APPLICATIONS OF THE DIRECT CORRELATION FUNCTION SOLUTION

THEORY TO THE THERMODYNAMICS OF FLUIDS AND FLUID MIXTURES

By

SYED WASEEM BRELVI

A DISSERTATION PRESENTED TO THE GRADUATE

COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL

FULFILLMENT OF THE REQUIRfEENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

1973

TO FARAH

ACKNOWLEDGMENTS

The author sincerely thanks the many persons who gave him

assistance during the course of this work.

Tracy Lambert, Jack Kalway and Myron Jones of the Chemical

Engineering Shop and Ed Logsdon of the Engineering Glass Shop willingly

and cheerfully provided assistance on request.

Lewis Moore's contribution to the early stages of experimental

design and construction was considerable. Continuing discussions

with Dr. M. S. Ananth enabled the author to unravel some complexities

of perturbation theory. Anthony DeGance's suggestions on computations

of thermodynamic properties were very timely.

Dr. Keith Gubbins elucidated some aspects of the molecular

basis of the compressibility correlation. Valuable advice on the

intricacies of experimental work and the loan of several pieces of

experimental apparatus from Dr. T. M. Reed are deeply appreciated.

The author is thankful to Dr. O'Connell for his guidance

through this work. His unlimited patience, easy accessibility and

unfailing optimism were tremendous assets, especially during periods

of low productivity. His encouragement, nay, insistence towards

perfection resulted in some of the positive achievements of this study.

Working with Dr. O'Connell has been an experience that will not be

soon forgotten.

Finally, the author expresses his thanks to his wife for her

support and devotion through these testing years. Her participation

in this effort made it all possible and worthwhile.

TABLE OF CONTENTS

ACKNOWLEDGMENTS ............................................

LIST OF TABLES..............................................

LIST OF FIGURES .............................................

NOMENCLATURE ................. ...............................

ABSTRACT..................................................

CHAPTERS:

1. THERMODYNAMIICS OF GAS-LIQUID SYSTEMS ..............

1.1 Phase Equilibrium in Gas-Liquid Systems.......

1.2 Isothermal Pressure Dependence of Thermo-

dynamic Properties............................

REFERENCES FOR CHAPTER 1..........................

2. DISTRIBUTION FUNCTION SOLUTION THEORY..............

2.1 Radial Distribution Function Solution Theory.

2.2 Direct Correlation Functions.................

2.3 Taylor's Series Expansions for Thermodynamic

Properties...................................

2.3.1 Solute Gas in Binary Solvent...........

2.3.2 Pure Component ........................

2.3.3 Gas in Single Solvent..................

2.4 Linear Composition Dependence of Direct

SCorrelation Function Integrals...............

REFERENCES FOR CHAPTER 2..........................

3. PERTURBATION THEORY FOR MOLECULAR DISTRIBUTION

FUNCTIONS OF DENSE FLUIDS..........................

3.1 First-Order Perturbation Theory..............

3.2 Reduced DCF Integrals.........................

Page

iii

vii

ix

xi

xv

1

1

5

7

8

8

13

15

16

18

18

21

27

28

30

33

TABLE OF CONTENTS (Continued)

Page

3.3 Multicomponent Systems ....................... 39

REI iRENCES FOR CI'APTER 3........................... 43

4. MACROSCOPIC STATE DEPENDENCE OF DIRECT CORRELATION

FULCIION INTEGRALS .................................. 44

4.1 Corresponding States Correlations for C22

and C 22

and C1 ........................................... 44

4.2 Generalized Isothermal Equation of State

for Liquids ................................. 57

4.3 Temperature and Pressure Dependence of DCF

Integrals...................................... 66

REFERENCES FOR CHAPTER 4.......................... 73

5. SOLUTION THEORY FOR SUBCRITICAL SYSTEMS........... 75

5.1 One-Fluid Theory............................. 76

5.2 Rules for Composition Dependence of DCF

Integrals..................................... 82

REFERENCES FOR CHAPTER 5.......................... 98

6. DETERMINATION OF EXPERIMENTAL THERMODYNAMIC

PROPERTIES OF GAS-LIQUID SYSTEMS.................. 99

6.1 Description of Apparatus..................... 99

6.2 Experimental Procedure....................... 103

6.3 Treatment and Analysis of Experimental Data.. 107

REFERENCES FOR CHAPTER 6.......................... 123

7. SOLUTION THEORY FOR GAS-LIQUID SYSTEMS............ 124

7.1 Correlations for Activity Coefficient

Parameters................................... 124

7.2 Detailed Analysis of Hydrogen-Benzene System. 135

7.3 Thermodynamic Properties of Gas-Mixed

Solvents Systems............................. 144

TABLE OF CONTENTS (Continued)

