Van der Waals in his time and the present revival opening address

Physica 73 (1974) l-27

VAN

DER

0 North-Holland

WAALS

Publishing Co.

IN HIS TIME OPENING

AND

THE

PRESENT

REVIVAL

ADDRESS

J. DE BOER Institute for Theoretical Physics, University qf Amsterdam, Amsterdam, The Netherlands

Synopsis In this opening address a survey is given of the influence which the scientific work of Van der Waals, concerning the equation of state of gases and liquids, the theory of mixtures, the principle of corresponding states and the theory of capillarity, had on the scientific development. In particular attention is paid to the position which his theory of condensation and of the critical point has in the modern theory of phase transitions.

1. Introduction. On the 14th of June 1873 Johannes Diderik van der Waals defended his thesis “On the continuity of the gaseous and the liquid state”“Over de continuiteit van den gas- en vloeistoftoestand”-at the University of Leiden. Today, one hundred years later, we start this Van der Waals’ Centennial Conference on Statistical Mechanics to celebrate this event, by trying to overlook the present status of the field of physics which was advanced and stimulated so much by Van der Waals. It seems appropriate to start this morning with considerations of a somewhat more historical nature which have the aim to place Van der Waals in the scientific context of his time and to sketch the present time developments which can be considered to have their origin in his ideas. I will touch very briefly the historical frame work, because they will be discussed in very much more detail by Professor Klein. At the time that Van der Waals got interested in the subject of his thesis, already considerable insight had been obtained in the properties of gases and liquids by the many investigations of Regnault, James Watt, Cagniard de la Tour, Joule, Andrews and James Thomson. The kinetic theory of gases had been given already a definite form by Clausius and by Maxwell. Clausius had published in the fifties his important articles: “On the kind of motion which we call heat”l) and “On the mean length of the path described by separate molecules of gaseous bodies”2), giving the foundation of the kinetic 1

IN HET OPENUAAR TE VERDEDICEN

GEBOREK

TE LEIDEN.

LElDEN, A.

W.

SJJTHOFF. $873,

Fig. 1. Cover page of thesis of Van der Waals.

VAN DER WAALS IN HIS TIME AND THE PRESENT

REVIVAL

3

theory of heat and OSthe heat conductivity in gases. Maxwell published in 1860 his: “Illustrations of the dynamical theory of gases”3), in which he gave the kinetic theory a statistical basis and derived the velocity distribution function. 2. The equation of state - 1873. With his derivation of the equation of state, Van der Waals really made a consistent extension of the kinetic theory of gases to the field of compressed gases and liquids, where the molecular collisions are no more incidental, but where, on the contrary, the molecular interaction determines the macroscopic behaviour. Basic were Van der Waals’ assumptions about the intermolecular forces: the molecules were assumed to be hard spherical bodies which exercise distant attractive forces on each other, having a range of a few molecular diameters and being the origin of what Van der Waals called the cohesive pressure holding liquids together against the heat motion of the molecules. In the calculation of the equation of state the two aspects of the interaction: the contact repulsive forces and the distant attractive forces were treated in a different way. Van der Waals chose as starting point the virial theorem, published just a short time before by Clausius4) in 1870, according to which for a system of particles the average kinetic energy equals minus one half of the average virial of the internal and external forces P(ui) acting on the system of particles with coordinates vi, velocities ui and masses mi.

(1) The average kinetic energy was assumed to be equal to +RT, also in the liquid state, an assumption which Maxwell in his review of Van der Waals’ thesis in Nature still called “somewhat too hasty” and the virial of the external forces was calculated to be -3pV, so that the virial theorem may finally be written as $RT = SpV + $(I: riif(rij)).

(2)

The attractive forces are accounted for by the second term on the right-hand side, in which the summation has to be carried out over all pairs of molecules with mutual distance rij and attractive force f (rid). This represents the virial of the cohesive pressure, now usually called internal pressure. With a very general argument - not depending on the way in which f (r) depends on r - Van der Waals showed that the internal pressure is proportional to the square of the density and can therefore be written in the form pi = a/V. So if the effect of the spatial extension of the molecules is neglected one finds with Clausius’ virial theorem

RT=

(p +pi)v.

The repulsive forces were assumed only to affect the so-called kinetic or thermal pressure RT/ I/ and not the internal pressure pi. The effect of the molecular volume on this kinetic pressure was obtained from free-path arguments. The result was, quoting Van der Waals5), “that up to certain degree of compression the external

