Johannes D. van der Waals

O H A N N E S

D .

V A N D E R

A A L S

The equation of state for gases and liquids

Nobel Lecture, December 12, 1910

Now that I am privileged to appear before this distinguished gathering to
speak of my theoretical studies on the nature of gases and liquids, I must
overcome my diffidence to talk about myself and my own work. Yet the

thought that you are entitled to expect that of me leads me to hope that you
will forgive me if I state my views in this field with utter conviction, even

in regard to aspects which are not yet universally known and which have
so far not achieved universal recognition. I intend to discuss in sequence:
(1) the broad outlines of my equation of state and how I arrived at it;

(2) what my attitude was and still is to that equation;

(3) how in the last four years I have sought to account for the discrepancies
which remained between the experimental results and this equation;
(4) how I have also sought to explain the behaviour of binary and ternary
mixtures by means of the equation of state.

(1) The first incentive to this my life’s work came to me when, after my

studies at university, I learned of a treatise by Clausius (1857) on the nature
of the motion which we call heat. In this treatise, which is now taught with
negligible modifications in every highschool in Holland, he showed how
Boyle’s law can very readily be derived on the assumption that a gas consists
of material points which move at high velocity, that this velocity is of the
order of that of sound and increases in proportion to the square root of the
absolute temperature. It was subsequently realized that this is only the root
mean square of the velocities and that, as Maxwell stated, although he er-
roneously considered to have proved it, there is a law for the distribution
of the velocities known as the Maxwell law. Boltzmann was the first to give
the correct proof for this velocity distribution law. Clausius’ treatise was a
revelation for me although it occurred to me at the same time that if a gas
in the extremely dilute state, where the volume is so large that the molecules
can be regarded as points, consists of small moving particles, this is obviously

still so when the volume is reduced; indeed, such must still be the case down
to the maximum compression and also in liquids, which can only be re-

E Q U A T I O N O F S T A T E F O R G A S E S A N D L I Q U I D S

255

garded as compressed gases at low temperature. Thus I conceived the idea
that there is no essential difference between the gaseous and the liquid state
of matter - that the factors which, apart from the motion of the molecules,
act to determine the pressure must be regarded as quantitatively different
when the density changes and perhaps also when the temperature changes,
but that they must be the very factors which exercise their influence through-

out. And so the idea of continuity occurred to me. I then also asked myself
how the solid state behaved. Although I have not as yet seriously gone into
that question I do think that in the amorphous state the close proximity of
the molecules impedes their mutual displacement. The crystalline state def-
initely behaves in a slightly different way. Actually I should still be silent
on this question. Nevertheless it can now scarcely be doubted that continuity
exists between the other two states of aggregation.

As you are aware the two factors which I specified as reasons why a non-

dilute aggregate of moving particles fails to comply with Boyle’s law are
firstly the attraction between the particles, secondly their proper volume.
Turning first to the second factor I should like to point out the following:
originally I had anticipated that simply the total volume must be decreased
with the total volume of the molecules to find the volume remaining for
the motion. But closer examination showed me that matters were not so
simple. To my surprise I realized that the amount by which the volume
must be reduced is variable, that in the extremely dilute state this amount,
which I have notated

is fourfold the molecular volume - but that this

amount decreases with decreasing external volume and gradually falls to
about half. But the law governing this decrease has still not been found. This
very point has proved to be the most difficult in the study of the equation of
state. Korteweg, Lorentz, Boltzmann, Jeans, and of my pupils, Van Laar,
my son, and Kohnstamm have subsequently worked on it. I had thought
that it was simply a matter of studying how the mean free path decreases
between the collisions of the molecules which are the consequence of their
extensiveness, i.e. of the fact that they must not be considered material points
but small particles with a real volume in common with all bodies known to
us, and so I arrived at this formula:

where

is fourfold the molecular volume.

(A)

256

1910 J.D. VAN DER WAAL S

Boltzmann has shown, however, that this is inadequate and later on Kohn-

stamm proved that the formula will be more complex and that for

b/b

quotient will be obtained of two series in which powers of

occur. It is

so difficult to determine the coefficients

α, β

etc. that Van Laar was compelled

to carry out fearfully long calculations when determining the second co-
efficient

by my method

(α

gave rise to far less trouble). This prevented me

from proceeding further. And here I have come to the weak point in the
study of the equation of state. I still wonder whether there is a better way.
In fact this question continually obsesses me, I can never free myself from it,
it is with me even in my dreams.

