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TER VERKRIJGING VAN DEN GRAAD VAN
DOCTOR IN DE WIS- EN NATUURKUNDE
AAN DE RIJKSUNIVERSITEIT TE UTRECHT,
OP GEZAG VAN DEN RECTOR MAGNIFICUS
DR. J. BOEKE, HOOGLEERAAR IN DE FACUL-
TEIT DER GENEESKUNDE, VOLGENS BESLUIT
VAN DEN SENAAT DER UNIVERSITEIT TEGEN
DE BEDENKINGEN VAN DE FACULTEIT DER
WIS- EN NATUURKUNDE TE VERDEDIGEN OP
MAANDAG 13 JUNI 1938, DES NAMIDDAGS TE 4 UUR

Het voltooien van mijn proefschrift biedt mij een wellcome
gelegenheid allen, die tot mijn wetenschappelijke vorming hebben
bijgedragen, mijn erkentelijkheid te betuigen.

In het bijzonder geldt dit U, Hooggeleerde Uhlenbeck, Hoog-
geachte Promotor. Dat ik als Uw assistent gedurende enige jaren
onder Uw directe leiding heb mogen werken, beschouw ik als een
niet genoeg te waarderen voorrecht.

Hooggeleerde Ornstein, het is ongetwijfeld van veel nut voor
mij geweest, dat ik een groot deel van mijn studietijd in Uw labora-
torium experimenteel gewerkt heb. Voor de belangstelling en steun,
die ik ook later steeds van U heb mogen ondervinden, ben ik U
ten zeerste dankbaar.

Hooggeleerde Kramers, door U ben ik het eerst in aanraking
gekomen met de moderne theoretische natuurkunde. Ik dank U voor
het vele, dat ik, zowel in Uw Utrechtsche tijd als daarna, van U
heb geleerd .

Hooggeleerde WoLFF, het bewerken van mijn proefschrift gaf
mij dikwijls gelegenheid met dankbaarheid aan Uw colleges in de
functietheorie terug te denken.

I am greatly indebted to Dr. M. H. Hebb for his help in improving
the presentation of my thesis.

In this introductory chapter we shall give a brief survey of the
different kinds of problems which arise when one tries to obtain a
molecular theory of the equation of state of gases and liquids.

First of all, one requires a theory that can explain the general
qualitative features of the behaviour of gases and liquids. The most
striking of these features is the existence of sharp phase transitions
(condensation and evaporation). This means mathematically that
the equation of state cannot be represented by one analytical
function but consists of several analytically different parts. Since
this property is common to all substances one would expect it to be
possible to give a very general explanation demanding no exact
knowledge of the interaction between molecules, since this is
different for every substance.

The first attempt at such an explanation was the theory of
Van der Waals. In this theory the molecules of a substance are
treated as mutually attracting elastic spheres. It is assumed that this
attractive force (the quot;Van der Waals forcequot;) acting between the
molecules, is for each pair a function of the distance of s.^paration
only, and is independent of the velocities or of the positions of
other molecules. These forces are therefore the same in every state
of the substance. Forces of this kind are said to have the property
of additivity because the total potential energy of a configuration
is the sum of the potential energies of all the pairs.

Starting from these assumptions and using the methods of
classical statistical mechanics one can show that for very small
densities the equation of state for 1 mol of gas is given by

R the gas constant and a and b are constants'which are related
respectively to the attraction forces and to the diameter of the
molecules. This equation is quite arbitrarily extrapolated by Van
der Waals to the famous equation

which is claimed to be vahd for all densities. This equation does
not show the desired properties. In fact it represents a completely
analytical connection between p, V and T. However, below a certain
temperature T^, the critical temperature, the isotherms are not
monotonie functions, as they are for Tgt;T^, but consist of two parts
where (ôp/ôy)7-lt;0, connected by a part where (ôp/ÔV^)rgt;0.
Now a state with ( dpjd V)t gt; 0 does not represent a state of stable
equilibrium. In order to remove this unstable part one introduces
as an experimentally known fact the states of coexistence of vapour
and liquid, giving a line p = const, which connects the two stable
parts of the isotherm. These stable parts are then interpreted as the
isotherms of the pure vapour and of the pure liquid. The value of
the pressure of the two-phase system is determined by a thermo-
dynamical consideration (Maxwell rule).

It will be clear that this theory cannot be considered as a real
molecular explanation of the transition phenomena but is rather a
semi-phenomenological description.

Only very recently a new and, as may be hoped, more successful
approach to a real theoretical explanation of the phase transition
has been put forward by Mayer and his collaborators i). The origin
of Mayer's theory lies in a quite different line of development.
Because of the fact that all efforts to derive a general exact
equation of state had been unsuccessful, Kamerlingh Onnes 2)
proposed to represent the empirical data on the equation of state
of gases by a development in inverse powers of the volume

Here the first term gives the ideal gas law, which holds at large
volume (small density), and the following terms represent the
deviations from the ideal behaviour when the volume is diminished.

vinal coefficient by Kamerlingh Onnes ²). For these coefficients
it is possible to derive exact expressions in classical statistical
mechanics, starting from the partition function (Zustands integral)
introduced by GiBBS. These expressions were obtained from the
first terms of a suitable expansion of thia partition function. In
order to justify this procedure Ursell 3) has given a more detailed
mathematical investigation of this development, which has been the
starting point of Mayer's theory. He tries to show that the develop-
ment (3) is only convergent when the density of the gas is smaller
than a certain critical value, depending on temperature, and that
when the gas is compressed further the theory gives automatically
a pressure independent of the volume, corresponding to the
coexistence of vapour and liquid. The isotherm of the pure liquid
remains unexplained in this theory.

In the third chapter of this dissertation we shall give an account
of Mayer's theory. There we shall show that it is possible to
extend the considerations of Ursell and Mayer to quantum
statistics and that there exists a close analogy between Mayer's
theory and an older theory of Einstein 4) where a condensation
phenomenon for an ideal BoSE gas was predicted. Furthermore we
hope that from a mathematical standpoint this treatment will be
more satisfactory than Mayer's original one.

The failure of this theory to explain the liquid isotherms may be
due to the fact that it is, just as the Van der Waals theory, an
approximation from the quot;vapour sidequot;. Perhaps a treatment starting
from the crystalline state would be better suited for this purpose.
There have been many attempts in this direction, but without com-
plete success.

1nbsp; One sees immediately from (1) or (2) that in the VAN der WAALS
theory B(T) =b — a/RT.

2nbsp; In reality kamerlingh OnNES used a polynomial of six terms instead
of an infinite series and gave the name virial coefficients to the coefficients of
this polynomial, which are not the same as the coefficients of the infinite
expansion. We shall, however, always mean the latter coefficients in speaking
about virial coefficients.

an exact knowledge of the intermolecular forces is not necessary,
but it is necessary if one wishes to find an exact quantitative
expression for the equation of state. The determination of these
forces however is not a problem in statistical mechanics but a
quantum mechanical one and will not be treated in this dissertation.
Moreover London 5) has recently given an excellent survey of its
history and present situation so that we will give only a brief
description.

Let us first consider the attractive forces. In the older pre-
quantummechanical theories the intermolecular attraction was
attributed to two causes. In polar substances there will be an inter-
action between the electric and magnetic dipoles (or higher poles)
of the molecules. This interaction is called the orientation effect of
Keesom since it depends on the mutual orientation of the molecules.
A priori there is an equal probability for a repulsive and an attractive
interaction, but due to the Boltzmann factor the positions giving
rise to an attraction will be preponderant. On the average therefore
there will be an attraction which however decreases to zero with
increasing temperature. This very unsatisfactory property of the
orientation forces led Debije to the consideration of a second effect
which gives rise to an attraction independent of temperature. This
is the so-called induction effect, which is due to the polarization
of the molecules by the fields of force of neighbouring molecules.

10. These forces are not additive but vary with varying states
of the substance. For instance the polarization force almost dis-
appears in a condensed state where each molecule is surrounded
uniformly by its neighbours. This is contrary to all experimental
evidence which shows that the intermolecular forces have always
the same order of magnitude.

20. For spherical symmetric molecules (like helium) there would
be no attraction at all.

These difficulties are removed in the modern quantum mechanical
theory of the Van DER Waals forces which is chiefly due to
London. Here the potential energy of two molecules at distance r
is calculated by means of the perturbation calculus. The attraction
appears then as a second order effect. This calculation can be inter-
preted as follows. One considers not only the interaction of the

static multipoles of ttie molecules, but also the interaction of the
rapidly varying multipoles corresponding to the possible transitions
between the states of a molecule. This effect is called the dispersion
effect because the characteristic quantities of these multipoles, the
oscillator strengths, also occur in the dispersion formula. It is found
that the interaction between dipoles gives rise to an attractive inter-
molecular potential which varies as the inverse sixth power of r.
The consideration of higher poles leads to potentials which vary
as the inverse of the eighth, tenth, etc. power of r. It is shown that
all these forces have the property of additivity which was assumed
in the Van der Waals theory. Moreover it has been shown that
even in most polar molecules they are preponderant over the
statical orientation and induction effects.

When two molecules are brought close together their attraction
is replaced by a very strong repulsion. The older theories did not
explain it but symbolized the repulsion by treating the molecules
as elastic spheres. In the new theory there are two reasons for a
repulsion when the charge clouds of the molecules overlap:

10. The electrostatic repulsion of the atomic nuclei which are
then not completely screened by the electrons.

20. The Pauli principle, which forces the electrons to move to
higher states. For the potential of this repulsion one finds the
approximate expression

The approximate magnitude of the dipole-dipole interaction has
been determined by London for a number of substances by a semi-
empirical method, using the experimental dispersion curve.

For the interaction of two helium atoms extensive calculations
have been performed by several authors. Slater 6) and Slater and
Kirkwood'') have calculated the exponential repulsion and the
dipole-dipole attraction. They find the potential

have been calculated by Margenau») and very recently by Page»),
Page's expression for the attraction is

The problem of finding an exact expression for the equation of
state when the intermolecular forces are known has as yet not been
solved. However, as we mentioned in § 1, in classical statistical
mechanics one can give exact expressions for the virial coefficients,
which are measurable quantities. In particular for the second virial
coefficient one obtains in the case of central forces

(5)

mol '

where N is the number of molecules per mol, r the intermolecular
distance in cm, V{r) the intermolecular potential and k Boltz-
mann's constant. Two problems may be attacked by means of (5):

1.nbsp;When one has a theoretical expression for V(c) one can
calculate B{T) from (5) by a numerical or graphical integration.
This gives the possibility of an experimental verification of the
theoretical calculation of ^(r).

2.nbsp;On the other hand, when B{T) is known experimentally, eq.
(5) gives an integral equation for V^(r). This fact has been exten-
sively used by Lennard-Jones 10) for an empirical determination
of the intermolecular forces. Because the integral equation (5) has
not been explicity solved, Lennard-Jones has assumed the following
expression for V(r):

Here the first term represents the repulsion and the second term
the attraction. It is obviously necessary that ngt; m. Lennard-Jones
found that it is possible to represent the second virial coefficient of
many substances by a suitable adaptation of the four arbitrary
constants in (6). This fact already indicates that the values of

B(T) in the regions of temperature used will not depend very
sensitively on the exact form of ^(r), since the expression (6) will
surely not represent exactly the actual intermolecular force. This is
confirmed by the fact that a good agreement with experiment is
obtained also by insertion of the Slater-Kirkwood potential (4)
into (5) 11).