Page

7.4 Vapor-Liquid Equilibrium in Multicomponent

Systems ................ ...................... 156

REFERENCES FOR CHAPTER 7.......................... 162

8. CONCLUDING REMARKS................................ 163

APPENDICES.................................................. 169

A. MULTICOMPONENT ORNSTEIN-ZERNIKE EQUATION.......... 170

B. RECURRENCE RELATIONS FOR DCF INTEGRALS............ 171

C. FIRST-ORDER PERTURBATION THEORY FOR THE

DISTRIBUTION FUNCTIONS OF A DENSE FLUID............ 174

D. TEMPERATURE DERIVATIVE OF LIQUID DENSITY AT

CONSTANT PRESSURE.................................. 179

E. EXTRACTION OF HENRY'S CONSTANTS AND ACTIVITY

COEFFICIENTS FROM VAPOR-LIQUID EQUILIBRIUM DATA... 181

F. PERCUS-YEVICK AND HYPERNETTED CHAIN APPROXIMA-

TIONS FOR TRIPLET CORRELATION FUNCTIONS........... 185

BIBLIOGRAPHY................................................ 187

BIOGRAPHICAL SKETCH......................................... 191

LIST OF TABLES

Table Page

3-1 Intermolecular parameters for simple fluids........ 38

3-2 Thermodynamic properties in subcritical systems

from perturbation theory........................... 41

4-1 Generalized correlation for liquid compressibility. 50

4-2-a Reducing volumes from isothermal compressibilities. 51

4-2-b Characteristic volumes determined from partial

molar volumes at infinite dilution................. 54

4-3 Partial molar volumes of gases at infinite dilution 58

4-4 Reduced integrals for isothermal equation of state. 63

4-5 Volumes of compressed liquids...................... 65

4-6 Pressures of compressed liquids.................... 67

4-7 Temperature and pressure derivatives of thermo-

dynamic properties................................ 72

5-1 Isothermal compressibilities of binary liquid

mixtures calculated by pseudo pure fluid

approximation ......................... ............. .78

5-2 Isothermal compressibility of ternary liquid

mixtures................................ ............ 80

5-3 Isothermal compressibilities of fluid mixtures..... 81

5-4 Second and third order DCF integrals in the

benzene-cyclohexane system......................... 86

5-5 Rules for composition dependence of DCF integrals -

calculations of thermodynamic properties in

subcritical systems................................. 88

5-6 Activity coefficients in binary liquid mixtures.... 96

6-1 Densities of pure and mixed solvents at 250C........ 108

6-2 Measured bubble pressures of gas-solvent systems... 110

6-3 Volumetric properties of constant composition

mixtures--coefficients for equation (6.1).......... 112

vii

LIST OF TABLES (Continued)

