• TABLE OF CONTENTS
HIDE
 Title Page
 Acknowledgement
 List of Tables
 List of Figures
 Abstract
 Introduction
 Mixture rules for the MIE (n, 6)...
 A mixture rule for the exponential-6...
 The relationship between the Mie...
 Saturated liquid properties from...
 Excess properties of the methane-perfluoromethane...
 Corresponding states for fluid...
 Estimation of excess properties...
 Appendices
 References
 Biographical sketch






Title: Intermolecular pair potentials in the theoretical description of fluids and fluid mixtures
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Permanent Link: http://ufdc.ufl.edu/UF00097604/00001
 Material Information
Title: Intermolecular pair potentials in the theoretical description of fluids and fluid mixtures
Physical Description: xiii, 143 leaves. : illus. ; 28 cm.
Language: English
Creator: Calvin, Donald William, 1945-
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 1972
Copyright Date: 1972
 Subjects
Subject: Fluid mechanics   ( lcsh )
Statistical mechanics   ( 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 140-142.
Additional Physical Form: Also available on World Wide Web
General Note: Typescript.
General Note: Vita.
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Bibliographic ID: UF00097604
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000577252
oclc - 13943639
notis - ADA4947

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Table of Contents
    Title Page
        Page i
    Acknowledgement
        Page ii
        Page iii
        Page iv
        Page v
        Page vi
    List of Tables
        Page vii
        Page viii
        Page ix
    List of Figures
        Page x
    Abstract
        Page xi
        Page xii
        Page xiii
    Introduction
        Page 1
        Page 2
        Page 3
    Mixture rules for the MIE (n, 6) intermolecular pair potential and the dymond-alder pair potential
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
    A mixture rule for the exponential-6 potential
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
    The relationship between the Mie (n, 6) potential and exponential-6 potential
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
    Saturated liquid properties from the Mie (n, 6) potential
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
    Excess properties of the methane-perfluoromethane system from the one-fluid Van der Waals prescription in perturbation theory
        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
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
    Corresponding states for fluid mixtures -- new prescriptions
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
        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
    Estimation of excess properties for various systems using the total geometric mean rule in the gas phase
        Page 96
        Page 97
        Page 98
        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
    Appendices
        Page 117
        Page 118
        Page 119
        Page 120
        Page 121
        Page 122
        Page 123
        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
    References
        Page 140
        Page 141
        Page 142
    Biographical sketch
        Page 143
        Page 144
        Page 145
Full Text















Intermolecular Pair Potentials
in the Theoretical Description of Fluids
and Fluid Mixtures











By


DONALD WILLIAM CALVIN


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA III PARTIAL
FULFILLMENT OF THE REQUIREMENTS OF THE DEGREE OF
DOCTOR OF PHILOSOPHY









UNIVERSITY OF FLORIDA


1972









ACKNOWLEDGMENTS


The author wishes to express his sincere appreciation to

Dr. T. M. Reed III, Chairman of his Supervisory Committee for directing

this research. He wishes to thank the other members of his Supervisory

Committee, Dr. J. U. Dufty, Dr. J. P. O'Connell and Dr. U. W. Menke,

for their cooperation in serving on the committee.

The author gratefully acknowledges the financial assistance of

the National Science Foundation and the Department of Chemical

Engineering, University of Florida. He is grateful to Professor Donald

Vives for supplying the subroutine BrEN for evaluating the second

virial coefficient for the Mie (n,6) potential, to Dr. A. U. Westerberg

for supplying the subroutine RMINSQ, a nonlinear least squares routine,

and to Dr. K. Rajagopal for supplying the subroutine PYCX for evaluating

the Percus-Yevick hard-sphere radial distribution function.

The author also wishes to thank the University of Florida

Computing Center as well as the Dow Chemical Company, Louisiana

Division for the use of their computing facilities. Special thanks are

extended to Mr. C. U. Calvin, Mr. C. E. Jones and Mr. C. A. Smith of

Dow, Louisiana, for their assistance with part of the computer work.

The author wishes to thank his fellow graduate students for numerous

helpful suggestions and wishes them the best of luck in their future

endeavors.

Finally, the author extends his thanks to his wife, Barbara,

and daughter, Sandy, whose active support made this work both possible

and worthwhile.










TABLE OF CONTENTS


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

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

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

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

CHAPTERS:

1. INTRODUCTION .......................................

2. MIXTURE RULES FOR THE MIE (n,6) INTERMOLECULAR
PAIR POTENTIAL AND THE DYMOND-ALDER PAIR POTENTIAL.

Introduction ..................................

Unlike-Pair Potential .........................

Semitheoretical Mixture Rules .................

Empirical Mixture Rules .......................

The Dymond-Alder Potential ....................

Conclusions ...................................

3. A MIXTURE RULE FOR THE EXPONENTIAL-6 POTENTIAL.....

Introduction ..................................

Mixture Rules .................................

Conclusions ...................................

4. THE RELATIONSHIP BETWEEN THE MIE (n,6) POTENTIAL
AND EXPONENTIfAL-6 POTENTIAL ........................

Introduction ..................................

Equivalence of Potential Parameters...........

Unlike-Pair Parameters........................

Conclusions ...................................


Page

ii

vii

x

xi



1


4

4

5

6

8

15

15

19

19

19

21


24

24

25

27

30










TABLE OF CONTENTS (Continued)


Page

5. SATURATED LIQUID PROPERTIES FROM THE MIE (n,6)
POTENTIAL.......................................... 32

Introduction.................................. 32

Barker-Henderson Perturbation Theory.......... 33

Liquid Properties from Best Virial Coefficient
(n,6) Potential............................... 35

(n,6) Potentials for Liquids.................. 41

Conclusions................................... 43

6. EXCESS PROPERTIES OF THE METHANE-PERFLUOROMETHANE
SYSTEM FROM THE ONE-FLUID VAN DER WAALS PRESCRIP-
TION IN PERTURBATION THEORY......................... 44

Introduction................................... 44

One-Fluid Perturbation Theory of Mixtures ..... 44

The Methane-Perfluoromethane System........... 47

Potential Parameters Independent of Choice
of Reference Fluid ............................ 58

Averaged Excess Properties .................... 61

Conclusions................................... 64

7. CORRESPONDING STATES FOR FLUID MIXTURES--NEW
PRESCRIPTIONS...................................... 66

Introduction.................................. 66

The Boyle Prescription (vcB).................. 67

Relation of the vcB Prescription to the vdW
Prescription.................................. 69

The vcB Prescription for (12,6) Systems....... 71

The vcB Prescription for Mixtures of
Molecules with Different (n,6) Potentials..... 72

