• TABLE OF CONTENTS
HIDE
 Title Page
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 Nomenclature
 Abstract
 Thermodynamics of gas-liquid...
 Distribution function solution...
 Perturbation theory for molecular...
 Macroscopic state dependence of...
 Solution theory for subcritical...
 Determination of experimental thermodynamic...
 Solution theory for gas-liquid...
 Concluding remarks
 Appendices
 Bibliography
 Biographical sketch














Title: Applications of the direct correlation function solution theory to the thermodynamics of fluids and fluid mixtures
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 Material Information
Title: Applications of the direct correlation function solution theory to the thermodynamics of fluids and fluid mixtures
Physical Description: xvi, 191 leaves. : illus. ; 28 cm.
Language: English
Creator: Brelvi, Syed Waseem
Publication Date: 1973
Copyright Date: 1973
 Subjects
Subject: Statistical mechanics   ( lcsh )
Statistical thermodynamics   ( lcsh )
Vapor-liquid equilibrium   ( lcsh )
Molecular dynamics   ( lcsh )
Kinetic theory of liquids   ( lcsh )
Chemical Engineering thesis Ph. D
Dissertations, Academic -- Chemical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis -- University of Florida.
Bibliography: Bibliography: leaves 187-190.
Additional Physical Form: Also available on World Wide Web
General Note: Typescript.
General Note: Vita.
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Bibliographic ID: UF00097564
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: alephbibnum - 000585164
oclc - 14196168
notis - ADB3796

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Table of Contents
    Title Page
        Page i
        Page ii
    Acknowledgement
        Page iii
    Table of Contents
        Page iv
        Page v
        Page vi
    List of Tables
        Page vii
        Page viii
    List of Figures
        Page ix
        Page x
    Nomenclature
        Page xi
        Page xii
        Page xiii
        Page xiv
    Abstract
        Page xv
        Page xvi
    Thermodynamics of gas-liquid systems
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
    Distribution function solution theory
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
    Perturbation theory for molecular distribution functions of dense fluids
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
    Macroscopic state dependence of direct correlation function integrals
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
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        Page 60
        Page 61
        Page 62
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        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
    Solution theory for subcritical systems
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
        Page 94
        Page 95
        Page 96
        Page 97
        Page 98
    Determination of experimental thermodynamic properties of gas-liquid systems
        Page 99
        Page 100
        Page 101
        Page 102
        Page 103
        Page 104
        Page 105
        Page 106
        Page 107
        Page 108
        Page 109
        Page 110
        Page 111
        Page 112
        Page 113
        Page 114
        Page 115
        Page 116
        Page 117
        Page 118
        Page 119
        Page 120
        Page 121
        Page 122
        Page 123
    Solution theory for gas-liquid systems
        Page 124
        Page 125
        Page 126
        Page 127
        Page 128
        Page 129
        Page 130
        Page 131
        Page 132
        Page 133
        Page 134
        Page 135
        Page 136
        Page 137
        Page 138
        Page 139
        Page 140
        Page 141
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        Page 144
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        Page 156
        Page 157
        Page 158
        Page 159
        Page 160
        Page 161
        Page 162
    Concluding remarks
        Page 163
        Page 164
        Page 165
        Page 166
        Page 167
        Page 168
    Appendices
        Page 169
        Page 170
        Page 171
        Page 172
        Page 173
        Page 174
        Page 175
        Page 176
        Page 177
        Page 178
        Page 179
        Page 180
        Page 181
        Page 182
        Page 183
        Page 184
        Page 185
        Page 186
    Bibliography
        Page 187
        Page 188
        Page 189
        Page 190
    Biographical sketch
        Page 191
        Page 192
Full Text





















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/

PV- = Bx i (2.20)
I ji- = -
J=1

-i t t t -1
x D = I ((B ) i i )/i B i (2.21)

where

B = I + x G (2.22)


A, D and G are matrices with elements ai, dj, and Gi, respectively,
=' ij ij ij
x is a diagonal matrix of mole fractions, xii Ni/N, i is a unit column

vector. The basic equation (2.10) relating chemical potentials to the

radial distribution function integrals is


A (B x)1 (2.23)


2.2 Direct Correlation Functions

O'Connell has shown5 that integrals of the direct correlation

function may be used in the distribution function solution theory. The

direct correlation function is related to the radial distribution

function by the Ornstein-Zernike integral equation2


g(r) 1 = c(r) + p f c(r,r'){g(0,r') l}dr' (2.24)


Like g(r), c(r) is also a function of the intermolecular potential and

the macroscopic variables of the system.

There is no conceptual advantage in relating thermodynamic

properties to integrals of the direct correlation functions rather than

to those of the radial distribution functions. However, on the

molecular scale, as r, the distance between molecular centers increases,








the function c(r) goes to zero much more rapidly than the function

g(r) 1.2,6 Hence, it may be possible that the integral c(r)dr

can be calculated more accurately from a molecular theory than the

corresponding integral J{g(r) l}dr. Anticipating these developments,

the further results of the distribution function solution theory are

presented in terms of integrals of direct correlation functions.

The spatially integrated form of the multicomponent Ornstein-

Zernike integral equation is (Appendix A)


G = C + G x C (2.25)


where C c (2) ,r )dr2 (2.26)
ij V ij (1 2 2

Then from (2.23) and (2.25) we have

-l
x A = B-1 = I x C (2.27)


and the equations for thermodynamic properties become


M
pv = {1 x C }/{ Cjk xxk} (2.28)
j=1 j=l k=l

and

/p j=l k=l
x Fi i
x D = I F (2.30)
= it F x i

where
F = I x C (2.31)


Equation (2.30) has elements of the form










Sxi {l+Cij xn(Cn+Cn )+ xx m(C C m-CijCn
6 [ i n nm

T,P,N n m nm
k nm

(2.32)

In principle, one can use currently available theories for the

distribution functions in the above expressions to calculate

thermodynamic properties. However, the volume integrals of the direct

correlation functions Cij, (and the radial distribution functions Gij),

depend on the macroscopic variables of the system. Their composition

dependence must be known before equations (2.32) or (2.18) can be

integrated to obtain the Gibbs free energy of the mixture.

However, this restriction is by no means absolute. As in

other branches of mathematical physics, one avenue of progress is via

functional expansions about those limiting compositions in the mixture

at which the distribution functions can be determined with some degree

of certainty, e.g., the pure components. Then, within reasonably small

ranges about these compositions, the thermodynamic properties can be

evaluated by treating the distribution functions as independent of

composition.


2.3 Taylor's Series Expansions for Thermodynamic Properties

The integrals Cij, equations (2.25) (2.32), are generally

dependent on the system temperature, density, and the composition. The

limiting composition dependence is determined from the behavior of the

pure fluid (solvent component) and the infinitely dilute solution

solutee component). To relate the discussions to the thermodynamics









of gases in mixed solvents, expansions are derived for the following

physical systems:

i) Pure fluid, i.e, pure solvent,

ii) Solute gas in single solvent--binary system,

iii) Solute gas in binary solvent--ternary system.

Component 1 is the solute, 2 and 3 are solvents, superscripts

and refer to solute at infinite dilution in single and mixed

solvent, respectively. The general expressions will be derived first,

i.e., case (iii), and the results for (ii) and (i) inferred as simplifica-

tions of the general case.


2.3.1 Solute Gas in Binary Solvent

The general form of functional expansion is:

2
W) W) +x1 axw x .
x l=x x=ax x =0 2
X1 1 1 3x2
x -0 1 x=0

(2.33)

where W is the thermodynamic property and xl is the solute mole

fraction. The derivatives ( [ are obtained from relations
1 1x

(2.28), (2.29) and (2.32), respectively. The integrals Cij are functions

of xl, and the derivatives include integrals of higher order direct

correlation functions. The relations between integrals of different

orders are presented in Appendix B. The relation used here for the

composition derivatives is equation (B.9). The final results are:


Eny f (2x x f ( x ) + ... (2.34)
f(2x 1 1 2 1 1











where

++ (1- t)2
2f+ = C ) + t)
2 11 (1 D)


+ + +
3f (1 -c C ) +
3 11 111


+ -+
1 + C {1 + u p (1 w)}
(1 D)


-+ -+ -+ -+
+ pvl[2(1 2t)(l pv ) 3r + pvl{3u + D PV (D +w))]

(2.36)


- -+ 2- -
pv = pv + x 2pV (1 pv )
1 1 1 11


-+ -+ +
+
{pv1[2u+t+D-pvDw( 1-C -r}
+ (1 D)


1
1 = (1
pKRT


-+ -+
- D) + x1[2(D t) u + wpvI D(1 pv )] + ...


2 + 2+ +
where D E x2C 2 + x3C33 + 2x2x3C23

+ +
t 2C12 + 3C13

2+ + 2+
u = x2C122 + 2x2x3C + x3C33
2122 23123 3 133

2+ 2 + 2 + 3+
S 222 + 3x 223 + 3x2x3C233 + x3C333

+ +
r x2C2 + x3C13
From equation (2.38) with x113 0

From equation (2.38) with x, M 0


1
PKRT
1=0


- (1 D)


relating (1 D) to the compressibility of the solvent mixture.


from (2.37) with xl = 0


(2.35)


(2.37)


(2.38)


(2.39)


(2.40)


Again,










(pv1)0 = (pV) l


S1 t
1 D


Thus (1 t) is related to the partial molar volume of the solute at

infinite dilution in the solvent mixture.


2.3.2 Pure Component


Setting xl = x3 = 0 in (2.38) gives


1 o
-RT = 1 C22
pKRT 1 22


(2.42)


O
relating C22 directly to the isothermal compressibility of the pure

solvent 2.


