- Email: [email protected]
The equation of state of gases and critical phenomena
23 N o v e m b e r 1937
P h y s i c a IV, n o 10
THE
EQUATION OF STATE OF GASES AND CRITICAL PHENOMENA by J. E. LENNARD-JONES The University Chemical Laboratory, Cambridge
1. Van der Waals tomes. It is a fitting tribute to v a n d e r W a a 1s' memory that those intermolecular forces, whose importance in the theory of gases and liquids he was the first to recognise, should now be universally known by his name. V a n d e r W a a 1 s' cohesive :forces have been the subject of continuous study since his equation of state was introduced and within recent years interest in them has again been stimulated by the discovery of their extensive applications in physics and chemistry. It is, however, only recently that it has become possible to study quantitatively the nature and magnitude of these forces. K e e s o m 1) was the first to give a precise derivation of the equation of state for particular forms of intermolecular fields and by a comparison of the theoretical formula with observed isotherms of gases to derive numerical values for the attractive forces. In all his molecular models, however, the repulsive field was represented by rigid spheres or by ellipsoids of revolution and the attractive field by inverse powers of the distance. In view of the obvious limitations of a rigid sphere model, a repulsive field of the inverse power law of the type Xr-', on which was superimposed an attractive field of a similar type, was considered by the author 2) and an equation of state deduced appropriate for gases of low density. This equation has been found adequate to explain in a surprisingly satisfactory way the observed isotherms of gases and from it information of interatomic fields has been derived 3) which so far as it has been tested has given a reliable indication of many properties of gases and solids. Other theoretical calculations of the equation of state for special types of intermolecular fields have been made by Z w i c k y 4), W o 11 1 5), - -
941
- -
942
j. E. LENNARD-JONES
Kirkwood and K e y e s 6 ) and shown in particular cases to explain the observations. These methods may be described as inductive in that they proceed from the observations and endeavour to find such representations of force fields as will explain the facts. Recently it has been found possible to calculate under certain conditions interatomic forces from the electronic structure of atoms. Attempts to construct such deductive methods were made by B o r n and L a n d ~ 7 ) , Deb y e 8) and others in terms of the older pictures of atoms and molecules but the advent of quantum mechanics has provided a deeper understanding of the nature of these fields and has permitted direct evaluation of their magnitudes. E i s e n s c h i t z and L o n d o n 9) were the first to carry through such calculations for the interaction at large distance of two hydrogen atoms and simpler or improved methods have since been given by L e n n a r dJones1~ H a s s ~ 11), A t a n a s o f f 1 2 ) , Slater and K i r k woodla), Margenau 14) and P a u l i n g and B e a c h 1 5 ) . These investigations were concerned mainly with hydrogen and helium atoms. Approximate methods of dealing with larger atoms have been given by L o n d o n l ~ ) , S l a t e r and K i r k w o o d 13) Kirkwood17), V i n t i 18) and H e l l m a n n 1 9 ) . Using a variation method B u c k i n g h a m 20) has recently given formulae for the v a n d e r W a a 1 s cohesive forces between atoms with any number of electrons and, using available wave functions of the rare gas atoms and alkali ions, has compared the results with those derived by the writer from the equation of state 21). While it must remain one of the objects of theoretical research to determine force fields in this direct way, actually the fields so found are complicated functions of the distance and are not very convenient to apply in other investigations. For the present, therefore, it seems desirable to be able to express force fields in terms of simple functions which lend themselves to mathematical analysis. The indication is that the interaction of neutral atoms at large distances can be represented by a potential function which varies as the inverse sixth power of the distance. At smaller distances the function is not so simple. Nevertheless it is convenient to adopt the asymptotic form of the function as valid over the whole range and to make any necessary modifications in the repulsive field which must be used in conjunction with it.
THE EQUATION OF STATE OF GASES AND CRITICAL P H E N O M E N A
943
There is no clear indication as yet as to the best form of the field at distances so small that the atoms repel each other. In the case of hydrogen atoms this repulsive field contains an exponential function multiplied by a series of powers of the distance and so it has become customary to express the repulsive field by an exponential function alone. This form has been used by B o r n and M a y e r 22) in their'work on crystals and found satisfactory in this connexion. S 1 a t e r 23) also found the exponential form to be a satisfactory representation of the calculated field exerted between two helium atoms. Yet it must be recognised that the adoption of the simple exponential foI:m is to some extent arbitrary and it is doubtful whether other forms should be discarded in favour of it. The inverse power law has much to commend it, p a r t i c u h r l y when the total field is expressed in some such form as Xr - ~ - - ~r -'~, for, as will be seen below, this function has simple and elegant properties, in that the force constants are easily expressed in terms of the equilibrium distance of two atoms and the value of their potential energy when in this position. In this paper, therefore, considerable use will be made of this type of field. The object of this paper is to show how force fields can be determined from observations of the equation of state at low pressures and then to describe a method by which the equation of state at high pressures m ay be derived. This leads to a calculation of the critical temperature of a gas in terms of the constants of the interatomic field and the formula seems to reproduce observed critical temperatures within a few degrees. 2. The equation o] state o] a gas at small c o n c e n t r a t i o n s - - D e t e r m i n a tion o / i n t e r a t o m i c ]orces. V a n d e r W a a 1 s' equation of state may be regarded as the first successful attempt to interpret the equation of state of a gas in terms of interatomic forces, and conversely as the first at t em pt to derive information about molecular fields from the equation of state. While m any other methods have since been devised to evaluate these fields, it is still probably true to say that the method of the equation of state is the most powerful and reliable y e t available. The physical arguments by which v a n d e r W a a 1 s derived his equation are suitable, however, only for the special molecular model with which he dealt, viz. that of a rigid sphere surrounded by
944
J . E . LENNARD-JONES
an a t t r a c t i v e field.Thus his m e t h o d had special reference to the conditions at the b o u n d a r y of gas and gave the pressure actually exerted on the walls of the containing vessel, so t h a t the t e r m a/v 2 in his equation is on this view the diminution in the rate at which m o m e n t u m is c o m m u n i c a t e d to unit area of the walls in unit time. A t t e m p t s to obtain equations of state for molecular models of more generality have been derived b y essentially different methods, and often refer to the uniform conditions in the interior of the gas. I t is not then always clear w h a t i n t e r p r e t a t i o n is to be given to them. While the pressure at t h e . b o u n d a r y of a gas is the same as t h a t of the interior, the word pressure has a different meaning in the two cases. At the b o u n d a r y , it is due entirely to the m o t i o n of the molecules and is described in terms of the m o m e n t a of the b o m b a r d i n g atoms whereas in the interior only part is due to the molecular motion. The remaining p a r t is due to the stress set up across a n y area in the interior of the gas b y the interaction of the atoms on one side with those on the other. Although at a n y given point it is a fluctuating function of the time, it has e v e r y w h e r e within the gas the same statistical m e a n value. This m a y be referred to as a statical pressure to distinguish it from the more usually understood d y n a m i c al or m o m e n t u m pressure. The equation of state for a gas of small c o n c e n t r a t i o n can be shown to be B p v = k N T 4- --(IA) 7)
or
pv = kNT
(IB)
4- B ' p
where B and B ' are functions of t e m p e r a t u r e depending on the molecular forces and the o t h e r symbols have their usual meaning. B and B' are given in terms of intermolecular fields b y the expressions O<3
B = B' kNT
= 2= N 2 k T f r 2 { 1 - - e x p ( - - ~ ( r ) / k T ) } o
dr
(2)
