Gas-liquid phase transition

Gas-liquid phase transition*

A C BISWAS

Tata Institute of Fundamental Research, Bombay 400035 MS received 3 November 1975

Abstract. A theorem has been proved giving a sufficient condition for a first order phase transition in a gas. The corresponding fugacity expansions for the isotherms, in the different phases have been derived.

Keywords. Partition function ; Massieu-Planck function ; thermodynamic stability ; grand partition function ; first order phase transition ; analytic continuation ; thermo- dynamic limit; isotherms.

1. Introduction

The problem of gas-liquid phase transition has been known to be one of the most fundamental problems in theoretical physics. T h o u g h a good advance has been made in tllis direction in the last decade or so, the basic problem still remains.

The question we ask ourselves here is " Given a many-particle system with an interaetioll function ~' ,~ (to) in equilibrium (i) can one give a sufficient condition that the system will undergo first order phase transition ? (ii) the condition being satisfied what will be the forms o f the isotherms in the saturated vapour phase and the liquid phase ?" We start with the pal tition function given by

1 f -/3 ~ 4'(r~)

Q(fl, W, V ) = ~.~ e , > , dr, . . . d r n. (1) We put the thermal wave length equal to unity and in (1), /3 = 1/KT, T being the temperature ; K the Boltzmann constant ; N and V respectively are the number o f particles and the total volume of the system.

We shall write (1) in terms of the generating function as derived by Mayer and M a y e r (1940) namely

o n

VL' bz (V, 7")~r 1 (~ e ~=1

Q~ --- Q (fi, N, V) = 2-~i - e ~,~-i d~

(2)

where bz (V, T) are the well-known cluster integrals and C is a closed contour

* A synopsis of this paper was presented at the International Conference on Statistical Physics at Budapest, •975. This paper is the revised version of an earlier paper published by the author (Biswas 1973) and subsequently criticized by Groeneveld (Groeneveld 1973). This paper is self-complete and free from the objections raised by Groeneveld about the earlier paper.

17 P--2

18

oo

around ~ = 0 in the complex ~-plane so that on C, ~' bz~! is convergent.

I=1

In dealing with the question of phase transition we must first enquire how the canonical Massieu-Planck function (Miinster 1969) should behave at the point of a transition if any. This is discussed in detail by Mtinster (1969). The answer is that the canonical distribution to be physically meaningful should be thermody- namically stable with respect to fluctuations. At the point of a phase transition this stability breaks down. In that case for a given temperature there exists a density p0 (/3) beyond which the canonical partition function does not make sense. In fact for densities p greater than go (/3) the canonical ensemble does not lead to the same asymptotic average as the micro-canonical ensemble and hence is physically meaningless and the corresponding canonical partition functions should be omitted from the sum giving the Grand-Partition function of the system. This fact must be taken into account in formulating a theory of phase transition. The Grand Canonical Partition function for a system of particles in a finite volume V is thus a polynomial of degree ~-o ---- Vpo where go is the density for which the canonical partition function becomes unstable. This number /V0 is not the close- packed number and in fact must be much less than the later. This observation will bring a point of difference of our formulation in the next section in terms of the Grand Partition function flora that of Lee and Yang (1952).

The second point of observation we make is that since the Grand-Partition function for finite V is a polynomial of degree ~o we must be able to find the Grand-Partition function for the whole complex fugacity plane before the thermo- dynamic limit (Ruelle 1969) is taken. This should be possible because if we know the polynomial in any part of the complex plane we can analytically continue it to any other part of the same complex plane. It is only when we take the thermo- dynamic limit that we should be able to see whelher the flee energy tends to different limits in different regions of the fugacity plane. The above observations will motivate the philosophy of the following analysis. We shall confine our discussion to systems which have (if any) only one phase transition.

2. The Massieu-Planck function Let us define the functions ~b z by

log Q (/3, N, V) : N ~ (fl, v, V) : V~, (/3, p, V). (2) We follow closely the notations of Mtinster. As has been discussed by Miinster (1969) in detail the canonical paltition functions Q (/3, N, V) has a domain of stability on the set {N}. This domain of stability is defined by the set {/9} such that N = Vp, where p satisfies

where

o O . 7 .

~2 bp2 < o (3)

~oo ___ L t v . ~ ~ (/~, N, V)

N ~ . = P .

(4)

The boundary of the domain of stability is given by

~2 ¢ . _ 0. _3p2 (5)

Let pc (fi) be the solution of (5). Then -~0 - Vpo gives the largest value of N for which the canonical partition function is stable with respect to statistical fluctua- tions. The existence of a finite pc (fi) is a necessary (but not sufficient) condition for phase transition. We shall assume in what follows that a finite pc (fl) exists.