2.3.3 Gas in Single Solvent


Setting x3 = 0 in (2.38) and (2.37) yields


1 o o o o 0 -o o -o
RT= (1 C 2) + x[2(C2 C0) C122 + C2 C (1 PV )]
pKRT 122 12 122 222 1 22


(2.43)


pv = pv + x


1 where


where


o o -oo o -o -o o -o 2 o
-C -C 2 +3pvC +2C v -p C (v) 2 (C +C0 )
11 112 1 12 122 1 1 22 1v 22 222
o
(1 C )
22


(2.44)



(2.45)


o
1 C
-o 12
pv
1 o
22


o
thus C12 is related to the partial molar volume of solute 1 at infinite

dilution in solvent 2. Then, with x3 = 0 in (2.35) and (2.36)

o 2
(1 Co2
o o 12
2f = (1 C ) + (2.46)
(1 C )
22


(2.41)










0 a -o 0
1 + C {1+ C p (l C )}
o 11 122 1 222
3f = (1 C C ) +
3 11 111 o
22


-o o2 -o o -o o o -o o o
+ pvl[2(l 2C2)(1 p) 3C + v{3C122 + 22 Pl(C22 + C222


(2.47)
From (2.34)

ainy
[ n 1-2f0 (2.48)
ax 2
.. x=0

o o
C is contained in f and is thus related to the limiting slope of
11 2
the composition dependence of the solute activity coefficient.

Equations (2.42), (2.45) and (2.48) dictate a hierarchy for the
o o
determination of the direct correlation function integrals C22, C12

and C from experimental data. It is:
11

pure solvent isothermal compressibility gives C22.
o o
C22 and p.m.v.a.id. give C12.

o o o
C22, C12 and limiting slope of activity coefficient give C11'


A similar hierarchy exists in the more general case of the solute in

binary solvent. With experimental values of (23nyl/3ax), V1 pcRT

one can calculate D, t and C1+ from equations (2.40), (2.41) and

(2.35), respectively.

However, no such identification can be made between the thermo-
o +
dynamic properties and the three body integrals Cik and Cijk. This is

a natural consequence of the Taylor's series expansion. The triplet

functions result from the composition and density derivatives of two




20




body integrals in the composition dependent expressions for thermodynamic

properties. Therefore, the C..k must be estimated from experimental data
lJk
by truncating the expansions after a certain number of terms.

It is appropriate to make some general remarks about the

expressions derived thus far. Being expansions about zero composition,

they should be valid only in the range of small solute concentrations.

To increase the composition range one must include terms from second

and higher derivatives of the direct correlation function integrals;

these will include triplet and higher order correlation function

integrals.

The expressions for isothermal compressibility and partial molar

volume indicate an important difference between these properties and

the chemical potential. At small concentrations where the linear

expressions in x1 are valid, the chemical potential depends on two-

body correlations only, i.e., f2, whereas the compressibility and

partial molar volume expressions include two and three-body terms.

Equation (2.42) has another important characteristic; C1,

which is related to the solute nonideality, appears in the linear

term for the partial molar volume. It can be determined from the

experimental partial molar volume in its linear range; i.e., from the

slope at zero concentration. This presupposes a reliable method for
o o
determining the three-body integrals C112 and C122. A generalized

method for doing so is described in a later chapter. (One may be

tempted to make the simple approximation of setting the C = 0,
ijk
but this is not always physically realistic.)









2.4 Linear Composition Dependence of Direct Correlation Function Integrals


It is possible to relate the thermodynamic properties of the

solute in the binary mixture to those of the solute in the pure solvents

by postulating simple rules for the composition dependence of the

quantities Cij. The solution theory itself does not provide any such

prescriptions. These must come from a molecular theory for mixtures.

The simplest such rule is a linear variation of the form:


+ o *
Ci C + x (C0 C.) (2.49)
ij ij 2 ij ij

where the superscript refers to the solvent composition X3 = 1,

and the superscript refers to the solvent composition X2 = 1. A

possible variation of the quantities Cij according to the rule (2.49)

is shown in Fig. 2-1. Quantities that can be determined from

experimental binary data alone are shown circled. For example,
O o
isothermal compressibility of pure 2 gives C22; C22 and partial molar
O o O
volume of 3 at infinite dilution in 2 gives C2; C22, C32 and limiting

slope of activity coefficient of 3 in 2 gives C3. The quantities
o *
C13 and C12 are different from the two-body terms encountered thus far.
o
C is the integral of the direct correlation function between one
13
molecule each of 1 and 3 when both are entirely surrounded by a medium

of type 2. It is not directly related to any thermodynamic property.

A naive approximation for C3 is a mean of C2 and C3 i.e., simply
13 12 320
the arithmetic mean.


C3 (C + C ) (2.50)
13 2 12 32

A more appealing approximation is





















+ +
C + C
32 23



C
+


11




CC+
C121





33
C33
-_ --------
^^~--12


^^^C22


__ _ __ _ _ ^ __ ___+


(C2)
32


(C )


(C
12

0
13



(C2)

(C3)
33


Linear composition dependence of the
integrals Ct. in system of solute in
binary solvent. Circled quantities
can be determined from binary data alone.


(C23)




*




(C13)





2C12

(C 22)


(c33)


Fig. 2-1.








0 0*
C13 C13 v /v
13 13 2 3


(2.51)


*
and is arrived at by the following argument. C13 is the 1-3 interaction

in an environment of type 3; C13 results when the environment is changed
13
to type 2 and this is achieved macroscopically by changing the volume

in the ratio of the pure component volumes. This assumes that the

change in environment is accurately described by size effects. When

the solvents 2 and 3 are chemically similar and are composed of nearly

equal sized molecules, a realistic assumption is


o o
C C
12 13


(2.52)


1
p

From relationships of

2 2
+ x2 x3
2 + 3 +
o O
P2 2RT p3 <3RT


the type (2.49) we get


2x23( 3C23
2x (1 x3C


o
x2x3(x2C22 x2C22



(pv1) (pKRT) 1 x2x3(C12 + C13



-* 2
+ 0* PV1
2f x2Cll + x3Cll + -1
2f2 2 11 3 11 (p + + K


- x2C)
2 23


o + *
- x3C33 +xC ) (2.53)

-o -*


S(2.54)
(2.54)


(2.55)


All quantities on the right side of these equations can be determined

from binary experimental data.

The above expressions describe the thermodynamic properties of

the solvent mixture over its entire composition range, i.e., from pure

2 to pure 3. The component activity coefficients are expressed in the

symmetric convention, i.e.,





24




2 2= 2 + RTn x2Y2


The superscripts and


again refer to pure 2 and pure 3, respectively.


Then


N3,T,P


1
x2


2Zny2
x2 T
N T,P


Using the relation


2
3N 2
N3


SN2
3x 2
N3


N2 N
N


N2 + N3
x3
3


in equation (2.32) leads to


1
[ 1 +
x3a


C 2xC22 2x C2 + x(C23C C)]
22 2 22 3 23 3 23 23 22 33


where


2 2
a = 1 x2C22 x3C33 2x2x3C23
2 22 3 33 2 323


(2.59)


(2.60)


At the compositions of 2 and 3 at infinite dilution in each other, we

get


Co + C 2C
33 22 23


( 0 0c 0 C0
- (C C 23 C23
22 33 23 23


(1 C22)



*
C22 + C33 2C23 (C22C33 C23C23)

(1 C33)


1RT X2
RT [ x
x2


(2.57)


a2 i
3


8any2
8x2


(2.58)


1
x3


any3
8x3


= -


- e3


a8ny2
ax2


x =0


3x=0


x2=0


(2.61)






(2.62)


(2.56)


= -e2









From equation (2.28)




S23 3 2 ,o
1 C



.*. C23 = 1 32R s 82

Similarly

C23 = 1 2/ RT = B3


Finally from equation (2.42)


33 = */p 3KRT a

C 1- 1/p22RT E ca
22 2 2


Then for a linear composition dependence of the


Cj we get


C22 = c3x2(1 a2) + x3(a3 33e2 (1 83)2)


C33 = a2x3( a3) + x2(a2 a2e3 (1 82)2)



C23 = x262 + x383


And, substituting in equations (2.29), (2.28) and (2.59), we get


2
1 3 2
pa RT = 1 x2a3(l a2) x2x (a3 a3e2 83 ) -

P 2 3 2
x3x2(a2 a2e3 1 82 ) 2x2x3(x3 3 + x282)


x2a2(1 a3)


(2.71)


S2 2- -2
pv2 = [ x2a3(1 2) x2x3[3 a3e2- (1 83) ] x3(x383+x2B2)]/a


(2.72)


(2.63)



(2.64)


(2.65)


(2.66)


(2.67)


(2.68)


(2.69)


(2.70)














1 1
3 x [(1 + (1 2x2){a3x2(1 a2) + x3[a3 a3e2
x 3 ax3 2 3 2 2 3 3 3 2


2 2 2
- (1 3)]}- 2x3(x33 + x22) + x3{(x33 + x )2 -


+ x3[a3 a3e2 (1 B2) ]}{a2x3(1 a3) + x2(a2 -

2
- 2 )]]


[a3x2(1 a2)



2e3


(2.73)


These expressions are used in Chapter 5 to study the thermodynamic


properties of liquid-liquid systems.


and


axny2
Sx2





27




REFERENCES FOR CHAPTER 2


1. J. G. Kirkwood and F. P. Buff, J. Chem. Phys., 19, 774 (1951).

2. J. S. Rowlinson, Liquids and Liquid Mixtures, Plenum Press,
New York, 1969.

3. T. L. Hill, Statistical Mechanics, McGraw-Hill, New York, 1956,
p. 113.

4. N. Davidson, Statistical Mechanics, McGraw-Hill, New York, 1962,
p. 274.

5. J. P. O'Connell, Mol. Phys., 20, 27 (1971).

6. J. K. Percus in Classical Fluids, edited by H. L. Frisch and
J. L. Lebowitz, Benjamin, New York, 1964.










CHAPTER 3

PERTURBATION THEORY FOR MOLECULAR DISTRIBUTION
FUNCTIONS OF DENSE FLUIDS


The molecular distribution functions in a real fluid are quite

complex functions of its molecular interactions and the macroscopic

variables of the system. No complete theory presently exists which

can accurately predict the radial distribution function (RDF) in a

real fluid in terms of its intermolecular potential and the temperature

and bulk density of the system. This situation is even more complex,

-and hopeless from the point of view of prediction, in a fluid with

many chemical species, and hence different molecular interactions.