where p(r) is the potential energy of two molecules at a distance r apart. These equations are, like t h a t of v a n d e r W a a 1 s, only first a p p r o x i m a t i o n s and valid only for dilute gases, for t h e y are derived on the assumption t h a t the atoms influence each other only in pairs. The t e r m B / v 2 represents the statical pressure in the interior of the gas 24) and m a y conveniently be d e n o t e d b y Ps. It appears from the expression for B t h a t the terms a/v 2 and 'b' of v a n d e r W a a 1 s
' r i l e EQUATION OF STATE OF GASES AND CRITICAL P I I E N O M E N A
945
equation arise respectively from the contributions of the attractive and repulsive parts of the field to the statical pressure. In the v a n d e r W a a 1 s molecular model the repulsive and attractive parts of the field are in separate regions of space and it is not surprising that their contributions to the statical stress should be separable. It is :not to be expected that in other models the effects of the repulsive and attractive fields can be so distinguished. In passing, it may be observed that often the internal pressure of a gas is defined by means of the thermodynamic equation
(3)
p, =
= T (~p/or)v - - p, but this is not the same as the statical pressure; in fact, it is clear from (3) that when equation (1) is valid
Pi= T\~T/
--P~= T2~
~ ~
(4)
At high densities the internal pressure can be very high as has been found by M i c h e l s 2 5 ) , even exceeding by many times the external pressure. When v a n d e r W a a l s equation is written in the form of equation (1B), it appears that
B' : b - - k N ~
(5)
whereas the corresponding formula derived from (2) for molecules which behave as rigid spheres of diameter ~, surrounded by an attractive field whose potential is ~r-" is 1)
B'
52 r~N a3 I 1
3 (~o/kT) [
~
(6)
where ~o =
v.e'-m
(7)
and is the potential energy of two molecules in contact. This formula for B' may be identified with (5) provided % is small compared with /~T and a comparison of the two gives the interpretation of a and b in terms of the molecular fields. A more general formula than (6) has been given ~) which corresponds to interatomic forces whose potential can be written as the sum of inverse power laws qo : Xr- ~ Physica IV
~r--~
(8A) 50
946
j. E. LENNARD-JONES
This function can be written in the more convenient from
where r 0 is the distance between two molecules in equilibrium u n d e r the field (8A), and ] % [ is the energy required to separate t h e m from this configuration. The formula for B' is t h e n B' = 52 ,~N r ao t bo/kT)
(9)
The function / is a polynomial whose coefficients depend only on n and m. Thus
/ ( % / k r ) = (m/n) 3/'~-m F(y) where
r
3v =
~,
n
/
(lo)
and
y----- (re~n)-c"l') (l--re~n) -1+<'/') (%/kT) l-I'/'0
(11)
Methods of determining X and [~b y a comparison of the theoretical formula for B' and the values of B' required in (1B) to fit the observe isotherms have been given elsewhere 2) 3). W h a t m a y be of interest to observe here is t h a t % can be d e t e r m i n e d easily from a knowledge of the B o y 1 e point, t h a t is the t e m p e r a t u r e for which B ' is found from the experiments to vanish. Denoting the B o y 1 e t e m p e r a t u r e b y TB and the root of F(y) = 0 (12) b y y . . . . t h e n we have %/kTB = (re~n) m/(n-m) {1 - - (re~n)} EY., m]n[(n-m)" (13) T y p i c a l values of y., m are given ill Table I with the corresponding values of kTB/% T A B L E I *) V a l u e s of Yn, m, [n, m, a n d ~o/kTB
1 11 12
1.1190 1.0997 1.0818
41044
4.07 3.71
0.4781 0.5049
3.43
0.5285
In some cases the 'observed' values of B' have a m a x i m u m and this p r o p e r t y is exhibited also b y the f u n c t i o n / . F r o m the value of *) T h e a u t h o r is i n d e b t e d to Dr. A. F. D e v o n s h i r e r e q u i r e d f o r t h i s a n d t h e o t h e r tables.
for his h e l p in the c a l c u l a t i o n s
THE
EQUATION
OF
STATE
OF
GASES
AND
CRITICAL
PHENOMENA
947
the maximum, r0 can be deduced for assumed values of n and m, using the maximum values of / (denoted by/~, ,) given in Table I. The most appropriate values of n and m can only be determined by a comparison of / and the whole range of observed values of B. For the inert gases and some molecules it has been found that m = 6 (corresponding to the theoretical value for v a n d e r W a a 1 s forces) and a value of n between 9 and 12 fits the observations satisfactorily. Values of X and tx have also been deduced in this way and from them the values of r0 and ?0 given in Table II "). These refer in all cases to n = 12, m = 6. TABLE Constants
of Interatomic
Fields
k Helium . . . . Neon . . . . Argon . . . . Hydrogen . Nitrogen . . Oxygen . . Carbonmonoxide Methane . .
. . . .
3.605 3.545 16.2 6.49 3.70
. . . . .
II
(Potential
p~ 10- TM 1 0 - x~ 10 - 1 ~ 10 - l o s 10 - 1 ~
1,17 8.32 1.034 1.05 1,40
. . . . ,
energy
r0 (/~-) 10-60 1 0 - s~ 10 - 5 8 10 - 5 ~ 1 0 - 58
2.917 3.049 3.819 3.276 4.174
=
),r-n--l~r-m) ( 1 0 _ ~ 0 e r g s ) . , ( c a l . / g rq~o .atom) 0.950 4.881 16.50 4.246 13.24 16.97 13.36 19.70
13.7 70.6 239 51.4 192 244.9 192.8 284.3
Values of ?0 for 02, CO and C H 4 a r e also included, derived from the observed B o y 1 e temperatures given in Table III, where a comparison of observed and calculated B o y 1 e points for the other gases is also given.
3. The equation o~ state o~ dense gases - - Critical phenomena. The equations of state given in the preceding paragraph take into account only binary encounters between the atoms of the gas and are found to be in agreement with the observations just where they would be expected to be, viz. when the concentration is small and the temperature not too low.They cease to be valid except under these conditions ~nd should not be used in the critical region or regarded as adequate explanations of critical phenomena. At higher concentrations the observations can sometimes be expressed by more extended series in 1/v or in p than are given in (1A) and (IB). It is possible to work out under certain limited conditions a theoretical formula for the coefficient of 1/v 3 for special molecular models, but not for force fields of a general type. General m~thods have been given by
948
j. E. LENNARD-JONES
U r s e 1 1 26) for the evaluation of the coefficients of the series, which correspond to the interaction of the atoms in groups of three, four, etc., but they have been not applied to particular force fields. They have, however, recently been used by M a y e r 2,~) to show that when carried far enough the equation of state thus derived does actually lead to critical phenomena. A method of obtaining an equation of state for condensed gases, when the atoms are assumed to be rigid spheres, has also been given by T o n k s 26b), based on the virial theorem. M i c h e 1 s 25) has found that at high pressures the series expansion of pv in powers of 1Iv or of p soon ceases to be useful except over limited ranges of temperature and pressure. It is even found that the coefficients B and B' found by fitting (1A) and (1B) to the observations at very small pressures are not consistent with those found by fitting more extended series to the observations at high pressures. It has been found necessary to use series of the type
pv=A
+ (B--~) + C
Z
D
E
F
(14)
where ~ expresses the difference between the coefficient of 1/v at high and low pressures. After extended trials to fit the observations at high pressures, M i c h e 1 s and his collaborators have come to the conclusion that isotherm data obtained at high densities cannot be expressed as ordinary power series. When the isotherms are expressed in this form it is even found that for large densities the series is not convergent. In view of this result it is doubtful whether any useful purpose is served by trying to build up a theory in terms successively of diatomic, triatomic and more general complexes. At high pressures when the density of a gas is approaching that of a liquid, each atom must be in the field of several other atoms not only occasionally but all the time. This is illustrated in the figure 1. The atoms, represented by dots are placed regularly on a crystal lattice at the average distance apart corresponding to the critical density, a 1,0, 0 section of a face centred cubic lattice being shown. The potential field of one atom due to another, as found in the preceding section, is represented by a series of contour lines. The dotted circles show the contours of negative energy, that is the attractive region, while the continuous circles show the repulsive region. The outermost circle corresponds to an
THE EQUATION OF STATE OF GASES AND CRITICALPHENOMENA 949 energy equal to (0.1) q~0, the second one to (0.2) ~0 and thereafter t h e y increase at equal intervals of (0.2) q0o. If now this picture of an atomic field is drawn on tracing cloth and moved over the lattice array of atoms, it is found t h a t whatever position it takes up the attractive field always overlaps three or four other atoms and in three dimensions it m a y well overlap double this number.
.."i --'- . . . . . . . I
l / I
I i I ,
"
I
i ~1.1
I
~
~
-"-
~
\ \
i / , ' f f ~ ~ \ ' , ' , ' ,
I.'lWt, f///
"
", ', \
~ ' ~ \ i \ ~ '.
"*qlll
\\\l(l.I
I i I Ili~llll ~-I 1 ~ I Ill~llll I !~ l lltli I ; t t,l\\\\\ ~'
".
