On the Statistical Theory of Phase Transformations of the First Order

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Geilikman, B. T.

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NATIONAL RESEARCH COUNCIL OF CANADA

Translation TT-139

ON THE STATISTICAL THEORY OF PHASE TRANSFORMATIONS OF THE FIRST ORDER

(K Statisticheskoi Teorii Fazovykh Prevrashchenii

Pervogo Roda) by B, T o Geilikman Translated by E. Rabkin 0 TTAVJA 1 June, 1950

T i t l e :

TRANSLATION TT-139

On t h e S t a t i s t i c a l Theory of Phase Transf ormat f ons of t h e F i r s t Order

By: B. T. Geilfkman

Reference : Doklady Akademii Nauk, U. S. S. R,

,

V o l . 69, No, 3 , 1949, p. 329

Translated by: E s t h e r Rabkin

Page

-

1 TT-139 Doklady Akademii Nauk, U , S , S . R .

Vole 69, Noo 3, 1949

Physics

ON THE STATISTICAL THEORY OF PHASE TRANSFORhSATIONS

OF THE FIRST ORDER

(presented b y M, A. Leontovich 31 Aug. 1949) Translated by Esther Rabkip.

The presently e x i s t i n g theory of the condensation of gases ('-') has, a s i s known, a number o f shortcomings ( f o r example the h o r i z o n t a l s e c t i o n of the curve p--v cannot be converted i n t o a curve f o r l i q u i d s , e t c . ), We w i l l show

t h a t , using t h i s theory, i t i s possible t o develop a general theory of phase transformations o f t h e f i r s t order and t h a t the d e s c r i p t i o n of the condensation i n ( q - 4 ) i s f a u l t y ,

If we express the sum o f s t a t e s of a system, con- s i s t i n g of mutually i n t e r a c t i n g p a r t i c l e s , i n t h e form

1

z

=

-

k3, Q, = h then, as i s known ( ' ) :

Page

-

2

TT-139

In the c l a s s i c a l case f o r short-range forces:

bl 1s expressed through ttirreduciblet' i n t e g r a l s

Considering that ( s e e (')) :

we can write ( 1 ) in the form:

where the summation with respect t o each

5

now can be carried out independently of each .other:

Page

-

3

TT-139

where [s]--is the whole range of S;

#(csvb ,N,N) does not have any poles (except

r:

= -) and branch points

(since i n

(4)

I n i s taken as the main branch), Similarly, f o r bl w e obtain f r o m ( 2 ) :

O f the formulae

,

used e a r l i e r and deduced d i f f e r e n t l y (= )

,

( 3 ) and

( 5 ) d i f f e r b y the f a c t t h a t N

#

-

(1

#

-) i n the expression

Equations ( 3 ) and

(5)

may be calculated b y the contour o r

b y the method of a generating function function may be

Page

-

%C

TT-) 39

Both methods give:

where eN 4 O., and z--is the root of t h e equation N 4 - 3

which gives the l a r g e s t modulus f o r e @ ( Z ~ ~ ~ ) / Z . he complex

and negative roots should be neglected as i s evident from (6),

e 1 $ ( y 9 P 9 1 ~ l - q ) + l e l

bl = 9

l ~ y . l - ~

where y i s the root of t h e equation:

giving the l a r g e s t modulus f o r e # ( Y , B ) / ~ ~

B y the method of contours we find: el

=

-

&

.In l/le

From (8) i t i s evident t h a t ( ') the radius o f convergence of the

z s e r i e s #(zpvbgo~,-) i f i equal t o ( f o r temperatures a t which they

converge # ( y g @ y ~ ~ ' ~ ) ) ~

whew y- = l f m y and of the function

C)X

are equal to:

Page

-

5 TT-739

A simple analysis shows t h a t i n ( 6 ) and(7) we can

proceed t o the l i m i t N =

-

( f o r z > 0 ) : I ) i f a l l bl > 0

-

only

a t z 4 [ $ ( z , v b , , ~ , ~ ) n N I n z/z a t z >

F];