The situation is quite different in the quantum theory. The
expression ioT B(T) which one obtains here (see Chapter IV, § 2)
is not directly évaluable in terms of V(r). Eq. (5) is the limit of
this expression for high temperature. There are three reasons for
deviations from (5) at lower temperatures;

1.nbsp;Due to the wave character of the molecules they will not
interact according to the laws of classical mechanics but diffraction
effects will occur. These will be large when the de Broglie wave
length of the molecular motion is large, compared with the diameter
of the molecules. This wave length is equal tonbsp;where h
is Planck's constant, m the mass of the molecule and E its kinetic
energy. The mean value of E is proportional to T and therefore
the mean de Broglie wave length is proportional to Ijl^mT. We
see therefore that the diffraction effects will be large for light
gases and low temperatures.

2.nbsp;The classical Boltzmann statistics has to be replaced by the
Einstein-Bose or the Fermi-Dirac statistics. It will be shown in
Chapter IV that this also gives deviations for light gases and low
temperatures,

3.nbsp;When the attraction between the molecules is strong enough,
discrete quantum states of the relative motion of two molecules will
exist or, in other words, loosely bound quot;polarization moleculesquot; will
be formed. This will have an effect on the second virial coefficient
at low temperatures. Since this effect depends on the magnitude of
the attractive forces, it may become important even for heavier
gases.

The calculation of B{T) with a known V(r) turns out to be
much more difficult than in the classical theory, and the solution
of the inverse problem is almost impossible.

In our treatment of the equation of state we shall use throughout
Gibbs' method of the canonical ensemble. We shall present this
method directly in its quantum theoretical form.

Let the system whose thermodynamical properties one wishes
to investigate have energy states E, with weights G,. A large
number of independent, identical systems is called an ensemble.
This ensemble is canonical when at the temperature T the
relative numbers of systems in the different states are given by
Gi exp {—EilkT). where k is Boltzmann's constant. It is assumed
then that the behaviour of the system in temperature equilibrium
is given by the behaviour of the canonical ensemble in the sense
that the mean value of a macroscopic quantity may be obtained by
averaging this quantity over the ensemble, while the mean square
fluctuation is given by the average of the square of the deviations
from the mean value. From this assumption it can be proved that
the Helmholtz free energy,

The sum in (1), which is to be extended over all possible states, is
called the partition function (Zustandssumme) of the system An
unessential arbitrariness is yet left in (1) because of the fact that
the weights are only relatively defined numbers. This may be
removed by any convention about their absolute value. It is usual to
give to each non-degenerate state the weight unity. Then G denotes

simply the multiplicity of the state With the determination of
W as a function of volume and temperature the problem of finding
the equation of state is solved. Other quantities may be obtained
immediately from W by using simple thermodynamical formulae.

We shall now proceed to the classical analogue of (1). This will
be done in two steps. Let the system consist of N identical molecules.
Then the number of possible states is restricted by the symmetrical
or by the antisymmetrical exclusion principle. These are the cases
of Einstein-Bose (E. B.) or Fermi-Dirac (F. D.) statistics. It
will be often convenient, however, to consider systems for which all
the states which are forbidden by one of the two exclusion principles
exist. This case, which is of course purely academical, is called the
case of Boltzmann statistics. For this case it is desirable to choose
another convention for the absolute values of the weights, namely
to give to each non-degenerate state the weight IjN!, so that

when Gi means again the multiplicity of the state With this
convention the expressions (1) and (2) approach each other for
high temperatures ¹).

The second step in the transition to classical statistical mechanics
is to replace the sum in (2) by an integral over the phase space.
For a monatomic gas we have then, since to each single quantum
state corresponds a volume /i^^in phase space.

Here r,, ...Tat are the coordinates of the N particles, p,,.. . pjv their
momenta and H(pi,...p^, r,,...ta^) the Hamilton function of the
system. The integration has to be extended for the momenta over
all values and for each r^ over the volume V of the vessel in which
the system is enclosed. Since

energy of the system, we can perform the integration over the
momenta and find then

The integrand exp (—VjkT) represents the relative density of
probability in configuration space for the canonical ensemble.

A function which plays the same role in the quantum theory can
be easily found. Consider namely the expression

where the summation has to be extended over all the normalised
eigenfunctions 9?, *).
We have

It is clear that the expression (6), which we shall call the Slater
sum represents the relative density in configuration space for
the canonical ensemble, since (p* 9;,- represents this density for each
state. The strict analogue of the classical Boltzmann factor
exp (—VjkT) is the function

In the case of BoLTZMANN statistics we shall, in conformity
with (2), define the Slater sum by

where now the eigenfunctions 99, are no longer restricted by an
exclusion principle.

The eigenvalues and eigenfunctions of the system and therefore
also the Slater sum are completely determined by its Hamiltonian
operator H{pi,...pN, r,,...rAr), where now p should be understood
as the operator hd/lmdr^^ ¹).

It is possible to write the Slater sum in a form which clearly
demonstrates this fact.

1nbsp; In tiie following the classical HAMILTON function will be written ƒƒ, while
the corresponding operator will be denoted by H.

2nbsp; This definition should be used with some care. When i and g are non-
commuting operators, for instance the parts of H corresponding to the kinetic
and the potential energy, then

We shall now introduce a new complete orthogonal set of normalized
functions (r,.... r^). Then each may be developed in terms
of the u,^:

As the set u„ is quite arbitrary, (11) expresses the Slater sum in
the most general way.

§ 3. Connection between classical and quantum theoretical
expressions. Examples 13).

One should therefore expect S^ to be the limitof 5 either for ft 0
or for r CO, for in the last case the high quantum states play the
principal role. Before discussing this correspondence in general we
shall illustrate it by some examples.

a. As the simplest system we choose a single mass point which
may move freely on a line segment of length L. Then (12) becomes

In order to calculate the Slater sum we observe that the
normalized eigenfunctions are

f 2

(Pr,

U-

For large L the exponential factor changes slowly with n, while for
X not in the neighbourhood of the boundaries the factor sin^innxjL)
changes rapidly. We therefore may replace the latter factor by its
mean value Yi and replace the sum by an integral over n. This gives
5 — 5e . At the boundaries however the different terms are in phase
and we must now replace the unchanged expression by an integral.
This gives

for small x and the same expression with L—x instead of x at the
other end of the segment ¹). We see therefore that 5(x) is equal
to 5c(x) except in regions of the order of magnitude I at the ends
of the segment, where it drops to zero. As 2 0 for /z 0 or for
T 00 this shows clearly the correspondence mentioned above.

For a particle moving in a cubical box of more dimensions the
Slater sum is simply the product of (14) for each dimension.

b. Let us now consider a linear harmonic oscillator of frequency
V. The Hamiltonian is

jmv
' h

In the quantum theory we have the eigenvalues
E„ = {n i)hv, n = 0, 1,...
and the eigenfunctions

c. A simple example of a system of several particles is furnished
by two mass points which move freely on a line segment of length L.
Let and Xg be the coordinates of the two particles. Then

In the quantum theory we shall treat the cases of BoLTZMANN, E. B.
and F. D. statistics separately. In each case the eigenvalues are

In Boltzmann statistics each state with n^ ng is degenerate,
having the two eigenfunctions

which is again equal to S^ except when one of the particles is near
a boundary of the segment ¹).

Now the exclusion principle removes the degeneracy by allowing
only one definite linear combination of Q9lt;quot; and qjP' . In E. B.

F. D. statistics the antisymmetrical combination (gsj^quot;^ —95®^)/|/2.
The originally non-degenerate levels with nj = n2 remain unaltered
in E. B. statistics and disappear in the F. D. case. Therefore the
Slater sums are

(19)

L, nigt;n2

' (20)

L /

1nbsp; Like (14) this expression holds for x^ lt; LI2, x'2lt;LI2, whereas in the
other half of the segment we have to replace x^ by L — and :gt;;2 by L — X2.
The same will be the case in the expressions which will be given afterwards.

5e.b.
sf.d.

ti=i

5m ^

— W'K'timkTL'

2quot; e

2L2

5E.B.

= Sb±

— e

For the discussion of (22) we first remark that for / 0 it goes
over into as we should expect. The term exp { — ^2)2/22 }
is less important since it has finite values only when both particles
are at the boundary. The essential term is exp {—— Xo)^!?.^}
which causes a maximum in the E. B. case and a minimum in the
F. D. case for x^^xo. Therefore there is an apparent attraction
between E. B. particles and an apparent repulsion between F. D
particles.

Almost everywhere 5 is equal to Se. Deviations occur, however,
along the sides, especially at the corners, and along the diagonal

d. We turn next to the consideration of an ideal gas. The
essential characteristics of such a gas are illustrated by the preceding
example. Let the gas consist of N mass points of mass m, moving
freely inside a cube of side L and let rk(xk,yk,Zk) be the coordinates
of the k'^ particle. It is immediately seen that in the classical theory

In the quantum theory the cases of BoLTZMANN, E. B. and F. D.
statistics must again be discussed separately. For BoLTZMANN
statistics one finds easily

It is of interest here to discuss how the equation of state which
follows from (24) differs from the ideal gas law.
We have

P —

we obtain

the ideal gas law.

pV=NkT........(25)

NkT NkT

P —

This deviation from the ideal gas law is in all practical cases com-
pletely negligible. In helium, for instance, 2 = 0.755 X IQ-^/l/r cm
and so for a volume of 1 cm3 and a temperature even as low as
1° abs. the value of Vquot;'/» A/2 is only 0.38 X 10-7.

When we now turn to the E. B. and F. D. gases we may expect
that the deviations from the ideal gas law due to the deviations of
the Sl.^ter sum from Sc at the boundaries of the vessel are again
negligible and the only important deviations will be due to the
deviations of S from Sc when some of the particles are near together,
corresponding to the deviations along the diagonal in fig. 1. This
is the reason why we shall not start from wave functions which
fulfill the true boundary conditions but from the running waves

where k^ is a vector whose components can have all positive or
negative integral values. The eigenvalues are then

where P is any permutation of the indices k, Pk is the index which
replaces k after the application of P and the summation is to be

1 ^

k=\

np is the number of these eigenfunctions. In P. D. statistics we have
the antisymmetrical eigenfunction

In computing the Slater sum we can eliminate, as in the transition
from (19) and (20) to (21), the disagreeable factors Hp by summing
for each particle independently over all values of the k^. The
summation over the permutations has to be extended then over all
the permutations and we have to divide the result by N!. Replacing
the summation over k^ by an integration over p^. we obtain

In these sums over all permutations the term corresponding to the
identity gives Sb whereas the other terms represent the deviations
arising from the exclusion principle.

For later use we shall write down for the case of E. B. statistics
the function (rj,... r^) which, according to (7) corresponds to
the classical BoLTZMANN factor:

It is clear that We.b. -gt; 1 when all the particles are far away from
each other, whereas We.b. gt; 1 in all regions of configuration space
where some particles are near together. This corresponds to the
apparent attraction between the particles of an ideal E. B. gas, which
was already mentioned in example c. One should point out that
this attraction can not be represented by a potential which has the
property of additivity.