\NN~.\\\ \ \
,';///I.,t/// , i,
k
, ~
/~li~lll , i iIIIm~ I,,
~
,.
~ i
, i
Illll~l, I;, ! I /lll'~fl I., i i I 1111(9,# I s I I ///r ~ / I I , //7'](II I i I i , ' / / ," '
t '"Ok\\\
"\ '. ~ \~}&\\\ , '\ \ "~T~k.~ ' '
-'Od
Fig. 1. The overlap of atomic fields at the critical density. This state of affairs is reached in the critical region and theoretical a t t e m p t s to devise an equation of state suitable for the cvaluation of critical constants ought perhaps to begin from this point of view. Interatomic fields, if correctly derived, should be the connecting link between the observed values of B in equation (l A) and the critical constants. An a t t e m p t has recently been made to effect this correlation 2~). The method begins not from the usual concept of a gas but from the supposition t h a t the average envirc L-
950
j . E. LENNARD-JONES
ment of an atom in a dense gas is more like that of an atom in a liquid or solid. An atom is regarded as imprisoned for most of its time by its immediate neighbours in a cell, just as it is in a solid, and though migration or diffusion will in practice take place from one cell to another, this possibility is neglected as being an infrequent event compared with the duration of time spent within each cell. This theory can only be regarded as a first approximation developed from a new point of view but it is probably better than attempts to deal with a dense gas by methods which depend essentially on binary encounters. When the density of a gas is increased to such an extent that any one atom is simultaneously under the influence of several others, the total field acting upon it will be of a very fluctuating kind and the first step in dealing with this field must be to replace it by some kind of statistical average. One method of doing this is to take the field exerted on one when the others are at their average distances from it. All the immediate neighbours m a y then be conceived as moving so that their centres remain on a sphere of a definite radius and the field created by them within tile sphere will, when averaged out, have spherical symmetry. Even so the field within the sphere will be of a complicated kind. It will depend not only on the field exerted by one atom on another but also on the size of the spherical cell. When this is large the attractive fields of the neighbouring atoms will overlap at the centre but not to a very great extent and direct calculation shows t h a t the region of lowest potential lies near to the circumference of the sphere. It is as though the atom were enclosed in a spherical box in which there is a strong adsorbing field at its surface. The enclosed atom will tend to be near the circumference and not at the centre of its cell. This state of affairs corresponds to the tendency of the atoms to form pairs. As the size of the sphere is decreased, so the region of lowest potential will contract and the potential barrier at the centre will diminish owing to the greater and greater overlap of the attractive fields. Finally a size of cell will be reached for which the position of the minimum potential energy will be at the centre and will remain there for all lower sizes, though indeed the absolute value of the potential at the centre will depend on the size. After reaching an absolute minimum it will rise for all higher ~ Not only the absolute magnitude of the potential energy, but also its form within a cell will depend on the volume and it must accordingly be denoted
THE EQUATION OF STATE OF GASES AND CRITICAL PHENOMENA
951
by a function of the type, q0 (r, v), where v is the volume of the cell and r is a vector denoting space coordinates within it. When the potential is replaced by a suitable average either by supposing the :nearest neighbours to take up all orientations with equal probability on a spherical surface or b y supposing them fixed and averaging the potential within the cell over all directions, then it can be expressed by a function of two parameters qb (r, v), where r is the distance from the centre of the cell. The partition function for a particle moving in this field when the energy zero is taken to be that of a particle at the centre is / (v,T)=(2r: m k T ) ' l , h - 3 f exp-J~{(~ (r,v)
+ ( O , v ) ) / k T } 4re
r2
dr
(15)
To obtain the partition function of the whole assembly, we could suppose that as a first approximation the particles might be taken to be moving independently of each other, as in E i n s t e i n's theory of the specific heat of solids. In that case we can write F (v, T) = [1 (v, T)] ~ exp (--@o ( v ) / k T )
(1 6)
where *0 is the mutual potential energy of the assembly when each atom is at the centre of its cell. If in this configuration the atoms form a face centred cubic crystal lattice, we have
O0 (v) = (N/2) {+ (o, v) + ~ (v)}
(17)
where + (o, v) is twelve times the mutual potential energy of each pair of atoms and ~ (v) is an extra term to take into account the interaction of particles which are not nearest neighbours. For a face centred cubic structure the interaction of such particles may increase the attractive term in the potential energy b y as much as 20 per cent 28). If, however, take into account the fact that atoms can change places (though maybe slowly), the correct partition function will be given by (16) multiplied by N N / N ! This factor will not affect the equation of state and will be omitted below, though it may be important in other connexions. The free energy of the gas (A = U - - T S ) is then given by the usual formula A = - - k T log F (v, T) : - - N k T log [ (v, T) + dP0 (v) - -
3NkT
T log(2 ~ m k-)--Nk logv
,
I18)
952
J.E. LENNARD-JONES
where 7. is so defined that Z v* is the volume, available for each atom, ~( (v, T) : ~ , j ' e x p - - { ( +
(r, v ) - - +(O,v))/kT}4rcr2dr
(19)
N k T ~Z
(20)
The equation of state is
NkT
P-
v
+ ~
~v
OaPo
~v
If the atoms do not exert forces on each other, then + (r, v) is independent of r, 7. is equal to unity, q% (v) is zero and the last equation reduces to the ideal gas law. Another method of obtaining equation (20), which while essentially equivalent to the above brings out more clearly the assumptions made in its derivation and prescribes more closely its range of validity, is by means of the equations
~p
~u
and U=O0+ Nf{q(r,v)+(O,v)} exp {(t~(r,v)--+(O,v))/kT}4rcr2dr f e x p - - {(+ (r, v) - - + (o, v))/kT} 4r~r2dr
= a)o - - N k ~(@/T) log Z
(22)
The integration of (21) for this value of U leads to (20) if the constant of integration be adjusted to give the perfect gas law in the absence of interatomic forces. It is evident that (22) gives the internal energy of the assembly provided that we can divide up the potential energy into two terms, one referring to the mutual potential energy of the particles when each is at the centre of its cell and the other referring to the average potential energy when each particle moves within its cell. From equation (20) for the pressure we get some inkling as to why the expansion of p as a power series in 1/v breaks down at high densities. While it m a y be possible under certain conditions to express *0 and Z as a simple function of v, it is probably incorrect to expand the quotient of two functions such as (1/~)~7./~v as a series when v becomes small. From its definition it would appear that 7. must approach a constant value as v becomes large and calculations in a special case indicate that 7. falls off very rapidly as v approaches a volume v0 of
THE EQUATION
OF STATE OF GASES AND CRITICAL PHENOMENA
953
the order of that appropriate to liquid densities so that it is probable that )~can be written in the form *) 1 ----A + ~ 0 - - ~ ) - ' + X Then we get
. . . . G 0 - - ; - - ~ -~.
--
1 dX _
/
v0
v /
f2
(2,,
\--s--lq
"'
(22)
: dv V2[A +
;and the term O*o/~v will in general fall off at least as quickly as (vo/v) 2, if the cohesive forces are of the usual v a n d e r W a a 1 s type. Substituting the above expression in the equation for p, we see that it gives the correct asymptotic form, viz. equation (1A), when v is large compared with Vo. As v approaches Vo, we get 1 Lt
~
dZ
)~ dv
1 --
s
v 1--vo/v
s
V--Vo
(23)
and this term is then the predominating one. Actual calculations have been made of X as a function of v/v o and T for the special case when the potential of the interatomic field can be expressed as the sum of two inverse power laws, viz. as given in (8A), the value of m being assumed to be 6 and that of n to be 12 **). The appropriate expression for Z then takes the form X . : (2~/2)r:/y 89exp c[kT[' [ q~~J -
(~).__'l(y) + 2 ( ~2 ) m(y)}ldy
(24)
where c is the number of nearest neighbours (taken to be 12), and **)
y = r2/a 2 l(y) = (1 + 12y + 25.2y 2 + 12y3 + y4) (1-- y)-,0 __1 re(y) = (1 -7 y) (1 - - y ) - 4
1
(25A) (25B) (25C)
r being measured from the centre of a cell and a the distance between nearest neighbours. This function can be evaluated numerically as a function of V/Vo for a series of values of (%/kT). Under the same conditions we have *) A c t u a l l y v0 is the v o l u m e of the a s s e m b l y if all the a t o m s are a r r a n g e d on a face centred cubic l a t t i c e and the distance between neighbours is r e. **) The n u m b e r of n e a r e s t n e i g h b o u r s is a s s u m e d to be 12 because this is the n u m b e r a p p r o p r i a t e to the solid s t a t e for a s p h e r i c a l field of the t y p e here considered (of. ~ref. 28).