2 ) i f bl are a l t e r -

n a t i n g i n s i g n (this corresponds t o y < 0 i n ( 9 ) )

-

f o r a l l

%-I

cases z 2 0 (however, f o r t h e most i n t e r e s t i n g case bl

=

y / l

z

_{) ;}

moreover f o r z we must use the a n a l y t i c a l continuation of the f'unction ~ o ( z , v b ) . It i s evident t h a t f o r bl > 0 a p a r t i c u l a r

point v ~ ~ ( z , b ) = # ( ~ ~ v b : ~ - ~ - ) PB found on the p o s i t i v e a x i s , and f o r a l t e r n a t i n g i n s i g n b1

-

on the negative axis, I n the

l i m i t N = -:

We w i l l determine the conditions necessary f o r phase t r a n s i t i o n s . Q is determined b y the r o o t zi ( 1 2 ) , which gives the smallest value f o r ze -.~o(z,b) _{" Zi} appear t o be fhxtiorrs

o f v, T , and i f i n one region v , T , the smallest value of ze -vGo 3 t h a t i s t h e configuration f r e e energy F = ~ k T ( l n z

-

vGO), gives

q

a r o o t z l ( v , T ) , then i n another region the smallest value f o r

F can give a root z2(v,T), Since the functional v a r i a t i o n of

9

z t and z2 from v , T 9 i s generally speaking, d i f f e p e n t , then Q ( ~ , T )

i n these regions w i l l be expressed b y d i f f e r e n t functions w, T,

which corresponds t o two d l f f e ~ e n t phase s t a t e s ( t h e gaseous phase corresponds t o %he r o o t of ( 9 2 ) which apppoaches zero a t v + a ) *

The condition of equilibrium of two phases appears t o be the e q u a l i t y of pressures ( i n a d d i t i o n t o the e q u a l i t y of

t e m p e r a t u ~ e s ) , Therefore, f o r the i n v e s t i g a t i o n of a phaee equilibrium i t i s more convenient t o go over f r o m the general

Page

-

6

TT-.4,39

sum of s t a t e s Z ( V , T ) t o z ' ( ~ , T ) , which may be achieved by in- cluding a movable p i s t o n (') :

I n csmespondence w i t h (6) z(v,T) may be written as a sum:

($(zi,vb,N,I?)

=

W30(zi,b) i n the reglon o f convergence Gh(z,b)),

which i n view o f the f a c t t h a t N

,,

2 , mag be reduced i n each

kegion v,T $ 8 a s i n g l e

-

l a r g e s t

-

t e r n ;

w

N [ ~ O O ( z i ) - l n zi-p~/'k~+eN]

2'' = d ~ ;

0 %

N >> *P, therefore the expT?esshsns under the i n t e g r a l must have a sharp maximum a t V ~ ( ~ , T ) , determined b y equating t o zero the derivative of the exponeaz with k e ~ p e c f , t o v::

[ ( l 5 ) eoincfdes with (=!3)], I n f a s t , the second derivative

Page

-

7

TT-139 Thus: Z u ( p , T ) =

I n each region p , T the l a r g e s t value f o r viOo -4 n z i-p~/kl'.

e gives one of the roots sip t h a t i s one term

of the sum (161, The equilibrium of two phaees is possible ordy

f o r the condition V ~ G C O ( Z I )

-

?In ze

.-

pvo/kT = V ~ G C O ( Z Z )

-

In Q

-

-

pvP/kT, t h a t is, f o r the general condition sf the chemical

p o t e n t i a l s being equal: pa = k (3n a9 = I n 2s). If N i s small,

then f o r a small lpegjlon near the phase curve (2, = z,), Z o would

be determined b y both terms of the sum ( -1 6 ) , t h a t f 8, the phase

t r a n s i t i o n would be somewhat diffused, Thus, the sharpness of the

phase t r a n s i t i o n 5s connected with very large values of N o

A necessary condition of the p o s s i b i l i t y of phase

t r a n s i t i o n s of the f i r s t order i s the existence o f more than one r e a l positive root sf equation (12), It i s evident t h a t Sf a t any