As regards the other cause underlying the non-compliance of real gases

and liquids with Boyle’s law, i.e. the mutual attraction of the molecules, the
situation is somewhat better although here again the last word has not yet

been spoken. Using the procedure adopted by Laplace in his capillarity the-

ory, in my continuous theory I have reduced this attraction, which acts in the
whole volume, to a surface force which acts towards the interior and thus,
together with the external pressure, holds together the moving molecules.

Laplace regards his liquid actually as a continuum; at the time he was still
unaware of molecules. And if we were dealing with stationary molecules it

would have been inadmissible to reduce the attracting forces in the interior
to just a surface force. Yet since the molecules are in motion each point in the
interior will certainly not be filled with matter at every moment. But space
may be regarded as continuously filled with matter of mean - normal - den-
sity. I shall return to this point however when I speak of my studies in recent
years. The above considerations brought me to the following formula:

which became universally known only as a result of Eilhard Wiedemann’s
efforts.

(2) Having reached the second part of my lecture I must now mention my

own attitude to this equation. It will be abundantly plain from my earlier
comments that I never expected this equation, with

and

assigned a con-

stant value, to give results numerically in agreement with experiment and
yet people almost always act as though that were my opinion. This astonishes

me as in my treatise of 1873 not only did I

expressly emphasize the variability

E Q U A T I O N O F S T A T E F O R G A S E S A N D L I Q U I D S

257

but also quoted a series of

values from Andrews' experiment in which

for small volumes the change in

with the volume is calculated. For carbon

dioxide I moved from the limiting value

0.0023 down to

b =

0.001565.

In this series of values the volumes of the liquid even go below the limiting
value of

for infinite volume.

Perhaps the reason why my opinion tha t

is variable has rarely been re-

garded as seriously intended is to be found in the manner in which I cal-
culated the critical parameters. In this calculation I had to assum e

as in-

variable. But that was because I believed that Andrews’ values justified still
assigning the limiting value

in the critical volume. But as I later

showed that was a mistake; the value o

in the critical volume has, of

course, slightly decreased. Yet to determine the critical volume whe

changes with the volume, the first and second differential quotients d

and d

also need to be known. To determine the critical volume, as I

later showed, we obtain the equation:

( C )

Only when we neglect d

and d

do we find v

And it can

again be seen from this equation that what I have termed the weak point
of my theory is actually responsible for the theoretical impossibility of cal-
culating accurately the critical volume. Using an approximation formula for

I was able to determine

as about 2.2

This equation for determining

the critical volume I gave and used in my paper in honour of Boltzmann;
it was repeated in part in my 1910 treatises. When I had realized this large
deviation in the critical volume I also feared a large deviation in the other
critical data, i.e. non-compliance with the formulae:

To my great joy, however, and not without astonishment, I found that the
two equations used to calculate

and

from the critical data remain un-

changed. That I was able to conclude from the results of Sydney Young’s
admirable experiments on the volumes of co-existing vapour and liquid

258

1910 J.D. VAN DER WAALS

phases and on the saturation pressure level at various temperatures. I showed
this recently in a treatise to which I shall refer again in the present lecture.
However I can explain it as follows: Sydney Young determines inter alia
the value of

(D´)

at the critical point and finds 1/3.77. If

and

have the same values as in

my theory, providing

is imagined to be constant, the difference from 3/8, the

value which I found for

pv/RT,

can be ascribed wholly to the volume. And

this, we can say, is entirely correct. The critical volume is not 3

but (3/ 2)

= 2.125

(approximately). And the product

pv/RT

is not 3/8 but 3/8

For the value of

my equation give s

a/pv

. A value of about 6 is found for this value

from Sydney Young’s experiments. Therefore fo

we have the value:

a/6vk

, and with v

= (3/2)b

the value

(1/27)

a/b

as was

originally found, is again obtained for

It will thus be seen that I have never been able to consider that the last

word had been said about the equation of state and I have continually re-
turned to it during other studies. As early as 1873 I recognized the possibility
that

and

might vary with temperature, and it is well known that Clausius

even assumed the value of a to be inversely proportional to the absolute
temperature. Thus he thought he could with probability account for the
equation

only the half being found when

and

are held constant. The foregoing

remarks signify that this value does not arise from a change in

with tem-

perature, but merely from the change i n

with the volume.

For a long time I searched for a definite characteristic to find whether just

making

variable is sufficient to bring about complete agreement between

E Q U A T I O N O F S T A T E F O R G A S E S A N D L I Q U I D

259

my formula and experiment, and in the case of remaining discrepancies,
whether perhaps it is necessary to assume variability o

and

with tempera-

ture; and now I come to my latest studies on the equation of state.