954
j. E. LENNARD-JONES
r
= ~ - c I ~o [
--
2.4
(26)
and we may note in passing that aPo has a minimum value of (0.72) Nc [P0[ when (Vo/V)2 is equal to 1.2 or v equal to (0.9) v0. In the special case of a face centred cubic arrangement here discussed and forces of the special type used, we note that aP0, which gives the energy required for sublimation of a crystal at the absolute zero, is 8164 %, if % is measured in calories per gram molecule; the distance between neighbours at the absolute zero (neglecting zero point energy) is about 3 per cent less than r0. It has been found that the isothermal which has the property of satisfying ~p/~v = ~2p/~v2= 0 is the one for which C[~o]/kT is equal to 9 approximately, so that
kTc = (4/3)1%1
(27)
The observed values of the critical temperatures of a few gases (taken from the International Critical Tables) are given in Table III, T A B L E III
Calculated and Observed Properties of Gases
TB Theory (.=12, m=6
Helium . Hydrogen Neon . . Nitrogen . Argon . .
calc. - 3.42 ~o/k
obs.
23 1) 1)
23.4 105 121
5.2 33.1 44.7 126 149.7 155
Oxygen
CarbonMonoxide Methane . .
333 3) 491 4)
Holborn Holborn
1) 2) 2) 2)
328 408
Uo calc.
obs.
109 123 323 410 423
~) 2) 3) 4) 5) 6) 7)
TC
134 190.5
~
calc. obs.
9.1 40.9 47.1 128 159.3
(164)
(13o)
(191)
TB/Tcj Uo/kTc
590 1860 2030 1890 2060
2.57
~
s) ~) I e) 7)
2090 7) 2700 7)
lt8 530 609 1655 2058
4.42 3.29 2.75 2.56 2.73 2.72 2.5 2.52
6.45
6.6 7.4 6.8 6.1 6.7 7.9 7.1
and O t t o , and O t t o ,
Z. Phys., 38, 365,(1926). Z. Phys., 33, 9, (1925). Bartlett, J. Am. Chem. Soc., 52, 163, (1930). Keyes and Burkes, J. Am. Chem. S o c . , 4 9 , 1403, (1924). C 1 u s i u s, Z. phys. Chem. B, 4, 1, (1929) corrected for zero point energy. Born, F., Ann. P h y s i k , 69, 473, (1922). London, Z. phys. Chem. B, 11, 240, (1930).
together with the observed B o y 1 e temperatures and heats of sublimation. Calculated values, using the formulae at the head of each column and the values of % in Table 12, are given in adjoining
T H E E Q U A T I O N OF STATE OF GASES A N D CRITICAL P H E N O M E N A
955
columns. For those gases for which ~0 has been derived from the B o y 1 e temperature and the theoretical formula (appropriate to n ---- 12, m ---- 6), the calculated values of the critical temperature are given in brackets. The agreement between calculated and observed values is seen to be surprisingly good and shows that the formula (27), connecting critical temperature with the force field, is approximately correct, whatever the defects of its derivation. The corresponding formula derived from v a n d e r W a a 1 s equation (using equation (6) and the relation kTc ~ 8a/27bN) is kTc = 8~0/27 and clearly does not correspond to the facts. Only in the lighter gases is the divergence between theory and observation appreciable, and this is to be expected, seeing that the theory has been developed in terms of classical statistics and has ignored the quantisation of energy levels within individual cells. It is satisfactory to find that it gives the order of magnitude in these cases In the same table are to be found values of the heat of sublimation. The ratio of Uo/kTc observed is given in the last column and the values are not markedly different from the theoretical value for the particular potential field for the field kr-12--~r -s. The results suggest that the theory is on the right lines and encourage further efforts to improve it. In view of the results obtained similar methods have been applied to a two dimensional gas, using the same force fields 2,). The theory predicts that adsorbed layers, which are able to move freely over a surface, should have critical temperatures which are only about one half of the value of those observed for three dimensional gases and vapours. There is as yet little information available about the critical properties of gases in adsorbed layers, but S i m o n s0) has observed that the specific heat of adsorbed argon is the same as that of freely moving particles at temperatures well below those at which the atoms would have condensed in the bulk phase.
Added in Proo/. The method described above has now been extended by the author and A. F. D e v o n s h i r e to give the boiling temperature. Using the same force fields gewin in Table II, the calculated values come out to be: neon 26.7 ~ (obs. 27.2), argon 85.8 (obs. 87.4), nitrogen 71.8 .(obs. 77.2). The boiling temperature cannot always be expressed simply in teams of ~0 alone, lut in many cases it appears to be given approximalety by k T = ~ ~o. Received October 22 th, 1937.
956
T H E E Q U A T I O N OF STATE OF GASES A N D C R I T I C A L P H E N O M E N A REFERENCES
1) K e e s o m, Comm. Leiden, No. 24b, 1912, Phys. Z. 22, 129, 643 (1921). 2) L e n n a r d - J o n e s , Proc. roy. Soc., A 106, 463, (1924). 3) L e n n a r d - J o n e s , Proc. phys. Soc. 43, 461,(1931); cf. alsoFowler Statistical Mechanics, 2nd edition (1936) Table 19, Chapter X. 4) Z w i c k y , Phys. Z. 22, 449, (1921). 5) W o h l , Z. phys. Chem. (B) 14, 36, (1931). 6) K i r k w o o d and K e y e s , Phys. Rev. 37, 832, (1931). 7) B o r n and L a n d ~ , Berl. Ber. 1048, (1918). 8) D e b y e , Phys. Z. ~1, 178, (1920). 9) E i s e n s c h i t z and L o n d o n , Z. Phys. 60, 491, (1930). 10 L e n n a r d - J o n e s , Proc. roy. Soc., A , 1 2 9 , 598, (1930). 11 H a s s 6, Proc. Camb. phil. Soc., 26, 542, (1930); 27, 66, (1931). 12 A t a n a s o f f , Phys. Rev., 31, 1232, (1930). 13 S l a t e r andKirkwood, Phys. Rev., 37, 686, (1931). 14 M a r g e n a u , Phys. R e v . , 3 8 , 747, (1931). 15 P a u l i n g and B e a c h , Phys. Rev., 47, 686, (1935). 16 L o n d o n , Z. phys. Chem. ( B ) , I I , 222, (1930). 17 K i r k w o o d , Phys. Z. 33, 57, (1932). 18 V i n t i , Phys. Rev., 41, 813, (1932). 19 H e l l m a n n , Aeta Physieochim. U.S.S.R., 2, 273, (1935). 20 B u c k i n g h a m , Proe. roy. S o c . , A , 1 6 0 , 9 4 , 113,(1937). 21) B u c k i n g h a m, P h . D . Dissertation, Cambridge (1937). 22) B o r n and M a y e r , Z. Phys. 75, 1,(1932). 23) S 1 a t e r, Phys. Rev., 32, 349, (1928). 24) L e n n a r d - J o n e s , Proc. Camb. phil. Soe., 22, 105, (1924). 25) M i c h e l s and others, Proe. roy. Soc.,A 160, 348, 358, 376, (1937); M i c h e l s V e r a a r t , Dissertation, A m s t e r d a m (1937). 26) U r s e 11, Proe. Camb. phil. Sot., 32, 685, (1927). 26a) M a y e r , J. chem. Phys. 5, 67, (1937) ; M a y e r and A c k e r m a n n , J. chem. Phys. 5, 74, (1937). 26b) T o n k s , Phys. Rev., 50, 955, (1936). 27) L e n n a r d - J o n e s and D e v o n s h i r e , Proe. roy. S o e . , A , (to be published shortly). 28) L e n n a r d - J o n e s and I n g h a i n , Proe. roy. Soc., A , 1 0 7 , 636, Table I, (1925). 29) D e v o n s h i r e , Proc. roy. Soc., A, (to be published shortly). 30) S i m o n and S w a i n , Z. phys. C h e m . , B , 28, 189, (1935).