Table Page

6-4 Volumetric properties of solvents.................. 115

6-5 Partial molar volumes of solution.................. 117

6-6 Excess free energies of equicolar solvent mixtures. 120

6-7 Henry's constants and activity coefficient param-

eters from experimental data........................ 121

7-1 Henry's constants and activity coefficients from

experimental data on binary systems................ 127

7-2 Vapor-liquid equilibrium in binary systems.......... 130

7-3 Binary vapor liquid equilibrium.................... 131

7-4 Activity coefficient parameters for binary systems. 134

7-5 DCF integrals in hydrogen-benzene system at

4130K, 50 atm....................................... 143

7-6 Direct correlation function integrals in ternary

systems............................................ 147

7-7 Henry's constants in mixed solvents................ 155

7-8 Vapor liquid equilibrium in ternary systems......... 157

viii

LIST OF FIGURES

Figure Page

2-1 Linear composition dependence of the integrals

C in system of solute in binary solvent........... 22

ij

3-1 Radial distribution function for Lennard-Jones

6-12 fluid......................................... 32

3-2 Reduced direct correlation function integral for

Lennard-Jones 6-12 fluid from perturbation theory.. 35

3-3 Microscopic correlation of isothermal compressi-

bility for Lennard-Jones 6-12 fluid................ 37

4-1 Isothermal compressibilities of pure liquids....... 46

4-2 Generalized correlation for isothermal

compressibility of liquids......................... 48

4-3 Generalized correlation for partial molar volumes

of gases at infinite dilution in liquids............ 56

5-1 Experimental DCF integrals in benzene (1) cyclo-

hexane (2) system at 298K......................... 84

5-2 Isothermal compressibility function of benzene

(1) cyclohexane (2) mixtures..................... 90

5-3 Partial molar volumes in benzene (1) cyclo-

hexane (2) system at 298K......................... 91

5-4 Activity coefficients in aniline (1) nitro-

benzene (2) system at 338K........................ 92

5-5 Compressibility function of aniline (1) nitro-

benzene (2) mixtures............................... 94

6-1 Experimental apparatus vacuum section............. 100

6-2 Experimental apparatus pressure section........... 102

7-1 Effective f0 for H -solvents systems............... 128

2 2

7-2 Effective f0 for CH -solvents systems............... 128

7-3 Experimental DCF integrals in H2-benzene system

at 513K..................................... ....... 137

LIST OF FIGURES (Continued)

Figure Page

7-4 Isothermal compressibility and partial molar

volume in H2-benzene system....................... 141

7-5 Isothermal compressibility and partial molar

volumes in H2-benzene system....................... 142

7-6 Isothermal compressibility of N2-benzene-octane

mixtures at 2980K.................................. 150

7-7 Partial molar volume of N2 in N2-benzene-n-octane

mixtures.......... ................. ............... 151

NOMENCLATURE

A = coefficient of Tait equation, equation (4.11)

A.. = partial derivative at constant temperature and volume

1J

of chemical potential of species i w.r.t. concentration

of species j, equation (2.9)

B = coefficient of Tait equation, equation (4.11)

B.. = element of matrix in expression relating chemical

potential derivatives to integrals of molecular

distribution functions, equation (2.11)

C.. = volume integral of pair direct correlation function of

species i and j, equation (2.25)

C.. = volume integral of triplet direct correlation function

13k

of species i, j, k, equation (5.13)

F = matrix in expression relating derivatives of chemical

potentials to integrals of direct correlation function,

equation (2.31)

G.. = volume integral using radial distribution function of

species i and j, equation (2.12)

H. = Henry's constant (reference fugacity) of component i

in solvent m, atm, equation (1.7)

I = identity matrix, equation (2.22)

N. = number of molecules of species i, equation (2.2)

P = pressure (atm), equation (1.4)

R = gas constant (cc atm/gm mole K), equation (1.4)

T = absolute temperature (K), equation (1.4)

V = total volumeof system (cc), equation (2.5)

X.

a

c(r)

ij (rl2)

ci

d

dij

f2

f3.

f (j)

i

g(r)

(2)

g. (

ij 12)

k

r

= mole fraction of component j at zero composition of

component 1, equation (2.39)

S diameter of rigid core in hard sphere intermolecular

potential (cm), equation (3.6)

= pair direct correlation function between molecules

distance r apart, equation (2.24)

= pair direct correlation function between molecules of

species i and j, distance r12 apart, equation (2.25)

= molecular concentration of species i (molecules/cc),

equation (2.5)

= reduced hard sphere distance, equation (3.6)

= partial derivative at constant temperature and pressure

of chemical potential of species i w.r.t. concentration

of species j

S coefficient in Margules expression for solute activity

coefficient, equation (2.34)

= coefficient in Margules expression for solute activity

coefficient, equation (2.34)

= fugacity of component i in phase j (atm), equation (1.3)

= radial distribution function between molecules distance

r apart, equation (2.24)