4

J. DE BOER

volume must be diminished by four times the molecular volume and for larger compression with a further diminishing multiple of this volume”. In this way Van der Waals arrived at his equation of state ---RT p=JLb

a V2’

where b is written for 4 times the molecular volume. It may be important to stress once more that the expression RT/(V - b) was considered to be only an approximation by Van der Waal@) only valid for V = 2b. In fact he continued for many years to search for a better expression for this effect of the volume of the molecules. It is important to note the very different treatment which is given by Van der Waals to the long-range attractive forces and the short-range repulsive force. Only the attractive fbrces are taken into account in the evaluation of the virial, whereas the repulsive forces are considered to affect only the kinetic- or “thermal”pressure RT/(V - b) in the equation of state. Lorentz’) was in favour of treating both repulsive and attractive forces in the same way in the evaluation of the virial, having the result that RT/(V - b) is replaced by RT(I/V + b/V2). Lorentz’ point of view was considered to be more logical or systematic, in particular when later on all attention focussed on the series expansion of the pressure in powers of the density. Only in the last decennium the very fundamental importance of treating short-range repulsive and long-range attractive forces in a different way has become clear and has contributed to a revival of Van der Waals’ ideas, as will be seen in section 7. It was a great satisfaction for Van der Waals that his equation, being of the third order in V, gives for a givenp at temperatures below the critical temperature T = T,, three solutions for V. The critical point at which the three solutions for V coincide, is given by RT c* =*

27b ’

v,, = 3b,

pm = &

.

(5)

He remarked that according to Maxwell’s book “Theory of Heat”8), which booklet - as he writes - “is certainly in hands of every physicist”, already in 1871 James Thomson had had the bright ideas) to connect both experimental parts of the isotherm with a loop in the coexistence region, following the Bakerian lecture which Andrews’O) gave in 1869, called “On the continuity of the gaseous and liquid states of matter” and in which he presented very precise measurements of the isotherms of CO2 near what he called the first time the “critical point”. Van der Waals was satisfied that his equation of state shown in fig. 2 gave for the first time a theoretical justification for this loop connecting the gas and liquid state. Two years later Maxwell proved that the vapour pressure at coexistence at each temperature T < T,, is determined by the equal area rulell). With his thesis Van der Waals had shown that it was possible to extend the kinetic theory of gases, using the molecular hypothesis, from the dilute gas to the condensed state: Van der. Waals equation of state showed that both the gaseous and the liquid state can be described by one and the same equation, covering the

VAN DER WAALS IN HIS TIME AND THE PRESENT

REVIVAL

5

-\(9 ~7 \

\

0.

4

JZ.&

3,’

‘?. ,

*

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‘...

‘... l.

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.... .... .... ‘...

x.

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-..,

1%.

Fig. 2. Reproduction of three figures from the thesis of Van der Waals. Left: theoretical isotherms of Van der Waals; lower right: instable part in isotherms according to J. Thomson (ref. 9); upper right: experimental isotherm according to Andrews (ref. 10).

whole field of gases, compressed and condensed gases and liquids, and allowing for a continuous transition between the one and the other. This was a great break through, the importance of which is difficult to overestimate in a time in which even the whole molecular picture was still under discussion. 3. The law of corresponding states - 1880. In 1880 appeared Van der Waals’ famous publication12) on the law of corresponding states. Starting point was his conviction that vapour pressure curves for all substances should be representable

J. DE BOER

6

by one and the same equation, apart from the constants which should depend on the constants a and b of these substances. In order to derive such a general equation for the vapour pressure, Van der Waals started to introduce “reduced” quantities Z- = pIpCr, p = V/VO,, 8 = T/TOP by dividing pressure, volume and temperature by the respective critical quantities. States of two substances which were characterized by the same values of the reduced pressure, volume and temperature, were called “corresponding” states. In terms of these reduced quantities the equation of state took the form of a law of corresponding states: n=p--

8 3g,--1

3 912’

(6)

From this so-called “reduced isotherm” Van der Waals then showed how to derive in principle a reduced vapour-pressure curve rrs = p,/pCr = n,(e) showing e.g. that for the two substances at corresponding temperatures the vapour pressures are proportional to the respective critical pressures. Similarly Van der Waals proved that for two liquid substances at corresponding temperatures the thermal expansion coefficients ale inversely proportional to the respective critical temperatures. In this way Van der Waals showed how to use this principle to calculate unknown quantities for a particular substance from those of any other substance. An interesting question remained about the influence of very probable corrections on the equation of states for values of V < 26 on the validity of the law of corresponding states. Van der Waals expected that the law most probably would survive, provided that “the changes for different substances would act in a corresponding way”. In fact it is true that replacement of the expression l/(V - b) by any other function of V and b would not affect the law and it was about this kind of correction that Van der Waals probably was thinking. Kamerlingh Onnes got very much interested in the law of corresponding states. Not only was this law found to be very useful for practical purposes in his research to extend physics to lower and lower temperatures, but also did he succeed one year afteP) Van der Waals’ publication to give the law a much more general foundation. In modern language one might formulate Kamerlingh Onnes’ approach as follows: the molecular interaction is assumed to be given by two constants, a diameter c and a force constant K, such that the force between two molecules on a distance r becomes f(r) = Kv(r/o). Thus for each substance it is possible to choose a characteristic mass, length and time, i.e. the molecular mass mi, the diameter ci and the time -ri = \/(mici/Ki). For every substance Kamerlingh Onnes defined in terms of these characteristic base units “corresponding states of motion”, being states of motion of the whole system scaled according to these “molecular” base units mi, CJ~and TV. Then for thermodynamic systems, assuming the temperature to be proportional to the average molecular kinetic energy, corresponding molar volumes, pressures and temperatures will have to be scaled according to the appropriate “molecular” units o!, m,/a,$ and rniaf/Tz. Thermodynamic systems for different substances, each being in the critical state, are just examples of systems in such corresponding states. The critical parameters may therefore be used to replace the molecular units in order to define again corresponding states,