P h y s i c a IV, n o 10
THE
EQUATION OF STATE OF GASES AND CRITICAL PHENOMENA by J. E. LENNARD-JONES The University Chemical Laboratory, Cambridge
1. Van der Waals tomes. It is a fitting tribute to v a n d e r W a a 1s' memory that those intermolecular forces, whose importance in the theory of gases and liquids he was the first to recognise, should now be universally known by his name. V a n d e r W a a 1 s' cohesive :forces have been the subject of continuous study since his equation of state was introduced and within recent years interest in them has again been stimulated by the discovery of their extensive applications in physics and chemistry. It is, however, only recently that it has become possible to study quantitatively the nature and magnitude of these forces. K e e s o m 1) was the first to give a precise derivation of the equation of state for particular forms of intermolecular fields and by a comparison of the theoretical formula with observed isotherms of gases to derive numerical values for the attractive forces. In all his molecular models, however, the repulsive field was represented by rigid spheres or by ellipsoids of revolution and the attractive field by inverse powers of the distance. In view of the obvious limitations of a rigid sphere model, a repulsive field of the inverse power law of the type Xr-', on which was superimposed an attractive field of a similar type, was considered by the author 2) and an equation of state deduced appropriate for gases of low density. This equation has been found adequate to explain in a surprisingly satisfactory way the observed isotherms of gases and from it information of interatomic fields has been derived 3) which so far as it has been tested has given a reliable indication of many properties of gases and solids. Other theoretical calculations of the equation of state for special types of intermolecular fields have been made by Z w i c k y 4), W o 11 1 5), - -
941
- -
942
j. E. LENNARD-JONES
Kirkwood and K e y e s 6 ) and shown in particular cases to explain the observations. These methods may be described as inductive in that they proceed from the observations and endeavour to find such representations of force fields as will explain the facts. Recently it has been found possible to calculate under certain conditions interatomic forces from the electronic structure of atoms. Attempts to construct such deductive methods were made by B o r n and L a n d ~ 7 ) , Deb y e 8) and others in terms of the older pictures of atoms and molecules but the advent of quantum mechanics has provided a deeper understanding of the nature of these fields and has permitted direct evaluation of their magnitudes. E i s e n s c h i t z and L o n d o n 9) were the first to carry through such calculations for the interaction at large distance of two hydrogen atoms and simpler or improved methods have since been given by L e n n a r dJones1~ H a s s ~ 11), A t a n a s o f f 1 2 ) , Slater and K i r k woodla), Margenau 14) and P a u l i n g and B e a c h 1 5 ) . These investigations were concerned mainly with hydrogen and helium atoms. Approximate methods of dealing with larger atoms have been given by L o n d o n l ~ ) , S l a t e r and K i r k w o o d 13) Kirkwood17), V i n t i 18) and H e l l m a n n 1 9 ) . Using a variation method B u c k i n g h a m 20) has recently given formulae for the v a n d e r W a a 1 s cohesive forces between atoms with any number of electrons and, using available wave functions of the rare gas atoms and alkali ions, has compared the results with those derived by the writer from the equation of state 21). While it must remain one of the objects of theoretical research to determine force fields in this direct way, actually the fields so found are complicated functions of the distance and are not very convenient to apply in other investigations. For the present, therefore, it seems desirable to be able to express force fields in terms of simple functions which lend themselves to mathematical analysis. The indication is that the interaction of neutral atoms at large distances can be represented by a potential function which varies as the inverse sixth power of the distance. At smaller distances the function is not so simple. Nevertheless it is convenient to adopt the asymptotic form of the function as valid over the whole range and to make any necessary modifications in the repulsive field which must be used in conjunction with it.
THE EQUATION OF STATE OF GASES AND CRITICAL P H E N O M E N A
943
There is no clear indication as yet as to the best form of the field at distances so small that the atoms repel each other. In the case of hydrogen atoms this repulsive field contains an exponential function multiplied by a series of powers of the distance and so it has become customary to express the repulsive field by an exponential function alone. This form has been used by B o r n and M a y e r 22) in their'work on crystals and found satisfactory in this connexion. S 1 a t e r 23) also found the exponential form to be a satisfactory representation of the calculated field exerted between two helium atoms. Yet it must be recognised that the adoption of the simple exponential foI:m is to some extent arbitrary and it is doubtful whether other forms should be discarded in favour of it. The inverse power law has much to commend it, p a r t i c u h r l y when the total field is expressed in some such form as Xr - ~ - - ~r -'~, for, as will be seen below, this function has simple and elegant properties, in that the force constants are easily expressed in terms of the equilibrium distance of two atoms and the value of their potential energy when in this position. In this paper, therefore, considerable use will be made of this type of field. The object of this paper is to show how force fields can be determined from observations of the equation of state at low pressures and then to describe a method by which the equation of state at high pressures m ay be derived. This leads to a calculation of the critical temperature of a gas in terms of the constants of the interatomic field and the formula seems to reproduce observed critical temperatures within a few degrees. 2. The equation o] state o] a gas at small c o n c e n t r a t i o n s - - D e t e r m i n a tion o / i n t e r a t o m i c ]orces. V a n d e r W a a 1 s' equation of state may be regarded as the first successful attempt to interpret the equation of state of a gas in terms of interatomic forces, and conversely as the first at t em pt to derive information about molecular fields from the equation of state. While m any other methods have since been devised to evaluate these fields, it is still probably true to say that the method of the equation of state is the most powerful and reliable y e t available. The physical arguments by which v a n d e r W a a 1 s derived his equation are suitable, however, only for the special molecular model with which he dealt, viz. that of a rigid sphere surrounded by
944
J . E . LENNARD-JONES
an a t t r a c t i v e field.Thus his m e t h o d had special reference to the conditions at the b o u n d a r y of gas and gave the pressure actually exerted on the walls of the containing vessel, so t h a t the t e r m a/v 2 in his equation is on this view the diminution in the rate at which m o m e n t u m is c o m m u n i c a t e d to unit area of the walls in unit time. A t t e m p t s to obtain equations of state for molecular models of more generality have been derived b y essentially different methods, and often refer to the uniform conditions in the interior of the gas. I t is not then always clear w h a t i n t e r p r e t a t i o n is to be given to them. While the pressure at t h e . b o u n d a r y of a gas is the same as t h a t of the interior, the word pressure has a different meaning in the two cases. At the b o u n d a r y , it is due entirely to the m o t i o n of the molecules and is described in terms of the m o m e n t a of the b o m b a r d i n g atoms whereas in the interior only part is due to the molecular motion. The remaining p a r t is due to the stress set up across a n y area in the interior of the gas b y the interaction of the atoms on one side with those on the other. Although at a n y given point it is a fluctuating function of the time, it has e v e r y w h e r e within the gas the same statistical m e a n value. This m a y be referred to as a statical pressure to distinguish it from the more usually understood d y n a m i c al or m o m e n t u m pressure. The equation of state for a gas of small c o n c e n t r a t i o n can be shown to be B p v = k N T 4- --(IA) 7)
or
pv = kNT
(IB)
4- B ' p
where B and B ' are functions of t e m p e r a t u r e depending on the molecular forces and the o t h e r symbols have their usual meaning. B and B' are given in terms of intermolecular fields b y the expressions O<3
B = B' kNT
= 2= N 2 k T f r 2 { 1 - - e x p ( - - ~ ( r ) / k T ) } o
dr
(2)