T a11 bl > 0, there e x i s t s only one r e a l root. Phase t r a n s i t i o n s

of the f i r s t order- are possible, therefom, only in the case when bl i s a l t e ~ n a t i n g i n sign, which correspon&s t o the existence not

o n l y of forces of a t t p a c t i o n but a l s o of forces of repulsion between the moleeubes,

I n the previous works ( q - 4 ) the mechanism of phase

t r a n s i t i o n s of the f i st order investigated i n them, condensation,

is assumed t o be completely d i f f e r e n t . Let a l l bl 0. Since

az/av < 0 ( ' ) ,

_-

then z increases when w d e c ~ e a s e s , A t some

v = v k

z = z ,

It can be e a s i l y shown t h a t z remains equal to

also a t a f u r t h e r decrease i n

v.

Therefore, a t v ( vk p =

TO^

(z)

.

Page

-.

€5

T6.?!-,.9 ,Tg

It was cons%dered t h a t e x a c t l y t h i s h o r i z o n t a l s e c t i o n of t h e

culrove p

-

IT corresponds t o the equilibrium region of two phases:

l i q u i d 8.3d gaseous, However, i t can be shown t h a t ap/av = 0

not onlg a t v -$ v b u t a l s o a t v = v],

+

0, t h a t i s , i n cont.raat

k 3

with e x p e ~ i m n t , th e Jump ap/av does not e x i s t . T h e parssibility

of a p / a p .

#

0 at

v

= vk

+

O 1 which was assumed i n ( ' 4 ) , f a l l s off w%t9h a c a ~ r e c t consideration of # ( z , v ~ , N , N ) , but not with a eon- aidepation of ~ ~ ( z , b ) i n ( 3 ) . I n addition, b y c a l c u l a t i n g the

d e n s i t y p ( r ) a8 a Axnction of coordinates i n a weak e x t e r n a l f i e l d , 0

i t can be shown t h a t a t z = z the subdivision surface between the 0

phases does not e x i s t . Thus, even a t z = z

( v

c

vk)

t h e system c o n s i s t s of one phase

-

the same a s i n a s i m l l a r case of t h e E i n s t e i n condensation of an i d e a l Bose

-

gas ( i n t h i e case

bl = ~ ~ - That, which was e a r l i e r assumed a s the point 1 .

0 % condenc;ation appears t o b e , t h e r e f o r e , a p o i n t of a transition

of %he t h i r d o r d e r

t h e r e f o r e , BC, = 0, e t c .

3

This t r a n s i t i o n m a t be i a v e s t i g a t e d i n more d e t a i l .

In t h e presence o f repulsion between molecules located a% small

d i s t a n c e s from each other, t h i s i s impossible a t high and,

as-

paraently, even a t f a $ r l y low temperatures. I t mst be noted that

a t bl a l t e r n a t i n g i n s i g n , t h e same as i n t h e case of an i d e a l

Femi- gas ibl =

-

z s equal t o

z,

does not

c o ~ r e s p s n d t o any phase t r a n s i t i o n (')

.

Sent t o the E d i t o r 16 August 49b9.

Page

-

9 TT-139

Bibliography

1 , J, Mayer and S o Harrison, Journ. Chem. Physa, 6,87 (1938); J, Mayer and H, Goeppert-Mayer, S t a t i s t i c a l Mechanics,

1940,

2775

2, A, C, Davydov, Zhurnal Experementalnoi i Teore ticheskoi Flaikf, f 0 ,

263, 9940,

3, Y o Born and KG f i c h s , PPOC. Royo Sot,, 166, 191 (1938).

40 B, K a h n and G, Uhlbnbeek, PWsf ca,

5,

399 (1938) a