(3) In 1906 before the Royal Academy of Sciences in Amsterdam I gave a

lecture entitled: pseudo association. At the time, however, I contented my-
self with an oral communication. But one of my pupils, Dr. Hallo, took it

down in shorthand and later Dr. Van Rij incorporated it in his inaugural
dissertation and developed the theme further. As the criterion whether the
variability of

with the volume would be sufficient to bring about agree-

ment between my formula and Sydney Young’s experimentally determined

liquid and vapour volumes I used in my lecture the Clapeyron principle
which, with

varying only with

leads to the equation

and not with

and with

constant,

To ensure that only parameters are involved which are directly determinable
by experiment, we may also write

All the parameters on the left of this equation were accurately determined
by Sydney Young for a series of substances. At the critical temperature the
value of the left-hand side is, of course, equal to unity; but what is the value
at temperatures that are only a fraction of

? Here it appears that as the

temperature decreases the value of the left-hand side increases, very rapidly
at first, then at an imperceptible rate.

260

1910 J.D. VAN DER WAALS

Sydney Young’s experiments only go as far as a value o f

T/T

= 2/3. The

right-hand side has then increased to 1.4 and seems to be approaching

asymptotically the value 1.5. The value of the right-hand side may very

accurately be represented by the empirical formula

where

T / T

= m.

At a temperature very close to the critical, e.g.

0.99,

the increase in

this expression is as much as 0.1 whereas if the square root of 1 -

did not

occur in the formula and the increase were only to be represented by

1 - m,

it would be less by a factor of ten.

I then had to examine whether the assumption that

is temperature-

dependent could account correctly for this increase of the right-hand side;
and were such not the case, whether another expression for the internal
pressure, which I have always written as

a/v

, could explain it. As a result

of this examination it was found that each temperature function for

and

compatible with such a rapid initial increase of the right-hand side, must
contain

m. Yet above the critical temperature that would give imag-

inary values for

and

and is obviously at variance with the whole behav-

iour of gases. It also appeared that no other assumption for the internal
pressure can lead to such a rapid initial increase as the one under discussion.

It should not be forgotten that the critical temperature is actually not a
special temperature. At this temperature the co-existing densities are equal
in magnitude. That alone gives this temperature a meaning which under
all other circumstances it does not have. An abrupt jump, a rapid increase

either in

would make the temperature quite a special one at any level

of compression and its determination would then be possible at any density

level. In fact, bluntly speaki ng, the result would be: an equation of state

compatible with experimental data is totally impossible. No such equation

is possible, unless something is added, namely that the molecules associate

to form larger complexes ; this year, therefore, I have given two treatises
at the Academy in Amsterdam on this possible association. I have termed it

"pseudo association" to differentiate it from the association which is of chem-
ical origin. The possible formation of larger molecular complexes, partic-

ularly in the liquid state, has frequently been emphasized and the finding that

the assumption is necessary to achieve agreement between the state equation

E Q U A T I O N O F S T A T E F O R G A S E S A N D L I Q U I D

261

and experiment will hence cause no surprise. Unfortunately my examination

is still incomplete. I have found it arduous. And I have had to make use of

every piece of evidence to derive something concrete. Nevertheless so much

has emerged that it will have to be assumed that a large number of single

molecules are required to form a new group which holds together and be-

haves as a new, larger unit in the molecular motion.

What is the origin of this complex formation, this pseudo association?

was compelled to assume it because it seemed to me the only way to make

an equation of state - whichever it is - compatible with the results of meas-
urements. However, as a result of a remark by Debye in last month’s

An-

nalen der Physik,

I remembered a phrase of Boltzmann’s. When a few years

ago I was privileged to have him with me for some days, among the many
matters which we discussed he told me in passing that he was unable to
reduce the attraction of molecules to a surface force. At the time the sig-
nificance of this remark of Boltzmann’s did not dawn on me - only now

do I think I appreciate what he meant. As far back as in my treatise of 1873
I came to the conclusion that the attraction of the molecules decreases ex-
tremely quickly with distance, indeed that the attraction only has an ap-
preciable value at distances close to the size of the molecules. At the time
this even prompted me to state that in the case of gases the collisions alone
are responsible for their exhibiting an attraction. And it must have been
Boltzmann’s view that it is only admissible to adopt Laplace’s procedure and
assume a surface force as a consequence of the attraction provided it does not