The problem of describing the RDF and relating it to the

intermolecular potential has traditionally been approached in two

distinct ways. The first method is the postulation of theories relating

the direct correlation function (DCF), the RDF and intermolecular

potential. These theories are postulated or derived by techniques such

as' the diagrammatic methods (HNC theory) or by functional differentiation

(Percus Yevick).l They provide a relation between g(r), c(r) and

*(r), where g(r) and c(r) depend on the temperature and density of the

system, whereas (r) is assumed independent of the macroscopic

conditions. These theories are moderately successful in describing
1,2,3
the dense liquid state.1,23

The other approach to the problem at hand is perturbation

theory. In this method the real fluid is modelled as a perturbed state

about a reference state3'4 whose properties can be well determined; the

perturbation is usually in the intermolecular potential, and the RDF is










determined as a function of the perturbation parameters. These theories
5
have been quite successful in determination of thermodynamic properties;

however, for close agreement, second order terms in the perturbation

parameters must be included.

In the present study we have derived a simple, but rigorous,

first order perturbation theory for a real fluid. For computational

convenience, the base fluid is taken to be the hard sphere fluid,

though this is not a requirement of the theory. A first order density

expansion is obtained relating the RDF's and DCF's in the real and hard

sphere fluids. Combined with the Ornstein-Zernike equation for the real

fluid, the perturbation expression provides a scheme for determining

the real fluid distribution functions. To reduce the solution of an

integral equation to that of computing a numerical integration, an

intuitive approximation is made about the nature of the real fluid

distribution function. The validity of this approximation is borne

out by the close agreement of the calculated RDF to that resulting from

computer experiments. The DCF of the real fluid is then calculated

from the known intermolecular potential and calculated RDF. It is

also numerically integrated to yield J c(r)dr, which has earlier been

shown to be related to the isothermal compressibility of the pure fluid.

The integral is computed as a function of reduced temperature and

density giving a table of values of the reduced compressibility

integral.

In multicomponent systems, the DCF and its integral depends on

the composition, and no theory is yet available to describe this

dependence accurately. Empirically, the composition dependence of the











DCF integral is represented as being through the macroscopic volume

of the system which depends on the composition. Then, using represen-

tative molecular parameters for the different intermolecular

attractions present, the DCF integrals in the mixture can be evaluated

at any liquid density. This method is used to determine the

thermodynamic properties in several mixtures of subcritical components

for the infinitely dilute composition states. The agreement between

calculated and experimental values is good enough to classify this

theory as acceptable, for the purposes for which it is used. Further

work in this regard is clearly warranted.


3.1 First-Order Perturbation Theory

The perturbation expression is derived by the method of

functional differentiation described by Percus; it is based on the

recognition that the molecular distribution functions are functional

derivatives of the molecular density at any point in the fluid with

respect to a change in potential on the fluid. This provides a precise

relation (hierarchy) between molecular distribution functions of

different orders.

Details of the derivation are presented in Appendix C. The

resulting integral equation is:

rf r g (rexp, r hs hs
f(r) - + rfIr) - ghs(r) chs (r)
hs [Bhs (r)]
g (r)exp


fgrf (s)exp rf (s)- hs(s)] hs hs
S -h (s)ex) + 1-g (s) + c (s)
*' ~g (s)


[grf(r') 1] dr'


(3.1)










where s r r' Now making the approximation that for terms

within the integral we can write



grf (r) g(r) (3.2)

we get


yrf(r) drfr) 1 = p fhs(s)yhs(s) + dhs(s)][e (r) ]dr
o
00
r [hs r rf(s) [yhs ,

+ p [d(s) + 1 e' ][ (r') 1] dr' (3.3)
a

where we are using Rowlinson's notation1


y(r) = g(r)e(r) ; c(r) = f(r)y(r) + d(r) ; f(r) = e-6(r) 1

(3.4)

Given an intermolecular potential function, and a value of the reduced
*
hard sphere diameter, a a/c, where a is the distance of zero

potential, the real fluid distribution functions can be calculated for

any thermodynamic state.

Calculations for the Lennard-Jones 6-12 potential have been

made at two thermodynamic states in the dense liquid region. The

hard sphere diameter was determined by the Barker-Henderson3 prescription,

and the hard sphere distribution function by the Walts and Henderson7

modification of the Percus-Yevick equation. The resulting RDF's are
8
shown in Fig. 3-1 and compared with molecular dynamics values of

Verlet. For one of the two states, we also show the RDF determined

from the usual first order Percus-Yevick theory. As anticipated, the

calculated g (r) follows the molecular dynamics result closely out to


















2.0- =0.85






1.0







C
=0.88
20 ~0o.85

2.0-




1.0


I I
1.0 2.0 3.0
r/a
Calculated
--- Molecular Dynamics
. Percus-Yevick
Fig. 3-1. Radial distribution function for Lennard-Jones
6-12 fluid.









large reduced distances in both magnitude and phase, and is clearly

an improvement over the Percus-Yevick result. The calculated first

peak of grf(r), which is most important in the determination of

thermodynamic properties is also much closer to the molecular dynamics

peak than is the Percus-Yevick peak. This indicates that thermodynamic
rf
properties determined from the g (r) calculated from (3.3) should

agree more closely with experimental (computer) results than do those

determined from the Percus-Yevick theory.

To obtain improved solutions of grf(r), equation (3.3) could

be used in an iterative solution scheme. It is expected that this

procedure will yield results in even closer agreement with molecular

dynamics values. In the same fashion, varying the hard sphere

diameter in ways other than that of the Barker-Henderson prescription

could lead to improved results.


3..2 Reduced DCF Integrals


The agreement obtained between the calculated RDF and Monte

Carlo results indicates that the perturbation theory developed here is

a satisfactory, if not completely accurate, model of the molecular

interactions in a Lennard-Jones 6-12 fluid. We now postulate that

within the limits of accuracy, the Lennard-Jones 6-12 potential

represents the interactions in a real fluid.

In spherically symmetric intermolecular potentials, the

compressibility equation of state, or equation (2.42) is



0
-- 1 AsJ c(r)r dr (3.5)

o









In terms of the reduced variables


3 kT a r
p T d = r -- (3.6)
Sa a

we have

-1
(1 p C)- (3.7)
a pd T


C 43 f c(r) r2dr (3.8)



The integral C has been calculated for state points covering the entire

liquid region, using the perturbation theory developed earlier.

Fig. 3-2 shows the variation of this quantity with reduced density at

several values of the reduced temperature including the critical

temperature and the triple point. At a fixed reduced density, this

integral is not a strong function of temperature, and in the dense

liquid region, it may be taken as approximately constant. It does

vary strongly with reduced density along an isotherm. This is not

surprising since the direct correlation function c(r), especially at

small r,is a strong function of density.

It is now a simple matter to calculate the isothermal

compressibility of the fluid at a given thermodynamic state. All

that is required are appropriate values of the potential parameters

a and E/k. However, before establishing a general rule for determining

these from some known macroscopic property such as a second virial

coefficient or shear viscosity, it is advantageous to examine further

macroscopic implications of the graphs shown in Fig. 3-2.

















-5






















'-25


Reduced direct correlation function integral
for Lennard-Jones 6-12 fluid from perturbation
theory.


---- -- T=.96
-----T = 1.2
















SI



\

\


Fig. 3-2.









The reduced compressibility < can be re-expressed as


1 = 1 +I (3.9)
p1 RT p

Since we know from macroscopic evidence that K depends on p and T,

we expect that 1 + - also depends on p and T. The calculated

quantity 1 + - _, where < is obtained from (3.7), is shown in
p <
Fig. 3-3. Over the entire liquid range it does not vary with tempera-

ture by more than 10% at a fixed reduced density, and may be

considered constant. This assumption makes isochoric lines on a P-T

plot linear.

The constancy of the reduced compressibility over the

temperature range provides the microscopic verification for a macroscopic

correlation of the isothermal compressibility of pure fluids. The

development of the macroscopic correlation will be discussed in detail

later. The general result for pure liquids is that (1 + ) is a
P universal function of a reduced density and is independent of temperature.

The reducing density is close to the critical density for nonpolar

liquids, and larger than the critical for polar liquids. These

correlating densities are used as the source of molecular parameters

in the microscopic theory. Existence of the correlation asserts that

macroscopic corresponding states is valid for this property, and the

representation of all molecular interactions is equivalent to postulating

corresponding states on the microscopic level. Then taking Argon as

the prototype or reference substance, i.e., as satisfying both

macroscopic and microscopic corresponding states, the molecular

parameters for the other fluids are determined from the relation













30 = 1.04
= .88

= phase boundary

O = .68 (triple pt.)









10 .










+





I I







.5 .7 .9


Fig. 3-3. Microscopic correlation of isothermal compressibility
for Lennard-Jones 6-12 fluid.









1/3
a = OAr (3.10)
Ar I
Ar

where v is the reciprocal reducing density for the macroscopic

correlation. Values of a for some substances are shown in Table 3-1.



TABLE 3-1

INTERMOLECULAR PARAMETERS FOR SIMPLE FLUIDS


Substance Ar CH4 N2 02 CO C2H4

v cc/mole 74.57 99.5 90.1 73.4 93.1 127.3

o A 3.405+ 3.749 3.627 3.387 3.666 4.069


H20 CC1 CH SO2 C3H8 nC410

46.4 276.0 255.0 115.0 200.0 255.0

2.907 5.267 5.130 3.934 4.752 5.130


CS2 CH30H NH3 Cyclohexane

165.0 101.5 65.18 311.0

4.437 3.774 3.756 5.481


From second virial coefficient data






The value of aAr is taken from second virial coefficient data.