= radial distribution function between molecules of

species i and j separated by a distance r12, equation

(2.4)

= Boltzmann's constant (erg/K), equation (2.9)

= radial distance (cm), equation (2.6)

= molar volume (cc/gm mole), equation (1.4)

= mole fraction in liquid phase, equation (1.4)

= mole fraction vapor phase, equation (1.4)

Greek Letters

y = liquid phase activity coefficient, symmetric convention,

equation (1.4)

*

y = liquid phase activity coefficient, unsymmetric

convention, equation (1.6)

6 = Kronecker delta

E/k = energy parameter in intermolecular potential (K),

equation (3.6)

K = isothermal compressibility (atm ), equation (1.14)

P )- chemical potential of species i in phase j (cals/mole),

equation (1.1)

p molar density (gm moles/cc), equation (1.14)

o size parameter in intermolecular potential (A),

equation (3.6)

{(r) = spherically symmetric intermolecular potential at

separation r (ergs), equation (3.1)

Superscripts

o

= standard state

o

infinite dilution of component 1 in 2

o

S pure 2 in 2-3 mixture

S pure 3 in 2-3 mixture

characteristic

xiii

L

= liquid phase

v

= vapor

M partial molar property

sat M saturation

+ = infinite dilution of component 1 in 2

hs M hard sphere

rf real fluid

r M reduced

E M excess

Subscripts

i component i

j M component j

m mixture

xiv

Abstract of Dissertation Presented to the

Graduate Council of the University of Florida in Partial

Fulfillment of the Requirements for the Degree of Doctor of Philosophy

APPLICATIONS OF THE DIRECT CORRELATION FUNCTION SOLUTION

THEORY TO THE THEPRMODYNAMICS OF FLUIDS AND FLUID MIXTURES

By

Syed Waseem Brelvi

March, 1973

Chairman: Dr. J. P. O'Connell

Major Department: Chemical Engineering

The formulation of a rigorous statistical mechanical solution

theory relates thermodynamic properties in a fluid mixture to

volume integrals of molecular distribution functions. These

distribution functions are complex functions of molecular interactions

and the macroscopic state of the system including the composition

although for many purposes the composition dependence of thermodynamic

properties can be represented by functional expansions about

limiting compositions in the mixture.

Molecular distribution functions of different orders are related

through density derivatives of the distribution functions. A first-

order perturbation theory has been developed to relate the distribution

functions of a real fluid to the similar functions in a hard sphere

fluid through differences in the intermolecular potentials. The

calculated radial distribution function for a Lennard-Jones 6-12 model

agrees well with molecular dynamics results. A one-parameter correspond-

ing states theory for the compressibility integral gives qualitative

predictions. for thermodynamic properties in subcritical systems.

xv

Corresponding states correlations involving only a single

characteristic parameter for each substance express the isothermal

compressibility of pure liquids and the partial molar volumes of gases

at infinite dilution as universal functions of the reduced solvent

density. A generalized isothermal equation of state reproduces

pressures and volumes in compression experiments of pure liquids up

to high pressures.

The isothermal compressibility of liquid mixtures is described

by a one-fluid theory with composition dependent parameters. A

postulated linear variation of direct correlation function integrals

between composition extremes gives accurate predictions of partial

molar volumes and activity coefficients at intermediate compositions.

The experimental triplet correlation functions in a representative

subcritical system did not agree with the Percus-Yevick or HNC

approximations.

An experimental determination of the volumetric and vapor-

liquid equilibrium properties of gas-solvent mixtures is reported.

Volumetric properties in the liquid phase are expressed by functional

expansions in the solute mole fraction, with a one-fluid model for the

solvent mixture. Generalized correlations are developed from experimental

data to relate the activity coefficient parameters of H2 and CH4 in

liquid solvents to the reduced solvent density. Several expressions

relating the Henry's constant of a gas in a mixed solvent to those in

the pure solvents are analyzed with experimental data. Vapor liquid

equilibrium calculations show that some of these expressions with the

generalized activity coefficients yield K-factors and bubble pressures

in good agreement with experimental data.

xvi

CHAPTER 1

THERMODYNAMICS OF GAS-LIQUID SYSTEMS

Our everyday experience presents a panorama of processes in

gas-liquid systems. They vary in nature from the complex transfer

of oxygen into the blood stream of animals, to the simple dissolution

of carbon dioxide in the Ubiquitous coke bottle. A major effort of

chemical and petroleum industries is contacting, treating and separating

gas and liquid streams.