fall off so rapidly.* Debye’s remark implied that Boltzmann had predicted

the formation of a complex. Thus, so I believe, the assumption of pseudo
association is justified from the theoretical standpoint. And now I think I
may state how I proceeded in my

latest treatise which appeared in November

of this year. Pseudo association differs from true association in that the latter
is the result of new chemical forces which arise only when molecules are
combined to form e.g. double molecules, whereas pseudo association must
be ascribed wholly to the normal molecular forces. Now, since this force
diminishes so quickly, it has two consequences. Firstly it results in the forma-
tion of a complex, but that is not all. Secondly it leads to a surface pressure,
although a lower one. And this is precisely what I was compelled to assume
in my calculations were I to have some prospect of accounting for the cited
differences. I thus sought to make this clear in the following manner. Let the
number of molecules that have combined into a complex be so large that it

* Cf. L. Boltzmann,

Gastheorie,

I I, p. 176.

262

1910 J.D. VAN DER WAALS

is possible to speak of a molecule at the centre surrounded by a single layer

containing almost as many other molecules as is possible simultaneously.
Then, for the surrounding molecules the attraction directed towards the

interior acts only to maintain the complex; and this part of its attraction is
lost for the surface pressure. Only the forces acting outwards from these
molecules can contribute to the formation of the internal pressure. But of
course, for pseudo association as for true association the number of formed
complexes increases with decreasing temperature and volume. At the critical
point, so I was compelled to conclude, only a very small part of the weight
is present as complexes.

If pseudo association exists in a substance, there are at least two types of

molecules, namely simple and complex. I say at least two types because it
cannot be assumed that all complexes are of equal size. But as a first step I

have assumed only two types, i.e. simple molecules and

-fold molecules.

For a really scientific treatment, of course, it would be necessary to assume
all values of

as possible and to seek the law of distribution for these values.

For the time being, however, I have confined myself to assuming only a

single type of complex. We then have a binary mixture. It was very for-
tunate that for many years I had made a serious study of the laws of binary
mixtures - and so I come to the fourth point of my lecture. Not to demand

too much of your attention, however, I promise to be quite brief.

(4) When I first conceived the idea of utilizing my equation to study the
properties of binary mixtures I can no longer say. But even 20 years ago,
at the insistence of my friend Kamerlingh Onnes, I was able to publish a
complete theory for binary mixtures. My "Théorie moléculaire d’une sub-
stance composée de deux matières différentes" (Molecular theory of a sub-
stance made up of two different constituents) appeared in the

Archives Néer-

landaises

for 1890. I had written it in Dutch but my esteemed friend Bosscha

undertook the difficult task of rendering it into French, a task which was all
the more exacting as I had written it in an extremely concise form and the
mathematical treatment led to particular points (plait points) and partic-
ular curves which at the time had rarely been examined in detail. Previous-

ly, however, my friend Korteweg, to whom I had communicated in broad

outlines the outcome of my examinations, had studied the mathematical

properties of these points and curves, a study which I have often found of
great use. The reasons why I hesitated so long over publication were many
and it would serve little purpose to enumerate them here. But one of the

E Q U A T I O N O F S T A T E F O R G A S E S A N D L I Q U I D

263

reasons of scientific importance was the question which I kept asking my-
self: is it any use, until the study of the equation of state has been completed,
applying it to mixtures ? I appreciated in advance that as long as I was ignorant

of the law governing the variability of

and hence had to assume that

did

not vary with the volume, the results for many parameters would yield
values exhibiting numerically large differences as compared with the values
of these parameters as determined by experiment. Nevertheless, the consid-
eration that even with

constant my theory had not been unimportant in

the case of a single substance gave me the hope that many phenomena would

be explained qualitatively provided that a suitable value for

and

were

introduced for mixtures. It was of great importance for me to be acquainted
with Gibbs’ treatises on the equilibrium of heterogeneous substances which
he had sent me immediately after their appearance. I made use inter alia of
his principle that for a given amount of substance equilibrium sets in if the
free energy is minimum for the given temperature and volume. In his
honour I named the equilibrium surface for a binary system the

-surface.

For, the free energy whose significance for the equilibrium he was the first
to recognize, he always represents by the sign

ψ.