For other substances, the value calculated from (3.10) is generally

smaller than that obtained from second virial coefficients. The

highly polar substances, water and methanol, have relatively much

smaller values of a.










The function C is assumed te'-. : ri Independent 2t a fixed

reduced density and no temperature riiastc .].eculir parameter is

necessary to characterize the microscopizc -,: of the fluid. C at

a given p is determined from Fig. 3-1.


3.3 Multicomponent Systems


In a mixture the quantities C.. depend on the molecular
1J
interactions and the macroscopic state of the system. No present day

molecular theory can describe the composition dependence accurately.

(The multicomponent perturbation theory of Barker-Henderson-Smith10

is a first attempt at describing the composition dependence of the

multicomponent radial distribution function.)

An intuitive method is proposed here to represent the dependence

of Ci on the system variables. It is based on two postulates, which

are:

1. All molecular interactions in the mixture, .ij, are of

the same type; i.e., they are representable by a unique reduced

potential function, the Lennard-Jones 6-12 potential.

2. The composition dependence of the Cij is completely

described by the macroscopic density of the system, which is

composition dependent. This assumption neglects the effects on Cij

of composition changes at constant density but emphasizes that the most

significant parameter for determining Cij is the macroscopic density.

On the basis of these two postulates, the general reduced

integral C.. may be written as

Cij F(PJ) F(Pij) F(p3) (3.
C F(p -j F(p )- F(pa )- (3.11)










where oij is the size parameter for the i-j int=rarcion. K:ere. it is

calculated from the arithmetic mean rule


a (i.. + a..) (3.12)
ij 2 ii + j


These equations form a method for determining the integrals C.., C..
-"- J
and Cij at any given liquid density from which the Lnernodynamic

properties can be calculated. The calculated thernodynamic prcoerries

for some binary systems are shown in Table 3-2 and compared with

experimental values as available. The composition dependent integrals

Cij have been calculated only at the state of infinite dilution of

component 1 in component 2. For all systems of simple molecules

Ar, 02 and N2, there is good agreement between calculated and

experimental isothermal compressibilities and partial molar volumes at

infinite dilution. This indicates that the molecular representation

assumed by the theory is quite valid for these systems. For the other

systems of more complex molecules, the agreement is only qualitative,

although the trends of all calculated quantities are in the direction

of the experimental values.

Comparison of the calculated C?. with the experimental values,
1j
where these are available, indicates that the calculated C.. are
ij
not negative enough, e.g., C22 in the n-pentane systems. The larger

negative values necessary are obtained only by calculating the quantities

at larger reduced densities (i.e., the o values are too small), or

by introducing a temperature dependent parameter into the formulation
o
of the theory, which would then allow somewhat larger negative C...

However, it is more likely that the assumptions of conformal solutions















SX






-4 ;




0I
Sx










I> *>
cz
0 r I
U


+
>4



O
z
E-





0
-4

















C/)
0



























C/CU
O







-L4
0
E-4








































0;e


0 r-4











-.4
0 -4
OU
J


r-D tr J)


I I I I I I I I I
cLl ci -i *) O





lA 0 0 0-' 0a 0 0 O -i ON 'a

I I I I I I I I I



m ro A .-4 -0 0 o L rH- c- CO 0


,-I mO in 0 Gn O r a -4 T r c)



Cl 00 Cr- 000 r- -
1- r- l -- -4 r-4


- Cl '0 00 A C 00 m ir) "M L --l





So (o4 ro o 0000 _; cy4 o r r-
CO- C- 4-4








'A0 000"0 % "c 0 0 0r CO





000.000 ACO000
T r-iq -i CO -0 -.1 'a.
('4 (C4 -4 r- ( n (- 4 ('4 (C 4 r-4 C) -4 CN









O0 r, On wr
a, 4 O -i o n' 'A L r- uI 0c
.-A ('4 I C4 ('4 C' (- (4 (0 (' (04 04 04





1 1 1 1 1
S00 0 0 LO 'A -0 r -- A
CP) (4 C'4 cl ('4 (C4 (04 Cl C
I I I I I I I I I I I I I



a' 0 'A u-I n C-I i-I 'n
r-I Cl 0 -7 Cl C 0 -' in r- 0 .-4 r-
C;o 1r- CO -; cc -3 r-1 CO a' 'A -1;
(S eN' (' 4 CO C CO u-- O o CO 0 0'





c 0 CO CO CO CO CO CO oO CO Cl C
CO CO wc O c o a' O O cc a -Z -a



XX x
C' -zr CM -c I--
c-4 1-1 r-i o Z 0
uma u- o u -40 0 % 0 %
S( Co o 0 c u u u ua '


x x



L) 'A 4. -.T WAZ U u U)
0 Z0 0 '.0 X D :;

< < r0r-t r. ( r0 l 0 0
2 <

aYI-











(potentials) and pairwise additivity break down for complex molecules

in dense systems.

Nevertheless, the above theory provides a basis for che

qualitative, and often quantitative, understanding of the state

dependence of the direct correlation functions and their integrals.

Two valid conclusions can be drawn:
o o o
1. The calculated integrals C11, C22 and C12 are approximately

in the correct ratio to one another.

2. When the calculated integral C2 is within 5-10% of the

experimental value, the calculated C0 and C0 are also within this
12 11
range of the corresponding experimental quantity. When the theory is

inadequate in predicting experimental data, it is due to the poor

modelling of the pure solvent state.





43




REFERENCES FOR CHAPTER 3


1. J. S. Rowlinson, Rept. Progr. Phys., 28, 169 (1965).

2. J. S. Rowlinson, Mol. Phys., 10, 533 (1966).

3. J. A. Barker and D. Henderson, J. Chem. Phys., 47, 2856 (1967).

4. J. D. Weeks et al., J. Chem. Phys., 54, 5237 (1971).

5. J. A. Barker and D. Henderson, J. Chem. Phys., 47, 4714 (1967).

6. J. K. Percus in Classical Fluids, edited by H. L. Frisch and
J. L. Lebowitz, Benjamin, New York (1964).

7. R. 0. Watts and D. Henderson, Mol. Phys., 16, 217 (1969).

8. L. Verlet, Physical Review, 165, 201 (1968).

9. F. Mandel et al., J. Chem. Phys., 52, 3315 (1970).

10. W. R. Smith and D. Henderson, Mol. Phys., 19, 411 (1970).










CHAPTER 4

MACROSCOPIC STATE DEPENDENCE OF DIRECT
CORRELATION FUNCTION INTEGRALS


In the preceding chapter we studied a molecular theory for the

state dependence of the DCF integrals. Although the generalized

microscopic method gave some adequate results for thermodynamic

properties, there are still areas of uncertainty in the molecular

approach, and no real clues are provided for improvements to the molecular

theory. In this chapter we study the available macroscopic data to

determine the state dependence of the DCF integrals.
o 0
A definite hierarchy exists in the calculation of C22, C12

and C01 from experimental data, and this order is followed in searching

for macroscopic correlations. Generalized equations for the temperature

variation of the pure solvent saturation pressure and liquid volumes

are combined with the previously derived correlations to derive the

temperature and pressure dependence of the Cj.


o o
4.1 Corresponding States Correlations for C22 and C12


The DCF integral C2 is related to the isothermal compressibility,
22
K, of a pure fluid by the equation


1 o 1- (4.1)
22 P20 RT
p2 2RT

It is experimentally derived from P-V-T measurements of compression

experiments. Commonly, the compression data are filled to analytic and fitted

expressions such as the Tait Equation (for most liquids) and the










Huddleston equation (liquid hydrocarbons). Experimentally, K for a

pure liquid is found to be a strong function of temperature, and very

weakly dependent on pressure. As the temperature approaches the

critical, the isothermal compressibility of the saturated liquid

becomes very large.

In Fig. 4-1, the experimental quantity 1 + l/pKRT is shown as

a function of p for several simple liquids; the plot for n-decane is

along an isotherm (3580K), i.e., a compressed liquid, while the

others are along the liquid vapor saturation line. The curves are all

similar in shape and one concludes that there exists a general correla-

tion of the form



1 + c = F() F(pv) (4.2)


Experimental compressibility data for Ar, 02, N2, CH4, CC14, nC10H22'

nC 2H26, nC14H30 and nC16H34 are used to derive an analytical

expression for this general relationship. The first five of these

substances are chosen among the primary set because they are simple

molecules. Further, their characteristic volumes can be taken as

the critical volumes to yield a set of curves for 1 + T that
I pRT
are coincident within experimental error. The function 1 +
1 pRT
then has a limiting value of unity as p pv 1; however, detailed

experimental data are not available in this lower reduced density range.

The reduced density range for this set of substances extends over

1.5 < p < 2.9. The long chain hydrocarbons are included in the primary

set to extend the reduced density range of the correlation to larger















nC 22
nC
10 22


nC 16
7 16


C6H6


.005 .010 .030
p gm moles/cc
Fig. 4-1. Isothermal compressibilities of pure liquids.


50


I .









values; their compressibilities are reported to be of a high degree

of accuracy ( 1%). The characteristic volumes for these hydrocarbons

are obtained by matching the experimental data to the curve for simple

substancesjin the region of overlap. The algebraic expression obtained

to represent the experimental data over the range 1.5 < p < 3.7 is


o 1 2 3
2 C22 = 1 + PRT = exp[-.42704(P 1) + 2.089(P 1) .42367(P 1)]


(4.3)

Fig. 4-2 is the graphical representation of equation (4.3) and

shows some representative experimental data to indicate the average

error of prediction. For all substances other than those listed above,

the characteristic volumes were obtained by fitting experimental

compressibility data to equation (4.3). For nonpolar substances, the

fitted characteristic volumes are estimated to be within the errors

of experimental and predictive methods for the critical molar volume.