where p(r) is the potential energy of two molecules at a distance r apart. These equations are, like t h a t of v a n d e r W a a 1 s, only first a p p r o x i m a t i o n s and valid only for dilute gases, for t h e y are derived on the assumption t h a t the atoms influence each other only in pairs. The t e r m B / v 2 represents the statical pressure in the interior of the gas 24) and m a y conveniently be d e n o t e d b y Ps. It appears from the expression for B t h a t the terms a/v 2 and 'b' of v a n d e r W a a 1 s
' r i l e EQUATION OF STATE OF GASES AND CRITICAL P I I E N O M E N A
945
equation arise respectively from the contributions of the attractive and repulsive parts of the field to the statical pressure. In the v a n d e r W a a 1 s molecular model the repulsive and attractive parts of the field are in separate regions of space and it is not surprising that their contributions to the statical stress should be separable. It is :not to be expected that in other models the effects of the repulsive and attractive fields can be so distinguished. In passing, it may be observed that often the internal pressure of a gas is defined by means of the thermodynamic equation
(3)
p, =
= T (~p/or)v - - p, but this is not the same as the statical pressure; in fact, it is clear from (3) that when equation (1) is valid
Pi= T\~T/
--P~= T2~
~ ~
(4)
At high densities the internal pressure can be very high as has been found by M i c h e l s 2 5 ) , even exceeding by many times the external pressure. When v a n d e r W a a l s equation is written in the form of equation (1B), it appears that
B' : b - - k N ~
(5)
whereas the corresponding formula derived from (2) for molecules which behave as rigid spheres of diameter ~, surrounded by an attractive field whose potential is ~r-" is 1)
B'
52 r~N a3 I 1
3 (~o/kT) [
~
(6)
where ~o =
v.e'-m
(7)
and is the potential energy of two molecules in contact. This formula for B' may be identified with (5) provided % is small compared with /~T and a comparison of the two gives the interpretation of a and b in terms of the molecular fields. A more general formula than (6) has been given ~) which corresponds to interatomic forces whose potential can be written as the sum of inverse power laws qo : Xr- ~ Physica IV
~r--~
(8A) 50
946
j. E. LENNARD-JONES
This function can be written in the more convenient from
where r 0 is the distance between two molecules in equilibrium u n d e r the field (8A), and ] % [ is the energy required to separate t h e m from this configuration. The formula for B' is t h e n B' = 52 ,~N r ao t bo/kT)
(9)
The function / is a polynomial whose coefficients depend only on n and m. Thus
/ ( % / k r ) = (m/n) 3/'~-m F(y) where
r
3v =
~,
n
/
(lo)
and
y----- (re~n)-c"l') (l--re~n) -1+<'/') (%/kT) l-I'/'0
(11)
Methods of determining X and [~b y a comparison of the theoretical formula for B' and the values of B' required in (1B) to fit the observe isotherms have been given elsewhere 2) 3). W h a t m a y be of interest to observe here is t h a t % can be d e t e r m i n e d easily from a knowledge of the B o y 1 e point, t h a t is the t e m p e r a t u r e for which B ' is found from the experiments to vanish. Denoting the B o y 1 e t e m p e r a t u r e b y TB and the root of F(y) = 0 (12) b y y . . . . t h e n we have %/kTB = (re~n) m/(n-m) {1 - - (re~n)} EY., m]n[(n-m)" (13) T y p i c a l values of y., m are given ill Table I with the corresponding values of kTB/% T A B L E I *) V a l u e s of Yn, m, [n, m, a n d ~o/kTB
1 11 12
1.1190 1.0997 1.0818
41044
4.07 3.71
0.4781 0.5049
3.43
0.5285
In some cases the 'observed' values of B' have a m a x i m u m and this p r o p e r t y is exhibited also b y the f u n c t i o n / . F r o m the value of *) T h e a u t h o r is i n d e b t e d to Dr. A. F. D e v o n s h i r e r e q u i r e d f o r t h i s a n d t h e o t h e r tables.
for his h e l p in the c a l c u l a t i o n s
THE
EQUATION
OF
STATE
OF
GASES
AND
CRITICAL
PHENOMENA
947
the maximum, r0 can be deduced for assumed values of n and m, using the maximum values of / (denoted by/~, ,) given in Table I. The most appropriate values of n and m can only be determined by a comparison of / and the whole range of observed values of B. For the inert gases and some molecules it has been found that m = 6 (corresponding to the theoretical value for v a n d e r W a a 1 s forces) and a value of n between 9 and 12 fits the observations satisfactorily. Values of X and tx have also been deduced in this way and from them the values of r0 and ?0 given in Table II "). These refer in all cases to n = 12, m = 6. TABLE Constants
of Interatomic
Fields
k Helium . . . . Neon . . . . Argon . . . . Hydrogen . Nitrogen . . Oxygen . . Carbonmonoxide Methane . .
. . . .
3.605 3.545 16.2 6.49 3.70
. . . . .
II
(Potential
p~ 10- TM 1 0 - x~ 10 - 1 ~ 10 - l o s 10 - 1 ~
1,17 8.32 1.034 1.05 1,40
. . . . ,
energy
r0 (/~-) 10-60 1 0 - s~ 10 - 5 8 10 - 5 ~ 1 0 - 58
2.917 3.049 3.819 3.276 4.174
=
),r-n--l~r-m) ( 1 0 _ ~ 0 e r g s ) . , ( c a l . / g rq~o .atom) 0.950 4.881 16.50 4.246 13.24 16.97 13.36 19.70
13.7 70.6 239 51.4 192 244.9 192.8 284.3
Values of ?0 for 02, CO and C H 4 a r e also included, derived from the observed B o y 1 e temperatures given in Table III, where a comparison of observed and calculated B o y 1 e points for the other gases is also given.
3. The equation o~ state o~ dense gases - - Critical phenomena. The equations of state given in the preceding paragraph take into account only binary encounters between the atoms of the gas and are found to be in agreement with the observations just where they would be expected to be, viz. when the concentration is small and the temperature not too low.They cease to be valid except under these conditions ~nd should not be used in the critical region or regarded as adequate explanations of critical phenomena. At higher concentrations the observations can sometimes be expressed by more extended series in 1/v or in p than are given in (1A) and (IB). It is possible to work out under certain limited conditions a theoretical formula for the coefficient of 1/v 3 for special molecular models, but not for force fields of a general type. General m~thods have been given by
948
j. E. LENNARD-JONES
U r s e 1 1 26) for the evaluation of the coefficients of the series, which correspond to the interaction of the atoms in groups of three, four, etc., but they have been not applied to particular force fields. They have, however, recently been used by M a y e r 2,~) to show that when carried far enough the equation of state thus derived does actually lead to critical phenomena. A method of obtaining an equation of state for condensed gases, when the atoms are assumed to be rigid spheres, has also been given by T o n k s 26b), based on the virial theorem. M i c h e 1 s 25) has found that at high pressures the series expansion of pv in powers of 1Iv or of p soon ceases to be useful except over limited ranges of temperature and pressure. It is even found that the coefficients B and B' found by fitting (1A) and (1B) to the observations at very small pressures are not consistent with those found by fitting more extended series to the observations at high pressures. It has been found necessary to use series of the type
pv=A
+ (B--~) + C
Z
D
E
F
(14)
where ~ expresses the difference between the coefficient of 1/v at high and low pressures. After extended trials to fit the observations at high pressures, M i c h e 1 s and his collaborators have come to the conclusion that isotherm data obtained at high densities cannot be expressed as ordinary power series. When the isotherms are expressed in this form it is even found that for large densities the series is not convergent. In view of this result it is doubtful whether any useful purpose is served by trying to build up a theory in terms successively of diatomic, triatomic and more general complexes. At high pressures when the density of a gas is approaching that of a liquid, each atom must be in the field of several other atoms not only occasionally but all the time. This is illustrated in the figure 1. The atoms, represented by dots are placed regularly on a crystal lattice at the average distance apart corresponding to the critical density, a 1,0, 0 section of a face centred cubic lattice being shown. The potential field of one atom due to another, as found in the preceding section, is represented by a series of contour lines. The dotted circles show the contours of negative energy, that is the attractive region, while the continuous circles show the repulsive region. The outermost circle corresponds to an
THE EQUATION OF STATE OF GASES AND CRITICALPHENOMENA 949 energy equal to (0.1) q~0, the second one to (0.2) ~0 and thereafter t h e y increase at equal intervals of (0.2) q0o. If now this picture of an atomic field is drawn on tracing cloth and moved over the lattice array of atoms, it is found t h a t whatever position it takes up the attractive field always overlaps three or four other atoms and in three dimensions it m a y well overlap double this number.
.."i --'- . . . . . . . I
l / I
I i I ,
"
I
i ~1.1
I
~
~
-"-
~
\ \
i / , ' f f ~ ~ \ ' , ' , ' ,
I.'lWt, f///
"
", ', \
~ ' ~ \ i \ ~ '.
"*qlll
\\\l(l.I
I i I Ili~llll ~-I 1 ~ I Ill~llll I !~ l lltli I ; t t,l\\\\\ ~'
".