3. Grand-Partition function

Let z = e ~'/Kr be the fugacity where/~ is the chemical potential of the system. In terms oft* and z equation (5) is equivalent to the existence of a z = zo such that

at z = Zo

~l~_ bp ~z bp -- 0. (6)

Then we define the Oland-Pmtiticn functicn as

Z ,, (fi, V, z) = .S Quz re= Z QNz u (7)

N e {/~} ^{N = o}

using (2).

(~)-~'0+1 co

1 I - - V 2: bg(V, DCt

e z=l d~

2°, v

c<~

Let Zo (V) be the radius of convergence of the series Z' b~ (V, fi) z z and we assume i=1

th at

(i) Ltv_~o, zo (g) = f:o > 0 exists.

O O _ CO

(ii) 2J b~o ~ and Z lb#o z are finite.

|--I |--i

where

/;z = Ltv_~,, b, (V, fl).

Then the function Zgr (/~, V, z) is analytic for ] z [ _< Zo in the complex z-plane Let us define ~----z/zo then

v 2: g~ (re, T)gz

1 _{~=1 d~}

Za, (/3, V, r:) = ~ 1 kg/_ z- e -~- (9)

is analytic in the complex ~-plane in the domain ] ~ I --< 1 ; on ~ i n t h e complex

~-plane [ ~ [ ~ 1,

g, (V, T) = b, (V, T) Zo'. (10)

4. T h e r m o d y n a m ' e pressure and density

We shall define the isothermal pressure p and the density p by the relations (Ruelle 1969)

= L t v . ~ l l o g Z , , (fl, V, Z )

p

₍₁₁₎

K T

(N) Ltv_~o 1 3

p = Ltv..>, --V = T" z ~ log Z o, (/3, V, z). (12) We have assumed above that the interaction X 4' (%) satisfies the stability criterion (Ruelle 1969) and the limit in (11) and (12) is realised in the sense of Fisher (Ruelle 1969) which is a smoothness condition on sequences imagined in passing to the limit of infinite volume.

5. First order phase transition

Definition 1. We shall define a first-order phase transition in the above-defined system if at a fixed temperature T the density p becomes a discontinuous function of the fugacity z wheleas the pressure p remains continuous. In this context we shall prove the following theorem.

TrtEOREM 1: The sufficient condition that a first order phase transition occurs in the system is that there exists a temperature T = T, > 0 below which all b~ (V, T)'s (except perhaps a finite number of them) are positive for V > V0, some finite value.

P r o o f - - T h e p r o o f of the theorem consists of the following steps:

Lemma 1 . - - F o r :~ < 1

Ltv+~ log Zo~ (fl, V, f:) = Z gzz ~

Z = I

(13) where

g~ = Ltv.,oo gz (V, T).

The relation (13) is well known (Mayer and Mayer 1960). We shall, however, rederive the same starting from (9) for the sake of completeness.

Proof.--We start from relation (9). We choose the real axis in the ~-plane such that the phase of ~ is zero, i.e., ~ = ] 2 I. Since 2 < 1, let us choose (7 to be the circle ] ~ ] = ~ and substitute ~ = ~:e ~° in (8) to get

(~)-,0+1

1 - - V~' gl~ r d~

e 1 = 1 - - - (14)

z,, (/3,

v , ~) = ~gi l ~1 =

i.e.,

z , , (/3, v, ~) = U~

_,~

1 - - e -~ ~7o+1~ 0 v 2: g:Z e,t0

1 - - e -~0 e z=x dO.

(is)

We now note that

1 ^-- e-*t~v~ +1) 0

Lt~,~,, 1 - - e-~0 ~ ~ (0)

where ~ (0) is the Dirac delta. This immediately gives L t v . ~ p log 1 Zo, (fl, V, ~).

= Z gz~ z ~ < 1 (16)

|=l

We assume that Ltv.->~o Z' gz ~-z -- Z' ~ £ . A sufficient condition for this inter- l--i

change of the ' limit' and ' summatiort' to be possible has been discussed by Katsura (1963).

Hypothesis 1. Let

i.e.,

3 3 V V bl ( V , T ) > O for a l l l ,

To Vo T<Tc V>Vo

7I 3 V V g, (V, T) > 0 for all l. (17)

TC Vo T<Te V > V o

Observation 1. If hypothesis 1 is true then the series Z ~z~ I must have a singu- oo

|HI

larity (Titchmarsh 1939) at ~. = 1 in the complex 2-plane and hence the relation (16) does not hold for 2 > 1. On the other hand since the series Z' gz2 ~ and

I=1

its derivatives are analytic withirt the circle 12[ = 1, according to definitiorL 1 the system does not have a phase-transition for [ 2 [ < 1 and at ~ = 1 equation (6) is satisfied defining the domain of stability of the system in the fugacity plane.