We shall now calculate in first approximation for the ideal E. B.
and F. D. gases the deviations from the ideal gas law. This first
approximation is obtained by taking only those permutations into
account which consist of the interchange of two particles. When
we observe that there are N{N~ 1 )/2 different simple interchanges
we find

Eq. (30) contains the well known expressions for the second virial
coefficients of an ideal E. B. and F. D. gas;

Comparison of (26) and (30) shows clearly the difference in
character of the deviations from the ideal gas law in the two cases.
In (26) the deviation is independent of the density of the gas, in
(30) it is proportional to it. The reason is that in the BOLTZMANN
gas the deviation is caused by the interaction of each molecule with
the walls of the vessel, giving, just as the main term, a contribution
proportional to N. In the E. B. or F. D. case, however, the deviation
is caused by the apparent interaction between pairs of molecules
which gives a contribution proportional to the number of pairs or
to N2.

In Chapter III, §§4 and 5, a treatment of the equation of state
of the ideal E. B. and F. D. gases will be given in which all the
permutations in the expressions (27) and (28) are taken into
account,

e. In the foregoing examples we have always treated the particles
as simple mass points. It often occurs however that the molecules
of a gas possess an intrinsic, constant angular momentum. We shall
not investigate here the influence of the small magnetic forces which
are associated with this spin, but only its influence on the weights
and symmetry properties of the different states.

Consider ^tyo identical particles with an angular momentum shjln.
This spin can have 25 1 different directions in respect to a fixed
axis, all with the same energy. In Boltzmann statistics, where the
two spins can choose their directions independently, each state is
(2s 1)2-fold degenerate and the partition function is simply
multiplied with this factor. Here the spin has no influence on the
thermodynamical properties of the system.

In E. B. and F. D. statistics on the contrary, the spin has a great
influence. First consider E. B. statistics. Without spin only states
occur whose wave functions are symmetrical in the coordinates.
When the particles have a spin, however, the wave functions must
be symmetrical in respect to a simultaneous interchange of the
coordinates and of the spin directions, whereas the orbital wave

functions need not be symmetrical. From the (2s1)2 different
spin functions in Boltzmann statistics one can form (s l)(2s 1)
symmetrica] and s{2s 1) antisymmetrical combinations. In order
to make the complete wave function symmetrical the first ones have
to be multiplied with symmetrical orbital wave functions, the
others with antisymmetrical ones. Therefore in the Slater sum the
states with symmetrical orbital wave function will appear with
weight (5 1) (2s 1) and the states with antisymmetrical orbital
wave function with weight s{2s 1). In this way we get

Here the upper index on 5 denotes the value of the spin.
In the same way one obtains for F. D. statistics

We remark that, when s is large, and are mixed in almost
equal proportions and we obtain practically the Slater sum for
Boltzmann statistics. One can express this by saying that the spin
has the tendency to diminish the influence of the exclusion principle.

For systems of more than two particles the consideration of the
spin becomes much more complicated. We remark only that the

§ 4. Connection between classical and quantum theoretical
expressions. General theory.

The examples in the foregoing paragraph all showed a correspon-
dence between the Slater sum and the classical expression in
such a way that the first went over into the second when the para-
meter which represents the de Broglie wave length corresponding
to a mean temperature motion, became small. For a general gas one
should expect the particles to behave almost classically when their
de Broglie wave length is small compared with the distances in
which their potential energy undergoes considerable changes. These

latter distances are clearly of the order of magnitude of the
dimension of the molecules, say d, where d is a sort of molecular
diameter. We therefore expect a development of the form

Kirkwood 15) has investigated the connection between 5 and Sc
for a non-ideal, monatomic E. B. or F. D. gas. We shall present his
treatment only for the E. B. case; the F. D. case is completely
analogous.

The theory is based on the general form (11) of the Slater sum.
The Hamiltonian of the gas has the form

Now a suitable choice for the set of wave functions has to be
made. In any case all the u„ must be symmetrical in the particles
for otherwise the inverse of the transformation of the eigen-
functions of the gas into the would not exist. We shall take the
symmetrical combination of running waves which was used in
example d of the foregoing paragraph. This means that the wall
corrections will be neglected. For the Slater sum one now obtains
(comp. eq. (27))

By differentiating (36) with respect to we see that F is a solution
of the so-called Bloch equation 16)

If in (35), instead of the operator li we had written the classical
Hamiltonian function H, we should have

which differs from S^ only by the occurrence of the apparent inter-
molecular forces due to the E. B. statistics. For V = 0 this is exact
and the same as (27). When we therefore put

w e

then w represents the other quantum effects. Introducing (38) into
(37) one finds for tv the equation

As we wish to see how S is approximated by it is appropriate
to develop w in powers of h:

We introduce (40) into (39) and compare equal powers of h. This
gives successively, after a simple integration,

If,

2 kZx

(41)

pi i N

3 f

1 / N

m \ k=i

8 71^ m

In this way all the may be obtained in succession. When we
now introduce this solution into (35) we find the following develop-
ment for 5:

•vj

•Pk

8 k' T'

k=i

In the case of Boltzmann statistics only the identical permutation
should be taken into account. Then

5B=5e 1

2- (Ai V-

(V. vy

T'kZi

2kT

This development has indeed the form (34) when we observe that
the first derivatives of V play the role of 1 jd. The further terms of
(43) will contain higher powers of h, higher derivatives and higher
powers and products of the lower ones.

This method of obtaining successive approximations for 5, starting
from Sc. shows a great resemblance to the W. K. B. method of
solving the schrodinger equation. But the latter method is not

directly applicable to our problem. In Chapter IV we shall however
encounter a case where either Kirkwood's method or the W. K. B.
method may be applied, both methods giving identical results.

The development (42) is of course only valid when the function V
is differentiable any number of times. It will converge rapidly when
/I Vt VjkT, PAk VjkT, etc. are small compared with unity, that is
to say, when V, measured in units kT, does not change much over
the distance X.

In this paragraph a property of the Slater sum will be discussed
which is fundamental for the theory of the next Chapter.

Consider the function Wa, = W (rj,.., r^^) as defined in (7) for
gases of resp. 1, 2, ... A^ ... particles in the same volume V. The
Wn are symmetric functions of the arguments r,,... r^^. Furthermore
Wi = 1 for a monatomic gas without external forces ¹).

The product property may now be stated as follows. When we
divide the particles into different groups containing a^, «2' •••
particles, then for configurations where particles of different groups
are so far away from each other that their interaction (including
the apparent interaction due to E. B. or F. D. statistics) vanishes,
we have

This property is a consequence of the fact that in these configura-
tions the Hamiltonian is separable into the sum of the Hamiltonians
of the different groups. In the classical theory (44) follows immedi-
ately from this fact. In the quantum theory it is easiest to consider
eq. (35) (or the analogue for F. D. statistics). For Boltzmann
statistics, where only the identical permutation must be taken into
account, our property is then again immediately clear. For E. B. and
F. D. statistics observe that the integration over p^^ leaves zero when
the points r^ and Vp^ are far away from each other. Therefore only
those permutations give a non-vanishing result where the k^^ and
the P/c'''particle are in the same group. But then the integrand of

1nbsp; As follows from § 3, example a, this is true exactly in the classical theory
and almost exactly in the quantum theory.

(35) may be written as a product of the analogous expressions for
each group and therefore also the integral has the product property.

It should be stressed that for this proof it is not necessary to
assume the additivity property of the intermolecular forces *). The
function W^ has in general not the property that it can be split into
factors, each containing only a pair of molecules, as is the case in
the classical theory when the forces are additive.

In this chapter we shall investigate how far the qualitative features
of the equation of state of gases and liquids may be understood by
means of statistical mechanics. The main properties in which we
are interested are illustrated by fig. 2 where two isotherms of a pure

substance are drawn, one for a temperature above the critical
temperature T^ and the other for a temperature below T^. For large
volumes (small densities) both curves are shaped according to the
ideal gas law. It is clear theoretically that this should be so, because
at small densities the interaction between the molecules may be
neglected. For smaller volumes deviations from the ideal behaviour
occur, due to the intermolecular forces. These deviations are quite
different for the two temperatures. For T gt; T^ the isotherm is
always a smooth curve while for Tnbsp;the curve consists of three

second horizontal part representing the saturated vapour in equili-
brium with the liquid, and a third one representing the liquid.

The problem can now be stated as follows: Suppose one has N
monatomic molecules of mass m in a vessel of volume V, then the
free energy W(V, T) is determined by:

(1)

where W(r,,...rj^) and / are defined by (11,7) and (11,5) and
where the integral over each r/, has to be taken over the volume V.

The question now is whether one can prove from (1) that at suffi-
ciently low temperatures p as a function of V consists of three
analytically different parts.

a.nbsp;As was already mentioned in the first chapter, the Van der
Waals theory does not give a solution of this problem, since here
the real stable isotherm is not derived directly from the integral (1).

b.nbsp;One might think perhaps that this stable isotherm cannot be
derived from the integral (1) without further assumptions. One
argues then that (1) gives the free energy for one homogeneous
phase. One should make a separate calculation for the case when
the system consists of two phases. For each volume one would obtain
therefore two values for the free energy, corresponding to one or
to two phases, and the real isotherm would be determined by the
lowest value of the free energy. In our opinion this argument is not
correct. The integral (1) contains all possible states of the system *)
and the W which one calculates from (1) will describe the most

probable state, which is the state of stable equilibrium. The real
stable isotherm should therefore follow automatically from (1)
without further assumptions.

c.nbsp;On the other hand, from the mathematical standpoint, it is
hard to imagine, how from (I) it can follow that W (and therefore
p) as a function of V consists of three analytically different parts.
It seems to us that this is possible because we are really only
interested in a limit property of W. The problem has a physical
sense only when N is very large. One may expect then that for a
fixed specific volume

for V ^ CO, N CO, VjN = v fixed, will exist. Strictly speaking the
property mentioned above has to be proved for xp as function of v.
It is not surprising that this function can consist of analytically
different parts.

d.nbsp;One may remark that the stable isotherm does not represent
all states of the system which are realizable. There is for instance
the well known phenomenon of supersaturation, which is represented
by the continuation of the vapour part of the isotherm beyond the
point of condensation. These states however are not states of stable
equilibrium, except in vessels which are everywhere so narrow that
capillary phenomena become of importance. Since we shall discuss
the integral (1) only for the

case that the vessel becomes large in
all directions, it is clear that we shall not obtain the states cor-
responding to the supersaturated vapour.

Ursell 3) has shown that it is possible to write the integral (1)
as a polynomial of the degree in the volume. His procedure was
based on classical statistical mechanics but can easily be extended

in order to include the quantum theory. One introduces a set of
functions t/(ri,r2,... r;) = Ui, which depend symmetrically on the
coordinates of / molecules in the volume V. They are expressed in
terms of the probabilities W^, W2, ■■■ W; by means of the relations:

and so on. The general rule is the following. We divide the I
particles which occur in Wi into a number of groups, and form the
product of the functions U, which depend on the particles of these
groups. Then Wi will be the sum of these products for all possible
ways of division of the / particles. The Ui in terms of W^, Wg,... W;
are uniquely defined by these relations. One finds for instance:

U(r,.r2, r,)= W(r„r„r^) ^(r,)-^(r^. r,) W{r,)- (5)
-W{r,. r,) W{r,) 2 W(r,) W{r,) W(r,). '

The rule which expresses the U in the W, is the same as the
rule which expresses the W in the U, except for a coefficient
(— ' (k—1).', when k is the number of groups into which the I
particles are divided. This will be proved at the end of this para-
graph.