\NN~.\\\ \ \
,';///I.,t/// , i,
k
, ~
/~li~lll , i iIIIm~ I,,
~
,.
~ i
, i
Illll~l, I;, ! I /lll'~fl I., i i I 1111(9,# I s I I ///r ~ / I I , //7'](II I i I i , ' / / ," '
t '"Ok\\\
"\ '. ~ \~}&\\\ , '\ \ "~T~k.~ ' '
-'Od
Fig. 1. The overlap of atomic fields at the critical density. This state of affairs is reached in the critical region and theoretical a t t e m p t s to devise an equation of state suitable for the cvaluation of critical constants ought perhaps to begin from this point of view. Interatomic fields, if correctly derived, should be the connecting link between the observed values of B in equation (l A) and the critical constants. An a t t e m p t has recently been made to effect this correlation 2~). The method begins not from the usual concept of a gas but from the supposition t h a t the average envirc L-
950
j . E. LENNARD-JONES
ment of an atom in a dense gas is more like that of an atom in a liquid or solid. An atom is regarded as imprisoned for most of its time by its immediate neighbours in a cell, just as it is in a solid, and though migration or diffusion will in practice take place from one cell to another, this possibility is neglected as being an infrequent event compared with the duration of time spent within each cell. This theory can only be regarded as a first approximation developed from a new point of view but it is probably better than attempts to deal with a dense gas by methods which depend essentially on binary encounters. When the density of a gas is increased to such an extent that any one atom is simultaneously under the influence of several others, the total field acting upon it will be of a very fluctuating kind and the first step in dealing with this field must be to replace it by some kind of statistical average. One method of doing this is to take the field exerted on one when the others are at their average distances from it. All the immediate neighbours m a y then be conceived as moving so that their centres remain on a sphere of a definite radius and the field created by them within tile sphere will, when averaged out, have spherical symmetry. Even so the field within the sphere will be of a complicated kind. It will depend not only on the field exerted by one atom on another but also on the size of the spherical cell. When this is large the attractive fields of the neighbouring atoms will overlap at the centre but not to a very great extent and direct calculation shows t h a t the region of lowest potential lies near to the circumference of the sphere. It is as though the atom were enclosed in a spherical box in which there is a strong adsorbing field at its surface. The enclosed atom will tend to be near the circumference and not at the centre of its cell. This state of affairs corresponds to the tendency of the atoms to form pairs. As the size of the sphere is decreased, so the region of lowest potential will contract and the potential barrier at the centre will diminish owing to the greater and greater overlap of the attractive fields. Finally a size of cell will be reached for which the position of the minimum potential energy will be at the centre and will remain there for all lower sizes, though indeed the absolute value of the potential at the centre will depend on the size. After reaching an absolute minimum it will rise for all higher ~ Not only the absolute magnitude of the potential energy, but also its form within a cell will depend on the volume and it must accordingly be denoted
THE EQUATION OF STATE OF GASES AND CRITICAL PHENOMENA
951
by a function of the type, q0 (r, v), where v is the volume of the cell and r is a vector denoting space coordinates within it. When the potential is replaced by a suitable average either by supposing the :nearest neighbours to take up all orientations with equal probability on a spherical surface or b y supposing them fixed and averaging the potential within the cell over all directions, then it can be expressed by a function of two parameters qb (r, v), where r is the distance from the centre of the cell. The partition function for a particle moving in this field when the energy zero is taken to be that of a particle at the centre is / (v,T)=(2r: m k T ) ' l , h - 3 f exp-J~{(~ (r,v)
+ ( O , v ) ) / k T } 4re
r2
dr
(15)
To obtain the partition function of the whole assembly, we could suppose that as a first approximation the particles might be taken to be moving independently of each other, as in E i n s t e i n's theory of the specific heat of solids. In that case we can write F (v, T) = [1 (v, T)] ~ exp (--@o ( v ) / k T )
(1 6)
where *0 is the mutual potential energy of the assembly when each atom is at the centre of its cell. If in this configuration the atoms form a face centred cubic crystal lattice, we have
O0 (v) = (N/2) {+ (o, v) + ~ (v)}
(17)
where + (o, v) is twelve times the mutual potential energy of each pair of atoms and ~ (v) is an extra term to take into account the interaction of particles which are not nearest neighbours. For a face centred cubic structure the interaction of such particles may increase the attractive term in the potential energy b y as much as 20 per cent 28). If, however, take into account the fact that atoms can change places (though maybe slowly), the correct partition function will be given by (16) multiplied by N N / N ! This factor will not affect the equation of state and will be omitted below, though it may be important in other connexions. The free energy of the gas (A = U - - T S ) is then given by the usual formula A = - - k T log F (v, T) : - - N k T log [ (v, T) + dP0 (v) - -
3NkT
T log(2 ~ m k-)--Nk logv
,
I18)
952
J.E. LENNARD-JONES
where 7. is so defined that Z v* is the volume, available for each atom, ~( (v, T) : ~ , j ' e x p - - { ( +
(r, v ) - - +(O,v))/kT}4rcr2dr
(19)
N k T ~Z
(20)
The equation of state is
NkT
P-
v
+ ~
~v
OaPo
~v
If the atoms do not exert forces on each other, then + (r, v) is independent of r, 7. is equal to unity, q% (v) is zero and the last equation reduces to the ideal gas law. Another method of obtaining equation (20), which while essentially equivalent to the above brings out more clearly the assumptions made in its derivation and prescribes more closely its range of validity, is by means of the equations
~p
~u
and U=O0+ Nf{q(r,v)+(O,v)} exp {(t~(r,v)--+(O,v))/kT}4rcr2dr f e x p - - {(+ (r, v) - - + (o, v))/kT} 4r~r2dr
= a)o - - N k ~(@/T) log Z
(22)
The integration of (21) for this value of U leads to (20) if the constant of integration be adjusted to give the perfect gas law in the absence of interatomic forces. It is evident that (22) gives the internal energy of the assembly provided that we can divide up the potential energy into two terms, one referring to the mutual potential energy of the particles when each is at the centre of its cell and the other referring to the average potential energy when each particle moves within its cell. From equation (20) for the pressure we get some inkling as to why the expansion of p as a power series in 1/v breaks down at high densities. While it m a y be possible under certain conditions to express *0 and Z as a simple function of v, it is probably incorrect to expand the quotient of two functions such as (1/~)~7./~v as a series when v becomes small. From its definition it would appear that 7. must approach a constant value as v becomes large and calculations in a special case indicate that 7. falls off very rapidly as v approaches a volume v0 of
THE EQUATION
OF STATE OF GASES AND CRITICAL PHENOMENA
953
the order of that appropriate to liquid densities so that it is probable that )~can be written in the form *) 1 ----A + ~ 0 - - ~ ) - ' + X Then we get
. . . . G 0 - - ; - - ~ -~.
--
1 dX _
/
v0
v /
f2
(2,,
\--s--lq
"'
(22)
: dv V2[A +
;and the term O*o/~v will in general fall off at least as quickly as (vo/v) 2, if the cohesive forces are of the usual v a n d e r W a a 1 s type. Substituting the above expression in the equation for p, we see that it gives the correct asymptotic form, viz. equation (1A), when v is large compared with Vo. As v approaches Vo, we get 1 Lt
~
dZ
)~ dv
1 --
s
v 1--vo/v
s
V--Vo
(23)
and this term is then the predominating one. Actual calculations have been made of X as a function of v/v o and T for the special case when the potential of the interatomic field can be expressed as the sum of two inverse power laws, viz. as given in (8A), the value of m being assumed to be 6 and that of n to be 12 **). The appropriate expression for Z then takes the form X . : (2~/2)r:/y 89exp c[kT[' [ q~~J -
(~).__'l(y) + 2 ( ~2 ) m(y)}ldy
(24)
where c is the number of nearest neighbours (taken to be 12), and **)
y = r2/a 2 l(y) = (1 + 12y + 25.2y 2 + 12y3 + y4) (1-- y)-,0 __1 re(y) = (1 -7 y) (1 - - y ) - 4
1
(25A) (25B) (25C)
r being measured from the centre of a cell and a the distance between nearest neighbours. This function can be evaluated numerically as a function of V/Vo for a series of values of (%/kT). Under the same conditions we have *) A c t u a l l y v0 is the v o l u m e of the a s s e m b l y if all the a t o m s are a r r a n g e d on a face centred cubic l a t t i c e and the distance between neighbours is r e. **) The n u m b e r of n e a r e s t n e i g h b o u r s is a s s u m e d to be 12 because this is the n u m b e r a p p r o p r i a t e to the solid s t a t e for a s p h e r i c a l field of the t y p e here considered (of. ~ref. 28).
