REFERENUHIC/66/56

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL

PHYSICS

EQUATION OF STATE OF REAL GASES

A. A. SABRY

1966PIAZZA OBERDAN

TRIESTE

i -^ :. •.r.-.q,t.;.pr'

IC/66/56

DJTEmTATIONAL ATOMIC BEERGT AGENCY

IITTSRMTIOttAL CSHTH3 FOR THSORSTICAL PHYSICS

EQpATIOlT OP STATS OP REAL GASES*

A . A . S a b r y * *

THIS STBMay 1966

* To "be submitted to "Physica"

** On leave of absence fxom the Univereity of Assyout, U.A.R.

ABSTRACT

Extensive work has been done to calculate the virial

coefficients which appear in the expansion of •==• in powers of rs-

In this work an expression of =r is obtained in a closedax ,form and hence oan be applied to very high densities of the gas.

The value of 5=7 at the critical point, as well as isotheriaals c

the gas, agree remarkably with experimental results.

- 1 -

EQUATION OP STATE OP REAL GASES

I . INTRODUCTION

A statistical study leading to the equation of state of a gas

usually starts ~by considering the virial of Clausius

f V - R T _ - l N V J r ^(jCrJ.ifTrr^r (l)

where P2( r) i-B e probability for two molecules to be at distance r

apart, ^ ( r ) the mutual potential between two molecules. The number JI

is the total number of molecules in one gram-mol of the gas.

The solving of the problem depends then in principle on a

knowledge of P , ( r ) . For the evaluation of this function, one canl)

refer to the work of BORN and GREE1J ' , in which they use a relation

suggested by KIRKWOOD y , relat ing the probability of relative

distribution Pi23^1 f **2* "*3^ o f "fc1nree molecules to the probabilitie

of relative distribution of two molecules p , P ^ , P-io*

The? Kirkwood relation was checked for the distribution of

molecules in the solid state, where the functions P 1?,, PT?

can be rigorously calculated on the assumption of small thermal

vibrations K The result was that the Kirkwood relation may be

taken to be an approximate relation and in no way a rigorous one

which can be reliable in all conditions.

It may be noted that the mean potential energy can be

expressed, on use of the same function po(r)» a s

(2)

- 2 -

There is a thermodynaraic relation between the derivatives

of ^ and £p , namely,

r A_ ft)

This relation is generally not satisfied, if jf> , tp are calculated

from equations (l), (2). This is because p2(r) is not known

rigorously.

In this paper, a different procedure is followed to obtain

the equation of state, without the use of the function p 2 ( r ) ,

and which conforms with the thermodynamic relations.

In the following, a short survey of the knotm eqxiationsstate of a gas will be given.

A)BOLTZIIA1JF HJ obtained the following expansion of

assuming the gas to oonsist of hard spheres

where b is four times the volume of the molecules. To obtain this

equation, he took into consideration the possibility that two

spheres of force surrounding the molecules intersect one another.

On adding to the above relation the Yan der HaalStermdue to the cohesion forces, one obtains}

\ . . (4)

This relation for small TT is, however, better than the

Yan der Waals equation. KALD&ERLIpJGti OHIJES ; applied this relation

to Helium and found that it agrees fairly well with experimental

results over a temperature range from 100° up to - 217°.

In order to obtain an expansion in density valid for all

gases, Kammerlingh Onnes assumed an empirical equation of state

- 3 -

- ,. ft. B + C

•where the coefficients A, B, C, D, E depend on the temperature as

an expansion in the form

(T, V being reduced teaperature and volume)

On giving numerical values to the 25 constants, he succeeded in

fitting observations of Amagat, Ransay and Young on a number of gases

(H2> 02, 1T2, C4H100 and C gH 1 2).

To obtain an equation of state correct for high values of the

density, CLAUSIUS assumed the empirical equation '

Although it fits the isothermals of COp for high densities, it is

found that for low densities the Van der WaalSequation fitted the

observations better. The equation, being partially successful -with

COp, was not equally successful with other gases. ClausiuB therefore

suggested a more general formula

(7)

where "*\ » A for COp and has sometimes to be taken as a function of T

to fit the observations.

- 4 -

7)There is no room to give all such empirical formula .

It is, however, obvious that there is no finality in any of these

formulae and it is possible to go on extending them indefinitely

without arriving at a fully satisfactory formula, as might bo

anticipated since they are purely empirical and not founded on any

satisfactory theoretical basis.