VAN DER

WAALS

IN HIS TIME

AND

THE

PRESENT

REVIVAL

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but now scaled according to the critical quantities as was done by Van der Waals. In this way Kamerlingh Onnes formulated his law of corresponding states in the form : 7? = ?i-(pl, e>

(7)

as a general law, no longer connected with the particular formula of Van der Waals. In fact Kamerlingh Onnes discovered that nature satisfied very much more the law of corresponding states in this general form than that particular mathematical relation which was derived from Van der Waals’ equation. In order not to be bound to one particular mathematical form for the equation of state Kamerlingh Onnes proposed 1ateP) - in 1901 - a general series expansion of the virial in powers of the density

B(T) -+v

C(T) v2

+...

1 ,

where the “virial coefficients” B(T), C(T), etc., are considered to be empiric functions of temperature. The principle of corresponding states then has the consequence that the “virial coefficients” take the form

B(T) = v,,K% C(T) = f%@> which was confirmed for many substances. The formulation by Van der Waals of the law of corresponding states had an enormous influence on the experimental research exploring the field of physics at lower temperatures, closely connected with attempts to liquefy the permanent gases. Thus Dewar, who liquefied hydrogen in 1898, said in his lecture for the British Association in 1902: “It is perhaps not too much to say that as a prolific source of knowledge . . . it would be necessary to go back to Carnot’s cycle to find a proposition of greater importance than the theory of the law of corresponding Onnes wrote in 1910, two years after succeeding to states”, and Kamerlingh guides for a liquefy helium, “Van der Waals theories have been indispensable large part of the research in the Leiden Laboratory. For everybody who overlooks the work in this laboratory, it will be clear that the ideas of Van der Waals were always considered as magic tools and that the cryogenic laboratory has developed Onnes conunder influence of his theories”, and in his Nobel lecture Kamerlingh fessed : “I was really happy to be able to show liquid helium to my respected friend Van der Waals, whose theory in the process of liquefying this gas, has been a guide up to the very end”. 4. The theory of mixtures - 1891, On the 23rd of February 1889 Van der Waals communicated at a session of the Royal Netherlands Academy of Sciences the broad lines of his theory of mixtures. The full article “Molecular theory of a system appeared in 1891 in the Archives Neercomposed of two different substances” landaises15). This theory of mixtures was based again on his expression for the equation of state in which, however, now the two constants a and b were assumed

8

J. DE BOER

to depend quadratically to the formulae

on the concentration

x specifying

the mixture,

according

a(x) = a,,(1

- x)” + 2a&l

- x) + Q2XZ,

(10)

b(x) = b,,(l

-

-

(11)

x)” + 2bl,X(l

x) + &.,X2,

where a,,, b,, and az2, bzz are the constants for the pure substances 1 and 2 and aI2 and b,, characterize the mixture. For b,, the simple arithmetic mean of the b-constants of the pure substances: b,, = i(b,, + b,,) allows b(x) to be written in the form of a linear relation b(x) = b,r(l

- x) + b,,x

(12)

but Lorentz7) had shown that the relation bt, = $(b$, + bi,), corresponding to the arithmetic mean of the diameters of the pure substances: o12 = &(oll + a.& represents a much more natural assumption. The equation of state for the mixture then becomes RT AT,

V, x) =

V -

a(x) b(x) - 7 ’

(13)

In order to decide which parameters couples V and x at a given temperature correspond to stable equilibrium of a homogeneous phase and which values lead to a separation into two or more phases, Van der Waals made use of the thermodynamical methods developed by Gibbs in the preceding fifteen years. It proved to be convenient to use (Helmholtz’) free energy which, following GibbP), was denoted by y(T, V, x). By integration y(T, V, x) was found to be y(T,

V, x) = -RTln +RT[x

[I’-

b(x)] -

In x + (1 -

a(x)/V x) In (I -

x)].

(14)

This expression was the starting point for his famous studies of the y-surface for a given temperature, as a function of V and x, in order to determine the behaviour of mixtures, in particular the conditions for the system to separate into two different phases. In analogy to the case of a simple substance where stable equilibrium requires the y(V) curve to have positive curvature as a function of V, the y( V, x) surface in the case of a binary mixture has to have positive curvature in all directions in the V, x-plane. Stable and unstable regions are separated by the so-called “spinodal” curve, defined by the condition

which on the y(V, x)-surface is the analogy of the inflection points a2y/aV2 = 0 An unstable region with negative in the y(V)-curve for a simple substance. curvature gives rise to a “plait” or “fold” (“Falte” in German) on the y-surface.