Mole-Fraction Averaged Excess Properties. 82









TABLE OF CONTENTS (Continued)


Page

Three-Parameter One-Fluid Theory ......... 84

The Virial Coefficient Least Squares (vcls)
Prescript-ion................................... 90

Conclusions.................................... 95

8. ESTIMATION OF EXCESS PROPERTIES FOR VARIOUS
SYSTEMS USING THE TOTAL GEOMETRIC MEAN RULE IN THE
GAS PHASE........................................... 96

Introduction................................... 96

Selection of Gas Phase (n,6) Potentials ....... 97

Mixtures of Molecules with Very Different
(n,6) Potentials.............................. 109

Conclusions ................................... 110

9. CONCLUSIONS........................................ 113

APPEINDICES: ................................................. 117

A. DETERMINATION OF (n,6) POTENTIALS FROM THE SECOND
VIRIAL COEFFICIENT....................... ............ 118

B. RESIDUAL THERMIODYNAMIC PROPERTIES................... 120

C. CALCULATION OF EXCESS FREE ENERGY................... 122

D. EXPERIMENTAL PROPERTIES OF THE CH, + CF4 SYSTEM!.... 123

E. A NEW APPROACH TO THE REFERENCE STATE FOR LIQUID
TRANSPORT PROPERTIES............................... 125

Introduction.............. ..................... 125

Two Current Theories.......................... 126

Hole Theory of the Liquid..................... 127

The Real Liquid............................... 129

Verification of the Proposed Reference State.. 133

The Glass Transition.......................... 137










TABLE OF CONTENTS (Continued)

Page

Conclusions.................................. .. 138

LIST OF REFERENCES.......................................... 140

BIOGRAPHICAL SKETCH......................................... 143









LIST OF TABLES


Table Page

1 Pure component parameters .......................... 9

2 Unlike potential parameters........................ 11

3 Cross virial coefficient 12 with the (n,6) Mie
potential.......................................... 12

s4 Cross virial coefficient B 2 with the Dymond and
Alder potential .................................... 16

5 Unlike-pair parameters............................. 22

6 Cross virial coefficient B_2....................... 23

7 Second virial coefficients of pure gases predicted
with (n,6) potential using exponencial-6 parameters 26

8 Exponential-6 potential parameters................. 28

9 Cross-term second virial coefficient............... 29

10 Comparison between Monte Carlo calculations and
perturbation theory. ............................... 36

11 (n,6) Potential energy parameters from second
virial coefficients................................ 37

12 Saturated liquid properties ........................ 38

13 Comparison of one-fluid van der Waals model with
Monte Carlo and multicomponent perturbation theory
calculations ....................................... 46

14 Gas phase potential parameters ..................... 50

15 Predicted and experimental potential parameter
ratios ............................................. 51

16 Excess properties of the CH, + CF, mixture at
1110K, P = 0, xI = x2 = 0.5, with (12,6)
potential.......................................... 53

17 Excess properties of the CH, + CF, mixture at
1110K, P = 0, x1 = x2 = 0.5 with various potentials
and reference liquids.............................. 55









LIST OF TABLES (Continued)


Table Page

18 Calculated properties of liquids at 111K, P = 0... 57

19 Potential parameters and excess properties
independent of reference fluid ..................... 60

20 Averaged excess properties (parameters independent
of reference fluid) ................................ 63

21 Comparison of one-fluid and two-fluid prescrip-
tions with Monte Carlo calculations................ 73

22 Comparison of excess free energy (G E) from vcB
and vdW prescriptions with Monte Carlo (MC)
calculations ....................................... 74

23 Comparison of excess enthalpy (H E) from vcB and
vdW prescriptions with Monte Carlo (MC)
calculations ....................................... 76

24 Comparison of excess volume (V E) from vcB and vdW
prescriptions with Monte Carlo (MC) calculations... 78

25 Comparison of the one-fluid vcB and vdW prescrip-
tions for equimolar mixtures of (12,6) gases ....... 80

26 Liquid phase potentials for CH, and CF,............ 83

27 Mixture properties and excess properties for the
CH, + CF, system with the (13.2,6) and (34.2,6)
potentials ......................................... 85

28 Averaged excess properties from vcB prescription... 86

29 Calculated mixture and excess properties for the
CH, + CF, system with the three-parameter vcB
prescription....................................... 91

30 Comparison of the one-fluid vcls prescription
and Monte Carlo calculations....................... 94

31 Gas phase values of n for various molecules found
from liquid mixtures with CH ....................... 100

32 Gas phase values of n for various molecules found
from liquid mixtures not containing CH ............. 101


viii









LIST OF TABLES (Continued)

Table Page

33 Gas phase potential parameters for various
molecules .......................................... 103

34 Like-pair and unlike-pair (12,6) potential
parameters estimated from gas phase (n,6)
potentials ......................................... 104

35 Estimated excess properties using (12,6)
parameters from Table 34........................... 106

36 Like-pair and unlike-pair (12,6) potential
parameters estimated from gas phase (n,6)
potentials......................................... 111

37 Estimated excess properties using estimated (12,6)
parameters from Table 36 ........................... 112

38 Comparison of predicted and experimental
viscosities........................................ 134

39 Best-fit parameters for equation (E-1)............. 136











LIST OF FIGURES


Figure Page

4n
1 Residual properties of liquid CF ................. 42

2 Excess free energy of the CH4 + CF4 system at
1110K, P = 0 .... .......... ..... ................... 87

3 Molar volume of the CH + CF, system at 1110K,
P = 0 ................. .... ... .................. 92








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


INTERMOLECULAR PAIR POTENTIALS
IN THE THEORETICAL DESCRIPTION OF FLUIDS
AND FLUID MIXTURES


By


Donald William Calvin

March, 1972

Chairman: Dr. T. M. Reed, III
Major Department: Chemical Engineering


In order to use modern molecular theories of fluids or solids

one requires information about the intermolecular pair potential. The

goal of the present work has been to provide such information for use

in the prediction of properties of fluids and fluid mixtures. General

features of empirical algebraic expressions for the intermolecular

pair potential in fluids have been investigated. Both the like-pair

potential energy (mutual energy of a pair of molecules of the same

species) and the unlike-pair potential energy (mutual energy of a pair

of molecules of different species) have been studied. The pairwise

additivity approximation for configurational energy is assumed through-

out this work.

Formulae for the unlike-pair parameters in terms of the like-

pair parameters for the Mie (n,6) potential energy model are examined

for their abilities to predict cross-term second virial coefficients.

The London dispersion formula and an assumption of geometric mean

repulsion energies are shown to apply only for cases wherein the









repulsion exponent n is not very different for the pairs of molecules.