The importance of the development (4) lies in the following
fundamental property of the functions Ui. When we divide the I
particles into different groups, containing /Sg, ... particles, then,
for configurations where particles of different groups are so far
away from each other that their interaction vanishes, we have
U, = 0. Less exactly one may say that Ui is different from zero only
when all the particles are near each other ¹).

The proof of this theorem follows from the product property of
the Wn which has been explained at the end of the second chapter.

1nbsp; Tills does not mean, however, that each particle is interacting with all
others, but only that all particles are linked together.

For U^ for instance it can be verified immediately from the explicit
expression (5). In this way one could give a general proof. It is
simpler however to consider the configuration mentioned in the
product property of W^. Develop both sides of (II, 44) according
to (4). The right hand side will then contain no U, referring to
particles of different groups. The sum of those terms on the left
hand side, which contain [// of this kind, must therefore be zero for
this configuration. By applying this argument successively to W»,
Wg, etc., one shows by induction that each Ui of this kind must be
zero, as the theorem requires.

A consequence of this theorem is that the integral of t/, over the
coordinates of the I particles will become proportional to the volume
V, when V is very large. To see this first perform the integration
over the coordinates of /—1 particles, keeping the coordinates of
the particle fixed. Because of the fundamental property of U, the
result will be independent of the volume and independent of the
position of the particle, provided that V is sufficiently large and
ill approaches zero sufficiently fast when the /— 1 particles are
separated from the particle. The integration over the coordinates
of the particle will then contribute a factor V to the integral.
We shall write:

(6)

It will be clear now that by integrating the development (4) for
Wn, one will obtain for Qjya polynomial of degree N in V. The
result can be written in the following form:

The mi are positive integers or zero. The summation sign means
that one has to sum over all sets of values of the m,, which fulfill
the condition:

To prove this, consider a definite partition of N in m^ groups of
one particle, mg groups of two particles, and so on. The m; will then
clearly fulfill (8). To a definite set of values of mi correspond many
terms in the development (4), due to the different ways of distri-
buting the N particles over the groups. All these terms will give the
same result after integration, namely:

since the permutation of particles in one group and the permutation
of groups of equal size will not give rise to new terms. By multi-
plying (9) and (10) and by summing over the m; one obtains (7).

Sometimes it is useful to write Ursell's development (7) in a
different form, namely

Here the a, are positive non-zero integers and the round summation
sign means that one has to sum over all sets of values of the a,-
which fulfill the condition:

To prove (11), first consider the partition of N into the two groups
ai and 02. Then ai and 02 fulfill (12) with k — 2. To this partition
corresponds again a number of terms in (4) which give the same
result after integration, namely

In order to obtain the contribution to Q^; of all partitions of N into
two groups we can sum over all integral values of a^ and «2 which
fulfill the condition (12) when we observe that each partition with
«1 ^ «2 will occur twice in this summation. The contribution of these
partitions to Q^ is therefore

The extension of this reasoning to partitions into more than two
groups gives immediately (11).

This means that Q^ is equal to the coefficient of t^ in the expansion
of (13) in powers of t ¹).

For the proof of (13) we may start either from (7) or from (11).
If we multiply each ^ in (11) by i«- then we may sum over al]
integral values of a,- independently and afterwards fulfill the con-
dition (12) by talcing the coefficient of t^ in the resulting expression.
The generating function is therefore

kgt; N give rise only to powers of t higher than the N^^ one. This
gives immediately the expression (13).

From (13) we can find the expression of b, in Q,, Q2, ... Q,
and therefore the coefficients in the expansion (5). The expression
of Q^f by means of (13) can be written as

One finds easily that this is equivalent to the following expression
for V bI, which is analogous to (11):

where the round summation sign means that one has to sum over all
sets of values of the /S, which fulfill the condition

Comparison of (16) and (11) leads immediately to the values of
the coefficients mentioned after eq. (5).

Mayer has shown how to derive from the development of Ursell
a general expression for the equation of state of the vapour phase.
We shall give here essentially his first derivation which, although
not rigorous, is very simple and gives the correct result. In § 6 an
exact proof will be given.

Suppose that all b, are positive, (which they probably are at suffi-
ciently low temperatures), then for large N one may approximate
the sum (7) by its largest term. To find the set of m, which gives
this maximum term, one proceeds in a way which is quite analogous
to the usual derivation of the Maxwell-Boltzmann distribution
law in statistical mechanics. One then finds, using Stirling's
approximation for m,.', that this maximizing set of m, is given by:

In order to find an explicit expression for the equation of state,
one has to eliminate z between the equations (a) and {b). Born i7)
and Mayer have given a formal solution of this problem. Their
result can be stated as follows ¹). Define a function

in such a way that P b, is equal to the coefficient of f'-i in the
expansion oi exp {Icp). This gives the bi expressed in terms of
the [iy. One can solve these equations successively for and thus
express the uniquely in terms of the With the help of the

(19)

(-3 18^2^3-20 bl) ...
This has the form of the expansion in virial coefficients (I, 3) with

(22)

These expressions for the virial coefficients were already known
in older theories and may be obtained directly from Ursell's
expansion ¹).

All the other thermodynamical quantities can easily be expressed
in terms of z and the b The following expressions are found:
Free energy

(24)

(25)

C = 2 (N—l) (2N—3) 2 (N—1) (N—2) 63
which coincides with (21) and (22) for large N.

(27)

Cy=T

(28)

1 = 1 d 1 ^

1 '=1

(29)

V\dp

The equations (a) and {b) show a remarkable analogy to the
equation of state of an ideal E. B. gas as given by Einstein 4). He
obtained:

(a,)

Eqs. (a) and [b) become identical with (a^) and [b,) by putting
^ =nbsp;= .......(30)

This analogy is especially of interest, since ElNSTElN has shown that
the equations (a^) and (b,) give rise to a condensation phenomenon.
Furthermore the case of the ideal E. B. gas furnishes an example
where the bi, which are characteristic for the behaviour of a real
gas, can be determined explicitly.

The equations (a^) and (bi) were derived by Einstein from
the E. B. velocity distribution law. According to this law the number
of particles with velocity components between | and f nbsp;and

In this form of the distribution law the quantization of the transla-
tional motion is neglected. The parameter A is determined by the
condition

. . . (32)

The equations (aj and are then obtained by developing the
integrands of (31) and (32) in powers of A exp {—ElkT), which
is possible for A lt; 1, and by integrating term by term.

An alternative derivation of Eqs. {a^) and (t^) will show more
clearly the origin of their analogy with (a) and (b). In chapter II
we have found the expression (II, 29) for the function W^ of an
ideal E. B. gas:

The integral Q^ can now be written in the same form as Ursell's
development (7). Observe namely that by integrating one term of
(33), corresponding to a definite permutation P, overdr^...dr one
obtains a power of V which is equal to the number of cycled into
which this permutation can be decomposed. The sum (33) is there-
fore analogous to the development (4) of in the U,. Suppose
that the permutation P can be decomposed into cycles of one
particle, m.^ cycles of two particles and so on. The m, will then
again fulfill the condition (8). To a definite set of values of m,
there will correspond many terms in (33), each of which gives the

N! n

This is different from (10) because only the / cyclic permutations
of the particles in one cycle will not give rise to new terms. To
obtain Qjv in exactly the same form as in (7) we must write V lb,
for the integral over the coordinates of the particles of a cycle of
length I. Therefore:

where ry.= |r,—The contribution of a term of (33) correspon-
ding to a definite set of values of m, will then be

The integral (35) can be performed straightforwardly¹). One
finds then for fc, the result (30) and we have already seen that with
this value of the equations (a) and {b) of Mayer become the
equations (a^) and (fe^) of Einstein.

The reasoning by which Einstein derived the condensation
phenomenon for an ideal E.B. gas from Eqs. (a^) and (fc^) is as
follows. For small values of the density the corresponding value of
A will be small. By increasing the density A will increase mono-
tonically. This goes on until for a finite value of the density, A
reaches the value one. Then:

For A gt; 1 the series (aj) and (bi) diverge. According to Einstein
Ijvc is the maximum density which can be reached. By further
compression of the gas, the superfluous particles will quot;condensequot;'
into the state of zero energy and will not contribute to the pressure
nor to the density so that the pressure will remain p,. We have
therefore indeed a kind of condensation phenomenon, which has
however some uncommon features, for instance:

In order to get a closer idea of this condensation phenomenon
we shall give the expressions for some of the thermodynamic
quantities in the vapour phase and in the region of equilibrium

1nbsp; See Note 3. Tiie same integral occurs in a paper of KRAMERS^®)
ferromagnetism.

'A3 V

4 r

3
'2

-0

The differences between the values of the thermodynamical quan-
tities for the two phases are obtained from the last group of formulae
by putting V=V^ and V = 0 respectively. We see that f is equal
for both phases, which is the thermodynamical requirement for each
phase transition. Furthermore this phase .transition is one of the
first kind since the volume and the entropy are different for the two
phases, the differences being V^ and 5 p, VdlT respectively. From
(38) follows

which is Clapeyron's equation. On the other hand we remark two
other points in which this phase transition differs from the con-
densation of a real gas, namely

1nbsp; The continuity of C^ has been remarked by london i»). His conclusion
however, that this would mean that we have to do here with a phase transition
of the third kind, seems to us incorrect when one uses the term in the sense given
by Ehrenfest''quot;).

have drawn some isotherms of the ideal E. B. gas, according to
(ai), (b,) and (38).

The reasoning by which Mayer first derived the condensation
phenomenon for a real gas is quite analogous to the argument of
Einstein. It can be expressed as follows. Suppose that the series
(a) and {b) have a certain finite convergence radius z, and that
they are still convergent for z = z ¹).

When in addition the bi are positive, then by increasing the
density the corresponding value of z will increase monotonically till
the maximum value z = Fis reached. For higher densities the series
(a) and {b) cease to have significance. Using a physical inter-
pretation of the equations, Mayer tries to show that by further
compression of the vapour condensation will occur, while the
pressure remains constant and equal to the value

1nbsp; In fact, Mayer tries to prove that for large I the bj become asymptotically

equal to bg'^/Pk, which would make the analogy with the ideal E. B. gas still
closer. This result of Mayer seems to us incorrect (see § 7); it does not affect
however his explanation of the condensation phenomenon.

These explanations of the condensation phenomenon are certainly
not yet complete. An instructive example is furnished by the ideal
FERiVlI-DlRAC gas. One shows easily that for this case the equation
of state is determined by

when A ^ 1. At first sight one might think that these equations will
also predict a condensation phenomenon. This however is not cor-
rect. Although the series (ag) and (62) are convergent only for
A^l, they represent analytic functions of A. which can be continued
along the positive real axis for all values of Agt; 1, so that the
pressure will be given by one analytic function of v for all values
of the volume. This can also be seen directly when one derives the
equations (ag) and (62) in analogy with Einstein's derivation of
(ai) and (fej. The formulae for N and e are obtained from (31)
and (32) by replacing the — 1 by 1 in the denominator. These
integrals, in contrast with (31) and (32), are convergent for all
positive values of A. Only for A ^ 1 they give rise to the series
(as) and (62). This is in contrast to the case of the ideal E. B. gas,
where the point A=1 is a singular point of the functions (a^)
and (bi). It can be shown (see § 6) that for the real gas also an
essential condition for condensation is that the functions (a) and
(b) have a singularity

Another objection against the reasoning of Einstein has been
raised by UhlenbecK 21). The Eqs. (a^) and {bx) are derived
namely by neglecting the quantization of the translational motion
of the molecules. Instead of the integrals (31) and (32) one should
write sums over all the possible discrete states. In the alternative
derivation this corresponds to the neglect of the influence of the
walls of the vessel in the calculation of b,. This neglect is justified
for small values of A, but it becomes dangerous in the neighbour-
hood or A = 1. In fact, when we regard the sum which should be
used instead of the integral (31), we see that the term for the lowest
state (which may always be taken as the zero for the energy scale)

becomes infinite for A t= 1 and therefore the number of particles
does not remain finite when A approaches 1, so that the above
reasoning fails. It remains true however that for densities larger
than 1/fc the isotherm will be almost horizontal, and that this will
be more pronounced the larger the volume is. Only in the limit
V ^ 00 will the isotherm consist of two different parts. When we
always understand the condensation phenomenon in the sense of
such a limit property (comp. remark c of § 1) then this objection
therefore loses its validity.