II. GHHERAL OUTLHTS OP THE METHOD

The equation of state of the gas, namely the pressure as a

function of T, V, will "be obtained from the free energy A

It may be noted that the entropy p is also given

hence the thermodynaniic relation (3) is automatically satisfied.

On the use of the Gibbs distribution for a fixed number of

particles the free energy A is obtained from

= -KTiog[Z. l^!KT ] (9)

where the summation is over all quantum states of the system. If 11^

be the Hamiltonian operator of the whole system, then we can express

e ^ as

(10)

- 5 -

where

(5= zrThe wave functions v describing tho whole system of molecules ccm

bo expressed as a product of a function U-j, describing tho

motion of the centre of mass of molecules, multiplied by a function

TfoC^ describing the internal structure of molecules. In a

similar way we express the total Hamiltonian ^ as a sum of II + Z

where E corresponds to the motion of the centre of masses of

molecules. The expression (10) hence becomes

-- TJ

where £rt. are the eigenvalues corresponding to the internal states

of a molecule i.

For a weak interaction between the molecules we can assume U

to be a plane wave

p ep (13)

The summation is taken over all permutations (of number lit) of

$>.,, £ 2, ..., and 6-p = 1 for Bose statistics, £+, = (-)

for Fermi statistics.

Summing "e n expressed by (12) over all states n, we

get for the partition function Z_ of the whole system

- 6 -

where

V £ L (15)is the partition function corresponding to one molecule. Also

z -- 5

(16)

whero the summation \__ over all momenta of the centre of mass

of molecules is replaced by V ) • '^ .It may also be noted that

the integration in (16) expressed by I is taken only over regions

in phase space corresponding to physical distinct states of tho body,

To esrtend the integral over all points of phase space, we have to

divide it by the number of possible permutations of the 1\T identical

molecules, and hence we get

N' ' • f« # (17)

where the integral now extends over all points in the phase space.

On substituting for U.4, from equation (13), we get for 21

J * >

(18)

- 7 -

The summation over the permutations P of the momenta is taken only-

over the first "j^. in the exponential £ ^ \. * -^

The Hamiltonian H describing the motion of the centre of

masses of molecules can be expressed as

where 0 is the mutual potential energy of the whole IT molecules

of the gas.

It will be the aim in the following to compute the partition

function Z of the centre of mass motion from equation (18).

Ill. TK3 PARTITION FUNCTION OP TRANSLATIONAL MOTION

In order to compute the partition function Z from (18),

we first carry out the operation c QT' » where II is given

by equation (19).

We s tar t from the equation

W e - t- e, ( 2 0 )

where

is the total energy of the centre of mass system.

Operating by H on both sides of equation (20), we get

= e S

- 8 -

where

7 ^W\ i

By successive applications of H, -we get

tHsnce it easily follows

If we assume

(24)

Then the expression (23) can be written in the form

(25)

Substituting from this equation into the expression (18) for X ,

we get .

(26)

where

y ^ - —•>

- 9 -

and the summation i s over a l l permutations of ^

We denote the quantity "between brackets in the integrandof equation (26) by I , i . e . ,

T- 1+ i t ^ - O E (*,

Tfe wish now to expand I in powers of 'k . This can be done, e i ther

direct ly by expanding f(S) in (28), or by using the following way ' ;

Differentiating (28) with respect to P , we get — ~ _ I

and, operating on both sides of (28) "by S, we get S3 = C £ )

hence

(29)

On expanding I in powers of K, and using the above equation, we can

easily find the coefficients of the expansion. The result can be

written:

O-

(30)

Substituting from this into (26), we get for Z

i^ 6p

I J

- 10 -

4. ZL. i

(31)

In this expression the term in the summation over all permutations

of particles for which all "Y]r = 0 is separated from other terras,

before the second integral in (31) is

only over permutations for which all V|r f* 0.

Integrating over all momenta i> , the first part in. i

equation (3l) "becomes

(32)

i *•'$the expression for 2, t o a n ^^ "simplified to

(32 a)

- 11 -

For the integration over j> in the second integral,we have f i r s t to

integrate over a l l direct ions in the 3H" space and then over the

magnitude \ f \ of the momentum, tfe set

3N-I* * & (33)

and use the following results of the integration over the solid

angle d Si in the 3U-dimensional momentum space (see Appendix i)

r p.