VAN DER WAALS IN HIS TIME AND THE PRESENT REVIVAL

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Within and in the close neighbourhood of such a plait the free energy can be minimized by splitting the system in two or more coexisting phases. The equilibrium conditions between the two coexisting phases read:

(2X= (3 ek=(Z), aY aY y-xax-v3 av 1 ax av 1 ( ( aY y--x--I/-

(16)

=

1

2

expressing equality of pressure and of the thermodynamic potentials for the two components. Consequently in order to find then the parameters for these coexisting phases a double tangential plane has to be constructed with two points of contact 1 and 2 specifying the two coexisting phases VI, x1 and V,, x2 connected by a double tangent. Following Cayley (1852) the points of contact were called “nodes” and by rolling the tangential plane over the y-surface, the pairs of nodes describe the “binodal”- or also “connodal’‘-curve, which separates on the y-surface the region where phase separation occurs from the homogeneous region. The plait may end in a “plaitpoint”, where the two points of contact coincide and which marks the end of the region of phase separation on the y-surface. These investigations of Van der Waals were very much stimulated by the strong interest which his colleague, the mathematician Korteweg, got in the general mathematical behaviour of plaits and plaitpoints on surfaces. Korteweg publishedl’) in the same year 1891 two interesting investigations on this subject, in which he studied their geometrical properties and systematized the various types of plaitpoints that may occur. The second article was specially devoted to plaits on Van der Waals’ y-surface. Van der Waals distinguished between two different cases, which correspond to very different behaviour of actual systems. In the jirst case there exists only a “transversal” plait on the y-surface, roughly parallel to the x-axis. The phase separation caused by such a plait corresponds to the gas-liquid phase separation for a pure substance. Obviously the plait will not extend at all temperatures over the whole region x = 0 to x = 1. When the temperature arises above the critical point of substance 1 the transversal plait ends in a plaitpoint. When the temperature is changed this plaitpoint moves along a critical curve, the “plaitpoint curve”, which connects the critical points of the two pure substances 1 and 2. This critical curve may of course have very different forms corresponding to the many different situations which are found for actual systems in nature, but this is not the place to discuss these in more detail. The second case, which is often realized in nature, is that in which also a “longitudinal” plait, roughly parallel to the V-axis occurs. In this case, in which in a certain region a2y/ax2 becomes negative, there exists a certain immiscibility of the two components, such that the system separates into two phases having very different concentration. Van der Waals studied in particular the situation which occurs when the longitudinal immiscibility plait penetrates into the transversal

10

J. DE BOER

gas-liquid plait. Then there exists the possibility of a three phase equilibrium between one gaseous and two liquid phases having different concentrations. Van der Waals investigated on the basis of the y(T, V, x) expression following from his equation of state under which conditions for a(x) and b(x), the plaits of various types may develop and to which p-x and p-T diagrams they give rise. In general Van der Waals assumed b(x) to be linear and if we limit ourselves here to mixtures of components with similar gas-liquid critical temperatures, it can be said that a longitudinal immiscibility plait only exists for small values of Q. No such plait will exist when a 12 > $(a,, + a,,), so that in that case the picture is dominated entirely by the transversal gas-liquid plait, which develops at low temperatures. It may be of historical interest to reproduce in fig. 3 a classical picture, a drawing of a model of a v-surface, which occurs in Van der Waals’ original articleP) and which shows from below a y-surface and the developable tangential surface

Fig. 3. Reproduction of drawing representing a p-surface with longitudinal and transversal plait showing three phase (liquid-liquid-gas) equilibrium given by Van der Waals (ref. 18).

VAN DER WAALS IN HIS TIME AND THE PRESENT

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Fig. 4a, b and c. Pictures of wooden w surfaces with double tangents used at lectures by Van der Waals. (a) Partial transverse plait. Temperature between the critical temperatures of the components. (b) Full transverse plait. Temperature lower than the critical temperature of either component. (c) Full transverse plait and second plait. Temperature lower than the critical temperature of either component; two liquid phases.

J. DE BOER

12

corresponding to this particular case. The y-surface shows in particular the crossing of a transversal and longitudinal plait and the coexistence between three phases resulting from this. In fig. 4 we give pictures of wooden models of v-surfaces used by Van der Waals. Van der Waals’ investigations on binary mixtures had also a tremendous impact on the experimental work. They made it possible to understand in principle a great variety of phenomena which had already been observed, and it stimulated very much new research on the critical properties of mixtures. In particular Kamerlingh Onnes and his co-workers started a very extensive programme on mixtures; the results were interpreted in terms of Van der Waals’ theoretical models and unexpected new phenomena excited Van der Waals to further investigations. A large group of young scientists such as Kuenen, Keesom, Verschaffelt, Reinganum, Hartman, a.o. concentrated in their doctoral thesis work in Leiden on binary mixtures of relatively simple substances making in principle a quantitative comparison with Van der Waals’ theory possible. As the mathematics ofdeterminingthe binodal curve on the y-surface was cumbersome Kamerlingh Onnes started for each mixture with constructing plaster models of the actual y-surface, such that the densities and concentrations of the coexisting phases could be determined on these models with geometric means. One of these models (fig. 5) gives a nice example of the transversal plait, found in the mixtures of CO, and CH,CI on which Kuenenlg) made in 1892 the famous discovery of the phenomenon of retrograde condensation. The results as presented by Kamerlingh OnneszO) are given

Fig. 5. Picture

of plaster

model of y surface made by Kamerlingh CO,-CH,CI. Comm. Leiden 59a (1900).