A geometric mean rule for each one of the three parameters is shown

to have far more general applicability and high accuracy in predicting

the cross-term second virial coefficient. The geometric mean rule for

the energy and distance parameters is also shown to perform well for

predicting the cross-term second virial coefficient for molecules

described by the Dymond and Alder potential energy function.

The set of geometric mean rules for all Mie (n,6) parameters

is called the total geometric mean rule. These rules may also be

adapted to predict cross-term second virial coefficients for the

exponential-6 potential. Relationships developed between parameters

for the Mie (n,6) potential and the exponential-6 potential show that

the three parameters in the latter (c ,r m,) can reasonably be equated

to three parameters (c ,rm,n) in the (n,6) potential. Furthermore, the

implied equivalence of these potential parameters suggests that the

unlike-pair parameters for the exponential-6 potential should follow

the total geometric mean rule. This set of rules predicts good values

for the cross-term second virial coefficients in six systems tested.

The perturbation theory of Barker and Henderson has been used

to test various Mie (n,6) intermolecular pair potentials for their

ability to predict liquid properties. It is shown that it is possible

to obtain a considerable improvement in predicted liquid properties

when n is allowed to vary from the value 12 usually assigned to this

parameter.

Two new prescriptions are developed for calculating mixture

potential energy parameters for use with the one-fluid or two-fluid










theories of mixtures. The van der Waals prescription is shown to be a

special case of one of these new prescriptions. It is further shown

that the total geometric mean rule found to work for the Mie (n,6)

potential in the gas phase can be used indirectly to predict the excess

properties of liquid mixtures in the one-fluid theory. The method

used obviates the empirical determination of the unlike-pair potential

energy parameters. A method is developed for obtaining both like-

pair and unlike-pair potential parameters which are independent of the

choice of reference fluids. The necessity is demonstrated for accounting

for the particular (n,6) potentials required for each molecule in liquid

mixture calculations, and two successful methods are proposed for doing

SO.


xiii









CHAPTER 1

INTRODUCTION


Statistical mechanical theories of fluids relate experimentally

observed thermodynamic properties to the potential energy between

pairs of molecules. These theories have reached a level of develop-

ment such that further refinements in their ability of these theories to

predict thermodynamic properties of fluids and fluid mixtures may

result mainly from the use of improved models for the intermolecular

pair potentials.

The present work is intended to demonstrate the benefits

derived from using a different pair potential characteristic of each

molecular species in accurate theories of fluids and fluid mixtures.

General features of empirical algebraic expressions for the inter-

molecular pair potential have been investigated. The first part of

this work (Chapters 2, 3 and 4) is concerned with models for accurate

intermolecular pair potentials in pure and mixed gases. The models

studied most extensively are the Mie (n,6) potentials. Those dealt

with less extensively are the exponeritial-6 potential and the Dymond-

Alder potential. Like-pair potential parameters for various molecules

are obtained from second virial coefficients of pure gases. For the

potential models studied methods have been developed for estimating

the unlike-pair potential parameters which characterize the interaction

between a pair of molecules of different species from the like-pair

parameters of the respective molecules. The resulting unlike-pair

potential parameters are used to calculate accurate values of cross-

term second virial coefficients in gas mixtures.











In the remaining chapters information gained in Chapters 2, 3

and 4 is used in the determination of effective pair potentials for

use in pairwise additive theories of pure and mixed liquids. Only

(n,6) potentials are used for the liquid studies. Effective pair

potentials have been found (Chapter 5) which when used in a perturbation

theory of the liquid give good estimates for the residual internal

energy and entropy of several liquids. In this part of the liquid

study parameters for the various (n,6) potentials are those determined

from the second virial coefficients of the respective species. In

general for a particular molecular species the (n,6) potential found

to give the best estimates of liquid properties is not the same (n,6)

potential found to give the best estimates of second virial coefficients.

It is demonstrated (Chapter 6) that the methods found to give

good estimates of the unlike-pair parameters in the gas phase can be

used indirectly to estimate unlike-pair parameters for use with the

liquid phase potentials. The importance in liquid mixture calculations

of accounting for the interaction of molecules with different pair

potentials is emphasized with reference to the particular case of the

methane + perfluoromethane system. A simple method is proposed for

accurately estimating the excess properties of such mixtures.

It is further shown (Chapter 7) that statistical mechanics pro-

vides relationships for calculating composition-dependent potential

parameters for use in the one-fluid and two-fluid theories of liquid

mixtures. The new prescriptions called the virial coefficient prescrip-

tions are shown to give accurate estimates of the properties of









mixtures of (12,6) molecules in both the gas phase and the liquid

phase. One of the new prescriptions, the virial coefficient Boyle

(vcB) prescription, is shown to be for real systems the analog of

the van der Waals prescription for van der Waals systems. Methods are

developed for using the vcB prescription to predict either accurate

estimates of the excess properties or accurate estimates of both

mixture properties and excess properties when the component molecules

obey different (n,6) potentials.

In Chapter 8 methods developed in previous chapters are

combined to demonstrate that it is possible with knowledge of only the

gas phase (n,6) potentials of pure components to make accurate estimates

of the excess properties of liquid mixtures. The mixtures studied

exhibit behavior ranging from nearly ideal to very nonideal. The

results provide an explanation for the deviation of the unlike-pair

energy parameter E.. from the geometric mean of the respective like-

pair parameters which is observed in mixture calculations where all

molecules are assumed to obey the same pair potential.











CHAPTER 2

MIXTURE RULES FOR THE HIE (n,6) INTER.IOLECULAR PAIR
POTENTIAL AI1D THE DYMOIND-ALDER PAIR POTENTIAL

Introduction


The Mie (n,6) model for the intermolecular pair potential

function is of the form


B A (i

r r


It has been studied in some detail for argon and nitrogen by Klein and
12 3
Hanlev and for methane by Ahlert, Biguria and Gaston.3 The repulsion-

term exponent n as a third adjustable parameter gives this model a
2
flexibility equivalent to that of other three-parameter models, in a

simple analytical form. The coefficients A and B in Eq. (1) may be

written in terms of the parameters E and o, the depth of the potential

minimum and the intermolecular separation at which .(r) = 0, respectively,

1
n n-6 EO6
A = (21
66 n-6

and
B n n-6- n

B 6 L -6

Alternatively the coefficients may be expressed in terms of c and rm,

where r is the intermolecular separation at which 4(r) = -c,

6
nc r
n-6


and









6c r
B n-6 (5)


The most familiar form of the Mie (n,6) potential is the Lennard-Jones

potential in which n is 12.