(34)

where the argument a of the Bessel funotion is given "by

The results after the integration over all directions of I) in

the second part of equation (3l) then "becomes

- 12 -

V « J

For the integration over the magnitude | fa \ of t> , we use the

general formula (see Appendix II)

[ (37)

The value of JJ - fluotuates about •g(*>"jp") t ani f o r "t110 least•g(*>"jp

possible volume V occupied by the hard spheres, the value of

-=- i and hence completely negligible, unless the—ir i =

temperature is extremely near the absolute aero. It follows then that

the oontribu

finally have

the oontribution Z ^ oan be neglected compared to 7-i , and we

- 13 -

IV. CHOICE OF TIE POTENTIAL

The evaluation of the 3N-dimensional integral in the q-space

for the partition function Z, given by equation (38), is a very-

difficult task. It is found possible to oarry out the integration

for a special choice of the mutual potential between the molecules.

The mutual potential ty . .(R. .) between any two molecules ifj is

supposed to be infinite within a sphere (of force) of radius CT , i.e

(39)

For R. . )> OT , the potential is left unspecified but, as is well

known, it tends to - *-7 as R. . becomes large.R. .

The total mutual potential energy $ » given by

- i 4. y.X^yJ (40)

can then be expressed as a sum of two terms

f -- + f • (41)

The part <£*• ' is a hard sphere potential which is zero

everywhere, except within the spheres of force where it is infinite.

It may be noted that a molecule as a hard sphere is of radius -r- ,

while the sphere of force around the same molecule is of radius C ;

hence the spheres of force may* overlap with each other.

On the other hand the part (£ 'is the cohesive potential

between the molecules, which can be expressed as

- 14 -

where the cohesive potentials <f) J(R. .) are only defined for

We start now to evaluate the classical part of1the partition

function Z, and postpone the quantum terms until later.

Substituting for $ from equation (41), we find that theclassical part of Z in (33) is

JO

"2 - J. . iT/VA

(43)

The cohesive potential 0 fluctuates about a mean value (p ,

Neglecting these fluctuations, we can take the part e

outside the integral in equation (43), and we are left with the

integral

If we now denote by B the average volume occupied by r spheres

of force, the above integral can easily be shown to be

( 4 4 )

Hence, the partition function Z in equation (43) can be expressedc

as

-O -(V-S,)V (45)

The average cohesive potential $^ 't and the average

volume B occupied by r spheres of force, can both be expressed in

terms of the radial distribution funotion P2(a) a s f o l l o w s :

- 15 -

The ave rage p o t e n t i a l (jr ' i s g iven by

(46)

where P_ (q*., ~q\) d q., d q. is the probability that two specified33

molecules i,j occupy the volume elements d q., d q. simultaneously.^ v

Since PO(Q.*» <lO depends only on the distance B. • between

the molecules, we can integrate the integral in equation (46) over

d q. and get

(47)

For big values of R the function Pp(*O ^ s constant and i s given

(43)

but for small values of R, the function p«(R) is a complicated

expression which depends also on the volume V and temperature T.

If we however assume P2(R) * T f o r a 1 1 Va-Lues of R > 0"

we obtain from (47)

•y (49)

where

O-(50)

- 16 -

More rigorously, a is not a constant and is defined "by

2.

For small densities, the quantity a can be safely taken as a constant,

but as the density becomes very high, a is no longer a constant and

depends on V, T.

In the following we shall regard a as a constant, and shall

find that the results accordingly conform with observation up to

densities as high as the critical point.

Uert we attempt to calculate the quantity B , the average

volume occupied by r spheres of force.

Assume A .(.R. .) be the volume

cut out by a sphere of force around

a molecule j from the sphere around

the molecule i. This quantity is easily

found to be

J ^ (52)

How, the total volume left from the sphere i is

and the total volume of all r spheres of force is

- 17 -

The average volume of r spheres is then given by

For low densities, we can substitute for pp(R) from equation (48),

and obtain

z r IS.- ioC rZJSr *- V" (54)

where -O -

z ^ r1 V

ii: (55)

when we substitute for -^(R) from (52).

In general 06 is a function of V, T,especially for very

high densities, and is given by

(56)

On substitution for Qi ^ Br fr<>m equations (49), (54) into

equation (45)» we obtain the classical part of the partitionfunction Z in i t s final form

V , TTri ^; 1 B,

- 18 -

V. THE EQUATION OF STATE

The free energy Af defined by equation (9), can now be

expressed, on use of equations (l4)» (15) and (57), as

3/

AM

vK V

plus another quantum terra whioh we have so far not taken into

account, Z_ being the partition function for one molecule.