In order to find the correct series expansion for the Free Energy for 12 ] > 1 we shall first find an analytic representation of the function Zg, (fl, V, 2) given by (9) in the region l i: ] > 1 in such a way that the thermodynamic limit can be conveniently taken. As already noted such an analytic representation is possible in the complex z-plane before the thermodynamic limit is taken since Zo~ (fl, V, 2) is then a polynomial in the ~:-plane.

Lemma 2. For given V a r d -No (may be very large but finite) (~__)/~'0+1

1 - - eO

VZgt¢ l d~

1 e l=1 T (18)

1

analytically represents Zo, (fl, V, ~) for I ~ 1 ~ 1 in the complex 2-plane.

Proof. (i) We first note that Zo,( fl, V, ~) and Z+(fl, V, ~) are analytic in the regions 1~]--<1 and I~1-->1 respectively.

(ii) On the circle [~ [ = 1

z~, (A v, e) =- z$ (A L e)

(iii) 1/~? is the zmique-function except for a phase factor which transforms homographically (Copson 1935) all points outside the circle

121=

1 uniquely to those inside and those on th~ circle onto themselves (the point at infinity is chosen to be transformed due to symmetry, to the centre of the re,it circle).

in the complex 2-plane. [To be precise, if I ~ I = I R (~) t is the radius of the contour in (18) then the statements ] ~ I >-- 1 and I R @) [ < 1 should uniquely imply one another and R (c@ = 0, this gives R (,9 = 1/~].

1

L e m m a 3. Ltv+~ V log Z + (fl, V, ~)

o o

Z

= 2 p o l o g £ + gz z '

1=1.

~ > 1

where

.g'o

po = Ltv-~= ~ . (19)

Proof. As in Lemma 1 we shall again choose the real-axis in the if-plane such that the phase of ~ is zero. We then put in (18) ~ = 1/~ d ° to get

~r V gt e aO

1

f

1 - - ( 2 2 e - i ° ) ~ +t ~=~ dO Z + (fl, I7, 2) ~ 1 --~," e -~O e

- - ' I t

f

1 f 1 ~(~2 e~O)~o+l

= 2~ 1 - - ~ " e ~o

--'tl*

V gt

, f

= U~r d o e ~.~

e

dO

x Rz

oos,0 '

| z j .

1 - - (z~" e-~°)~°+l"~ { ~ ( 1 ) t 1 - ~ 7 i0 J + s i n V g~ sinl0

l--1

× I. (1 - v e-'O~o+q- I (20)

~ ._--2 "~ e-t0 J_[

(R~ = Real part and I,, = Imaginary part so that x = Rzv + I,.x).

Since cos 10 = 1 - - 2 sin s 10/2 from (20) we have after a bit of manipulation,

o o

s (8, v, 2) = z~ +, (/~, v, 2) (2)-,~,o e ,=,

where

o o

i f

= ~ G (0) e z=,

--71"

dO

G(O) = cos [Vf(O) -1- -No0] - - ~ c o s 1 [Vf(O) + (/Vo + 1) 0]

(21)

- - \~o",1 cos [Vf (0) - - O] + \~-o2 } j

(22) with

f ( O) = g, sin I0

I = : 1

F r o m (21)

1 log Z + 2No

= - V log 2 +

o r

c o

g~(~)z -{-~1 l o g J ( ~ o , V )

1--1

Lt~_>oo I log Z ~

+ Ltv.~oo - l l o g J (~/o, V) (23)

Proposition I. J (~/o, V) is a non-zero positive quantity for any finite -N0 and V.

Proof This follows from the identity (21). Since Z + is a non-zero positive quantity as it is a polynomial consisting of a sum of positive definite terms for

real 2 > 1 and finite No and V, see (18), Proposition 2.

Ltv.~oo ~ log J(,~0, V) = O. 1

Proof We first note that for finite -No and V, G (0) is a continuous function o f 0 in the interval - - r r < 0 < 7r. Using the integral mean value theorem we have

ql"

J(Aro, V ) - - G ( 8 ) d O e - 2 V g, - sin'

277" 1--1

--71"

where 3 is a value of 0 in the interval [ - rr, 7r] and a function of A?0 and V.

Due to proposition 1, G(3) is non-zero positive.

We write

s (~o, v ) = o(~) u (v)

where

(24)

(25)

Since gz

for all

Heace

o r

71"

f E (-~)' ^,0

1 --2V gt si n2

U ( V)---- ~ d o e ,=~ T

0

> 0 for all l, we have (for V > Vo)

gl ~ sin2 2 - - < g~ 12 --4

l = 1 1 = 1

O, and

c o

E ('d

_gl sin ~ -~ >-- gl } s m ~ > ~ ^{,o , . ~o}

o0~

l = l

f

- - r r < O < rr, g~ ~ = b 1 = 1 .