The phenomena are really not complex in the case of a simple substance

and it is an easy matter to obtain a general picture of them. It is therefore
very surprising that in a mere binary mixture they become so complex that
they have often been compared with a labyrinth. This is particularly so
where three-phase pressure can exist. And it has now become clear that they
are at least qualitatively in agreement with the "Théorie moléculaire, etc."
from which they can be derived and indeed often predicted. Owing to the
many experimental studies to which it has given rise I have frequently found
the opportunity of discussing it closely in special communications. Thus in
the years 1907 to 1909 about 15 treatises of this nature have appeared in the
proceedings of the Academy at Amsterdam. It would be an impressive
number were I just to mention the names of the physicists and chemists who,

guided by this theory, have studied and still are studying binary mixtures.
It begins with Kuenen and ends with Dr. Jean Timmermans of Belgium
whose studies have still to be completed but which have already been pub-

lished in part.

And if now I may be permitted to look back on the way I have come, I

must confess that it may fairly be called a detour. Immediately the necessity

of assuming association became clear to me I extended my equation of state

264

1910 J.D. VAN DER WAALS

analogously to the formula for a binary mixture and introduced a new para-
meter, the degree of association. This degree of association is determined by

means of Gibbs’ equation to which I have referred. The degree of association
thus determined must then be introduced into the equation of state. And I
confess that this is a detour. Perhaps there is a direct way. That this way is
seriously being sought I know from those in direct contact with me. In the
search for this way Gibbs’

Elementary Principles of Statistical Mechanics

will be

a necessary guide. In this lecture, I have only given the history of the origin
and further elaboration of my theoretical studies and therefore had to speak

of the difficulties that had to be surmounted. Consequently I could not, or
only meagrely, discuss how they have assisted in correctly understanding the

phenomena. One of the main conclusions, which I have termed the "Law

of corresponding states", has, I may say, become universally known. Nor
have I discussed how this law was a potent contributory factor in Dewar’s
determination of the method of liquefying hydrogen, and particularly in
Kamerlingh Onnes’ determination of the method of liquefying helium. I
have also forgone detailed discussion of the temperature at which, to use
Regnault’s nomenclature, a gas starts to behave like a "pluperfect gas" (gaz

plus que parfait), which temperature has been found to have a value of
(27/8)

- besides the

temperature at which it may be stated that the Joule-

Kelvin effect reverses: its value is found to be (27/4 )

Neither have I spoken of my perhaps somewhat overhasty efforts to

determine the equation of state of the molecule itself. I have even omitted
to say why I thought I had to go to such trouble to determine the relation
between

p, v

and

for a substance. The formulae of thermodynamics are

effective and can actually be applied in all problems, even to determining
thermal parameters, only when this relation is known, otherwise they can
be regarded as just

one

equation between two unknowns. I have explicitly

emphasised this in the book in commemoration of Kamerlingh Onnes pub-

lished in 1904.

But it was not my intention to discuss all that and I would not have had

sufficient time. Yet it does not seem to me superfluous, perhaps it is even
necessary, to make a general observation. It will be perfectly clear that in
all my studies I was quite convinced of the real existence of molecules, that
I never regarded them as a figment of my imagination, nor even as mere
centres of force effects. I considered them to be the actual bodies, thus what we
term "body" in daily speech ought better to be called "pseudo body". It is
an aggregate of bodies and empty space. We do not know the nature of a

E Q U A T I O N O F S T A T E F O R G A S E S A N D L I Q U I D S

265

molecule consisting of a single chemical atom. It would be premature to
seek to answer this question but to admit this ignorance in no way impairs
the belief in its real existence. When I

began my studies I had the feeling

that I was almost alone in holding that view. And when, as occurred already

in my 1873 treatise, I determined their number in one gram-mol, their size
and the nature of their action, I was strengthened in my opinion, yet still
there often arose within me the question whether in the final analysis a

molecule is a figment of the imagination and the entire molecular theory
too. And now I do not think it any exaggeration to state that the real
existence of molecules is universally assumed by physicists. Many of those
who opposed it most have ultimately been won over, and my theory may
have been a contributory factor. And precisely this, I feel, is a step forward.

Anyone acquainted with the writings of Boltzmann and Willard Gibbs will
admit that physicists carrying great authority believe that the complex

phenomena of the heat theory can only be interpreted in this way. It is a
great pleasure for me that an increasing number of younger physicists find
the inspiration for their work in studies and contemplations of the molecular

theory. The crowning of my studies by the esteemed Royal Swedish Acad-
emy affords me satisfaction and fills me with gratitude, a gratitude which I
cannot call eternal and in my old age I

cannot even promise that it will be

of long duration, but perhaps for this very reason my gratitude is all the
more intense.