The last sum in this expression can bo expressed as an

integral, assuming the number N to be very big,

(53)

where \,y are constants given by

(60)

On use of this expression for A, we can obtain the pressure p, on

use of equation (8). I t will however be more convenient to use the

diraensionless quantity

pressure takes the form

Riinstead of pf and the equation for the

h R (61)

where we have used the constant b, already mentioned (equation (4l))»

and related to B. by

2 b

The diraensionless quantity F is given by

I -,

(62)

(63)

The value of17 W n e n

constant —p , but since V

to zero as R — > 0 , we expect

function f

tends to 1 as R

yoo and tends

than 1 for

CT <^ R <f 2 0" . Hence the value of 0C as defined by equation (56)17must be greater than -w . Por the moment we shall leave dL as an

unspecified constant. However, we shall see later that there is a

neat method for the numerical determination of the constant oC > as

we study the critical data determination of the gas.

It may be noted that since oL y it n e constants A,yu

defined by equation (60) are complex. The expression for -^ defined

as a function of real variables consequently takes the form

- 20 -

When F y 2, we must replace tan~ (—j • A " ) by the expression2 "I7T - tan" (—r* o"""") s 0 "that no discontinuity appears as C

passes 2.

We now first examine the equation of state (6l) when the densityv> 2 b

is small. Expanding in powers of f = - jr- , we find

RT V bftTr 3 ' V

17 b 2 5 b 2

For OC - -rsr . the term in —«• becomes * —w . which is identical32 V2 8 v?

to the Boltzmann result '.

We next examine the critical data* Prom equation (6l), we

find that -^ « 0 when S? satisfies the cubic equationdv s

In this equation we have used a reduced temperature x defined by

x - - HP (67)a

For the critical isothermal, equation (66) has two'repeated roots,

the condition for which is

( 6 8 )

The roots of this algebraio equation define the critical temperature,

It is clear that we must have only on© physically allowed root of

this equation.

- 21 -

If we substitute in equation (68) the value of o( = -rr , we find

that the four roots are

yx = 0.1864 y2 - 0.1923

y3 = 5-9334 y 4 - - 0.3125

The critical F corresponding to y, is negative, hence the rootsy3* y 4 a r e n ° * P)33rsically allowed, but still we have two physically

allowed critical temperatures corresponding to y1 and y?.

To evade this unnatural situation, we try to fix the numerical

value of U- r such that equation (68) has two repeated roots.

To do this we consider equation (68) to be a quadratic

equation in oi. . The condition -that 0( has two equal roots is found

to be y a -p. , and the two equal roots of oC are then

(X - § (69)

For this value of c< , equation (68) factorises to

(TO

whioh clearly shows that there is only one oritical temperature

corresponding to y, = y.? =» •?- .

Substituting 0 t » - | ^ a n d y » ^ x - - | inequation

we find that it factorisee to

Hence there is one critical volume defined by

4 0

The corresponding oritical temperature is given from otxo - - or

i RT - — (72)a c 27

- 22 -

To oomplete the critical data, we substitute in equation (64) for

d f £ o, T o and obtain

° 3 2 7 2 (73)

It may be noted that the modified value for oC - 0.54 given by17

(69) is a little greater than the value oC « -rk • 0.531,

assuming constant f ., a result which has already been anticipated.

A more elaborate investigation is furnished by the assumption

that o£ is a function of the temperature of the form

Oi. - * 0+ J r (74)

where t is a small constant. Substituting for this value of oL

in the condition (68) for equal roots of equation (66)t we find

that the critical temperature is obtained as a root of an equation

of the fifth degree in the temperature* One finds that the condition

that the equation for the critical temperature should have two equal

roots is

and the two repeated roots are given

I • (76)

Prom equations (74)» (75)> (76), one finds that the critical temperature

- 23 -

and the critical volume remain the same;

To = T

Alao the corresponding value of cL at xc ia

(78)

Hence the critical isothermal i8 not affected by the assumption (74),

"but only the other isothermals are affected. The higher the temperature

of the isothermal, the "bigger the effect.

A good way to choose the parameter £ for a certain gas is

to consider its "behaviour for small densities. On use of the expansion

(65) for small density, and substituting for oi from (74), we

get

«T

We can then try to give a numerical value to € for a

given gas, such that the above equation agrees with observation as

closely as possible.

VI. QUAMTUM EFFECT

We shall calculate the small quantum terms proportional to

X in the partition function, which we have so far omitted.