Unlike-Pair Potential

From the leading term in the London theory of dispersion forces

the unlike-pair attraction coefficient A.. (Note that the double

subscript "ij" will refer to the unlike-pair intermolecular interaction

and the single subscript "i" or "j" will refer to the like-pair

intermolecular interaction.) may be written as


A.. = (A.A.) 1/2 (6)

where
2(1.1.)1/2
= (7)
If (I. + I .)

and I is the ionization potential.

The theory for the repulsion interaction is not well developed;

however, one combining rule has been proposed by Amdur, Mason and

Harkness5 based on molecular beam scattering results. Mason and co-

workers use a purely repulsive potential of the form

= B.rni (8)
1 1

to represent the intermolecular interaction at small separations. For

the unlike-pair repulsion interaction they suggest that


ep = ( rep r. 1/2 (9)
21J qi 4' /











or B1/2
B.. BB
1 = i (10)

r r r


With this assumption dimensional considerations require that


n.. = (n. + n.)/2 (11)
1j 1 J


and therefore,
1/2
B.. = (B.B.)/2 (12)



Abrahamson6 has made theoretical calculations of the interatomic

repulsion interaction of both like and unlike inert gas atoms. These

calculations were based on the Thomas-Fermi-Dirac statistical model

of the atom and show Eq. (9) to be satisfied to within a few percent.


Semitheoretical Mixture Rules


Using Eq. (6) for the unlike attraction energy and Eq. (9) for

the unlike repulsion energy, the appropriate mixture rules for the

parameters e, n, and r (or a) may be derived for the Mie (n,6) potential.

Using Eqs. (12), (11) and (5)
1/2
n.. n. n.
6c.. r 13 6c. r 6c. r
ij mi] = m. 3 m
B.. = -1- = I (13)
ij n.ij 6 n 6 n. 6


or 1/2


E 1/2 1 mi nm (n 6) ((14)
1r 1 ( nij [(ni-6)(nj-6)]12
r j
SimilarlwithEq.(6)andEq.()

SimilarJ1" with Eq. (6) and Eq. (4)









6 6 6 1/2
n6.. .,r c .n.r 6 .n.r
j m 1 1im. 3 3j m.
A = -= fl (15)
Aij n. 6 n. 6 n. 6 f


or 6
= i)/ ] .1/2 (16)
Sr r
E1/2 ff (n, 6) m i mJ
13 n [(ni-6) (nj-6)1] 1/2 rm


where
1/2 (n 1/2
-- = 1 1 (17)
n n i (n. + n.)
ij 2 J

Elimination of c.. between Eqs. (14) and (16) yields after algebraic

manipulation

1 1
n.-6 n.-6 2(n.i.-6) 6-n. .
r = rm m 1n (f f) 1 (18)


Note that for the very special case where f = 1 (n. = n.) and f = 1
n 1 3j
we have from Eq. (18)


r = (r r )1/2 and (o.o.) 1/2 (19)
m.. m.

and from Eq. (16)

1/2 (20)
Cij = (Ei j) (20)


These rules, Eqs. (19) and (20), are those proposed by Lehman and
8
later by Good and Hope. It should be mentioned that due to the

relationship between o and r these two quantities will obey the same

mixture rule only when ni = n..

The performance of the semitheoretical mixture rules for cij










n.. and rm given in Eqs. (11), (16), (18) has been tested in seven

binary gas mixtures. Pure component parameters (Table 1) were

determined from a fit of the second virial coefficient (see Appendix A).

The cross-term second virial coefficient was calculated using the semi-

theoretical values of c.., n.. and a.. (Table 2) and compared with

experiment. The results can be interpreted as a test of the assumptions

given in Eqs. (6) and (9) within the framework of the Mie (n,6)

potential. These results, shown in Table 3, are poor in all but two

cases, the Ar + CH, and Ar + N2 mixtures. Both of these may be regarded

as special cases in which f and f are nearly equal to 1.
n I "

Empirical Mixture Rules

As an alternative to the unsuccessful rules proposed above

purely empirical mixture rules were found which could be applied in

all cases including those in which n. and n. differ greatly. The

limited success of the semitheoretical rules suggests that in case

ni = n. and I. = I the unlike-pair energy and distance parameters

should be the simple geometric mean of the respective like-pair

parameters. Such simplicity while appealing is hardly a basis for

choosing these forms. However, results of a least squares fit of

the cross-term second virial coefficient for the CH, + CF, system

shown in Table 2 indicate that the best-fit results are reasonably

reproduced by the geometric mean c and a. If then the geometric mean

is retained for these two parameters, the choice of a proper mixture

rule for the n's is all that remains.

The semitheoretical rules lead to the unlike repulsion exponent






















0








E
f-4


u C-1
0



E E
>d u


0
W


0

*o E








0


00 0' 0 4- 0 .-- 0 C -
3- r Q. 0 ( -3 .-I 0

O 0 LiO .--4 -4 O .-4

0
c
o


0)


r- 0 0 -4 0
) ,-0 0 -0 1 0



03 o


(.
0u CM 0 0

E 0 m 0
0 r- Ln rJ r, CN
W o r- Cr- C" M



w --I -D %l LA
%0o in r-- .-4 %D






o *-4 C', -m
D l 0 0 r-,


CN --I CN C r-I


Ln CM Co r. 'f
4 Cm e D m o
OC CM LA Li An %o






0 0' c-i Co 00
L r C CM









o o Cn u 0mm 0


10 10
%Dl 'D






r-I CI
- C-4 C0

Cn r-) CO



C0 Co



iLA -
10 LD0
* *



n0 n 0






a% CO 00
00 r~l oo

;-I CM CM

rin en M
in 0 1 e

,-1 -4 -4-






Co- m -i
-4 r-l C

o4 c' c-i





-4 -4 -4
0- 0D 0'






CM r- 0 o
_r -3




&J e 0 e
0 en i

in r~. coo
a) c-i LA



0)
-o
i-'-
0 Li
C 0

.W --I

0 Cu
0 E <
C 0 0E
Cu Li .C
CO O- C
li 0

0 *-1
c. > -i u-i
0 -i r-
0 0 3
2 &- C/)


co
0

LA



,-4
cu



0









-4
i-i











r4-
0















0
>,


4-.
S-,
U]







--1
0










































cl
C




















Cu
0-





.c

U-








4'-4
Oi
*-



01


Cu









-D
0
*i








.0-
01




iM



0U





Ol


-J


r-4


0
1-"
C-,
-4











0
C




(n







-a Ln
C
.--


mi






-
0

*.








U] r-. -3

Cl
o m

0 1-
CO ^


Cu--4
0 5




-4
-4 t
Cu


0 E -
ca a




C- DC .