Thus, in the absence of experimental compressibility data, the

critical volume of a nonpolar substance or a weakly polar substance

may be used as the characteristic volume. For polar substances,

however, e.g., water and methanol, v as fitted from equation (4.3)

is found to be less than vC. (On the molecular scale, v is related

to the distance parameter in the angle averaged or effective

spherical potential.) There is no clear and lucid explanation of this

relationship, however, that would permit the determination of v

from other macroscopic data. Therefore, this parameter must be

obtained from at least one experimental point.




















I I -


SC16H34
Sn C10H22

o isopentane
o methanol
w Ar


1 0
F1. I.




Fig. 4-2.


2.0 3.0


Generalized correlation for isothermal
compressibility of liquids.


100l


r-

a)


10-








The results of compressibility calculations with equation (4.3)

are shown in Table 4-1 for nine representative substances. 6, the
1
fractional root-mean-square deviation in 1 + is always less
pKRT
1
than 0.08; the largest single error in 1 + encountered over an
pKRT

aggregate of 70 state points is 11%. Characteristic volumes for all

the substances included in the correlation are presented in Table 4-2;

the list contains liquids of all types, polar, nonpolar, long chain

and branched hydrocarbons, alcohols, esters, inorganic liquids, etc.

In general, for substances with known characteristic parameters, both

polar and nonpolar, equation (4.3) provides a value of < with an

average deviation of 6-8% at p greater than 1.5.

In a binary solution, C is related to the partial molar
12
volume of component 1 at infinite dilution in component 2. (When

component 1 is a gas, whose critical temperature is appreciably lower

than the temperature.of the solution, the infinitely dilute state is a

thermodynamically conceptualized state.) The partial molar volume is

determined experimentally by observing the change in volume of the

mixture as the component composition is changed. It is then extracted

from the experimental volume by the general procedure for partial molar

properties and extrapolated to zero composition. If the solution is

assumed to be ideal, the partial molar volume at infinite dilution

(pmv aid) can be obtained from the change in solute solubility with

pressure. Eqn. (2.45) is

pv = (1 C )/( C(4.4)
1 12 22
-o o 0 o
.. v v2(l C1)/(l C2) (4.5)
1 2 12 22


C2 was calculated from equations (4.5) and (4.3) with experimental
12














0)



CO a)
0
r04





40



ca

(d

0U -.3 Lri Cl \o Cl CN .t (' \o


+
1.0 \%0 -1
'H 1C CM
000 C q 00 0











-: C: 0 0 0 0
- O r- 0 r- Co l -I












%.0 ol 4 u
C r-.- ON 0 -Z r- Cl C)
r-4 -4


U



41
























0
(


*o
3

ca














-
(U


CLn --T -zr U)




M c 00 r-i n
N C' r r-I N.
CI T. C" cn C"


0-H-
0 M C 00
C CO U O )
o s c a )
4 -.1 o o T 3 4-o
.- CM CMJ 'H =7 A4 03


0


o .-
O 0


0





p u


0

4-4





o u
o 0
,.-










03 0
u
0a a




a



4 -1
o
4-I




(0 X

4- ).
6 u

+ *+


0 0 0 0


I i I I
\D0 C

'H o o
coC 0 0 o










REDUCING VOLUMES


TABLE 4-2-a
FROM ISOTHERMAL COMPRESSIBILITIESa


Substance

CC14
nC12H26

nC14H30
nC16H34
C3H8

nC4 10
nC5H12
iso C5H12

nC6H 14
nC7H16
nC8H18
nC9H20
nC H
C11 H24
nC13H28
nC H32
15 32
nCl7H36
nC18H38
nC20H42
nCH
nC30 62
C6H 5CH3
C6H5NO2

C6H5NH2
C6H 5 C
C6H5Br
O-xylene
m-xylene
Cumene


Temp. Range
oK

250 343
298 358
318 358
298 358
310 327
310 344
273 344
223 298
273 444
273 473
403 553
303 473
298 473
323 473
333 408
323 523
333 408
323 573
373 573
223 298
298 358
298 358
298 358
298 358
298
298
298


# Data
Points

11
5
4
4
6
6
5
4
6
7
6
5
3
3
3
3
3
6
3
4
4
4
4
4
2
2
2


RPMS
Dev.
%

4.7
6.0
5.0
5.8
5.5
8.6
6.2
3.5
6.1
3.1
3.2
2.9
5.75
2.9
2.5
3.6
2.58
3.4
2.82
3.7
3.3
2.78
1.0
2.68
3.3
2.6
2.1


v
cc/g mole

276b
730
845
970
200b
255b
309
308b
369b
425b
489
541b
670
78-3b
915
1035
1100 b
1225
1880
312
321
285
306
321
363.5
362.5
298


Data
Reference

1
2
2
2
6
7
3
11
3
1
3
3
3
3
4
3
4
3
3
11
8
8
8
8
9
9
9


a
Also for use in partial molar
Within .5% of experimental or


volume correlation
predicted critical volume.













Substance


TABLE 4-2-a (Continued)
RMS
Temp. Range i Data Dev.
OK Points %


v
cc/g mole


Data
Reference


Mesitylene
Tetrahydronaph-
thalene
CH2C12
C2H OOCH3
CHC1
Cyclohexane
Isopropyl ether


Tetrahydro-
furan
Ethyl Ether
(CHCL2)2
(CC12)2
(CH2Cl)2
CHC1 CC12
(CHC1)2
CH3CHC12
CH3COCH3
PSU 25c
PSU 19c
PSU 174c
PSU 528c
PSU 532c
PSU 537c
PSU 175C
PSU 12c
PSU 574c
PSU 575c
PSU 122c
CS2


298

298
303
303
298


2 1.9


- 323
- 323


3.0
.06
7.5
.55
1.21


298 348
323


303 -
273 -
298
298
298
298
298
298
298
310 -
310 -
352 -
333 -
333 -
333 -
333 -
333 -
333 -
333 -
373 -
298 -


323
413


408
408
408
408
408
408
408
408
408
408
408
358


6.9
6.5


3.6
2.7
3.7
8.3
5.9
5.1
1.1
1.8
3.7
5.4
.75
2.0


423


430.2
168.8
283
219.6
311
415.8

237
281.2
286.7
303.6
226
258.9
208.6
233
200.2
1550
1465
1332
785
933
1116
1452
1368
800
873
1708
165


CComplex hydrocarbon. See reference 14 for formula.





53



TABLE 4-2-a (Continued)


Substance

Methanol
Propanol
NH3


Temp. Range
oK

273 323
298
253 313


# Data
Points

3
4
16


RMS
Dev.


2.7
2.5
S.8


v
cc/g mole

101.5
160.3
65.18


Data
Reference

9
9
17










TABLE 4-2-b
CHARACTERISTIC VOLUMES DETERMINED FROM PARTIAL
MOLAR VOLUMES AT INFINITE DILUTIONab


Substance

CO

CO2

H2
2H


- CF4

C2H4

CHZ
C2 2

(CH3)20

CH3 C

SO2


# Data Points

10

6

19

4

2

3

3

3

3

3


# Solvents

3

2

5

4

2

3

3

3

3

3


v cc/g mole

93.1c

80

51.5

158

139

127

112.6c

169.7

136.5

115


aAlso for use in compressibility correlation.

Sources of experimental data quoted in Table 4-3.

CAssigned to be critical volume.

dEffective critical volume. See reference 28.


__










data on a wide variety of solute (1) and solvent (2) pairs. In cases

where the total experimental pressure had not been reported, the

volume of the pure solvent was taken as that of the saturated liquid.

Hypothesizing a general dependence of the DCF integrals on the reduced

solvent density, the variation of C was found to be similar, but not
12
identical, for all solute-solvent pairs. A factor accounting for the

relative molecular sizes, as represented by the ratio of the

characteristic volumes, was introduced. The empirical relationship

obtained for C2 is shown in Fig. 4-3, with some representative systems.

The analytic expression for this relationship is



Zn -C2 --0 = -2.4467 + 2.12074 P2 ;2.0 < p 2 2.785
v
~o ~o2 ~o
= 3.02214 1.87085 p2 + .71995 2 ;2.785 < p < 3.2
2 2 2 8-


(4.6)

-o
Calculation of v1 for a solute-solvent pair of known characteristic

parameters thus requires only a knowledge of v2, the pure solvent

volume. C1 is obtained from (4.6), C22 from (4.3) and v from (4.5).
-o
p2 is the reduced solvent density and is used in both correlations.

The numerical coefficients in equation (4.6) were generated
-o
using vI for seven gases in different solvents. For Ar, N2, 02 and

CH4 the characteristic parameters had been determined earlier from

compressibility measurements. For CO and C2H2, the choice of the

critical volume as the characteristic parameter proved satisfactory;

similarly the effective critical volume was chosen for H2.

Equation (4.6) provides a satisfactory representation of






















60- S *d
SH2 C2H2 2
N2 o

SD CH4


(00
0 CT- 0



U S Solvents
H20, CH3OH, C6H6

10- / nCo, nC6, nC7, nCg

.C H5CI, CC14, Ar
6 -

2.2 2.6 3.0

Q/q(

Fig. 4-3. Generalized correlation for partial molar
volumes of gases at infinite dilution in
liquids.











experimental partial molar volumes. Since the experimental variable

in the correlation is the solvent density, which is a function of both

temperature and pressure, the variation of the volumetric properties with

pressure is partially accounted for by equation (4.6). Partial molar

volumesfor varied gas-solvent systems as calculated from equation (4.6)

are shown in Table 4-3, and compared with experimental data. Also shown
29
are volumes calculated from the generalized correlation of Lyckman,2

which is based on a cohesive energy density for the solvent. The

accuracy of the two correlations is similar for saturated systems at

low temperatures. However, since the correlation of equation (4.6)

is based on the reduced solvent density which is a function of both

temperature and pressure, it can account for the pressure effect on

volumetric properties to some extent, and for the rapid change in solvent

volume at temperatures approaching the critical. This is clearly seen

in the hydrogen-benzene system where a 30% change in partial molar

volume with pressure is well predicted, and the high temperature value is

correlated closely.