From equation (32), the quantum contribution to

- 24 -

the partition function is given by

This, added to the classical part Z defined by equation (57), gives

the whole partition function of the translational motion of the

molecules up to the second power i n t ,

On use of the expansion (40) for |> , we find that

(81)

(82)

Substituting from these into equation (8o)» and carrying out by

the same procedure as for the classical part Z , we obtainc

(83)

Por simplicity we assume for ©(E) (E > CT ) a van der Waal potential

(84)

l>/ /A ftThe result of the integral is —=• , on neglecting —••— compared

kto 1. Hence

< V

On use of the approximate relation

?r - N It(36)

we find that

(87)

vhere if , x are given by (63), (67). Since ^ is a small5 Zc

quantity, we get

hence the quantum contribution to the free energy A given by

equation (58) is

Consequently the quantum contribution to the pressure is P » - rj=jr»

and hence we must add the following quantum correction to the

equation of state (61):

It C89)

- 26 -

For the gas nitrogen, the numerical value of the factor

Jl '-I; A — is about 0.5 x 10 , and the quantum contribution

to at the critical point £ - -*- , X Q « ?pr is .roughly (for

(90)

which is of the order of experimental error. This is the reason that

the quantum terms were not included from the beginning in the

determination of the equation of state.

VII. NUMERICAL RESULTS AND COMPARISON WITH EXPERIMENT

The relation between •?= and £ was computed numerically,

taking d = .54 .

The results are shown in Table I and Figure I, for ten5 10 50 ^

isothermale x = ^ , •£=• , •.., -gsr . The argument W is taken

from zero to 2.0 with steps 0.2 .

Theoretically, U can be as great as •*-?— TT = 5.9238,

a value corresponding to close packing of the hard spheres.

Experimentally^however,, this is never achieved, for even subjecting

a gas(like C0?) at the critical temperature to a pressure up to

1000 atmospheres, the value of C is only about 2.

At first glance at the theoretical isothermals, one sees a

deviation to observations, starting at t about 1.5 and becoming

very strong as C increases* to 2.

In fact the oomputed values of •*•=• tend to a very big negative

value as T increases beyond 2, while, according to observations,

the value of ^ tends instead steadily to a very big positive value.

- 27 -

The deviations for high density could be anticipated from

the fact that the quantities a and aL are taken as constants in

the formula (61) for *~ . For high densities the radial distribution

function Po^T^ * a s a^-rea<^3r mentionedjcannot be assumed as simply — -

but must be some complicated function of T, V. Consequently the

quantities a, at can no longer be regarded as constants for high

values of

Nevertheless we find that for values of £ up to 1.5 there

is a very remarkable agreement with observations. Thus at the criticalp v

point the value of „ ,c, .c, was found near enough to the observed0 7)

values, given in the ROWLINSON article '.

We now try to illustrate the effect of the parameter 6 ,

introduced in equation (74)> in the equation of state. Since this

parameter does not affect the critical isothermal , and its effect

increases as the temperature of the isothermal increases, we

consider the case of a high temperature isothermal corresponding

to x = 1.851 (five times the critical temperature). Taking extreme

values for the parameters •£ , £ « + 0.1 , we find strongdeviations at high densities of the isothermals corresponding to

t » +_ 0.1 from the isothermal corresponding to € = 0. This

is clearly shown in Table II and Figure II.

In fact, ۥ should be given a lower value to fit the

isothermals of a given gas, in the way already described in tlie text.

A comparison between the theoretical and experimental criticalisothermals of C0o is shown in Figure III. The observed values of |=r

& Jrii

for COp are taken from the "International Critical Tables". Forcomparison with theory, the experimental values of =7 are

y- 2bretabulated in Table III as a function of y = ™ . Since the

value of b is difficult to determine experimentally, a value of

b consistent with the theory will be taken b = •§• £ v • -s vC c c y c

(according to viscosity measurement, the value of b for CCU wasv *•

found experimentally to be ..-P., )1.83

28 -

ACKNOWLEDGMENT

This work has been completed during a visit to the International

Centre for Theoretioal Physics in Trieste. The author is grateful to

Professor Abdus Salam and the IAEA for the facilities provided by the

Centre and to fruitful discussions with the members of the Plasma

Group.

He also wishes to thank the staff of the computing centre in

Cairo for carrying out many numerical caloulations during the

preparation of this work.