954
j. E. LENNARD-JONES
r
= ~ - c I ~o [
--
2.4
(26)
and we may note in passing that aPo has a minimum value of (0.72) Nc [P0[ when (Vo/V)2 is equal to 1.2 or v equal to (0.9) v0. In the special case of a face centred cubic arrangement here discussed and forces of the special type used, we note that aP0, which gives the energy required for sublimation of a crystal at the absolute zero, is 8164 %, if % is measured in calories per gram molecule; the distance between neighbours at the absolute zero (neglecting zero point energy) is about 3 per cent less than r0. It has been found that the isothermal which has the property of satisfying ~p/~v = ~2p/~v2= 0 is the one for which C[~o]/kT is equal to 9 approximately, so that
kTc = (4/3)1%1
(27)
The observed values of the critical temperatures of a few gases (taken from the International Critical Tables) are given in Table III, T A B L E III
Calculated and Observed Properties of Gases
TB Theory (.=12, m=6
Helium . Hydrogen Neon . . Nitrogen . Argon . .
calc. - 3.42 ~o/k
obs.
23 1) 1)
23.4 105 121
5.2 33.1 44.7 126 149.7 155
Oxygen
CarbonMonoxide Methane . .
333 3) 491 4)
Holborn Holborn
1) 2) 2) 2)
328 408
Uo calc.
obs.
109 123 323 410 423
~) 2) 3) 4) 5) 6) 7)
TC
134 190.5
~
calc. obs.
9.1 40.9 47.1 128 159.3
(164)
(13o)
(191)
TB/Tcj Uo/kTc
590 1860 2030 1890 2060
2.57
~
s) ~) I e) 7)
2090 7) 2700 7)
lt8 530 609 1655 2058
4.42 3.29 2.75 2.56 2.73 2.72 2.5 2.52
6.45
6.6 7.4 6.8 6.1 6.7 7.9 7.1
and O t t o , and O t t o ,
Z. Phys., 38, 365,(1926). Z. Phys., 33, 9, (1925). Bartlett, J. Am. Chem. Soc., 52, 163, (1930). Keyes and Burkes, J. Am. Chem. S o c . , 4 9 , 1403, (1924). C 1 u s i u s, Z. phys. Chem. B, 4, 1, (1929) corrected for zero point energy. Born, F., Ann. P h y s i k , 69, 473, (1922). London, Z. phys. Chem. B, 11, 240, (1930).
together with the observed B o y 1 e temperatures and heats of sublimation. Calculated values, using the formulae at the head of each column and the values of % in Table 12, are given in adjoining
T H E E Q U A T I O N OF STATE OF GASES A N D CRITICAL P H E N O M E N A
955
columns. For those gases for which ~0 has been derived from the B o y 1 e temperature and the theoretical formula (appropriate to n ---- 12, m ---- 6), the calculated values of the critical temperature are given in brackets. The agreement between calculated and observed values is seen to be surprisingly good and shows that the formula (27), connecting critical temperature with the force field, is approximately correct, whatever the defects of its derivation. The corresponding formula derived from v a n d e r W a a 1 s equation (using equation (6) and the relation kTc ~ 8a/27bN) is kTc = 8~0/27 and clearly does not correspond to the facts. Only in the lighter gases is the divergence between theory and observation appreciable, and this is to be expected, seeing that the theory has been developed in terms of classical statistics and has ignored the quantisation of energy levels within individual cells. It is satisfactory to find that it gives the order of magnitude in these cases In the same table are to be found values of the heat of sublimation. The ratio of Uo/kTc observed is given in the last column and the values are not markedly different from the theoretical value for the particular potential field for the field kr-12--~r -s. The results suggest that the theory is on the right lines and encourage further efforts to improve it. In view of the results obtained similar methods have been applied to a two dimensional gas, using the same force fields 2,). The theory predicts that adsorbed layers, which are able to move freely over a surface, should have critical temperatures which are only about one half of the value of those observed for three dimensional gases and vapours. There is as yet little information available about the critical properties of gases in adsorbed layers, but S i m o n s0) has observed that the specific heat of adsorbed argon is the same as that of freely moving particles at temperatures well below those at which the atoms would have condensed in the bulk phase.
Added in Proo/. The method described above has now been extended by the author and A. F. D e v o n s h i r e to give the boiling temperature. Using the same force fields gewin in Table II, the calculated values come out to be: neon 26.7 ~ (obs. 27.2), argon 85.8 (obs. 87.4), nitrogen 71.8 .(obs. 77.2). The boiling temperature cannot always be expressed simply in teams of ~0 alone, lut in many cases it appears to be given approximalety by k T = ~ ~o. Received October 22 th, 1937.
956
T H E E Q U A T I O N OF STATE OF GASES A N D C R I T I C A L P H E N O M E N A REFERENCES
1) K e e s o m, Comm. Leiden, No. 24b, 1912, Phys. Z. 22, 129, 643 (1921). 2) L e n n a r d - J o n e s , Proc. roy. Soc., A 106, 463, (1924). 3) L e n n a r d - J o n e s , Proc. phys. Soc. 43, 461,(1931); cf. alsoFowler Statistical Mechanics, 2nd edition (1936) Table 19, Chapter X. 4) Z w i c k y , Phys. Z. 22, 449, (1921). 5) W o h l , Z. phys. Chem. (B) 14, 36, (1931). 6) K i r k w o o d and K e y e s , Phys. Rev. 37, 832, (1931). 7) B o r n and L a n d ~ , Berl. Ber. 1048, (1918). 8) D e b y e , Phys. Z. ~1, 178, (1920). 9) E i s e n s c h i t z and L o n d o n , Z. Phys. 60, 491, (1930). 10 L e n n a r d - J o n e s , Proc. roy. Soc., A , 1 2 9 , 598, (1930). 11 H a s s 6, Proc. Camb. phil. Soc., 26, 542, (1930); 27, 66, (1931). 12 A t a n a s o f f , Phys. Rev., 31, 1232, (1930). 13 S l a t e r andKirkwood, Phys. Rev., 37, 686, (1931). 14 M a r g e n a u , Phys. R e v . , 3 8 , 747, (1931). 15 P a u l i n g and B e a c h , Phys. Rev., 47, 686, (1935). 16 L o n d o n , Z. phys. Chem. ( B ) , I I , 222, (1930). 17 K i r k w o o d , Phys. Z. 33, 57, (1932). 18 V i n t i , Phys. Rev., 41, 813, (1932). 19 H e l l m a n n , Aeta Physieochim. U.S.S.R., 2, 273, (1935). 20 B u c k i n g h a m , Proe. roy. S o c . , A , 1 6 0 , 9 4 , 113,(1937). 21) B u c k i n g h a m, P h . D . Dissertation, Cambridge (1937). 22) B o r n and M a y e r , Z. Phys. 75, 1,(1932). 23) S 1 a t e r, Phys. Rev., 32, 349, (1928). 24) L e n n a r d - J o n e s , Proc. Camb. phil. Soe., 22, 105, (1924). 25) M i c h e l s and others, Proe. roy. Soc.,A 160, 348, 358, 376, (1937); M i c h e l s V e r a a r t , Dissertation, A m s t e r d a m (1937). 26) U r s e 11, Proe. Camb. phil. Sot., 32, 685, (1927). 26a) M a y e r , J. chem. Phys. 5, 67, (1937) ; M a y e r and A c k e r m a n n , J. chem. Phys. 5, 74, (1937). 26b) T o n k s , Phys. Rev., 50, 955, (1936). 27) L e n n a r d - J o n e s and D e v o n s h i r e , Proe. roy. S o e . , A , (to be published shortly). 28) L e n n a r d - J o n e s and I n g h a i n , Proe. roy. Soc., A , 1 0 7 , 636, Table I, (1925). 29) D e v o n s h i r e , Proc. roy. Soc., A, (to be published shortly). 30) S i m o n and S w a i n , Z. phys. C h e m . , B , 28, 189, (1935).