The characteristic parameters for several substances other than

the seven previously listed, were obtained by fitting experimental
-o
partial molar volumes to equation (4.6). In each case, v1 for the gas

in several different solvents was used.. They are shown in Table 4-2-b.


4.2 Generalized Isothermal Equation of State for Liquids


Macroscopic experimental evidence, equation (4.3),shows that
1
the function 1 +-- is a universal function of reduced density for all
prRT
liquids. Substituting for pK from equation (4.3) into equation (1.15)









TABLE 4-3


PARTIAL MOLAR VOLLUMES OF GASES AT INFINITE DILUTION

v cc/gm mole
Data
Solute Solvent T,OK s Calculated Experiment Lyckman Reference

CH4 nC7 300 2.86 56.6 55.4 58.2 17
CF4 CC14 300 2.845 69.1 79.7 66.3
CF4 nC7 300 2.86 88.7 86.4 73.7
H2 C H6 298 2.865 31.1 35.0 37.1 18
403 2.496 57.4 58.3 52.7
2.521 53.9 53.7
2.55 51.4 49.7
433 2.375 66.9 72.1 60.6
2.412 63.0 64.0
2.443 56.7 57.5
473 2.197 92.9 104.4 64.7
2.251 81.6 85.6
2.296 69.0 72.5
H2 nC8 403 2.646 68.4 63.6 59.8
2.683 62.8 58.0
2.716 57.55 53.3
473 2.355 104.6 110.3 67.9
2.422 86.2 87.6
2.474 77.1 72.7
CH4 nC6 298 2.776 59.1 60.2 62.5 19
C2H6 76.8 67.9 72.
C3H8 88.8 93 93.2
3 8
nC4 102.5 104.9 107.3
Ar 50.4 51.5 53.4
N2 55.8 62.8 59.1

02 49.7 55.7 52.7
H2 CC14 298 2.838 33.6 38 39.2 20
Ar 41.8 44 43.5
CH4 CC14 298 2.838 49.2 52.4 51.2 21
C2H6 64.7 65.9 67.1
N2 43.1 53.0 48.3
C2H4 57.1 60.8 61.7 22
C2H2 53.0 53.6 57.3
(CH3)20 67.9 66.7 63.2








TABLE 4-3 (Continued)


v1 cc/gn nole


Solute Solvent T,K


Calculated


Experiment


Data
Lvckman Reference


298 3.033








298 2.865


CH3C1

SO2
C2H6
C2H
C 2H2




SO2
C2 6
SC2 4
C2H2

(CH3)20
CH3Cl
SO2
CH
N2
CH4



CH



CH
CH4




H2
N2

CO2
CO
02


C6 H5C1








C66











nC10



nC8



nC7
nC6



CH30H


323
273
248
223
248
273
233
223
248
298
273


2.998
3.162
3.242
3.257
3.172
2.726
3.156
3.094
3.00
2.812
2.566


59.4
53.6
57.4
54.6
47.0
60.1
52.9
47.7
60.1
56.9
49.4
62.8
55.2
49.8
43.2
43.1
58.4
52.6
46.1
40.8
48.4
75.2
46.2
44.5
47.9
56.2
35.1
48.6
45.4
49.6
43.2


54
53.2
63.4
57.8
49.3
65.3
52.8
47
66.1
60.4
50.3
67.1
54.1
46.8
52.0
52.6
53.0
49.0
46.0
43.0
47.0
59.0
44.0
45.5
48.0
53.5
35.
47.
40.
47.
42.


58.8
61.4
63.3
58.0
53.8
73.0
63.2
57.6
64.7
59.3
54.9
74.3
64.6
58.9
48.8
45.9
59.1
-


52.6
58.0
87.6
55.2
54.1
59.8
73.3
28.2
25.3
39.5
37.2
31.7








TABLE 4-3 (Continued)


Solute Solvent T,OK p


CH4
H2
N2
CO2
CO
02
CH4
H2
N2
CO2
CO

02
CH4
H2
N2
CO2
CO

02
CH4
H2
N2
CO2
CO

02
CH4
H2
N2
CO2
CO

02
CH4


CH3OH








CH OH








H20








H20








H20


298 2.500








323 2.428








273 2.577








298 2.569








323 2.547


v cc/gm mole
Calculated Experiment Lyckman

51.6 47. 39.4
39.1 35. 30.4
54.0 52. 26.8
50.5 43. 41.8
55.1 51. 39.5
48.0 45. 34.0
57.3 52. 41.7
44.2 42. 32.9
60.7 54. 28.7
56.7 46. 44.5
61.9 56. 42.2
54.0 50. 36.6
64.4 55. 44.4
24.9 24. 21.2
34.8 41. 29.4
32.4 32. 32.2
35.5 37. 30.0
30.8 31. 24.6
37.0 36. 32.2
25.2 26. 21.8
35.2 40. 30.1
32.8 33. 32.9
36.0 36. 30.7
31.2 31. 25.3
37.4 37. 32.9
26.1 34. 22.7
36.4 38. 30.9
34.0 33. 33.8
37.2 32. 31.5
32.3 32. 26.1
38.7 38. 33.7


Data
Reference


24








24








24








24








24









TABLE 4-3 (Continued)


v- cc/gm mole


Solute Solvent T,K p


Calculated


Experiment


Data
Lyckman Reference


nC8H18 426
455
483
512
Ar 83.78

02 77
02 83.78
90
Cyclo- 298
hexane 310.9
310.9
344
NH3 273


2.501
2.374
2.218
2.029
2.671
2.765
2.689
2.617
2.858
2.819
2.697
2.448


109.3
138.4
189.9
292.0
40.7
30.1
32.0
35.7
37.0
52.3
65.2
48.8


99.6
128.8
175.0
267.3
35.4
35.6
28.7
29.5
53.0
57.0
65.0
43.6


80.5
88.2
94.9
99.6
35.6
31.1
31.0
34.2
53.3
55.5
62.7









P2 2
do r

0 O<
Pi 0l

P2
(P9 P ) r
-v = [f(o) l]dp (4.8)
HTf
P1
where


f(p) = exp[-.42704(p 1) + 2.019(p 1)2 .42367(p )3] (4.9)


Equation (4.8) relates the isothermal pressure change corresponding to

a density change, and holds over the same reduced density range as

(4.3). For calculational convenience, the definite integral is

re-expressed as

2 P1
r r
(P P ) [f(p) l]dp [f(p) lld (4.10)
2 1 RT J, L
Po Po

where p is an arbitrary base density. The integral from p to p

has been numerically evaluated, and is shown in Table 4-4. (The

integrals are tabulated at integrals of Ap = 0.1; they were evaluated

for Ap = 0.002.)

Equation (4.8) can be used to determine unknown densities or

pressures. Given Pl, i and p2, P2 is found easily from the difference

of the integrals of equation (4.10). Given PV, P2 and pl, one looks

for a reduced density p2 such that the corresponding integral of Table 4-4

satisfies equation (4.10).

The characteristic parameters used in the following calculations

were those obtained from the experimental compressibility data.

Considerable manipulation of the experimentally measured volumes is










TABLE 4-4

REDUCED INTEGRALS FOR ISOTHERMAL EQUATION OF STATE


S[f(p-l)]dp



Po


f [f(p-l)]dp


1.4


0



2.0


.5705



2.6


1.5


.0214



2.1


1.6


.0603



2.2


1.7


.1237



2.3


1.8


.2209



2.4


1.9


.3641



2.5


.8631 1.2739 1.8469 2.6423 3.7427


3.0


5.2596 7.3427 10.1895


14.0564 19.2680 26.2246


j [f(p-l)]dp



p
P


f [f(p-l)ldp
PO


3.2


3.4


35.4028 47.3454 62.6364 81.8564 105.5160










necessary before compressibilities are extracted, e.g., fitting to an

equation of state such as the Tait equation. Thus the use of these

characteristic parameters in the integrated relation to predict

volumes constitutes a severe test of both the accuracy of the

parameter and the validity of the generalized equation of state. Also,

the integral in Table 4-4 increases very rapidly at high reduced

densities.

The results of compression calculation:on liquids with equation

(4.8) are reported in Tables 4-5 and 4-6. In most cases, the initial

state corresponds to the saturated liquid. Then given either the volume

or pressure of the compressed state, the other is calculated. Where

Tait equation parameters were available, calculations were made using

this equation too. The Tait equation is a two-parameter equation

usually written in the form

B + P2
v v 1 C Bn B+ ~i (4.11)


The constant C generally has a value of .11 for all liquids, whereas

B varies with temperature.

Volumes calculated from equation (4.8) agree to within 1% for

all the substances considered, over the entire pressure and temperature

range. The Tait equation with one fixed constant C (as for NH3) gives

results which are slightly improved over those of equation (4.8).

However, when both Tait contents B and C are varied with temperature

and fitted to experimental data as for nC 0H22, the Tait equation gives

decidedly better results than the generalized one-parameter equation

.(4.8).





