TABLE I

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

x = 0.185

£lRT

1.00000

0.56653

0.14793

-O.25271

-0.63189

-0.98655

-1.31708

-1.63663

-1.99382

-2.49604

-3.23960

xo = 0.370

1.00000

0.83653

0.68793

0.55728

0.44810

0.36344

0.30291

0.25336

0.16617

-0.06604

-0.53860

x - 0.555

RT

1.00000

0.92653

0.86793

0.82728

0.80810

0.81344

0.84291

0.88336

0.88617

0.74395

0.36139

x =. 0.740

21RT

1.00000

0.97153

0.95793

0.96228

0.98810

1.03844

1.11291

1.19836

I.24617

1.14895

0.81139

x = 0.925

£LRT

1.00000

O.99S53

1.01193

1.O432S

1.09610

1.17344

1.27491

1.38736

1.46217

1.39195

1.03139

0

0.2

0.40.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

x = 1.111

RT

1.00000

1.01653

1.04793

1.09728

1.16810

1.26344

1.38291

1.51336

1.60617

1.553951.26139

x - 1.296

RT

1.00000

1.02939

1.07364

1.13585

1.21953

1.32773

1.46005

1.60336

1.70902

1.66967

1.38996

x = 1.481

mRT

1.00000

1.03903

1.09293

1.16478

1.25810

1.37594

1.51791

1.67086

1.78617

1.756451.48639

x = 1.666

RT

1.00000

1.04653

1.10793

1.18728

1.28810

1.41344

1.56291

1.72336

1.84617

1.82395

1.56139

x = 1.851

•DVRT

1.00000

1.05253

1.11993

1.20528

1.31210

1.44344 j

1.59891 :1.76536 I1.89417 1

1.87795 :1.62139

1

- 30 -

TABLE I I

0

0 .2

0.4

0 .6

0 .8

1.0

1.2

1.4

1.6

1.8

2 . 0

6 =* -o.i

o( = 0.756

1.00000

1.04890

1.10135

1.15127

1.18744

1.19253

1.14594

1.03308

0.85591

0.63319

0.38947

e - oai = 0.54

1.00000

1.05253

1.11993

1.20528

1.31210

1.44344

1.59891

1.76536

1.89417

1.87795

1.62139

€ m + 0 . 1

ot = 0.324

i?

1.00000

1.05619

1.13910f

1.26447

I.4668Oi

1.84737 j

3.08925

1.73563

0.79152

0.34128

0.07130

- 31 -

TABLE III

Critical isothermal of carbon dioxide (ezperimental)

50.488900.52848

0.573550.63033

O.7OOO7

0.78442

0.89772

0.96476

1.03040

1.11793

1.15708

1.24330

1.34634

. 1.49339

1.52391

1.61331

1.719851.80898

1.89827

1.96183

2.021452.08392

mRT

0.57402

0.54802

0.51913

0.485480.44750

0.40629

0.35887

0.33457

0.313510.288950.279320.25988

0.24017

0.21804O.2148O

0.207150.20454O.210840.223350.239720.26150

0.29471

- 32 -

APPBEDIX I

To carry out integrations over the solid angle in a space of

n « 3 M dimensions, we use the spherical coordinates in n dimensions,

^H * IP

The element of volume dp is easily found, using the Jacobian of

transformation »

v-l i M-7.

¥e first consider the integral

JOn use of the above transformation

I,

+- \ < 5 ^ wo

Carrying out the integrations first over 6^ and then successively

over 0t d»,..., up to ^ _, ve find that the result is

where

- 33 -

c 1

In the derivation of the successive integrals, use has been made of

the relations

In order to obtain the other two integrals in (34) we consider

and differentiating with respect to h|

In a similar way

Hence

- 34 -

APPBKDIX II

On use of the asymptotio expression for J (z) for large n

one findB that (z - |r

t0

Integrating by parts successively, we easily arrive at the required

result

- 35 -

RBPEEENCES

1 M. BORN and H.S . GRE2N, P r o c . Roy. Soc. A JL88, 10 and _l8_2, 103

(1946)

2 J . G . KIHKWOOD and B.H. BOGGS, J . Cham. Phys . 10t 394 (1942)

3 A. SABRY Math, and Phya, S o c of Egypt 4 , No. 4 (1952)

4 L. BOLTZMAM, Vorlesungen tlber Gaatheorie - I I , X 51

5 KAMMERLINGH ONNES, Communioations from the Phys i ca l Laboratory

of Leiden, 102 a (1907)

6 SARRAN, Compt. Rend. JLL4, (1882)

7 J.S. ROWLINSON, Handbuoh der Physik, Band XII, p. 48 (1958)

8 L.D. LANDAU and E.M. LIPSHITZ, Statistical Physios, p. % (1959)

- 36 -

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