0 '4 -
co *






U] 0 w

St4
-'-4 .C ~
- -U
r-4












0 t.
0) E 0














CC
U) w


*
4 -4l

-4
LO 2- C








C
CO






0 *-
c o



o w-0
.-i u,4 --
3 c/i



o ^








-'-4 0-
0 I0
a n








-. -a
0
0) .*


0 0) i--)
e 1--1 1-1

Ci 3..


3"i -1
*.Q
03




3 -0 C


0
CD
0'
i-4


ca
0
O



O0

o
C:
0
c
*o









ca
CL
-






0' u













4 0.
CO


0















Cu Cu-


u)






U]
0)

0




co
U)








r-.
Ca
-4



0









-4

-im






0
+1
Oc.

U Cl











- y-
. .0 .








00 -


S> >
Q .Q a
0 0
-c CU
3 H 01
> 0 -
Ca -~ 1.-1
c .c cj m

co 4I ca
a -i ) o
-O



-1 C/ i rL

r-1 w a

j 0 0 r
-4 Eu
a)a
0 U] -
> : 3 Li 3
0 0 U
L i--4
Lz 0Cu
S EU
-0C Cu




S: *O co
C Li-i1

SCul. 3

C '4-1 x

U] 0 ,--

m E
*0-Li0
U UU]-4

*- co
-: 3 U)
*<-i L-


0 4-l 0 CO.


ac
CL
0
r





01

(0

--



















*0
4 r-.






0 0





0 1
1 r-











C' -












uj 0
*o 0
Li :I














CO



A -4
00
00 -|
-au 0







0)











o C
0 0







tL 4 L"




4U) a-) .0
0m -
0U -i

SC C
U] l
1 C


0 0




0 a
J L4L-



0)

E ])





































U,
Ln
an

r--



me




C7
C,*1
-A


l 0





j. co
\n *








0 e
a V. 0




E- -4 c c


(n in (
P~ C *0


- ) 1 r-
3o > E
















4-1 co (a' t
c E S
0 2 l





Q) M M


C"
Cl 0)
.-1 -r- j C






co a co


*0 2c
Cu. CI CB 0;








cu .-1
*-
) Cu
I!


oa <
ca -


1-1
cu
Cl -1 ,

o e
Q ifl


Q~i -0



Cl -

T-I Cu C
C1 Cu Cu



') U) U)



-: -i 2









Table 2. Unlike potential parameters.


System .

CH4 + Ar



CH + C2H6



CH4 + C(CH3 )4
4 i-4


CH4


+ CF


CH4 + SF



N2 + Ar


N2 2+ 6


semitheo
set A
set B

semitheo
set A
set B

semitheo
set A
set B


a
4 a best fit
semitheo
set A
set B

6 semitheo
set A
set B

semitheo
set A
set B


semitheo
set A
set B


c../k (OK)

189.41
192.40
192.40

268.95
244.32
244.32

154.90
376.99
376.99

289.74
238.28
297.06
297.06

180.73
325.26
325.26

151.88
150.48
150.48


198.82
191.09
191.09


aDetermined by least squares fit of cross-term second virial
coefficient. Root mean square deviation = 0.09 cm3/mole.


3.399
3.392
3.392

3.992
4.054
4.054

5.427
4.533
4.533

3.847
4.097
3.846
3.846

4.806
4.211
4.211

3.370
3.375
3.375


4.001
4.033
4.033


nij
2ii

20.50
20.49
20.50

19.37
19.30
19.37

66.45
48.47
66.45

59.54
80.67
58.41
80.67

79.84
53.97
79.84

19.66
19.65
19.66

18.53
18.51
18.53








Table 3. Cross virial coefficient B with the
(n,6) Mie potential.


Mixture
components

CH4+Ara

f = .9962


CH +CH6
4 26
f = .9981



CH +C(CH3 )4

f = .9939
I


CH4+CF,

fI = .9887


Temp.
(*K)

142.60
176.70
239.80
295.00
rms dev

273.20
298.20
323.20
rms dev

303.2
323.2
333.2
343.2
353.2
363.2
383.2
403.2
rms dev

273.16
298.16
323.16
348.16
373.16
398.16
423.16
448.16
473.16
498.16
523.16
548.16
573.16
598.16
623.16
rms dev


3
B (cm /mole)
Semi-
theoretical
Eqs. (11), Empirical rules
Exptl. (16), (18) Set A Set B


-138.6
- 86.7
- 48.1
- 26.9



-111.9
- 92.0
- 75.6



-165.0
-138.0
-132.0
-118.0
-113.0
-106.0
- 93.0
- 78.0


-135.1
- 89.2
- 46.3
- 26.5
2.4

-127.6
-106.6
- 89.5
14.75

42.2
52.4
57.0
61.3
65.3
69.0
75.9
82.0
181.5


- 62.07 32.13
- 48.48 20.54
- 37.36 11.08
- 28.31 3.20
- 20.43 3.45
- 13.98 9.15
- 8.33 14.08
- 3.21 18.38
1.02 22.17
4.94 25.53
8.28 28.54
11.39 31.24
14.10 33.67
16.55 35.88
18.88 37.90
22.90


-138.3
- 91.5
- 47.8
- 27.6
2.4

-110.0
- 90.9
- 75.4
1.27

-167.7
-144.7
-134.6
-125.2
-116.6
-108.5
- 94.0
- 81.3
4.2

- 67.19
- 52.81
- 41.19
- 31.60
- 23.57
- 16.74
- 10.87
- 5.76
- 1.28
2.68
6.21
9.37
12.21
14.79
17.13
2.94


-138.3
- 91.5
- 47.7
- 27.6
2.4

-109.6
- 90.5
- 75.1
1.61

-144.3
-123.4
-114.1
-105.5
- 97.6
- 90.2
- 76.9
- 65.3
15.9

- 56.95
- 43.67
- 32.92
- 24.04
- 16.59
- 10.25
- 4.79
- .04
4.13
7.82
11.10
14.05
16.71
19.11
21.30
3.56


Ref.