<0 oI

C
0


u O
WiI ( U


v > -4 ---
(3













0.
> -u



4
-0 1 (U


(U











-1
0

0





0

r-1 A


























U

03
cO


ul r--( 3--
* ** V
OON il0
CT000


*0








0000
0* *
n I rI 1
C C) 0




0000






I* 1" 1* 1"


r^. r C oo Co
o0 oo 0 oo
0000


0000



I I





" *,

--1 1-4- r-


-1 0 0 0

I + r
+


om
NcJ


Cm
("C0,


MO l
0 in

O0 C4
IT -


r 3- -C)o
0000



nO -To



00r 0
> C3 3
.... c
mclN

0440 o


CCC -4*1 0O l 0 -
- CNN in 3- -


\00

* *






m-


r-r0
Nh
(NJ(N


a cn










00
om-
Hnr-


0 C 0

So-T

i-4 CN CM4 CM4


()-4
NMo

coo
S00
0 0


('40


0 0 .n
oo \oin
C* 0 *


C, oNJ
0-1 CM4

Coi
r"rU


CLO
nO

-4r-
* 4c


04 00

-T0 00
IAe
hQ)H


00ch C









00000
00000

-1 -1 r-


0000 00000

....f.....


tn in in
-4 -4 -4


00J-1
C14 m


In n Ln in

o-4 0--1 G r-I
fl, a% r rl

r~N C14 CM f 17r^
ClMCM0en-


ir- CM rO-



04 CM0CM


000m
04 03
coHm







(-4 o0 CMN
-1 00 04






NOcl
0 a;

. .1 1
CI I(


0000
0000




0 0 o



S-I r-4
NC O" (









Pressure calculations are shown in Table 4-6. The errors of

calculation of equation (4.8) and the Tait equation are of the same order

of magnitude, but generally the Tait equation gives better agreement

with the experimental pressure.

Where extensive volumetric data for a specific liquid are

available, it is preferable to extract the Tait constants and use

equation (4.11) for compression calculations. Equation (4.8), however,

has the appeal both of generality and a firm basis in molecular theory.

Its dependence on a single characteristic parameter reinforces the

simplicity of use. Where there is limited experimental data, the use

of equation (4.8) will give results of adequate accuracy for compression

calculations. Further, it is equally possible, although it remains

uninvestigated, that a characteristic parameter extracted directly

from volumetric data, i.e., equation (4.8) rather than from

compressibilities, will yield results of accuracy equivalent to those

of a particular equation of state such as the Tait or Huddleston

equations.


4.3 Temperature and Pressure Dependence of DCF Integrals

o 0
The corresponding states correlations for C22 and C12 describe-
22 12
their variation with the reduced solvent density. The solvent density,

however, is a unique function of the system temperature and pressure.

Knowledge of the temperature and pressure behavior of the DCF integrals

directly yields the state dependence of the related thermodynamic

properties. We have, in the 1-2 binary,

1 o 412)
1 + C = F (p) (4.12)
PKRT 22 12






67























o r



C r-,i Cn
'0
*C *








I

o l P4 T o o
CM














Sr --4 -- o
o- in r~


en















0 00oo -0T
Sen r- 00o a%
:3 ca in -3T Cn CA





























S tO
cn E C) oooo
cO (^) 41 X C 0000




P r-1 o -1 r-1 v -1
r Ho n o C:o





0O rN r-

Ccn 4 E Ln 0



c0i
0 M C O CC

-i CM CA ID r, o

r- E no C, cMm
?> 0r a .

04 CA CA





Eo enoom






0)




c/) z




68




Taking a derivative w.r.t temperature at constant pressure


C
22
3T


o
2
3T


(4.13)




(4.34)


0 0 1 /
=- 2 RT /(


ap
aP


0C
22
3T


dF
S /[F 1]
dp2J


(4.16)


Now, writing the isothermal equation of state, equation (4.8),with
S
P2' the saturation pressure of the solvent as the base pressure,


P P
RTp2


[F1(P2) l]dp2


(4.17)


P2
=j [F(l2) l]d -


[FI 2) 1]dp
2o
2


F 6(2) F6 (2)


3F6
3T


P P
2 *
RT p2


P2


F [F6( P)]
T 6 2 p2


Now


(4.15)


then


1
RTp2


aT
\J1


(4.18)



(4.19)


s
dPT
dT


(4.20)


3FI

2


BTe









s
s s dF p )
S" T o dT 2 dT
RTp o RT P2
2 p

LP = RTp 2 dT + d (4.22)

p2

Substituting (4.34) in (4.29)



= /[F(p) ] RTp -
L T do R22 RT2 dT dT
dp RT1 1 2

(4.23)

Then, to determine the temperature derivative of C22, we need

(i) the compressibility correlation, equation (4.3)

(ii) an equation (relation)for the saturation pressure for

the solvent as a function of temperature; we shall

use one of the generalized equations suggested by

Reid and Sherwood, e.g., Antoine equation, or Miller-

Thodos equation.

(iii) an equation for the density of saturated liquid; here

we use the generalized Rackett equation.32

(iv) the generalized isothermal equation of state, equation

(4.8).

Differentiating (4.12) w.r.t. pressure


3C22 dF( 2 2
I- d= 2 I (4.24)
T d2 T

dF (P2)1 T o
dP2 RTAF(P -1] (4,25)









Similar expressions can be obtained for the state dependence of C12
12


3 .62
0 v2
- C12 -
1


= F2(P2)


(4.26)




(4.27)


C12 dF2 v 62
T dp RT d2) 1] T
P 2 2 P2


o 62
12 T2 1
and =P T d -
v2


- 1
RV (F 2) 1]


(4.28)


We can now write the pressure and temperature variation of the isothermal

compressibility and partial molar volumes using equations (4.23) through

(4.28).


K2
o
(1 C 2)

S- 22

o
22

-o
v

(1 Co)
22


2 2
T P2


P2 2 P


(4.29)


C22 2 ;2
jP T 2 2 p T


!C22
3P T


1
P (l C22)
222


(4.30)


o -o
312 T 1 2
T p0
2 T


1
o o
P (1 C2)
2 22


o0
C12
T


(4.31)
-o
v S P21

P 0 T P
2


(4.32)


The quantity


I p2
T


dF(P )
is obtained from dT in equation (4.23).
dT


The details of this derivation are in Appendix D.


[ <2 1
JT



3P T
T


-o


-o
I 1T J 1


-o0
v

(1 C2 )
22


C 0
22
9T




71




Calculations for general gas-solvent systems are shown in

Table 4-7. These are order-of-magnitude estimates of the pressure and

temperature derivatives of the compressibility and partial molar

volume. Ultimate accuracy of prediction depends on close correlation
o O
of C2 and C .
22 12













0D o


I > I o 0 .







H c0 -i r' C

I 00 O-0 0 0
o iI . .

> 0 I I I I I I I

'--0- U
U


X4-
--4

w
0

U)
u
--4




0




H

0


I -
1-4












*Pj


0 -4 %0


o c
v






0 *-1


(-I
0. Ln







E1
C 4 I




,-.0N




S00
,' 00

0 v
















0 '0







c




1 0T







Hco
r- 4> M -
01 %-
s-^


n( ~3-


\O r-









r-4 CM4

I IC




0 0
I I


























0 0
C o C
o- o
* **


I I























O o







00
00
000



































C.) Ca)
i-< 1-

(-> (-
OI u


'0 rN. 'h0 '0

,C- 1 C4 4unt C4
%0nco cl) o IT





' O N 03










CM CM 0 0
o 0 0 0
* *0% c






0H-0 000
000 000

Ill III




u- m r-I r0 00 r%
mn r- m \ o\N a

CIM CN CM




0% 0% CO CO
* *
0 0 00 0







%0 0








co co

co co s


33 Z X










REFERENCES FOR CHAPTER 4


1. J. S. Rowlinson, Liquids and Liquid Mixtures, Plenum Press, New York,
1969.

2. P. S. Snyder and J. Winnick, "The Pressure, Volume and Temperature
Properties of Liquid n-Alkanes at Elevated Pressures," 5th Symposium
on Thermophysical Properties, ASME (1970).

3. A. K. Doolittle and B. Doolittle, AJChE.J., 6, 157 (1960).

4. W. G. Cutler et al., J. Chem. Phys., 29, 727 (1959).

5. R. C. Reid and T. K. Sherwood, The Properties of Gases and Liquids,
McGraw-Hill, New York (1966).

6. B. H. Sage et al., I.E.C., 26, 1218 (1934).

7. W. B. Kay, I.E.C., 32, 358 (1940).

8. R. E. Gibson and 0. H. Loeffler, J. Am. Chem. Soc., 61, 2877 (1939).

9. P. W. Bridgman, Proc. Am. Acad. Arts & Sci., 69, 389 (1934).

10. L. G. Schornack and C. A. Eckert, J. Phys. Chem., 74, 3014 (1970).

11. F. I. Mopsik, J. Chem. Phys., 50, 2559 (1969).

12. G. A. Holder and E. Whalley, Trans. Far. Soc., 58, 2095 (1962).

13. D. W. Newitt and K. E. Weale, J. Chem. Soc., London, 3092 (1951).

14. D. A. Lowitz et al., J. Chem. Phys., 74, 3014 (1970).

15. J. S. Rowlinson, Liquids and Liquid Mixtures, Butterworths,
London (1958).

16. K. Kumagai and T. Toriumi, J. Chem. Engr. Data, 16, 293 (1971).

17. R. H. Schum and 0. L. I. Brown, J. Am. Chem. Soc., 75, 2520
(1953).

18. J. F. Connolly and G. A. Kandalic, C. E. P. Symposium Series, 59,
8 (1963).

19. Y. Ng and J. Walkley, J. Phys. Chem., 73, 2274 (1969).

20. J. E. Jolly and J. H. Hildebrand, J. Am. Chem. Soc., 80, 1050
(1958).

21. J. C. Gjalbaek and J. H. Hildebrand, J. Am. Chem. Soc., 72, 1072
(1950).










22. J. Horiuti, Scientific Papers of Institute of Physical and Chemical
Research, 17, 125 (1931).

23. J. P. Kohn, "Volumetric and Phase Equilibria of Methane-Hydrocarbon
Binary Systems," A.I.Ch.E. Meeting, Houston (1967).