Onnes for the system

VAN DER WAALS IN HIS TIME AND THE PRESENT

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Fig. 6. Original drawing of Kamerlingh Onnes?) giving the projection of isochores y-x plane for the model of fig. 5 (the dotted curves are ay/ax = const.).

13

on the

J. DE BOER

6

Fig. 7. The development of a longitudinal plait according to Korteweg”) in a theoretical symmetrical model a,, = uz2, uI2 < (19/27)a,,, b,, = b,, = b,, from low to high temperatures.

VAN DER WAALS IN HIS TIME AND THE PRESENT REVIVAL

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in fig. 6 giving the projection of the ~-surface in the critical region on the y-x-plane, with the isochores. An example of the behaviour of a system with a longitudinalplait, can be found in the original publication of Kortewegl’) in 1891 for the so-called symmetrical model in which a,, = az2 = a and b,, = b,, = b,, = b. In this model, coming from high temperatures, a longitudinal plait starts to develop when the temperature falls below a certain “critical temperature for complete miscibility” r, = (I - K)a/bR which for values of K = alz/a < 19/27 is higher than the critical temperature T,,= 8 a/27bR of the pure components. Below this temperature a phase separation occurs of two phases having different concentrations and showing a certain immiscibility of the two components, which increases when the temperature falls. At a lower temperature a three phase liquid-liquid-gas equilibrium becomes possible, which continues to exist to the lowest temperatures. The phenomenon of a separation into two fluid phases well above the critical pressure-in what is usually called the gas phase - was predicted by Van der Waalszl) in 1894 and by Kamerlingh Onnes and Keesomz2) in 1907 who coined the word gas-gas immiscibility for this phenomenon and predicted this in particular for very unsymmetrical mixtures in which one of the components has a very low critical temperature. Van der Waals distinguished between two types of gas-gas immiscibility, as shown in the p-T diagrams of fig. 8. In gas-gas immiscibility of the second kind the critical curve goes through a minimum corresponding to a “double plait point” which does not exist in the gas-gas immiscibility of the first kind. It would last about fifty years until a Russian physicist, Krichevskii23) in 1941 discovered this so-called gas-gas immiscibility of the first kind in mixtures of N, and NH,. After the war a Russian group under Krichevskii and Ciklis24) has found about 20 more binary systems exhibiting this behaviour. In the last decennium also other groups got interested in the subject, in particular also a group under Trappeniers and Schouten at the Van der Waals Laboratory. As an example of the first kind of gas-gas immiscibility I present the measurements of De Swaan Arons and Diepen25) on the He-Xe mixtures in fig. 19 and as an example of the second kind the measurements of Schouten and Trappeniers?) on Ne-Kr mixtures in fig. 10. I have given much attention to this work of Van der Waals, because I think it is a piece of work of great beauty, which developed in a relatively short period as a result of very fruitful collaboration between theoreticians and experimentalists. Van der Waals appreciated this collaboration with Kamerlingh Onnes very much as is made very clear in the dedication to Kamerlingh Onnes of the second part of his book: “Die Continuitat des gasfiirmigen und fliissigen Zustandes” which appeared in 1900 and in which he writes : “. . . Ich erachte es als eine Pflicht der Rechtfertigkeit, und zugleich die Befriedigung eines Dranges der freundlichen Gefiihle, welche mich mit Ihnen verbinden, mit dieser Widmung auszusprechen, dass die gliickliche Zusammenwirkung zwischen Theorie und Experiment, von der die nachfolgenden Seiten ein Zeugnis sind, in erster und vorztiglichster Stelle Ihnen zu verdanken ist”.

16

J. DE BOER

04

Fig. 8. P-T curves showing gas-gas immiscibility of first (a) and the second kind (b) according to Van der Waals. F and G are the critical points of the pure substances. G-E is the start of the gas-liquid critical line, interrupted by E-D the liquid-liquid-gas three-phase equilibrium line. D-H-F is the plait point curve for the liquid-liquid or gas-gas equilibrium. The figure is reproduced from De Swaan Arons and Diepen ref. 25, fig. 4, p. 2324.

Van der Waals’ activities stimulated also very much the research in this field at the University of Amsterdam, where he occupied the first physics chair since 1877. He was there together with the Nobel-laureates the chemist Van ‘t Hoff and the biologist Hugo de Vries. In 1896 Bakhuis Roozeboom, and two years later Van Laar, came to Amsterdam and in particular Van Laar has contributed enormously to the further development and the diffusion of Van der Waals’ theories in physical chemistry. Of particular importance has been for Van der Waals the close collaboration with his colleague Ph. Kohnstamm. This resulted in the wellknown “Lehrbuch der Thermodynamik”57) m t wo in 1908 in which on thermodynamics, on the in particular on mixtures in a systematic way. 5. Theory of capillarity - 1893. When in 1908 Van der Waals retired at the age of 70, his pupils and colleagues paid homage to him and presented as a tribute to his great discoveries a tablet in the wall of the physics auditorium on which were perpetuated four of his great discoveries: The equation of state, the law of corresponding.states, the theory of binary mixtures and the thermodynamic theory of capillarity.