Mixture
components

CH +SF6

f = .9819





N 2+Ar

f = 1.00
I







N2+C H6

f = .9898
I


Temp.
(K)

313.2
333.2
353.2
373.2
393.2
rms dev

148.2
173.2
198.2
223.2
248.2
273.2
298.2
323.2
rms dev

277.6
310.9
377.6
444.3
510.9
rms dev


Table 3 (Continued)
B (cm /mole)
Semi-
theoretical
Eqs. (11), Emp
Exptl. (16), (18). Set


- 85.0
- 68.0
- 57.0
- 45.0
- 33.0



- 81.6
- 59.1
- 44.0
- 32.6
- 23.7
- 16.4
- 10.9
- 6.2



- 65.4
- 38.6
- 20.1
- 3.8
5.9


19.3
27.1
33.8
39.7
45.0
91.0

- 82.2
- 59.6
- 43.9
- 32.4
- 23.6
- 16.7
- 11.2
- 6.6
0.4

- 66.3
- 49.1
- 25.2
- 9.4
1.7
6.1


Ref.


irical rules
A Set B


- 81.6
- 68.7
- 57.6
- 47.9
- 39.4
3.5

- 81.1
- 58.7
- 43.1
- 31.7
- 23.0
- 16.2
- 10.7
- 6.1

0.6

- 61.7
- 45.0
- 21.0
- 6.4
4.4
3.6


- 65.6
- 53.9
- 43.8
- 35.0
- 27.3
13.3

- 81.1
- 58.6
- 43.1
- 31.7
- 23.0
- 16.2
- 10.6
- 6.1
0.6

- 61.6
- 44.9
- 21.6
- 6.3
4.5
3.6


aThe lowest temperature point has not been included as there appear to
be errors in the calculation of the reported B 2 for this point in
Footnote b.
bG. Thomaes, R. van Steen Winkel, and W. Stone, Mol. Phys. 5, 301 (1962).

CR. D. Gunn, M.S. thesis, University of California, Berkeley, 1958.
dSee Footnote j, Table 1.
eS. D. Hamann, J. A. Lambert, and P. B. Thomas, Australian J. Chem.
8, 149 (1955).
fJ. Brewer and G. W. Vaughn, J. Chem. Phys. 50, 2960 (1969).











being the arithmetic mean of the two like repulsion exponents (Eq. 11).

Once again, however, a glance at the least square mixture parameters

for CH4 + CF4 reveals that the value of 59.54 for the unlike repulsion

exponent is very nearly the value 58.41 obtained as the geometric mean

of the two like repulsion exponents. As a result two sets of empirical

rules were examined.

Set A Set B

i = i 1/2 C =1/2


ij (C.)1/2 1= ( /2
a. = (v..) a.. = (a.a.)
ij 1 J 1J 1 J
= ,1/2
n.. = (n.n.) 2n.. = (n. + n.)/2
ij 1 J 13 1 J


The results of calculating B 2, the cross-term second virial coeffi-

cient, for the two sets of empirical rules are reported in Table 3.

The superiority of the total geometric mean rule, set A, is apparent.

This set of rules appears to be generally applicable to all of the

mixtures tested. The root-mean-square deviations of 0.6 to 4.2 cm 3/mole

appear to be within the accuracy to which the experimental values are

known. This is quite significant in the light of the great differences

in molecular character represented by some of the mixtures.

Other sets of rules have been tested, though not reported here.

One such set of rules is one in which each of the unlike parameters is

the arithmetic mean of the respective pure component parameters. This

set of rules performed remarkably well for the CH, + CF system with
a root-mean-square deviation of only 1.06 cm3/mole. 4

a root-mean-square deviation of only 1.06 cm /mole. However, this










particular set of rules failed to exhibit the general applicability

found for the rules given in set A.


The Dvrmond-Alder Potential


For molecules which obey the same form of the reduced pair

potential energy function (r ), where = ./c and r = r/r the

empirical rules always require the unlike-pair energy and distance

parameters c and r (or a) to be the simple geometric mean. For the

Mie (n,6) potential this is the case where n. = n.. This result,

however, is not restricted to the Mie (n,6) potential. The Dymond
9
and Alder numerical potential energy function for argon has been

shown by Reed and Gubbinsl0 to perform well for the other inert gases

and for 02, N2 and CH4 but not for CF The potential energy parameters

(c, 0) for the other molecules are obtained from the argon parameters

by corresponding states relationships. Results of predicting the cross-

term second virial coefficient for four binary gas mixtures using the

geometric mean rule for c and c and the tabulated reduced second

virial coefficients for the Dymond and Alder potential are presented in

Table 4. The results are excellent.


Conclusions


Within the framework of the Mie (n,6) potential the assumption

of a geometric mean unlike repulsion energy, Eq. (9), and a corrected

geometric mean attraction energy, Eq. (6), predicts the correct cross-

term second virial coefficient only for those cases where these rules

are equivalent to the geometric mean rule for all parameters. The










Table 4. Cross virial coefficient EB with the
Dymond and Alder potential.


Mixture
components

CH, + Ar






N2 + Ar












Ar + 0


Temp.
(K)

142.6
176.7
239.8
295.0
rms dev

90.0 b
148.16
173.16
198.16
223.16
248.16
273.16
298.16
323.16
rms dev

90.0


N2 + 02


3
B (cm /mole)

Geometric
Exptl. mean rule

-138.6 -138.1
- 86.7 91.2
- 48.1 47.4
- 26.9 27.3
2.3


-213.9
- 81.58
- 59.13
- 43.96
- 32.56
- 23.73
- 16.43
- 10.88
- 6.19



-233.3

-222.2


-212.72
- 82.07
- 59.32
- 43.61
- 32.13
- 23.41
- 16.57
- 11.08
- 6.51
0.50

-235.3

-216.9


aSee Footnote b, Table 3.
bThese experimental values of B,, were calculated from the excess
1
second virial coefficient E = B (B1 + B22) reported in Footnote
d with E11 and B22 calculated from the Dymond and Alder Potential.
The authors Knobler et al. of Footnote d originally calculated B11
and B,2 using the Lennard-Jones (12,6) potential.
CSee Footnote f, Table 3.
C. M. Knobler, J. J. M. Beenakker, and H. F. P. Knaap, Physica 25,
909 (1959).


Re f.

a






c, d












d

d










set A of rules, in which each parameter (c, o, n) is the geometric mean

of the respective pure component parameter, works well for all of the

binary gas mixtures studied. The significance of the results may be

seen in that sets of the three Mie parameters for like-pair inter-

molecular interaction allow the prediction of binary mixture properties

without any knowledge of mixture properties. This is in contrast to

the use of a two-parameter potential for the like-pair interaction

which has been shown by Eckert, Renon, and Prausnitz to require the

use of a third parameter (equivalent to f ) obtained from binary mixture

data to correlate mixture data.

The empirical rules in set A or set B suggest that when

molecules obey the same form of the pair potential the mixing rules

for the energy and distance parameters should be the geometric mean.

