and diverges as \(\tau\to0^+\).

Loops, metastability and Maxwell’s rule

For the general reader. Below the critical temperature a van der Waals isotherm has a wiggle: over part of it, squeezing the fluid would lower its pressure, which is impossible for a stable state. Real fluids avoid the wiggle by splitting into liquid and vapour at a fixed pressure, the vapour pressure. Maxwell showed how to find that pressure from the curve itself.

Stability

Mechanical stability requires \((\partial p/\partial V)_T<0\). The points where \((\partial p/\partial V)_T=0\) form the spinodal. Between the two spinodal points the fluid is unstable and separates spontaneously. Between the spinodal and the coexistence curve it is metastable: superheated liquid or supersaturated vapour, which exist in practice until a nucleus of the other phase forms. Van der Waals recognized these metastable branches in the thesis.

Proof. Coexisting phases have equal temperature, pressure and molar Gibbs energy \(\mu\). Along an isotherm, \(d\mu=V\,dp\), so \(\mu_g-\mu_\ell=\int V\,dp\) along the loop. Integrating by parts, \(\int V\,dp=p_\sigma(V_g-V_\ell)-\int_{V_\ell}^{V_g}p\,dV\), which vanishes exactly when the areas are equal. Equivalently, the Helmholtz energy \(F(V)\) has a common tangent at \(V_\ell\) and \(V_g\), and the tangent’s slope is \(-p_\sigma\). The loop itself is an artefact of assuming a uniform density; Mark Kac, George Uhlenbeck and Per Hemmer (1963), and Joel Lebowitz and Oliver Penrose (1966), proved that a model with weak, long-range attractions gives the van der Waals equation with Maxwell’s construction exactly.

Corresponding states and liquefaction

For the general reader. If every fluid obeys the same equation once pressure, volume and temperature are measured in units of their critical values, then knowing one fluid tells you about all the others. Van der Waals stated this in 1880. Low-temperature physicists used it to predict how cold they would need to go to liquefy hydrogen and helium.

The law does not depend on the precise form of the van der Waals equation. Statistical mechanics shows it holds exactly for any set of substances whose molecules interact by a pair potential of the same shape, \(u(r)=\varepsilon\,\phi(r/\sigma)\), with only \(\varepsilon\) and \(\sigma\) differing. In 1945 Edward Guggenheim plotted reduced coexistence curves for neon, argon, krypton, xenon, nitrogen, oxygen, carbon monoxide and methane and found them on a single curve, though with \(\rho_\ell-\rho_g\propto(1-t)^{1/3}\) rather than the van der Waals exponent \(\tfrac12\) (Lesson 10). Quantum effects make helium and hydrogen deviate.

Virial coefficient and the Boyle temperature

Expanding at low density, \(pV_m/RT=1+B(T)/V_m+\cdots\), the van der Waals equation gives

\(B\) vanishes at the Boyle temperature \(T_B=a/(Rb)=\tfrac{27}{8}T_c\), where the gas is nearly ideal over a range of pressures.

Joule–Thomson inversion

A gas expanding through a throttle at constant enthalpy cools if the Joule–Thomson coefficient \(\mu_{JT}=(\partial T/\partial p)_H\) is positive. At low pressure \(\mu_{JT}\approx\frac{1}{c_p}\big(T\frac{dB}{dT}-B\big)=\frac{1}{c_p}\big(\frac{2a}{RT}-b\big)\), which is positive below the inversion temperature

A gas must be precooled below its inversion temperature before throttling liquefies it. Hydrogen’s inversion temperature is about 200 K, so it had to be precooled with liquid air; James Dewar liquefied it in 1898. Helium’s is a few tens of kelvin, so Heike Kamerlingh Onnes at Leiden precooled it with liquid hydrogen. Guided by corresponding-states estimates of its critical point, he liquefied helium on 10 July 1908. The presentation speech for van der Waals’s Nobel Prize in 1910 named that achievement as a success of his theory.

Binary mixtures and the ψ-surface

For the general reader. Most fluids in nature and industry are mixtures. A mixture can separate into two liquids, boil over a range of temperatures, or behave strangely near its critical point, condensing when the pressure is lowered. In 1890 van der Waals extended his equation to mixtures of two substances and showed that all of this follows from the geometry of one free-energy surface.

Mixing rules

For a mixture with mole fraction \(x\) of component 2, van der Waals treated the mixture as a single fluid with

quadratic in composition because both terms come from pairs of molecules. The cross parameters are often taken as \(a_{12}=\sqrt{a_{11}a_{22}}\) and \(b_{12}=\tfrac18\big(b_{11}^{1/3}+b_{22}^{1/3}\big)^3\). These one-fluid rules are still the basis of the cubic equations of state used in chemical engineering.