In all of these instances, the equilibrium distributions of

material in the two phases follow simple thermodynamic relations. This

is not to imply that equilibrium conditions obtain in these processes,

for then there would be no transfer of materials. The equilibrium

distributions reflect the thermodynamic constraints on the system.

In the engineering domain, these equilibrium distributions are a key

factor in the selection of processing conditions and the design of

process equipment.

Thermodynamic properties of matter in bulk are the manifestation

of the molecular interactions between the different species present

in the system. Statistical mechanics provides the formal link between

the molecular characteristics of a physical system and its bulk

thermodynamic properties. This work describes the use of a statistical

mechanical solution theory based on molecular distribution functions

for correlating the thermodynamic properties of gas solvent systems.

1.1 Phase Equilibrium in Gas-Liquid Systems

Generally, for thermodynamic equilibrium between the phases of a

multicomponent system, the chemical potentials of each species are

equal in all phases, i.e.,

2 = 2) for all i (1.1)

i i i

V Jis the chemical potential, or partial molar free energy of

species i in phase j, and depends on temperature,pressure and composition

of the phase.

For any phase, solid, liquid or gas, the chemical potential of

the individual species can be expressed in terms of the fugacities, fi,

2

as

P i RTZn f /f (1.2)

Ui and fi are the chemical potential and fugacity in the standard

state. Of these two quantities, one can take on an arbitrary value,

but they are not both independent. The fugacity, fi, also depends on

temperature, pressure, and composition. The fundamental equilibrium

relation (1.1) reduces to

(1) (2) f(3) (1.3)

f f (1.3)

1 1 1

The central problem of phase equilibrium thermodynamics is the

description of the component fugacities in terms of temperatures,

pressures, and compositions. For vapor-liquid equilibrium,where the

species are liquids at the temperature of the system, a suitable

description of equation (1.3) is2

P -L

v ref oL I v

Si yP = xYi (P ) f exp dP (1.4)

pref

which relates the vapor and liquid phase fugacities. D. is the vapor

phase fugacity coefficient and represents the deviation from ideality

of the individual components in the mixture, and the nonideality of

v oL

mixing. For an ideal mixture of ideal gases, i is unity. f is the

fugacity of the liquid in the standard state; this is the pure

component at the temperature of the system, and under its own saturation

pressure (Pref = Pat) i(Pref) is the activity coefficient in the

liquid phase; it represents the composition dependence of the chemical

potential at the reference pressure. (The importance of this quantity

in the thermodynamics of liquids cannot be over emphasized.) For an

ref

ideal solution, yi = 1 at all xi. In a nonideal mixture, Yi(P ) is

normalized such that

Yi 1 as x. 1 (1.5)

The activity coefficient is unity when the component is in its standard

state. The exponential term corrects for the change in fugacity of the

liquid from the reference pressure to the pressure of the system.

The choice of the pure component as the standard state for

fugacities serves very well in the description of liquid mixtures

wherein each species can exist in the pure state at the system

temperature. However, this definition is unsuitable in considering

gas-liquid systems, for which the lighter components are super-

critical at the mixture temperature. Here, the solute at infinite

dilution in the solvent can be chosen as the standard state; the

solvent may be a pure, or mixed, solvent. The liquid phase fugacity

for the noncondensable components becomes2

P -L

L ref ref I vi

f. = xii (P )H (P ) exp R- dP (1.6)

1 i iM RT

pref

where the reference fugacity H. commonly called the Henry's

constant, is

L

lim fi = H (P ref (1.7)

x 10 x m

iO i

ref

The activity coefficient Yi(P ) is now normalized such that

ref

Yi(P ) 1 as xi 0 (1.8)

ref

where P ref is usually the bubble pressure of the solvent at the

ref ref

temperature of the system. The product xiYi(P ) Hm (P ) describes

the variation of the liquid phase fugacity with composition.