This is supported by the agreement between calculated and experimental

cross-term second virial coefficients of mixtures of molecules which

obey the Dymond and Alder numerical potential.
12
Sherwood and Prausnitz computed values of the third virial

coefficient for two special cases of the Mie (n,6) potential with

parameters (c, o) determined from least squares fits of the second

virial coefficient. These are the Lennard-Jones (12,6) potential and

the Sutherland (=,6) potential. Where the third virial coefficient

of a pure system has been calculated with both the Lennard-Jones and

Sutherland potentials, the values predicted by the (12,6) potential

are higher than experimental data, while the values predicted by the

(-,6) potential are lower than experimental data. The (n,6) potentials






18




reported in the present chapter for pure systems have values of n

ranging from 17.74 for ethane to 138.68 for sulfur hexafluoride. These

potentials will predict values for the third virial coefficient which

fall in the range where the experimental values lie, between those

predicted by the Lennard-Jones and Sutherland potentials.










CHAPTER 3

A MIXTURE RULE FOR THE EXPONENTIAL-6 POTENTIAL


Introduction

It was shown in the previous chapter that for the Mie (n,6)

intermolecular pair potential the unlike-pair parameters (c. .,o.,n..)
ii ij 12
are the geometric mean of the respective like-pair parameters for second

virial coefficients. It is shown below that these mixture rules can be

extended to define mixture rules for the exponential-6 potential,



(r) = (-6/) exp[a(l r/r )] (r /r) (21)
(1-6/c) 1 I m



Here r is the value of r at which '(r) = -c. The value of r at which

iq(r) = 0 (i.e., r = a) can be determined from r by solving the following

equation numerically.13



n(o/r ) = -(1/6){2n(6/a) + a(l-o/r ) (22)
m m



Mixture Rules


Since c and a in the exponential-6 potential have the same

meaning in the Mie (n,6) potential it is reasonable to assume that

these parameters will obey the same mixture rule for both potentials.

An estimate of the third parameter cL for the unlike pair may be

obtained in the following way.

In the (n,6) potential the repulsive energy is of the form


.(r)rep = Br-n (23)











where B is constant. From (23) we have


d rn 4 ep.
d nr


For the exponential-6 potential the form is


,(r)rep- = Ke-br

and

d n = -br .
d nr


(24)


(25)


(26)


The results of the (n,6) mixture rule study (geometric mean for nij

give


d(ne ire_,rp)/dunr = {(dn- rep_ /d-nr)(dn rep/d nr)) 1/2
which for the eponentia potential can be written as


which for the exponential-6 potential can be written as


(27)


b. .r = (b.b.)1/2 r
ij 1 J


From Eq. (21) we see that b = c/r which implies
mo


U.
r
m.,
13


Cl OL.
1 3
r r
m. m.
1 J


(28)


S1/2


(29)


aij = 3(eiej)1/2


and r = y-,(r r )1/2
.. m. m.
lj 1 3


where y is to be determined.


(30)










Substitution of Eq. (30) into Eq. (22) along with
1/2
0.. = (o.i.) from the previous chapter gives us an equation for y

in terms of known parameters of the pure fluids (i.e., a., r a.).
1
Resulting values for -, are near one (Table 5) and thus a.. is nearly

the simple geometric mean. Values of unlike-pair parameters for three

binary mixtures for which exponential-6 like-pair parameters are

available are given in Table 5. These are based on a geometric mean

rule for E and a and Eqs. (30) and (22) for ':. The like-pair parameters
12
were taken from Sherwood and Prausnitz. Results of the prediction

of the cross-term second virial coefficient with these parameters

are given in Table 6.


Conclusions

In two of the three cases the root-mean-square deviations for

the exponential-6 potential with the mixture rules proposed are lower

than the deviations for the Mie (n,6) potential with the same rules.
3
In the Ar + N mixture the deviation of 1.8 cm /mole is greater than

0.6 cm 3/mole found with the Mie (n,6) potential. However, either

potential model fits the cross-term second virial coefficient within

experimental error with the proposed mixture rules. The predictions

of the cross virial coefficient show that mixture rules obtained

previously for the Mie (n,6) potential can be extended to the

exponential-6 potential.

















Table 5. Unlike-pair parameters.


System

CH4 + Ar

Ar + N2

CH, + CF
4 4


E. ./k (K)

185.14

156.05

301.68


3 .

3.431

3.373

3.848


r (X)
1]

3.752

3.665

3.983


20.77

23.21

83.77


Y

0.9996

0.9990

0.9872









Table 6. Cross virial coefficienta B 2.


System


CH, + Ar






Ar + N2











CH4 + CF
4 4


Temperature
(K)


142.6
176.7
239.8
295.0
rms dev

148.2
173.2
198.2
223.2
248.2
273.2
298.2
323.2
rms dev

273.16
298.16
323.16
348.16
373.16
398.16
423.16
448.16
473.16
498.16
523.16
548.16
573.16
598.16
623.16
rms dev


B12 (cm /mo

Experimental


-138.6
- 86.7
- 48.1
- 26.9



- 81.6
- 59.1
- 44.0
- 32.6
- 23.7
- 16.4
- 10.9
- 6.2



- 62.07
- 48.48
- 37.36
- 28.31
- 20.43
- 13.98
- 8.33
- 3.21
1.02
4.94
S.28
11.39
14.10
16.55
18.88


le)

Calculated

-137.0
90.7
47.3
27.2
2.2

79.5
57.2
41.7
30.4
21.8
15.0
9.6
5.1
1.8

59.00
45.44
34.48
25.43
17.85
11.39
5.83
1.01
3.24
6.98
10.33
13.32
16.02
18.47
20.69
2.30


aExperimental data used for comparison are the same as that in Chapter 2.











CHAPTER 4

THE RELATIONSHIP BETWEEN THE MIE (n,6) POTENTIAL AND
EXPONENTIAL-6 POTENTIAL


Introduction


Hanley2 and Klein2,1 have recently shown that five of the

common three-parameter pair potentials are essentially equivalent

with respect to the ability to predict the second virial coefficients

of pure gases. It has been shown in the previous chapter that for

two of these potentials, the Mie (n,6) potential


(r) 6= {(r /r)n (n/6)(r /r)6} (31)
(n-6) m m


and the exponential-6 potential


4(r) = -(6)- {exp[a(l-r/rm)] (a/6)(r /r)6} ; (32)


though the forms of the repulsive energy differ, a relationship exists

between the parameters of the two potentials. In the (n,6) potential


d nre = -n (33)
d inr


For the exponential-6 potential

d nrep
d n' = -(a/r )r (34)
d mnr m


As n and a approach infinity both potentials become identical to the

Sutherland (-,6) potential. Thus in this limit the parameters c and r

for the two potentials would be the same when found from fitting the










same set of second viral coefficient data. This suggests that for

large values of n and a the parameters c, as well as r may have

essentially the same value in the two potential models.