24. I. Kritchevsky and A. Illinskaya, Acta Physicochemica U.R.S.S.,
20, 327 (1945).

25. S. D. Chang and B. C. Lu, Can. J. Chem. Engr., 48, 261 (1970).

26. V. Berry and B. H. Sage, J. Chem. Engr. Data, 4, 204 (1959).

27. M. Orentlicher and J. M. Prausnitz, Can. J. of Chem., 45, 595 (1967).

28. J. M. Prausnitz, Molecular Thermodynamics of Fluid Phase Equilibria,
Prentice-Hall, Englewood Cliffs (1969).

29. E. C. Lyckman et al., Chem. Engr. Sci., 20, 685, 703 (1965).

30. A. V. Itterbeek et al., Physica, 29, 742 (1963).

31. E. H. Amagat, Ann. de Chim et Phys., 29, 68-136, 505-574 (1893).

32. C. F. Spencer and R. P. Danner, J. Chem. Engr. Data, 17, 236
(1972).










CHAPTER 5

SOLUTION THEORY FOR SUBCRITICAL SYSTEMS


In previous sections we discussed the pressure and temperature

dependence of thermodynamic properties, and of the corresponding DCF

integrals for systems of constant composition, either pure solvent

or a solute at infinite dilution. Here, we consider the prediction

of thermodynamic properties in subcritical systems of variable

composition. The temperature or pressure of the systems may be fixed

or variable in accordance with the Gibbs phase rule.

The corresponding states correlations for the DCF integrals

derived earlier for limiting compositions can be extended to mixtures

by the concept of a pseudo pure fluid, analogous to a one-fluid

macroscopic theory. Here the mixture properties are represented by

those of a pure fluid whose critical parameters depend on the

composition of the mixture. This rule is similar in concept to Kay's

rule of pseudo critical, and Prausnitz and Chueh's2 mixing rule.

The mixing rules for the critical parameters are necessarily empirical,

unless they are based on a rigorous molecular theory. This approach

will be used to study the compressibilities of binary and ternary

liquid mixtures.

Finally, when faced with a lack of experimental data in the mid-

composition range of a solution, one can postulate (on the basis of

empirical knowledge) rules for the composition dependence of the DCF

integrals. Two of the simple rules are the linear arithmetic mean and

the linear harmonic mean. Expressions for the thermodynamic properties

with the linear rule have been presented earlier.










5.1 One-Fluid Theory

Leland and Chappelear1 discuss several methods of applying

corresponding states theory to mixtures. The microscopic principle

rests on the specification of the parameters of a pure fluid pair

potential function which generates the properties of the mixture at

the same temperature, pressure and density. On the macroscopic level,

the problem becomes one of defining a relation between the critical

properties of the pure components and the pseudocritical properties

of the hypothetical pure fluid. One of the simplest examples of these

is Kay's rule, and an equally simple rule is used here.

It is postulated that the properties of the fluid mixture

correspond to those of a pseudo pure fluid, whose characteristic

parameters depend on the composition of the mixture. The pseudo pure

fluid follows macroscopic states and its thermodynamic properties are

described by the correlations derived earlier.

The macroscopic correlation for compressibility, equation (4.3),

contains only one characteristic parameter for each substance, the

characteristic volume. For the mixture, this is defined to be the

simple mole fraction average of the pure component volumes

*
v = x.v. (5.1)
mix 1 1

Then, for the mixture


1 + 1I = F(Pi) (5.2)
KpRT mixture


where F(p ) is the same generalized function of p as equation (4.3).
mix











Equation (5.2), with the rule of (5.1), has been used to

determine the isothermal compressibilities of the binary mixtures of

benzene-cyclohexane, methyl acetate-water, nitrobenzene-aniline, and

of the ternary mixture n-hexane, n-heptane, n-octane. The results are

shown in Tables5-1 and 5-2. In all systems, there is good agreement

between the calculated and experimental compressibilities over the

entire composition range. This clearly demonstrates the efficacy of

the approximations and the composition mixing rules of (5.2) and (5.1).

In the binaries of the hydrocarbons, and of nitrobenzene-aniline, there

is a small, but consistent deviation between the experimental and

calculated compressibilities. However, this probably results from an

incorrect prediction of the pure compressibilities by equation (4.3)

and should not be immediately attributed to a failure of the mixing

rule. A possible remedy is to adjust the characteristic parameters of

the pure components such that equation (4.3) predicts their experimental

compressibilities correctly. The results of such calculations are

shown in Table 5-3, and reveal marked improvements in the benzene-

cyclohexane and aniline-nitrobenzene systems. This stratage has a

mixed effect in the water-methyl acetate system, which is a possible

reflection of the complex interactions in this mixture.

The volume of the binary mixture is required in all of the

above calculations and has been assumed an experimentally accessible

quantity. This may not always be the case and the simplest approximation

under these conditions is one of ideal mixing. The effects of this

assumption on the predictions of the one-fluid theory are shown in

Figs. 5-4 and 5-5.










































Co c \ooc M r
0 co) r, %0 \0 Ui


1-



o ci
r-v

C) 0


m -3-T



0CM nf in D
I -T. 3'r T
c'-IJo3~\

U,\O;~


H O







H






O




HO
rk-40
X0 ;D







>
> T-1
f^


0,0




r-ir-I
inin
'-" --l


in %.

* T

'-4-i
0o ,


H-i


U')
a


* 0






oul
oM
AM


o- -I r,-

-4 o 0

i-l


r- 100


'-I-00
* *





OmN

*4 *
0M

00 r-I



co oi oT -IT
nn 0r
a'Ou3Qu
CI\0(Nl


-(NJ
'-l4 C



< C







00"
* "


ai0
0o0
;00

T V
-4 1-4


Mr- .-4 0










00004
0 U) r~- C

(Ni (Ni C4 (NI


'.00


o r 00 o i -0
ST . -T
o ~J~
IAJIJtlf






Q3QJlDt if


-4-4
cWc


r-- CM'-T

-4c --
* *

OO.


0 o0 r,%.o m



C14 a%\o CM o
0-mj-ClC
e0a''a

Cllo iNi.


' -4'-4
onr


aNa%
*- *q
ooi


0 r-



CM1N
a'a'y


mcomo
we me)
****


'HH
X-i 40
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> m
a) U







































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r-I00-

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00




-4





00






00
00 -I


O r-I in
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o






000


CM CM C>


00 0

o\ CTh


0o


r-1 r-C %-I


ON
00







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CM

r-10
r-1 r-


CO C
C>N c




000










r-I
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CM 0 r M
0H <7 T
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On 0 in

O r(




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C0

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x







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> L


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a%

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r.-.






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ro *<


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r-i -. .- 4










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Nc CN










,7% CM

a3 -T
r-i U-)


P:


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U.,
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P-l
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nen n
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tI) 00 -3
* H H
r-l-l ll


0 0











olm




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.- ,.

'.0
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H- -

co oo

CM Cl









m cm










r1 r-

0 un


C*4


Wl Co Cl C
r4-- c e
C cn cn


,T H-- 00






r-I -T cn
\ r-l Cl>
IJl 3-en


+

.1







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co
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0)

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E-4 eNCl










































w






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O
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n
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U,


Ca4


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1-4










-41
xe







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a 6
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:u'E
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0


Uu


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c S3 o
r- ,-)m


Q 00

NNNN


ci
** r-

i-\o


unl in
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000 r


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r- .,T ooC


0 N %
1-4


04i 04 a\










e) 00 a\aC
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r(NNN


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0 0 cl


CM i 1-
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mei
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co co a)
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Uc o r -




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0-i C M



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-T r~, c0
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CscC0
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Oc







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S^hl -









5.2 Rules for Composition Dependence of DCF Integrals

The thermodynamic properties of a binary system are completely

defined by the three integrals C11, C12 and C22 though the expressions


1 2 2
pR 1- 1C 2C22 2X2C12 (5.3)
pKRT 1 11 2C22 1212



(1 xC1 x2C2) (5.4)
RKRT 1 11 2 12

and
any 1 + C 2x C 2x C + x 2(C C C C
1 __ 11 1 11 2 12 + (C12C12 C11C22)
axJ x2 2 2 2 xC
T,P 1 xlC11 2C22 2x C12

(5.5)


The quantities C11, C12 and C22 vary with the composition of the

mixture. At any composition, they can be determined only through a

simultaneous knowledge of all three thermodynamic properties.

However, experimental data are often available only at the

composition extremes, and the variation of the Cj over the composition

range must be hypothesized to calculate mixture properties at other

compositions. Equations describing this composition dependence may be

proposed and the calculated thermodynamic properties compared with

experimental data. Good agreement for a wide range of systems is

evidence for the accuracy of the composition rules which can then be

(confidently) applied to other systems. The two composition rules

studied are the arithmetic mean (5.6) and the reciprocal mean (5.7)

o *
Cij = C.. + Xl(Ci Ci ) (5.6)
ij ii 1 ij ij









x 1 x
1 Xl 1- x1
+ (5.7)
C o
ij Cij C
i 1ij

0
The superscripts refer to quantities at x2 = 0, while refers to

quantities at x = 1. The above relations are then substitution

into equations (5.3), (5.4) and (5.5) to yield expressions for thermo-

dynamic properties of the mixture in terms of DCF integrals at the

limiting compositions, which in turn are calculated as follows


1 o
= C (5.8)
oo 11
p 1KRT


-2 1
-0 0
V2 1- C12
o o



ny (1 C )
2 o 12
TP (1 C ) (5.10)
ax 22 o
2 2 (1 C )
IT,P 11

where the symbols have their usual meaning. Similar expressions
*
relate C11, C12 and C22 to properties at the compositions x2 = 1.

The benzene-cyclohexane and aniline-nitrobenzene systems were

studied initially. Complete thermodynamic data are available for these

systems.3 '4,9,5,6,0 The composition dependent C1, C12 and C22 in

the benzene-cyclohexane system were determined by simultaneous solution

of equations (5.3), (5.4) and (5.5) and are shown in Fig. 5-1. Although

this binary system is not very nonideal, the experimental C. 's show a
ij
quadratic dependence on composition. This is consistent with the

quadratic and higher order composition dependence of the thermodynamic

properties.




84



1I I



-55 -










-45

C22


C..j




-35 012





C11




0.2 0.8
x1

Fig. 5-1. Experimental DCF integrals in benzene (1) -
cyclohexane (2) system at 2980K.




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