An important relation exists between the Henry's constant

and activity coefficient of a gas dissolved in a mixed solvent and the

corresponding quantities for the gas dissolved individually in each pure

solvent. It is

N M

n B. x nH i x(lim lim nym (1.9)

-,m j=1 X j e) ij im

j=1 j=l xln1 x ni1

where the multicomponent solvent mixture consists of N solvents with

mole fraction x j=l,...,N. H. and H. are the Henry's constants

Si,m i,j

of the gas in the multicomponent solvent and the jth solvent,

*A *

respectively; yim and yi are the respective activity coefficients.

Then if the solute interaction with the individual solvents is known,

equation (1.9) can be used to calculate the solute nonideality in a

multicomponent solvent mixture from the Henry's constant, which is

an ideal solution property. Conversely, if the solute nonideality in

the multicomponent solvent can be approximated by some average of the

nonideality in the individual solvents, the solubility in the multi-

component solvent can be determined from those in the individual

solvents. However, one must recognize, the important and unavoidable

assumption, made in equation (1.9), that the composition dependence of

the activity coefficient as determined from dilute solution (or low

concentration) behavior can be extrapolated to the physically

unrealizable limit of xi = 1.

1.2 Isothermal Pressure Dependence of Thermodynamic Properties

The fugacity of a component in a mixture depends on the

temperature, pressure and composition. Its pressure dependence is

given by the partial molar volume

BDnf. v.

-- = (1.10)

TP i,x

The derivative is taken at constant temperature and composition; the

volume changes to satisfy the general equation of state of the system.

(The variance of the intensive quantities is, of course, determined

by the phase rule.) For a finite pressure change,

P2 -

n fj Zn fi] = dP (1.11)

SRT

2 1 P1

In most liquid systems, the partial molar volume does not vary

appreciably with pressure and (1.11) is simplified to

i n fi RT (1.12)

nfi P2 P I

2 1

When the vapor and liquid phases are assumed ideal, and the partial

molar volume is that at infinite dilution, we have

ref -o

SH (P = 0) viP

1 --= P exp R (1.13)

which gives the pressure dependence of the solubility in an ideal

liquid solution.

The effect of pressure changes on a fluid mixture is given by

the isothermal compressibility, K, which is defined as

= P T,n (1.14)

where n denotes the constant composition of the system. In integrated

form, equation (1.14) is

P2

-AP P2 1 P f(1.15)

2P-P -P1- J

P1

Given < as a function of density, this equation can be used to relate

corresponding changes in pressure and density. Equation (1.15) is an

isothermal equation of state.

7

REFERENCES FOR CHAPTER 1

1. K. G. Denbigh, Chemical Equilibrium, Cambridge University Press,

Cambridge, 1966.

2. J. M. Prausnitz, Molecular Thermodynamics of Fluid Phase

Equilibria, Prentice-Hall, Englewood Cliffs, 1969.

3. J. P. O'Connell, A.I.Ch.E.J., 17, 653 (1971).

CHAPTER 2

DISTRIBUTION FUNCTION SOLUTION THEORY

A distribution function solution theory for equilibrium

properties of multicomponent fluid systems was first proposed by

Kirkwood and Buff.1 By rigorous statistical mechanical methods,

they related thermodynamic properties of a mixture to volume

integrals of the radial distribution functions. The shape of the

radial distribution functions in a fluid mixture result from

intermolecular attractions and depend in a complex, and, as yet,

incompletely determined, way on the macroscopic variables of the

system and on the intermolecular potentials of the different species

present. Currently available theories for predicting the distribution

2

functions, albeit approximate, can provide an adequate basis for

determining thermodynamic properties through the use of statistical

mechanical relations to be presented here. The properties

considered are the isothermal compressibility, the partial molar

volume of each component, and the composition derivative of the

chemical potentials.

2.1 Radial Distribution Function Solution Theory

The framework of the grand canonical ensemble is used in

the development of the solution theory. Relations are obtained between

density fluctuations and integrals of the radial distribution functions,

and again between density fluctuations and the composition dependence

of the chemical potentials. Elimination of the density fluctuations

leaves the desired relationships.