We shall start from Ursell's development, but instead of making
the approximations of § 3, we shall now determine strictly for
N 00, V ^ 00, V/N — v finite

which according to (1) and (3) is equal to Pexp(—ylkT) ¹). Using
CaucHY's theorem of residues the expression of Ursell's develop-
ment by means of the generating function (13) can be written in
the form

where the integral has to be taken around the origin of the complex
^-plane, excluding the singularities of xi^)- The function x(t) is
the fundamental function of the problem.

The first method which presents itself for treating the integral
(41 ) is the method of steepest descents **). We have found another
method, however, more appropriate for the discussion of the
limit (40).

1nbsp; The Q^ are of course all positive, since they are integrals over the pro-

where the different Q;v have to be taken for the same value of
VjN — v¹). According to the well known theorem of Cauchy the
limit (40) is now just equal to the inverse of the radius of con-
vergence R of F{x). On the other hand the series (42) defines an
analytic function of x. and we can therefore find the convergence
radius R by determining the singularity of this function which is
nearest to the origin.

Another expression of F(x) can be found as follows. Introduce
(41) into (42); then the summation and integration can be inter-
changed when X is so small that on the whole contour

It is always possible to find such an x because \^{t)\ is bounded on
the contour. The summation of the geometric series gives then

The integral can be calculated by means of the theorem of residues
The only poles of the integrand within the contour are f = 0 and
the zeros of the function l~xi{t). For sufficiently small x this
iunction^ has only one simple zeropoint. One sees this from
Cauchy's integral. The excess of the number of zeros over the
number of poles of 1—within the contour is given by

2.711

1nbsp; The series (42) is completely different from the series (15) where all O
were taken for the same value of V.nbsp;v ; c an

The integral has always a certain finite value, so that n — p can be
made as small as one pleases by taking x small enough. Therefore
n — p must be zero, since it can assume integral values only. Now
1—x^{t) has one simple pole within the contour, namely ^ = 0,
and therefore also one simple zero, say at ^ = ^o• The evaluation

If we start from the origin and move along a definite path in the
x-plane, the equation (46) will determine t^ as an analytic function
of X. Along the corresponding path in the fo'Plane ¹) /(fg) will be
again an analytic function of f,, and therefore of x. In this way (45)
defines an analytic function of x, which for small x coincides with
the power series (42), and which therefore represents its analytical
continuation.

We have now to determine the singularity of F(x) which is
nearest to the origin. One needs to consider only real positive values
of x, because of the fact that the series (42) has real and positive
coefficients, so that the real positive point on its circle of convergence
will be a singularity of F(x) 22). Qne sees from (45) and (46) that
the possible singularities of F{x) are the values of x which cor-
respond to the zeros of the denominator 1—and which
correspond to the singularities ofnbsp;Whether these values of x

are actually singularities of F(x) and which of them is nearest to
the origin depends on the properties of the function /(tg) and on
the value of v. We know that for small values of to. /J^o) ^ ^o- We
shall assume further:

has a singularity z on the real positive axis; i may be
greater than or equal to r. The latter case will occur for instance
when all the b' are positive.

c. X (z) and / (z) are finite; the point 2 will therefore be a branch
point of ;t;(?o).

^oZ'(^o) is monotonically increasing on the real axis between
tQ — 0 and to=z*); this will again be the case when the b, are
positive.

To find the singularities of F(x) we shall start from the origin
and move along the real positive axis in the x-plane. When v is
large enough one sees that the first singularity of F(x) which one
meets will be determined by the zeropoint of the denominator in
(45). Let us call this zeropoint to=z, so that

The corresponding value of x, and therefore the radius of con-
vergence R, according to (46) will be

This is therefore the inverse of the limit (40), from which one
immediately obtains the expression for the pressure

The equations (47) and (48) are identical with the Eqs. (a) and
{b), which are now therefore rigorously proved when v is large
enough. It should be pointed out that for this proof none of the
assumptions a—e are necessary. Because of assumption e¹) the
equations (47) and (48) will remain valid, until

In the case that z gt; r, and for values of z between r and z, the
equations (47) and (48) are no longer identical with the series
(a) and {b), but represent their analytical continuations. Foi; all
these values of v the pressure remains a smooth function of the
volume.

1nbsp; When this assumption is not fulfilled, so that tox'(ia) has at least one
maximum (say at fo = then for a value of v corresponding to z =7, (dp/dp)j.
becomes infinite (comp. eq. (29)). This has been pointed out by BORN and
FUCHS=^®). It gives the physical reason for the assumption e.

is smaller than v^. When we again move along the real positive
x-axis, we shall reach the point corresponding to tQ!= z before
meeting a zero of the denominator of (45). Because of assumption d
this value of x will be a singularity of F(x). In this case therefore

When z—r this is identical with equation (39) of Mayer. The
pressure as a function of v consists therefore of two analytically
different parts, namely the curve represented by (47) and (48) for
vgt;v^ and the horizontal line (50) for vCv^.

10. In § 3 we have introduced the quantities /S^ in a formal way
in order to perform the elimination of z between the equations (a)
and (6). Mayer and BoRN were led to these quantities by the
consideration of the integrals defining the in the case of classical
statistical mechanics. They showed that these integrals can be split
up into the sums of products of certain quot;irreduciblequot; integrals, which
are immediately related to the /S,. We have not been able to
generahze this physical interpretation of the for the quantum
theory.

20. Mayer and Born have tried to derive from the expression
of the bi in terms of the /S„ certain general properties of the
characteristic function In particular Mayer has tried to make
plausible that in a certain region of temperatures below the critical
temperature, b; behaves asymptotically for large I as

One would obtain this by writing the expression of fe; in terms of
the /Sr in the form

and applying the method of steepest descents to this integral. A
consequence of (51) is that {dpldv)T becomes zero at the con-
densation point v = just as in the case of the ideal E. B. gas.

These considerations seem to us very doubtful, since they are
based only on the formal expression (52) for the b,. The physical
interpretation of the is nowhere used. Since the quantities ^^
can be defined uniquely by means of (52) for each arbitrary set
of quantities b,, this would mean that each infinite set would behave
asymptotically like (51). This is of course nonsense. It is clear that
one can hope to make a further advance only by going back to the
physical meaning of the or the A.. In particular it seems to us
impossible to say anything in general about the behaviour of
(dpldv)T near the condensation point. This will depend on the series

which may be divergent or convergent. In the first case (dp/dp )r
will be zero (or v=v^ while in the latter case it will have a finite
value.

30. From the further investigation of the integrals representing
bi must follow especially the properties a—e (§ 6) of x{z). which
are necessary to explain the condensation phenomenon. An essential
difficulty seems to us to lie in the fact that even with the assumptions
a—e oi it is impossible to obtain the third part of the isotherm,
corresponding to the liquid state. The reason is that for all v lt; v,
the singularity of F(x) which is nearest to the origin, is determined
by the singularity z of x{z), which is independent of y. Therefore
the isotherm will remain horizontal for all v

It is clear that the origin of this difficulty has to be found in the
neglect of the dependence of the b, on the volume V. It is true of
course that for every finite / the quantity b, has a definite limit
for V 00. We have assumed however more than this, since the
properties of ;;(z) depend on the behaviour of fe, for large I. There
is clearly a double limiting process involved and it may be that the
solution of the difficulty will be found by a more correct treatment
of these limits.

The foregoing chapter was chiefly devoted to an explanation of
the qualitative features of the equation of state. However we have
also shown that the isotherms in the gaseous state are exactly given
by the development of Kamerlingh Onnes and we have obtained
expressions for the virial coefficients in terms of the integrals b,,
such that the nquot;' virial coefficient contains the b, up to /1= n. This
gives us the possibility of treating the last problem mentioned in
the first chapter, namely the investigation of the intermolecular
forces from the equation of state. For, in order to calculate the nquot;'
virial coefficient, we do not have to consider the whole gas but only
systems of at most n particles. If this calculation in terms of a
certain assumed intermolecular potential is possible, comparison
with experiment can decide whether the assumed potential is a good
approximation. For this purpose the second virial coefficient is best
suited, because its calculation is simplest and the most reliable
experimental data are obtained for this coefficient. Many investi-
gations have been performed on these lines, most starting from the
classical expression for the second virial coefficient. It will be our
principal purpose to investigate the influence of the quantum theory.
According to the general considerations of the second chapter, this
influence will be largest for low temperatures.

In chapter III, eq. (21) we found that the second virial coefficient
was equal to

where V(r) is the potential energy of two particles at distance r.
We now introduce the coordinates of the centre of gravity and the
relative coordinates of the two particles. Integration over the first
gives a factor V, For the relative motion we introduce polar
coordinates. Then the angular integration may be performed and
we get the well known expression

This expression has the great advantage that it can always be
evaluated, at least numerically, when the potential V{r) is known.
This is not the case with the quantum theoretical expression which
will be derived now.

can be written as two times the partition function of an ideal
Boltzmann gas of two particles. Let us first consider the Boltzmann
gas. Then (comp. (11,2))

where the first term refers to a system of two noninteracting
particles. Both terms can be written as the product of two partition
functions, the first referring to the translation of the centre of
gravity, the second to the relative motion. The first factor for both
terms is equal to l^l^Vjl^ (the factor lquot;!-' arises from the fact that
the mass of the whole system is 2m) and therefore

where now the partition functions refer to the relative motion only.
We shall introduce polar coordinates r, lt;fgt; and group the states
according to the radial, azimuthal and magnetic quantum numbers,
n, I, m. The states which differ only in m have the same energy,
so that each state has a (2Z l)-fold degeneracy. Therefore

where Vnijr and f® /r are the radial parts of the wave functions. In
addition to these equations there is a boundary condition for the
wave functions, because the particles are enclosed in a finite vessel.
Since the final result for the virial coefficient is always obtained as
a limit for infinite volume we may choose the boundary condition
arbitrarily, if only it is the same for (5) and (6). As the simplest
choice we shall assume the v„i and ff/ are zero for r = R,
where R is large compared with the range of V(ir). Without such
a condition the spectrum of (6) is entirely continuous. But the
boundary condition at r = i? will make of this a finely spaced
discrete spectrum. On the other hand, eq. (5) may have some
discrete eigenvalues when V(r) has a strong enough attractive part.

corresponding to the formation of quot;polarization moleculesquot;, but the
largest part of its spectrum will be continuous. With the condition
at r ƒ? this part will also be discrete ¹). It will be clear that this
discrete spectrum is completely determined by the values of the
wave function for large r.