The ψ-surface

At fixed temperature, van der Waals plotted the molar Helmholtz energy \(\psi(V_m,x)\) of the uniform mixture as a surface over the \((V_m,x)\) plane, adding the ideal entropy of mixing \(RT[x\ln x+(1-x)\ln(1-x)]\) to the van der Waals free energy \(-RT\ln(V_m-b_x)-a_x/V_m\) and a function of \(T\) and \(x\).

Rolling a tangent plane across the surface traces pairs of coexisting states; where the two tangent points merge, the mixture has a critical point, which Josiah Willard Gibbs called a plait point. Van der Waals’s theory accounts for liquid–liquid immiscibility, azeotropes and the retrograde condensation observed experimentally in the 1890s. Philip Kohnstamm’s experiments and the two-volume Lehrbuch der Thermodynamik (1908–1912) developed the programme further.

Capillarity and the diffuse interface

For the general reader. Laplace pictured the surface of a liquid as a sharp boundary. Van der Waals argued in 1893 that the density changes smoothly across a thin layer, and that the surface tension is the extra free energy stored in that gradual change. Near the critical point the layer becomes thick and the tension vanishes. The same idea was rediscovered in 1958 by John Cahn and John Hilliard and underlies modern phase-field models.

Derivation. The Euler–Lagrange equation \(m\rho''=W'(\rho)\) has the first integral \(\tfrac m2\rho'^2=W(\rho)\), since both sides vanish in the bulk phases. Then \(\Omega=\int2W\,dz=\int m\rho'^2dz\), and changing variable from \(z\) to \(\rho\) with \(dz=d\rho/\rho'\) gives the integral over \(\rho\).

Near \(T_c\), \(W\) takes the double-well form \(W\approx A(\rho-\rho_g)^2(\rho-\rho_\ell)^2\), with \(\Delta\rho=\rho_\ell-\rho_g\propto(1-t)^{1/2}\). The profile is

and the surface tension is \(\gamma=\sqrt{2mA}\,(\Delta\rho)^3/6\propto(1-t)^{3/2}\). Van der Waals obtained both exponents. Experiment gives about \(-0.63\) and \(1.26\): the same pattern, with the different exponents of Lesson 10.

Van der Waals forces and mean-field theory

For the general reader. Van der Waals did not know what causes the attraction between neutral molecules. Twentieth-century physics found three sources, all now called van der Waals forces. His way of handling them, by replacing the pull of all neighbours with an average, became the first mean-field theory. It is qualitatively right everywhere and quantitatively wrong only very near the critical point, which is why it is still taught first.

Molecular forces

Between neutral molecules at distance \(r\), the long-range attraction has free energy proportional to \(-C/r^6\), with three contributions: thermally averaged dipole–dipole interactions (Willem Hendrik Keesom, 1921), dipole–induced-dipole interactions (Peter Debye, 1920), and dispersion forces from correlated quantum fluctuations of electron clouds (Fritz London, 1930). For nonpolar molecules such as argon, only London’s term exists. In the Lennard-Jones potential \(u(r)=4\varepsilon\big[(\sigma/r)^{12}-(\sigma/r)^6\big]\), the \(r^{-6}\) tail is the van der Waals attraction.

Critical exponents

Measured values for simple fluids are \(\beta\approx0.326\), \(\gamma\approx1.24\), \(\delta\approx4.8\), the same as for the three-dimensional Ising model. Mean-field theory neglects the large fluctuations in density near \(T_c\), the cause of critical opalescence. The renormalization-group theory of Kenneth Wilson in the early 1970s explained why fluids and magnets share the same non-classical exponents and why van der Waals’s values become exact above four dimensions or for interactions of very long range, as in the Kac–Uhlenbeck–Hemmer limit of Lesson 06.

The Amsterdam school, 1877–1908

Van der Waals taught at Amsterdam for thirty-one years. With van ’t Hoff in chemistry, de Vries in botany and the mathematician Diederik Korteweg, the young university became known abroad. Korteweg took his doctorate in 1878, the first awarded by the new university, with van der Waals as advisor. Among van der Waals’s later doctoral students were Philip Kohnstamm (1901), who became his closest collaborator, and Willem Hendrik Keesom (1904), whose thesis measured isotherms of mixtures of oxygen and carbon dioxide and who later worked at Leiden on low temperatures.

His wife died of tuberculosis in 1881, at 34. He did not remarry. His eldest daughter, Anne Madeleine, ran the household; Jacqueline Elisabeth became a history teacher and a known poet; Johanna Diderica taught English. His son, Johannes Diderik van der Waals Jr., became professor of physics at Groningen in 1903.

The great papers of these years were the law of corresponding states (1880), the theory of binary mixtures (1890) and the theory of capillarity (1893). He kept close contact with Heike Kamerlingh Onnes at Leiden from the 1880s, and Leiden’s cryogenic laboratory became in part an experimental test of his theory. His work was taken up abroad: the second volume of Ludwig Boltzmann’s Vorlesungen über Gastheorie (1898) gave a detailed kinetic treatment of van der Waals’s equation. From 1896 to 1912 he served as secretary of the Royal Netherlands Academy. He retired in 1908, at seventy, and his son succeeded him in the Amsterdam chair.