VAN DER WAALS IN HIS TIME AND THE PRESENT

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Palm

I T

1600 -

BOO -

600 -

200 -

200 -

0’

I

I

1 __

._

20

40

, __

60

I _^

(Iv

._

100

Fig. 9. P-x curves for He-Xe mixtures showing gas-gas immiscibility of the first kind, according to De Swaan, Arons and Diepen, taken from ref. 25, fig. 7, p. 2328.

Still, this last theory which Van der Waals published in 189327), may not have obtained so much attention as it really deserves. The problem of capillarity had always intrigued Van der Waals, as may be clear from the very first words of the introduction to his thesis: “The choice of the subject matter of my thesis results from my desire to understand better a quantity which in the theory of capillarity, as it has been developed by Laplace, plays a peculiar role. It is the quantity which

18

Fig. 10.

represents surface

J. DE BOER

P-x

curve for Ne-Kr mixtures, according to Schouten and Trappeniers, showing gas-gas immiscibility of the second kind, taken from Schouten, ref. 26, thesis p. 71.

the

. . .“.

cohesive

pressure

on the

surface

of a liquid,

bounded

by a flat

VAN DER

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proportional to the square of the density gradient, where c is a constant of a similar nature as the constant a of Van der Waals’ law. Integration over the coordinate z perpendicular to the surface leads directly to the surface density of the free energy being equal to the surface tension. The actual shape of the functional dependence of the density p on the coordinate z perpendicular to the surface is determined by minimizing this surface density of the free energy subject to appropriate boundary conditions. This theory led Van der Waals to an expression for the surface tension

(17) which,

using

the

proportionality

pr -

20

J. DE BOER

7. Van der Waals “internalfield” concept. Of major importance was another concept introduced by Van der Waals in the derivation of his equation of state: the internal j?eld. This fundamental idea to represent the average effect of the molecular attractive forces by an “internal” or “cohesive” pressure pi = a/V2 has found further applications in other fields. Thus Weiss presented in 1907 a theory forferromagnetic substances which was very similar to that of Van der Waals. In fact the law of Curie-Langevin: M/M0 = L(M,H/RT) is typical for a paramagnetic substance without interaction in which only saturation effects are taken into account. For ferromagnetic substances, however, this interaction is no longer negligible and to take them into account Weiss introduced an “internal” field Hi called “molecular field” by Weiss3s), which was assumed to be proportional to the magnetization : Hi = AM. This internal field then has to be added to the external applied field to give the total field which is acting according to CurieLangevin’s law on individual magnets, just as the internal pressure has to be added to the external pressure to counterbalance the kinetic pressure due to the thermal motion. The similarity between the two theories becomes more apparent if we express Van der Waals law in parametric form by considering the density p = l/V to be a function of the “local” or “kinetic” pressure plOCwhich consists of two contributions, the external pressure p and the internal, or cohesive pressure pi = ap2. In this way the situation becomes analogous to the case of a ferromagnet, where the magnetization M is a function of the local field H,,,, which also consists of two parts: the external field H and the internal field Hi = AM. For the Van der Waals fluid the relation between p and plOCis the inverse of the relation plOC= RTp/(l - bp) proposed by Van der Waals. Comparison between the Van der Waals fluid theory and the Weiss-Langevin theory for the ferromagnet then gives :

Fluids : bp = f@p,,lR7’) =p + up2

Plot

Van der Waals:

x = bp,JRT

f(x) = x/(1 + x) x << 1 f(X) % x - x2 + . . . x>>l f(X)%1 - l/x

Ferromagnets

:

M/M, = WfoH,,,IRT) H,,, = H + AM Langevin

:

x = M,H,,,IRT

L(x) = coth x - l/x x<> 1 L(x) F3 1 - l/x

It is well known that this similarity leads to a large number of similarities between the isotherms, the heat capacities and between the compressibilities and the susceptibilities. Many years later, when Bragg and Williams40) published in 1934 their theory of order-disorder phenomena, also this theory appeared to be built on exactly the same mathematical formalism. At present all these internal-field theories are placed under the same heading: the kind of approximation to obtain these theories from Gibb’s statistical mechanics is the same and this so-called mean-field approximation constitutes a convenient

VAN DER WAALS IN HIS TIME AND THE PRESENT REVIVAL

general starting for calculations any system which the force plays predominant role41s42). should be that the to consider der Waals’ theory rests on a in which short-range the long-range forces are in a way and initiated by der Waals. terms of partition function as follows: the configurational function one total energy of forces as sum of and attractive forces exp -@[@‘“” This may

transformed

Qattr] dr,

..

21

as a repulsive in fact may be the repulsive

(18)

into

= QzP(exp

(19)

QgP is partition function the fluid with short-range forces only where (exp represents the of exp /3CPttr in this same system. The mean-field approximation then consists essentially in replacing (exp -@Pttr) by exp -B(Ptt’). One so obtains for the pressure

in which the two terms correspond closely to the two terms in the equation of Van der Waals: the kinetic pressure N/CT/( V - b) and the internal pressure pi = a/P, respectively. Obviously to obtain Van der Waals equation one has to approximate :

Qz" m (V - b)N,

(Qattr) = -a/V.