For large r this solution is proportional to

sin kfj r

( (0) ' ^

Eq. (5) is for large r identical with (6). Its solution for large r is
therefore a linear combination of the two fundamental solutions of

The quantity tj^in) represents the phase shift of the wave v„,
compared with the free wave yW. It is determined by the condition
that v„i becomes zero at least as r for r = 0. The quot;continuousquot; eigen-
values of (5) are determined by

1nbsp; For convenience we shall refer to this set as to the quot;continuousquot; spectrum
in contradistinction to the original discrete states which are not affected by the
boundary condition.

Now we can transform (4) by changing the summation over the
quot;continuousquot; part of n by an integration. Observing that, because of
(8) and (10), in the first term of (4)

gt;. . . (12)

The negative quantities are the discrete eigenvalues of (5).
We shall now consider the E. B. gas. Instead of (3) we have now

The first term is the same as in (3), but the second is different. The
separation into translational and relative motion is the same as
before. In the summation over the different states of relative motion
we must take only those states into account whose eigenfunctions are
symmetrical in the two particles. Now interchange of the particles
means replacement oi q) hy cp n and of ^ by n — ■d. As is well
known, the angular part of the wave function remains unchanged

under this substitution for even I, whereas it changes its sign for
odd I. Instead of (4) we get therefore

If the gas, for which we want to calculate B, were an ideal E. B. gas,
then we should have to replace the in the second sum of (15)
by But in this case we already know the expression for B
(Ch. II, eq. (31)). Therefore (15) may be written as

The general formulae (11), (12), (16) and (17) for the second
virial coefficient have been found by Uhlenbeck and Beth 24).

We may generalize these expressions to include the case where
the particles possess an angular momentum shjln. In Boltzmann
statistics, as we have seen in Ch. II. the spin has no influence. For
E. B. and F. D. statistics we obtain easily from the considerations
in Ch. II, § 3, example e

where and are the expressions (16) and (17). For large s
both (18) and (19) reduce to Bb.

The expressions (16) and (17) for B consist of three parts which
each have a definite physical meaning. The first term represents the
effect of the apparent attraction in an E. B. gas or of the apparent
repulsion in a F. D. gas. In the second term the part containing
•B/, dis^T. shows how the pressure is lowered by the diminishing of the
number of independently moving particles, due to the formation of
polarization molecules. It is in fact possible to derive this term
directly according to this interpretation by means of the formulae
for dissociative equilibrium. The part in the second term which
contains Bisect, represents the effect of the collisions between the
molecules. As we have seen, it depends only on the asymptotic
behaviour of the wave functions of the relative motion of two
particles, represented by the phase shifts t]^{k).

The calculation of B consists therefore in the calculation of the
discrete eigenvalues E„i and of the phase shifts »^/(fc). These latter
quantities also play a fundamental role in the theory of atomic
collisions, and it is in connection with this theory that the existing
calculations on the f]j{k) are performed. For this reason we shall give
in the following paragraph the fundamental formulae of the collision
theory, together with the formulae for the transport phenomena in
gases, which are closely related to them.

When a beam of particles, moving with a velocity v, falls on a
particle at rest, many of them will be deflected in different directions.
Suppose the density of the beam to be such that an area of 1 cm2
perpendicular to the beam is traversed by one particle per second.
We shall call /(^) dœ the number of particles which is scattered
per second into a solid angle dm in a direction which makes an
angle with the original beam. The number of particles scattered
between ê and ê dê will be called the differential cross-section

The problem of collision theory is to calculate ƒ(#) and a as
functions of the velocity when the potential between the interacting
particles is known.

We shall suppose that the scattering and scattered particles have
the same mass m but are nevertheless distinguishable for the
moment *). The expression for the differential cross-section will be
given for the system where the centre of gravity of the two inter-
acting particles is at rest. In this system, which is moving with a
velocity vjl, the two particles move before and after the collision
with the equal and opposite velocity y/2. The angle of scattering in
this system, d, is twice the corresponding angle -d' in the original
system, as may be seen from a simple geometrical consideration.
Hence the results obtained for the differential cross-section in the
moving system can be transformed to the original one. The total
cross-section is of course the same in the two systems. In the original
system the angle between the directions of the two particles after
collision is 90°.

Plicos 0) are the Legendre polynomials and r]j(k) is the phase shift
in the radial part of the wave function of the relative motion of the
two particles, as defined in eq. (9). From (22) one obtains for the
total cross-section

We shall now give the formulae for the case when the exclusion
principle is taken into account. Because the two interacting particles
are indistinguishable the cross-sections will now refer to the pro-

bability of finding a scattered particle, and it will not be possible
to conclude whether this particle is the incident particle or the one
originally at rest. For particles without spin one obtains

For particles with spin s, one has (if the spin directions of both
particles are isotropically distributed)

The values of the transport quantities in gases can be expressed
in terms of the collision cross-sections 27), One finds in first
approximation for the coefficient of viscosity in E. B. and F. D.
statistics

From these results we see that in the theory of collisions and of
transport phenomena the problem is the same as in the theory of
the second virial coefficient, namely the calculation of the phase
shifts ri^{k). It is therefore possible in principle to correlate the
experimental data on the second virial coefficient with the results
of colhsion experiments and with the measurements on viscosity
and heat conduction.

For a general potential V{r) it is not possible to give an exact
analytical expression for the quantities rj^ik) and we have to make
use of numerical or graphical calculations or of approximation
methods. First we shall try to get a general idea about the shape of
as a function of k. This function was defined as the difference in
phase between the asymptotic solutions of the one dimensional
Schrodinger equation

(35)

both solutions being determined by the condition that they become
zero at least as r for r 0. Because is a term in the phase of the
wave function, and the sign of the wave function is arbitrary, one
may always add an arbitrary multiple of n to its value. Of course
this arbitrariness has

virial coefficient and for the collision cross-sections. Therefore this
arbitrary constant may be fixed in any convenient way. We shall
do it always in such a manner that r/^ becomes a continuous function
of k. When the potential V(r) is finite everywhere it is possible for
instance to fix rj^ in such a way that it approaches zero for k large,
because then the disturbance of the wave function by the field of
force becomes small.

Before we examine a V(r) of the type which actually occurs
between two molecules it will be useful to give the values of r]^ when
the molecules are considered as noninteracting elastic spheres of
diameter q. In this case rj^ can be calculated exactly. The potential
is now

The solution of (34) will be that linear combination of the two
fundamental solutions of this equation for V(r) = 0 which is
determined by the condition of being zero for r = £gt;. The funda-
mental solutions are

The actual potential V(r) consists of a strong repulsion when r
is small which for larger r goes over into a short ranged attraction.
The effective potential in equation (34) is for Z 0 the sum of this
V{r) and of the potential of the centrifugal force. This effective
potential has therefore the shape of fig. 4. We can now find the

first the case I ^ 0. When k is very small the wave function will
be negligibly small in the region where V'(r) is large, because the
centrifugal repulsion acts like a potential barrier. The wave function
is therefore practically undisturbed by V{r) and with a suitable
choice of the arbitrary constant it becomes zero for A: = 0. In fact
we shall show in the next paragraph that tj^ik) behaves as the
(2 / power of k for small k. When k is somewhat larger the
wave function will be perturbed by the attractive part of V(r). This
will make positive because in regions where V(r) is negative the
wave function oscillates more rapidly than the corresponding wave
function for V(r) = 0, which means an increase of the phase. When

k becomes so large that the wave function has considerable values
in the region where V(r) is positive, will begin to decrease and
finally becomes negative. For very large k¹) the tj^ will be entirely
determined by the repulsion and behave approximately as the 37 (A:)
of an elastic sphere of diameter ro, where rg is the classical distance
of closest approach.

For / = 0 the behaviour for large k is the same, whereas the
discussion for small k is a little more complicated. One can show
however in a way indicated by Fermi 28) thatnbsp;behaves like k

for small k, in conformity with the result for / i^é 0. The asymptotic
solution of the wave equation for Z = 0

must for small k go over into the asymptotic solution for k = 0,
which is clearly

and therefore

It will be of interest for later discussion to investigate the case
where the potential field possesses a discrete or virtual level close
to zero. For simplicity we shall assume V(r)=:0 for r gt; r^.
Consider first the (exceptional) case where this level lies just at
energy zero. In this case the wave function for k = 0 outside the
field of force is a constant or, in other words, the tangent of the
wave function is horizontal at r = rj. Now eq. (41) breaks down
and has to be replaced by f]o{k) = n!2 for small k. When the
attraction is a little stronger so that there exists a discrete level close

1nbsp; That means here that the energy is large compared with the attractive
part of V{r). It must not be so large that the molecules can be excited for then
their description as simple centres of forces becomes invalid.

to zero, then the wave function for A: = 0 will be a little more curved
and will have a small negative derivative (taking the wave function
itself as positive) at r = r^. This means that a is large and negative.
Because of (41) fj^ik) will therefore decrease very rapidly with
increasing k till it reaches the value —nj2 ¹ ). On the other hand,
when the attraction is a little less, there will be a virtual level and a
will be large and positive. One can easily see that in both cases the
value ± nj2 of rj^ will be reached for an energy of the order of the
energy of the virtual level or of the absolute value of the energy of
the discrete level. For a virtual level one can see this as follows.
The level is defined as the energy corresponding to the minimum
ratio of the amplitudes of the wave function outside and inside the
field of force. This is obviously the case when the wave function
has a horizontal tangent at r = r]. But then the wave function
outside has the form

so that the phase, because of the smallness of kr,, is practically
equal to 7i/2. When there is a discrete level of energy —h^ kx^j'in^m,
its wave function will be A exp (—k^r) for r gt; r,^. The wave
function for k lt;= 0 will have practically the same derivative
—k^Aexpi—fciTi) for r = rx, so that a = —ir,— l/Ar^ ^—l//ci,
which means that Voi^) 'S of the order unity for k ^ k^.

After this qualitative discussion we shall try to calculate the 7]^{k)
by approximation methods and shall use the results to calculate the
second virial coefficient. There are two standard methods available,
namely that of Born 29) and that of Wentzel, Kramers and
Brillouin (W. K. B. method) so).

The approximation of BoRN consists in treating the potential
V(r) as a small perturbation in equation (34). We may expect that

1nbsp; If one prefers to have tjoik) for k^ Q change continuously with varying
field of force one has to put J?o(0) = n instead of »?o(0) =0 when there is one
discrete level, or, more generally, »?o(0) = nn when there are n discrete levels
with 1 = 0. Analogously one has to take »?; (0) as many times ji as there are
discrete levels with azimuthal quantum nimiber I. For a V(r) which is finite
everywhere this choice of j?,(0) will give obviously »?,(A:) =0 for oo.

this method will yield reliable results when V(r) is small compared
with the total energy in those regions where the solution of the
unperturbed equation has considerable values. As we saw in the
foregoing paragraph, this condition is only fulfilled fornbsp;and

small A:. A calculation of the second virial coefficient by using Born's
method throughout can therefore not be expected to yield good
results. We can however make the following, rather academic,
statement. Suppose that V{r) is small for all r (contrary to the
actually existing potential fields), then for high temperatures the
energies of the particles will be large compared with V{r), and we
may apply Born's approximation. But for high temperatures the
classical theory is valid, so that we can expect that the results
obtained with Born's approximation will be the same as those
obtained with the classical formula (1) when one introduces the
approximation corresponding to V(r) fcT. This statement is
confirmed by the calculation.