Nobel Prize, last years, legacy

The 1910 Nobel Prize in Physics went to van der Waals for his work on the equation of state for gases and liquids. He was 72. In his Nobel Lecture of 12 December 1910 he reviewed the theory’s history candidly, describing the open problem of the volume dependence of \(b\) and calling one of his own lines of argument a detour. He also discussed what he called pseudo-association, transient groups of molecules held by ordinary molecular forces, as a way of explaining the deviations near the critical point.

With Kohnstamm as editor he published the Lehrbuch der Thermodynamik (1908–1912). His daughter Jacqueline died in 1922. Van der Waals died in Amsterdam on 8 March 1923, aged 85. Among his honours were an honorary doctorate from Cambridge, foreign membership of the United States National Academy of Sciences (1913) and corresponding membership of the Institut de France and the Berlin Academy.

Legacy

• Equations of state: the cubic equations used to design every gas-processing and refrigeration plant, such as Redlich–Kwong and Peng–Robinson, are modified van der Waals equations.
• Phase transitions: the idea that a single equation can describe two phases, a critical point and metastable states; the first mean-field theory, the model for Pierre Weiss’s theory of ferromagnetism and Lev Landau’s general theory.
• Interfaces: the square-gradient theory of capillarity, revived as Cahn–Hilliard theory and the basis of phase-field methods.
• Molecular forces: the attraction between neutral molecules, from the cohesion of liquids to gecko adhesion and layered materials held by van der Waals bonding.
• Low temperatures: the law of corresponding states guided the liquefaction of hydrogen and helium.

Connections

People linked to Johannes Diderik van der Waals across this collection. A name without a link has no page here yet. See Johannes Diderik van der Waals in the network of all 337 lives →

Teachers and advisors

• Pieter Rijke 1873

Students

• Diederik Korteweg 1878
• Philip Kohnstamm 1901
• Willem Hendrik Keesom 1904

Influenced by

• James Clerk Maxwell 1874-1875
• Josiah Willard Gibbs
• Pierre-Simon Laplace
• Rudolf Clausius
• Thomas Andrews

Influenced

• Heike Kamerlingh Onnes

Colleagues and collaborators

• Hendrik Lorentz
• Jacobus Henricus van 't Hoff 1878-1896
• Ludwig Boltzmann

Family

• Johannes Diderik van der Waals Jr.

Also mentioned in this life

• Robert Boyle
• Peter Debye
• Leonhard Euler
• Hermann von Helmholtz
• Christiaan Huygens
• James Prescott Joule
• Joseph-Louis Lagrange

Fields

• thermodynamics 12 lives
• statistical mechanics 8 lives

On CoursesHub

• Thermodynamics
• Statistical Mechanics

Sources and further reading

Biography

1. Nobel Prize, Johannes Diderik van der Waals: Biographical, and Award ceremony speech, 1910.
2. Johannes Diderik van der Waals, Wikipedia.
3. Johannes Diderik van der Waals, Knowino (P. Wormer).
4. University of Amsterdam Album Academicum, P. A. Kohnstamm.
5. Huygens Institute / KNAW, W. H. Keesom (biography).
6. A. Ya. Kipnis, B. E. Yavelov, J. S. Rowlinson, Van der Waals and Molecular Science, Clarendon Press, 1996.

Physics

1. J. D. van der Waals, Over de Continuïteit van den Gas- en Vloeistoftoestand, Leiden, 1873; English edition: On the Continuity of the Gaseous and Liquid States, ed. J. S. Rowlinson, North-Holland, 1988.
2. J. D. van der Waals, The equation of state for gases and liquids, Nobel Lecture, 12 December 1910.
3. J. C. Maxwell, Van der Waals on the continuity of the gaseous and liquid states, Nature 10 (1874); On the dynamical evidence of the molecular constitution of bodies, Nature 11 (1875).
4. T. Andrews, On the continuity of the gaseous and liquid states of matter, Phil. Trans. R. Soc. 159 (1869).
5. J. S. Rowlinson, Translation of J. D. van der Waals’ “The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density”, J. Stat. Phys. 20 (1979).
6. J. S. Rowlinson and B. Widom, Molecular Theory of Capillarity, Clarendon Press, 1982.
7. M. Kac, G. E. Uhlenbeck, P. C. Hemmer, On the van der Waals theory of the vapor–liquid equilibrium, J. Math. Phys. 4 (1963).
8. E. A. Guggenheim, The principle of corresponding states, J. Chem. Phys. 13 (1945).
9. Van der Waals equation, Wikipedia.