(21)

We thus see that Van der Waals equation can be seen to be really the equation obtained in the mean-field approximation adding the further approximations given above. From this modern approach to derive Van der Waals’, equation in its original form from the mean-field approximation of statistical mechanics it will again be clear how fundamentally this is connected with Van der Waals’ treatment of the repulsive and the attractive forces. 8. Influence of Van der Waals’ theory on modern scientific developments. Several problems originating in Van der Waals’ theory of the equation of state have been sources for new and further developments even up to the present time. I would like to mention here the following items: a) the problem of the nature of the unstable parts (the “Van der Waals’ loop”) in the isotherms and Van Hove’s theorem;

22

J. DE BOER

b) the independence of Van der Waals’ theory on the dimensionality of the system and the rigorous basis for Van der Waals’ equation with infinite-range forces; c) the behaviour of Van der Waals’ equation at the critical point. These three points will be discussed in somewhat more detail in this section. 8.1. The unstable parts of the Van der Waals’ isotherm. The first problem arising from Van der Waals’ theory, i.e. the nature of the Van der Waals’ loop in the isotherms, became of central interest when Mayer43) published in 1937 for the first time a systematic theory of the equation of state giving in principle rigorous expressions for all coefficients of Kamerlingh Onnes’ virial expansion in powers of the density. Mayer’s theory was presented by Born44) at the previous Van der Waals Conference held in Amsterdam in 1937 just a few months after its publication and this gave rise to a very lively discussion about whether one might ever expect to obtain with Gibbs’ ensemble theory and the partition function an experimental isotherm with a section in which the pressure is independent of the density or whether one would obtain isotherms with loops of the Van der Waals type. Of course already the discontinuities in the derivatives of the real isotherm seemed to be difficult to obtain from a partition function Qs(V, r), which should be analytic throughout, but it was Kramers who pointed out that we are really interested not in the partition function itself, but in the thermodynamic limit of the free energy per particle and that this limiting process may be able to introduce a nonanalytic behaviour at certain densities and temperatures. However, a further difficulty was the fact that the horizontal part of the isotherm corresponds to a region where the system is no more homogeneous. The possibility was considered that in approximate calculations implicit homogeneity assumptions are incorporated, which lead to a Van der Waals loop in the isotherms. There was general agreement, however, that in principle the partition function should always give maximum weight to the most probable state, even if this is not homogeneous, so that any rigorous calculation should lead in the coexistence region to the horizontal section in the isotherm and not to Van der Waals loops. This was indeed confirmed in 1949 by the well-known theorem of Van Hove. 8.2. A rigorous basis for Van der Waals’ equation. A second and very serious difficulty of the Van der Waals’ theory which came up after the last war, was its independence of the number of dimensions. In fact the whole reasoning of Van der Waals can also be applied to a two-dimensional system of hard disks and a one-dimensional system of hard rods, which exercise attractive forces on each other. In general for a d-dimension space the constant b turns out to be 2d-1 times the volume of the molecules, making this constant two times the total area of the hard disks in two dimensions and equal to the total length of the hard rods in the one-dimensional case. In fact this is even exact in the one-dimensional case. However, it has been proven rigorously by Takahasi (1942)45), Rushbrooke and Ursell (1948)46), Gursey (1948)47) and Van Hove4*), th.at in one-dimensional systems with hard core and finite interaction range no phase transition can occur.

VAN DER

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The central question now became the following: Looking at the success of the Van der Waals theory - or more in general the mean-field approximation - in the statistical-mechanical treatment of interacting many-particle systems, one should ask: under which conditions is this approach exact? The answer came from the one-dimensional systems where the existence - or not - of a phase transition was at stake. In 1961 Baker and Kac4s) showed independently that for a one-dimensional Ising model with interaction of the form &Jr exp --xrl between the spins at the lattice sites k and I, the energy and heat capacity approach that of the Weiss theory when the interaction range approaches infinity. Two years later Uhlenbeck, Kac and Hemmerso) proved a similar theorem for a one-dimensional fluid model of particles with hard-core diameter 0 and attracting each other according to the exponential law p(xij)

= --cry exp --xij,

xij > u,

(22)

such that the integral J p;(x) dx = --c( is independent of the constant y, defining the range. When, after taking the thermodynamic limit N, L + co; L/N = I, the constant y is put equal to zero, i.e. the range goes to infinity, then the equation of state becomes the Van der Waals equation ---kT p=&@

cc 12’

(23)