This will be introduced in eq. (11) for the second virial coefficient
in Boltzmann statistics, where we may neglect the contribution
from the discrete states. This gives

(43)

Ti-'NP r

The formula (43) for Bb follows indeed from (1) by expanding the
exponential. The introduction of (42) into the Eqs. (16) and (17)
forfiinE.B. andF.D. statistics also gives rise to closed expressions.
One finds

Finally we shall show that the behaviour of rj^{k) ior 1^0 and
small k. as stated in the foregoing paragraph, follows from Born's
approximation, which is valid in this case. Consider that in (42) not
only k, but also kr can be considered as small because, due to the
rapid decrease of V(r), large values of r do not contribute to the
integral. We may therefore insert the series expansion for the
Bessel function by means of which we obtain an expansion of
Viik) in odd powers of k, starting with the {21 I)quot;' power.

In the second chapter we have shown the connection between the
classical and the quantum theoretical expressions for the partition
function. In particular we have seen that the two expressions
become identical in the limit of high temperatures. By means of the
method of Kirkwood it was possible to approximate the quantum
theoretical partition function successively, starting from the classical

partition function. This method can be apphed directly 3i) to an
approximate calculation of the deviations of the second virial coeffi-
cient from its classical values. These calculations will be performed
in §§ 8 and 9. In this and the following paragraph we shall show
that the same result can also be obtained by applying the W. K. B,
method to the calculation of the r]j(k).

(46)

2ni

2nt

By substitution of (46) and (47) in (34) and comparing equal
powers of h¹), the terms of (47) can be obtained successively by
simple integrations. In this way one obtains the two independent
solutions

Vl =

32 Km 2

and

2ni

exp

Vi =

(E-Qfl^

1nbsp; In order to get a good approximation to classical mechanics, one has to
disregard in this comparison the fact that Q(r) contaiins PlANCK's constant,
since Q(r) must be considered as a classical quantity, namely the sum of the
potential energy and the energy of the centrifugal force.

The conditions of validity for this approximation will be discussed
later on. Here we remark only that (48) is certainly'invalid when
Q{r) = E. When there are several regions of r, separated by points
where Q{r) = E (the limit points of the classical motion), then the
problem arises how to determine the linear combinations of the two
solutions (48) (now with fixed integration constants) in the different
regions in order that the W. K. B. solution in these regions may
approximate the same exact wave function. This problem has been
solved by Kramers. When in this way the W. K. B. solution is
determined, it is easy to find the value ofnbsp;We shall develop

where the successive terms arise from the successive terms in the
exponent of (48). In this paragraph we shall be concerned with the
first approximation only.

As may be seen from fig. 4 there can exist one or three limit
points of the classical motion (except for special values of E where
there are two). Consider first the case where there is one such point
which will be called iro(A:). We have now to determine the W. K. B.
solution for r gt; Tq which approximates the exact solution which
becomes zero at the origin. From Kramers' connection formulae
one finds for this solution

(50)

(51)

In order to obtain ri^(k) this must be compared with the free particle
wave function. It is more consistent to make the comparison with
the W. K. B. solution for the free particle rather than with the

exact function ¹). The W. K. B. solution for the free particle is
found from (51) by insertingnbsp; I )/r2 and performing the

When there are three limit points of the classical motion, rilt;r2lt;r3,
one obtains

The following theorem can now be proved. Let there be for fixed I
so many discrete levels E„i that the sum over n in the formula for
the second virial coefficient may be replaced by an integral. Then,
using the first W. K. B. approximation for the calculation of E„i
and ïj^ik), Vfe obtain

We shall prove (55) first in the case where / is so large that /(r)
is monotonie for all r. In this case no discrete levels are present, so
that Bi = Bi,cont.. By introducing (53) in (12) one gets

1nbsp; For the calculation of drjjdk the use of the exact and of the W. K. B.
solution for the free particle give the same result.

This integral has to be transformed in such a manner that the
integration over k can be performed. When we do this ¹) we obtain

11/1(1 1)

(57)

Now the integration over x can be easily performed by taking
X—f{r) as a new integration variable. The result is

Xl-'NP

df 1)

We may now consider the case that f{r) is not monotonie but still
always positive. In this case also there are no discrete states. But
now in a certain region of values of k there are three limit points of
the classical motion and there we have to apply eq. (54) instead of
(53). A calculation similar to the foregoing confirms (55) for this
case too.

When there are regions where /(r) is negative, there may be
discrete states. Nownbsp;is not given by the right hand side of

(55). We shall give the calculation for Z = 0 only. The case 1^0 is
quite analogous. Again the integral (56) has to be calculated, but
now /(ir) = 47i2m V(r)//i2. Instead of (57) one finds

gt;(58)

,-xlko-

We shall now add the contribution from the discrete states which,
after replacement of the summation by an integration, can be
written as

-II

Introducing (61) into (59) and substituting E^h^xlin^m (x is
here negative), we find

.-xlko^

\yx-f(r)

when we interchange summation and integration. For E. B. and
F. D. statistics we have to sum over even, and over odd, values of I
respectively and to multiply by a factor two. The sum

cannot be calculated exactly. However, Mulholland 32) has given
the following asymptotic expansion for this sum, which furnishes an
approximation for high temperatures.

(64)

^ k! (ko'r'Y^---

and the Bernoullian numbers B„ are defined as the coefficients of
xquot;ln! in the expansion of — 1). The same expansion is obtained
for twice the sum over even or odd Z. By inserting the first term of
this expansion in (63) one obtains the classical expression (1). It
is premature to introduce here the second term of Mulholland's
expansion in order to obtain the first deviation from the classical
formula, since the second approximation of the W. K. B. method
will give a contribution of the same order of magnitude.

(65)

(k'-ffl

The difficuky is that this integral diverges. An examination of the
derivation of Kramers' connection formulae shows however that
the integral (65) has to be interpreted as the half of a complex
integral in the r-plane, taken along the contour of fig. 5. Hada-

MARD33) has shown that this can be written in real form as the
quot;principal partquot; of (65). This means one must integrate (65)
partially so many times that the resulting integral becomes con-
vergent while the infinite terms of the integrated parts have to be
dropped. By using Hadamard's symbol for this principal part we
find after a partial integration that one can write for the second
approximation to rj,{k) ¹)

(k^-ffl'

(67)

(k^-ffU

1nbsp; Insertion of /(r) =/(/ l)/r2 in (66) gives, as shown in Note 5,
nf\k) = —Jtll6[/l{l T). It is therefore more consistent to substract this value
from the expression (66). This has of course no influence on the value of

• (68)

By means of (67), (68) and the obvious extension of (67) for a
non-monotonic f(r) we can now prove ¹) that the second term in
the approximation of B, is given by

This has to be introduced in the formulae for the second virial
coefficient. For the sums over I which now occur we can give
asymptotic expressions for high temperature by differentiating (64)
with respect to Ijk^'^f^. In this way we obtain

3kT

We have to add here the result which is obtained by introducing
the second term of (64) in (63):

drre-^'l''^ V.

By partial integration of the first term we see that the first and the
last term cancel. Therefore, finally

The results of the last two paragraphs can be obtained with much
less labour by applying the method of Kirkwood to the Slater
sum of the relative motion of two particles. We have nevertheless
thought it worth while to give the calculation with the W. K. B.
method because the W. K. B. expressions for 7i^(k) may be of some
use for practical calculations and because this method shows more
explicitly the contribution of the discrete levels, which in the method
of Kirkwood is completely obscured.

In order to find the development for B, we have to apply
Kirkwood's method to the partition function of the relative motion
of two particles after separation of the angular coordinates. We
start therefore from eq. (4) which we shall write as

As was already remarked in chapter II the fact that this set does
not satisfy the boundary conditions is not important. We have now

2 Jim

d'Q

(73)

W2 =

dr' 'nbsp;ml dr

(74)

we obtain, by introducing the solution (73) into (72) and performing
the integration over p

-QlkT

^ e

X[/2

(75)

. g-Q/tr gt; _

Sf (r) =

\2\/2nkT

The corresponding functions for the free particles are obtained by
putting Q(r) = /j2/(/ l)/47i2mr2 in these formulae. The terms in

the development of B, which corresponds to (74) are obtained by
introducing the solution (75) and the corresponding solution for
free particles into (71). In this way one easily finds that B® is
given by the expression (55), fij') =0, while Bf) becomes after a
partial integration identical with (69).

We have already seen that an expansion for B can be obtained
by introducing the expression for B, into the formulae for B and
using the asymptotic expansion (64) of mulholland for the sum-
mation over /. The resulting expansion for B can however be
obtained directly by applying Kirkwood's method to the partition
function of two particles without separation of the angular coor-
dinates ). This procedure has the advantage that now one can
also give the expansion for E. B. and F. D. statistics. This was not
possible by starting from B, because of the fact that the expansion
of Mulholland remains unaltered for the sums over even or odd I
which occur in these statistics.

Since we have already obtained in Ch. II the general expansion
of the Slater sum for a gas of N particles, we shall start from the
expression for B before the separation of the motion of the centre
of gravity, i.e. from eq. (2). Here we shall introduce eq. (II, 42)
and then introduce relative coordinates in the integral.

Let us first consider for simplicity the case of Boltzmann
statistics. Then, from (2) and (II, 43)

R - ^
Bb =

When we now introduce the variables r,.r = r,—r2, instead of
r,, rj, and make use of the fact that V depends only on r so that

1 1

24 ne T^

2kT

r2 V'

y'2

3kT

a. The validity of the approximations used. We must investigate
now how far the methods represented in the foregoing paragraphs
are suited to a practical calculation of the second virial coefficient

when the intermolecular potential is known. We have already seen
that for potentials actually occurring the BoRN approximation can-
not be used throughout. It can be used only for the calculation of
rj^{k) when Iz^Q and k is small.

For theW. K. B. method and the equivalent method of Kirkwood
the condition of validity was given in chapter II. The change over
a distance k of the potential, measured in units kT, must be small
at least for values of r where V(r)lkT is not large. In order to
check this condition we must know the function V(r). For that we
refer to the quantum mechanical calculations mentioned in the first
chapter. The theoretical potential which has been discussed most
in connection with the second virial coefficient and the transport
phenomena is that of Slater and Kirkwood for helium, eq. (I, 4).
This potential, when inserted into the classical formula (1) for B,
gives a good representation of the experimental values n). We
shall see however that one cannot attach much significance to this
correspondence. From (1,4) we find that for r = 2.6A, where

so that we may expect the W. K. B. method or the method of
Kirkwood to give a good approximation only for T gt; 70° K. For
lower temperatures it is therefore certainly not permitted to use
the classical expression for B and the agreement with experiment
has to be considered as accidental. On the other hand, for T = 70°
we have kT gt;=96 X 10—16 erg, whereas the minimum of V(r) is
—12.6 X 10-16 erg. For T

gt; 70° therefore inaccuracy in the
attractive part of V(r) will not have much influence on the resulting
value of B. We can conclude only that the Slater-Kirkwood
potential gives a good value for the atomic diameter of helium.