1 }

1 - 2 V gt 02 dO

2re e

0

1 f 2 VO 2

_ _ - - - ~ - dO

< U ( V ) < • e

0

)

1 t

7-~ 0 2V gz l z

1=1

1 0 Q r ~ / ~ )

< u(v)_< U~ v'2-P

(26)

(27)

(28)

(28),

(29)

(30)

where 0 ( x ) = .( e-"'dx is the well-known error function.

o We now note t h a t

(i) G (3) is a uniformly bounded non-zero positive quantity for all -Vo and V.

(ii) The error function 0 (x) -+ 1 as x ~ co.

These immediately give from (30) and (25),

Ltv_~ j / l o g J ( ~ ? 0 , V)---- O. 1 (31)

Hence lemma 3 is proved.

We can now derive expression for the isotherms from (13) and (19) using (11) and (12).

We get

P __ ~ ~t~ ~ KT

2 < 1 (32)

o o

= > l~z~, z p

l - - 1

o o

P - - 2 p o l o g 2 + gl KT

l - - 1

O = 2 0 0 - l~,

l = 1

2 > 1. (33)

To get the isotherms at ~ = 1 we continue the isotherms from both sides to /: = 1 by Abel's theorem (Whittaker and Watson 1963) and then make use o f relation (6).

O 0

~ = ~

o . ~ o o

Pc = Z t ~ < p < p z = 2 p o - - Z l ~

l ~ i l--I

1

^{2 = 1. (34)}

F r o m (32), (33), (34) we see that afirst order phase transition does take place and Theorem 1 is proved.

A typical isotherm for T < T, is shown in figure 1. The regions 2 < 1, $ -= 1 and ~ > 1 characterise respectively th~ gas, satvrated vapour and the liquid phases.

pG and PL are respectively the gas and liquid specific densities of coexistence at equilibrium with the saturated vapour phase. The critical temperature is defined by varfishing of the order parameter Pz--Pc, i.e., by the equation

p--3

P

<l/J /

_I

I I

/

~>1

o p

Figure 1. A typical ;SOt',lerm for T< T o .

o o o o

2 / g , ( L ) = 2p0 (r0) - - X / g ~ (r0)

o r

l~z (T~) = Po ( T , ) = Pc (35)

where Pc is the critical density. We remember that P0 is the solution & e q u a t i o n (5).

6. Discussion

The above considerations are valid for any potential ~' ~ (to) which is stable, i.e., there exists a B ~ 0 such that

dp O%) >_ - - n B for all n > 0 . (36)

i > J

It has been shown by Ruelle that under condition (36) the series ~' b~z z is con- vergent at least up to

z < e - ~ -~ [C (fl)]-i (37)

where

C (fi) = I d k I e-6q~(~' - - 1 ]. (38)

This means that the system does not have a phase transition in the region defined by (37). The region (37) defines the gas region. This also means that there is no phase transition at high temperature, since for small/3 the region (37) extends to the whole of the positive z-axis (see Ruelle 1969).

What we have shown above is that if below a certain temperature T = Tc all b~'s are positive for large V, the system undergoes a first order phase transition (we have not discussed the case when a finite number of b~'s do not have a definite sign. However the generalisation to that case is not difficult).

The above discussion is qualitative and is not supposed to give a final answer to the problem. There remain the detailed investigations of the analytic proper- ties of b{s themselves which we have assumed to be tiue. This should be done in terms of some definite potential function satisfying (36). A quantitative test of the theory is possible. F r o m (34) we have

Pz + Pc = 2to (fl) (39)

where po (]9) is determined by (5). F o r a realistic potential (39) should quanti- tatively describe the law o f rectilinear diameter (Cailtetet and Mathias 1886) when it holds good. One can also derive the following pressure-density relalion fen a liquid from (33)

P - - 2to log ~0 + 2to ~ " .~t~ ~+~

K r

+ ~ - - ~ fl,,h TM (40)

k

where /S' k are the irreducible Mayer cluster integrals, b = 2 p o - - p , p being the density o f the liquid; ~0 is as defined in section 3 (i). One can quantitatively test (40) with a realistic potential for a given liquid.

Acknowledgement

My thanks are due Groeneveld for his interest in this work and useful discussions.

References

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Copson E T 1935 Introduction to the theory of f tractions of a complex variable (see Secs 8.3 anti 8.31)

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Katsura S 1963 Adv. Phys. 12 391

Lee T D and Yang C N 1952 Phys. Rev. 87 404

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Whittaker E T and Watson G N 1963 A Course of Modern Analysis (Cambridge University Press)