In the coexistence region one obtains rigorously the horizontal isotherm as indicated by Maxwell’s rule and not the Van der Waals loop. The phase transition thus only occurs in the limit y -+ 0, so that no collision with Van Hove’s theorem, requiring finite interaction range, occurs. In 1964 Van Kampensl) proved that a similar theorem holds in three dimensions. He divided the system into a large number of cells, such that each cell is large enough so as to contain a large number of particles, but on the other hand such that the cell dimensions are small compared to the range of the attractive forces. The distribution of the molecules over the cells is then determined by minimizing the free energy. The result is again Van der Waals’ law (4) for those values of density and temperature for which a homogeneous distribution of the molecules over the cells was found to minimize the free energy. In the coexistence region the free energy was found to be minimized by inhomogeneous distributions over the cells leading directly to the horizontal section in the isotherm obtained from Maxwell’s rule. In 1966 Lebowitz and Penrose52) improved this theory and in fact proved rigorously that for any number of dimensions the Van der Waals’ equation including the Maxwell rule results in the limit y ---f 0 for any system of particles interacting with short-range repulsive forces and integrable long-range attraction of the type q(rij)

= -or”+,

r > 0,

v(rij)

= a,

r < ff,

in the limit y -+ 0, provided again that the thermodynamic limit is taken properly before taking the limit y + 0. This then shows exactly under which conditions Van der Waals’ equation is exact and in fact has given, after nearly one hundred

J. DE BOER

24

years, a new understanding of the reason why Van der Waals’ approach gave such a very good approximation to the true behaviour of fluid systems and strongly interacting systems in general. 8.3. The analytic behaviour near the critical point. The third problem arising from Van der Waals equation was connected with the behaviour near the critical point. Obviously for this equation all derivatives of the pressure and free energy with respect to density and temperature exist for V > b, in particular also around the critical point, defined by (@/al’), = (a2P/aV2) = 0. This makes it possible to derive with a Taylor expansion relations between measurable quantities which should be valid closely round the critical point. Taylor expansion of the reduced Van der Waals equation of state 7~ = n(v,, 0) close to the critical point gives up to terms of the third order in de and dp, d7r = 448 -

6d&lp,

-

$l$

+ 9dedv2,

(25)

where d7~, np and 110 are the deviations of the reduced pressure, volume and temperature from the critical value 1. Thus one obtains for the critical isotherm (Ae = 0) An =

--gAv3

(26)

and for the compressibility temperature : &r/&p = -648.

at the critical

density

(Ap, = 0) above

the critical

(27)

The analysis by Mrs. Levelt Sengers a.o. of the experimental data56) indicates that the exponent 6 characterizing the behaviour of the isotherm, is much closer to 5 than to the value 3 required by Van der Waals and that the exponent y for the compressibility at the critical isochore seems to be closer to 1.25 than to the value 1. These experimental facts already show that Van der Waals’ equation is not able to describe this region close to the criticai point. Consequently the Taylor expansion which was the starting point for (25) and which necessarily follows from the analytic character of Van der Waals’ equation, is clearly precluded by the experimental facts. Some degree of nonanalyticity is introduced when one uses below the critical temperature the Maxwell construction to obtain the pressure and the densities of the coexisting phases. In fact closely below the critical temperature the densities of the coexisting phases are given by vi1 -

vi’ = 4(1 -

e)*,

(28)

whereas there appears a discontinuity in the heat capacity at constant volume coming from the discontinuity in the second derivative of the pressure with respect to temperature at the critical density. AC,

= 3R.

(29)

Already at the beginning of this century it was shown and stressed in particular by Verschaffelt that the exponent /I = 4 in the powerlaw for the coexisting densities

VAN

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25

much critical density very to a power C, A&” with 0 < u < 0.12. The set of exponents following from Van der Waals’ equation - usually referred to as the “classical” set of scaling exponents - is not really specific for Van der Waals’ law , lone. In fact they follow from any equation of state which behaves analytically at the critical point and they apply also to a great many more cases than just the equation of state of a fluid system, e.g. magnetic systems, lattice gases, order-disorder systems and binary mixtures. It was also a great surprise values for this variety of systems showed very similar that the “experimental” deviations from the “classical” values. The deviations from this “classical” set of coefficients which the experiments show, thus point much deeper than only to deviations from one particular equation of state. This situation has given rise to the great outbreak of interest in the critical phenomena as shown also by the many papers presented at this conference in this particular field. Tremendous progress had been made since Widomso), FisheP4) and Kadanoff55) between ten and five years ago made a real start with the theoretical exploration of the critical region, but its evaluation is not the object of this lecture. We only have wanted to point out the relation which exists between these modern developments and the work of Van der Waals. Summarizing one may say that Van der Waals’ investigations on the kinetic theory of fluid systems: gases, compressed gases, liquids of pure substances and their mixtures, not only had a tremendous impact on the physics of his time, but also were a stimulus for further research at the present time. Many of the features characteristic for Van der Waals’ theory has been a source for further inspiration. Let us mention only the description of gas- and liquid-state by one single equation leading to an instability region in the isotherms and the different treatment of short-range repulsive and long-range attractive forces which was closely connected with the internal-field concept. The discovery that for certain model systems the equation of Van der Waals including the Maxwell equal area rule is exact, gave rise to a real revival of interest and also to a feeling of admiration for the depth of Van der Waals’ considerations which may not always have been understood. In the critical region the analytic approach of Van der Waals really broke down, but it is a nice feature of the historical development in physics that just this failure has given rise to very important new developments, which to my feeling belong to the most fundamental in statistical mechanics since many years.

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