In order to get an idea of the order of magnitude of the different
terms in (77) we have calculated some of them for the Slater-
KiRKWOOD potential for a temperature of 10° K. One finds for
instance for one mol of gas

IjiiV jquot;

-itznJ

We see therefore that, while the deviation due to E. B. statistics is
small, the other quantum correction is much larger than the classical
value itself.

b. Behaviour of B at very low temperatures. According to the
classical formula ( 1 ), B will go exponentially to — oo when T —gt; 0
for every potential field V(r) which is somewhere negative. In the
quantum theory the situation is not so simple. We shall take T so
small that in eq. (16) we have to consider only the contribution of
I — 0. We can make an estimate for which temperatures this is
permitted. For 1 = 2 the function f{r) for the Slater-Kirkwood
potential has a maximum of 0.189 Â—2 for r = 4.6 A. When there-
fore /co2 lt; 0.189 Â-2 the value of Bj, cant and all B,,eonf. with I gt; 2
will be small. This value of k^^ corresponds to a temperature of
2.28° K so that our discussion strictly speaking will be valid only for
temperatures from about 2° downwards. In this region we have
therefore, since hehum is an E. B. gas

1. Let there be no discrete levels and let the energies of all
virtual levels be large compared with kT. Then in the calculation
of Bo cont we can, according to the considerations at the end of § 4
and because k^^ is small, replace drjojdk by the constant a. Therefore

Eq. (79) represents the beginning of an expansion which can be
obtained by inserting in the general eq. (16) for drjjdk its expansion
in powers of k, which contains even powers only. The expansion
obtained has therefore the form

Here the first term, the second virial coefficient of an ideal E. B.
gas, is known. The second term depends on the value of a. The
absolute value of a could in principle be determined experimentally
from the cross-section for slow collisions between two helium atoms
which, according to (27) is equal to \6na^, or from the values of
the coefficients of viscosity and of heat conduction at very low
temperatures, according to (31) and (32). The existing experimental
material is however not sufficient for this determination.

2.nbsp;In the case when there are discrete levels whose energies in
absolute value are }) kT, and also the energies of the virtual levels
are }}kT, we have only to add their contribution to the expression
for B in the foregoing case. Therefore, in addition to the
expansion (80), we have one or more terms which go to —oo as
— exp{cjT)jT'U for T -gt;Q.

3.nbsp;Let there now be no discrete levels but one virtual level with
energy {{kT. As we saw in § 4, rjo(k) will now increase rapidly
from zero to njl in an interval where exp (— k^jk^^) is practically
constant and then change more slowly. The integral in the
expression (12) for Ba^^ont. will therefore be practically equal to
this increase ofnbsp;or to nil. Therefore

In order to calculate the next approximation, which depends on the
ratio of the energy of the virtual level to kT, one would have to
know more precisely the shape oi rig{k).

4.nbsp;When there is a discrete level whose energy in absolute
value is {{kT and the energies of all virtual levels are )) kT, then
10 W will decrease rapidly from 0 to —njl (or from n to njl).
Therefore Bo, co„f. will be the negative of the Bo, cont. in case 3. In the
contribution from the discrete level we can replace the exponential

by unity and we then find that B is again given by (81). It is
therefore not possible by using this first approximation to decide
experimentally between the cases 3 and 4.

The theoretical calculations of the VAN der Waals forces for
helium (see Ch. I, § 2) all exclude the cases 1 and2.The expressions
of Slater and Kirkwood, Maroenau, and Page all predict the
existence of a discrete level of about 0.5 X 10—16 erg, corresponding
to a temperature of some tenths of a degree. On the other hand the
attraction deduced by london from the dispersion curve of helium
is smaller and probably gives rise to a virtual level close to zero.

The experimental values of B for helium are also decidedly in
favour of case 3 or 4, i.e. they make it probable that there exists a
discrete or virtual level close to zero. If this were not the case, B
would be determined by (79)*) with a ((2 for sufficiently low
temperatures. Let us take for instance a = 1 A. Then for T = 3° we
get B = — 43.9 cm^/mol, whereas from (81) one would obtain
B = —120.3. The experimental values of Keesom and Kraak 34)
are —117.2; —96.5; —74.7 for 7 = 2.58°; 3.09°; 4.22°. It is clear
that these values agree better with (81) than with (79).

A formal derivation of eq. (11,20) is obtained by applying
Poisson's summation formula 35) to (II, 19), which gives

are given, to find the function y{x) in the form of a power series.
This problem can be solved by using the following theorem, which
is a simple specialization of the theorem of LagrANGE36):

When ƒ (f) is analytic within and on a contour C which surrounds
the origin, and the value of z is so chosen that on C

Conversely, when (5) is given, the expression of z in x is, for values
of z which fulfil (3), given by (4).

Now (1) can be put in the form (5) by introducing a function
in such a way that l^bi is the coefficient of in the expansion
of exp (llt;p{^)), or

has one solution within C, namely

■ Jn-l

i=o

00 2quot;
n=i nl

(6)

(7)

(8)
(9)

f=0

(1)

where nj = | r, — r^ [ and the integral has to be extended over all
possible values of the variables. Consider first the integral

Take the coordinates of the first particle as origin and the direction
from the first to the second particle as x-direction, then

we can calculate I by successively integrating over r2, r^, etc.,
always using (3). In this way we obtain

It is not permissible simply to interchange the integrations over r
and k. One has instead to proceed in the following way. Fix a value
of r which is so large that [{r) is practically equal to Z(Z l)/r2
for rgt;R, or

parts (fig. 6). We shall calculate the integrals over these parts
separately in the limit R-^ co. The contribution of part 1 is

(2)

Z(Z l)/r2. The integral over r is just r^ so that the contribution of
this part vanishes ¹). Finally, the integral over part III is

We can again replace ƒ (r) by 1(1 1 )/r2 and perform the integration
over r. We get then

b. For 1 = 0 the calculation is simpler. We may perform the
integration over r to a large value R and then let R go to infinity.
A calculation, similar to the derivation of (2) gives

1= Lim

R-^ cc

(k^-i^f

According to the definition of the principal value of an integral we
must integrate (2) partially till the resulting integral becomes
convergent and drop the infinite parts of the integrated terms. We
obtain then, because the finite parts of the integrated terms are all
zero

(3)

•'2

drjr-.

(k'-fV^

We shall again divide the integration region into three parts as in
the foregoing note, so that

■'2

(k'-ffl'

f'2

•'2

First we remark that lu is zero. For here we may replace f(r) by
/( / 1 )/r2and therefore the integral overr is proportional todrjf) jdk
for a free particle which, according to (3), is zero. By successive
partial integration and by dropping the infinite terms we can write
Il in the form

Ii= Jdkke-'^'l'quot;^-

k(R)

r(R)

(5)

Here the first two terms would give infinite results. They can
however be compensated by Iju which after two partial integrations
can be brought into the form

f'iR)

I,11 =

k{r)

'=ƒ

(7)

Afquot;'fquot; 3f

■//3

f'3

■M

/cm fquot;2\

From (4) and (11) we immediately obtain eq. (69) of the text by
inserting the expression (35), (50) for f{r) and by replacing the
first term of (11) by the negative of the second term with
f(r)==/(/ l)/r2.

We shall not give here the calculation for a non-monotonic f(r)
since it presents no special difficulties.

J. E. Mayer and Ph. G. Ackermann, ibid., 5, 74, 1937.
]. E. Mayer and S. F. Harrison, ibid., 6, 87, 1938.
S. F. Harrison and J. E. Mayer, ibid., 6, 101, 1938.

3)nbsp;Ursell, Proc. Cambr. phil. Soc. 23, 685, 1927. Comp, also R. H. FOWLER,
Statistical Mechanics, Ch. 8.

Van der Waals forces will be found.
6) J. C. Slater, Phys. Rev. 32, 349, 1928.
■7) J. C. Slater and J. G. Kirkwood, Phys. Rev. 37, 682, 1931.

13)nbsp;Comp. g. E. uhlenbeck and L. gropper, Phys. Rev. 41, 79, 1932; g. E.
uhlenbeck, J. Math. Phys. 14, 10, 1935.

15)nbsp;J. G. Kirkwood, Phys. Rev. 44, 31, 1933.
18) F. Bloch, Z. Phys. 74, 295, 1932.

22)nbsp;Comp. J. HadamARD, La série de Taylor et son prolongement analytique,
Paris 1901, p. 20 (or 2nd ed. p. 41).

23)nbsp;M. Born and K. Fuchs, Proc. roy. Soc. London, in press.
21) E. Beth and G. E. Uhlenbeck, Physica 4, 915, 1937.

28) Comp. N. F. MOTT and H. S. W. MaSSEY, The theory of atomic collisions,
Chs. 2 and 5.

1933; E. A. Uehling, Phys. Rev. 46, 917, 1934.
28) E. Fermi a.o., Proc. roy. Soc. London, 149, 522, 1935.
28) Comp. réf. 26, Ch. 2, § 2.

H.nbsp;P. Mulholland, Proc. Cambr. phil. Soc. 24, 280, 1928.
J. hadamard, Le problème de Cauchy, Paris 1932, p. 184.

W. H. Keesom and H. H. Kraak, Comm. Leiden, N». 234 e, 1934
^S) Comp. R. Courant and D. Hilbert, Methoden der mathematischen
Physik, I, 2nd ed. p. 64.

§ 2. A transformation of the SLATER sum . .nbsp;11
§ 3. Connection between classical and quantum

theoretical expressions. Examples ....nbsp;12
§ 4. Connection between classical and quantum

§ 7. The W. K. B. method, second approximationnbsp;72
§ 8. Approximation of B, by means of Kirk-

De wijze, waarop Mayer en Harrison de volume-afhankelijkheid
der grootheden bi behandelen, is onjuist.

De overeenstemming van de door van Itterbeek en Keesom
gemeten waarden voor de viscositeitscoëfficiënt van helium met de
door UehliNG theoretisch berekende, kan niet als een experimentele
bevestiging van de quantumtheorie der viscositeit beschouwd worden.

Het feit, dat op grond van de door FermI gegeven theorie der
straling berekende kernkrachten veel te zwak zijn, is geen beshssend
argument tegen de samenhang van de kernkrachten met de /3-radio-
activiteit.

De wijze, waarop FrÖhlich, Heitler en Kemmer de krachten
tussen protonen en neutronen en de magnetische momenten van
deze deeltjes berekenen, is niet consequent.

De wijze, waarop van Laar de vergelijking de = CvdT IdV
integreert, is nodeloos omslachtig.

De door Uhlenbeck en Beth gegeven sommatieformules zijn
een direct gevolg van de formule van Clebsch.

Uit de bepaling van de levensduren en van de verhouding der
activiteiten van twee isomere ;S-radioactieve atoomkernen kan een
bovenste grens worden afgeleid voor de waarschijnlijkheid van de
overgang tussen deze kernen.

De bepaling van de bovenste grens van een continu |S-spectrum
met behulp van absorptiemetingen geeft geen betrouwbare resultaten.

De metingen van Alichanian en zijn medewerkers over de vorm
van het |8-spectrum van Rs E en Th C rechtvaardigen niet de door
deze onderzoekers getrokken conclusie omtrent een van nul ver-
schillende massa van het neutrino.

A. i. Alichanian, A. i. Alichanow, B. S. Dzelepow
en S. J. Nikitin, Phys. Rev. 53, 766 en 767, 1938.

Bij geschikte ontladingen in moleculaire gassen kan men uit de
optische bepaling van de rotatie-energieverdeling van het molecuul
en van het molecuulion besluiten of de ionen hoofdzakelijk in aan-
geslagen dan wel in de grondtoestand gevormd worden.

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•si:: i-s:- 'nvigt; :■:gt; ; ;

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