The system of interest, with volume V, is considered to be in

a large heat bath and open with respect to molecules in the system.

The bath provides a reservoir of heat at temperature T and of molecules

at chemical potentials Ul, U2' "'' uM. The system is then characterized

by the thermodynamic variables T, V, ui, i = 1, M. (M is the number

of chemical species present.) There are fluctuations in the number of

molecules, Ni, of each species within the system. The average singlet

and pair densities and the fluctuations in the density of each species

are related through equations of the form

1P ()rl)drl = (2.1)

a a av

S (r,r2)dr dr = - < > (2.2)

a 12 2 1 -2 a 6 av g- a av av

P (2)r r (1)(r )p(1) d dr

M 6 (2.3)

a av a a 8 av a8 a av

(2)

Here p (r r ) is the average pair density for molecules of species

(1)

a and 8 and p (r1 ) is the singlet density for molecules of type a,

is the average number of molecules of species a within the volume V.

(2)

The radial distribution function g ( (r,r) or g (r), is simply

a$ -z-1 o-72 as

related to the average pair and singlet densities:

(2) (,r2 (1) (1) (2)

Paa 1'-2 a P r -2 aB l'-2) (2.4)

(1)

c = P (r ) = /V (2.5)

Sla -1 a av

r l r2l (2.6)

c is the bulk concentration of species a; r is the scalar difference

(2)

of r and r2. Equations (2.4) and (2.6) assume that g (r) depends

-1 -2 as

only on the relative distance between r and r2, and not on their

(2)

exact locations within the system. g (r) is generally a function of

density, temperature, composition and the intermolecular forces between

molecules of species a and B.

Equations (2.4) and (2.3) together yield the desired relation

between density fluctuations in the grand ensemble and the integrals

of the radial distribution functions

f (2 - 6

(2) a p av a av B av aS

{g( (r) l}dr = V
j a8 c

a av 8 av a

where the integral extends over all sets of relative coordinates of the

(2)

pair a and p. It is instructive to note that g (r) as written

represents systems of monatomic fluids only, where the molecules have

only translational degrees of freedom. If the system under study

consists of molecules with structure, these must be accounted for in the

(2)

expression for g (r) and the integration extended over all possible

a$

(2)

configurations. In the present discussion, g (2)(r) is considered to

dp

include the averaged effect of all degrees of freedom of the molecule.

In the grand canonical ensemble the relation between density

fluctuations and the composition dependence of the chemical potentials

3

is

= A I/JAI (2.8)

a av a av 8 av a

where

Sa k-T (2.9)

1 aJT,V,N

Y

and IA I is the cofactor of A in the determinant det/A/; y is the

chemical potential per molecule of species a. Then, eliminating the

density fluctuation between (2.8) and (2.7), we get

a 1 B I/v B (2.10)

kT N aN T,V,N aal

where

B I c 6 + c c G (2.11)

G J {g 2)(r) l}dr (2.12)

Equation (2.10) deals with fluid mixtures at constant volume and

temperature. To obtain relations at constant pressure, we use the

Gibbs Duhem equation

M /apJ N

[ a= 0 (2.13)

T,P,N

and a mathematical relation4

S8u i v v

L 01 a N l J + a ( 2 1 4 )

ST,V,N B T,P,N KV

where v is the partial molar volume, per molecule, of species a, and

< is the isothermal compressibility of the mixture. Use of the additive

property of the partial molar volumes

M

SNv = V (2.15)

a=l

results in

KkT= IBi/I cacBIBaI (2.16)

a B

v = ca lBaI c}/ cc BIB aB (2.17)

a a

(I kT aIB aB )( cBaBI)

aN V[ IBaI -1 (2.18)

B T,P,N VIB cc B a

Ba

Equations (2.16), (2.17) and (2.18) are the key equations of the solution

theory. For ease of manipulation, they are conveniently expressed in

3

matrix form. Redefining

N i (2)

G {g(2) (r) l}dr

aB V a -

and further

d kT aN

aB N kT la T,P,N

aS kT 3N

a aa = N I IT,V,Ny

equations (2.16), (2.17) and (2.18) are written as

v/