Equivalence of Potential Parameters

In Chapter 3 it was pointed out that Eqs. (33) and (34) implied

that the quantity c/r in the exponential-6 potential should obey the

same mixture rule as the parameter n in the (n,6) potential. If,

however, it is assumed (a) that the values of r in Eqs. (31) and (32)

are the same value, (b) that -(r m) by Eq. (31) is equal to *t(r ) by

Eq. (32) and (c) that dinj../dZnr at r by Eq. (33) is equal to that by

Eq. (34) then the following equivalence of parameters is obtained:


(n,6) -E (exp-6) (35)

r = r (36)
(n,6) (exp-6)

n (n,6) = (exp-6) (37)


This suggests that where sets of the three-potential parameters

are available for one potential they may be used for the three parameters

in the other potential model. In order to test this equivalence,

exponential-6 parameters for six pure gases, determined by either
12 1
Sherwood and Prausnitz or Klein from fitting second virial coeffi-

cients, have been used with Eqs. (35) to (37) to predict (n,6) potential

parameters for the same gases. The results of predicting the second

virial coefficients with the (n,6) potential using these parameters

are given in Table 7. In general the results are almost within the











Table 7. Second virial coefficients of pure gases predicted
with (n,6) potential using exponential-6 parameters.


Number of
Temperature experimental m Data
Gas range (K) points dev (cm /mole) ref.

CF, 273.16-623.16 15 0.11 b

C(CH ), 303.16-548.16 16 2.2 c,d,e

C2 H6 220.0 -500.0 11 2.4 f

N2 400.0 -700.0 4 1.4 f

CH4 273.16-623.16 15 1.61 b

Ar 81.0 -600.0 14 7.7 f



aThe root mean square (rms) deviations are the deviations between
calculated virial coefficients and the experimental data from the
reference indicated. The experimental data in some cases are not the
same as thoseused by others for the determination of the e;-;ponential-6
parameters in Table 8. However, in all such cases the experimental
data in Table 7 do cover the same temperature range as that used by
the original authors to determine the parameters.
bD. R. Douslin, R. H. Harrison, and R. T. Moore, J. Phys. Chem. 71,
3477 (1967).
cJ. A. Beattie, D. R. Douslin, and S. W. Levine, J. Chem. Phys. 20,
1619 (1952).
S. D. Hamann and J. A. Lambert, Australian, J. Chem. 7, 1 (1954).
CS. D. Hamann, J. A. Lambert, and R. B. Thomas, Australian J. Chem. 8,
149 (1955).
fJ. H. Dymond and E. B. Smith, The Virial Coefficients of Gases
(Clarendon, Oxford, 1969).










experimental uncertainties. As might be expected the worst results

are obtained for argon which has the lowest value of o equal to 18.

The potential parameters used are given in Table 8. These findings,

along with those of Hanley and Klein, indicate that for the second

virial coefficient of most molecules there is little real difference

between the (n,6) and exponential-6 potentials.


Unlike-Pair Parameters

The implied equivalence of the potential energy functions

suggests that the mixture rules for the exponential-6 potential

parameters should be even simpler than those suggested in Chapter 3.

The exponential-6 mixture rules could be taken to be the same as those

for the (n,6) potential in Chapter 2.


= (Ec )/2 (38)
1/2
1.. = (o.o.) (39)


= ( .)1/2 (40)


These mixture rules have been used with the exponential-6

potential and the parameters in Table 8 to predict the cross-term

second virial coefficients (Table 9) of six binary gas systems previously

studied with the (n,6) potential. The predictions of the rules in

Eqs. (38) to (40) with the exponential-6 potential are similar to those

with the (n,6) potential in Chapter 2. The predictions for the

C2H6 + N and CHl + C(CH )4 systems are not as good with the exponential-6

potential as with the (n,6) potential, but they are much better than











Table 8. Exponential-6 potential parameters.



Component c/k (K) rm a (or n) Ref.

CF4 403.6 4.209 300.0 b

C(CH3)4 635.4 5.980 100.0 b

C2H6 377.93 4.502 30.0 c

N2 160.2 3.695 30.0 b

CH, 225.5 3.868 24.0 b

Ar 152.0 3.644 18.0 b




aFor the (n,6) potential a, the value of r where O(r) = 0, is found from
1
I6 In-6
0 = r -
m n


for the exponential-6 potential it is found from numerical solution
of Eq. (22) in Chapter 3.
bA. E. Sherwood and J. M. Prausnitz, J. Chem. Phys. 41, 429 (1964).

CM. Klein, J. Res. Natl. Bur. Std. A70, 259 (1966).











Table 9. Cross-term second virial coefficient.a


System

CH4 + Ar

CH4 + C2H6

CH4 + CF

CH4 + C(CH3)4

N2 + Ar

N2 + C2 6


rms dev
(exp-6)

2.2

1.24

2.85

7.42

1.88

4.71


(cm 3/mole)
(n,6)b

2.40

1.27

2.94

4.2

0.6

3.6


aCross-term second virial coefficient data used are identical with
those in Chapter 2.
previously reported in Chapter 2.











the predictions of the (12,6) potential with any of the eleven sets

of mixture rules tested by Good and Hope.

It should be pointed out that for a given set of E, r ,

n (or a) the value of o(n,6) does not equal to o(exp-6). A given

mixture is characterized by e., e., r r and n., n. (or equivalently
i m. m.
ci and a.). Using Eq. (39) for the exponential-6 potential and for the

(n,6) potential does not lead to the same r for both potentials.

This result is, of course, inconsistent with Eq. (36). It would have

been consistent if the following,


a(n,6) = (exp-6) (41)


were chosen in place of Eq. (36). This choice could have been made in

the first place. In fact, calculations based on Eqs. (35), (37) and

(41) give about the same results as reported in Table 7. This indicates

that the effect of the inconsistency referred to above is small. To

further illustrate this fact the case of the CH4 + C(CH3)4 system is

examined. The pure component parameters for this system are given in

Table 8. If these are taken to be (n,6) parameters, the rules in

0
Eqs. (38) to (40) would predict r = 4.787 A. If taken to be
m..
10
exponential-6 parameters, the same rules predict r = 4.782 A. This

difference would lead to a difference of only 0.3,% in the predicted

virial coefficients.


Conclusions

The (n,6) and exponential-6 potentials are sufficiently alike

with respect to the prediction of second virial coefficients that




31




sets of the three exponential-6 parameters can be used for the three

parameters in the (n,6) potential with very good results in the

prediction of second virial coefficients. The mixture rules shown

previously to work with the (n,6) potential give similar results with

the exponential-6 potential.










CHAPTER 5

SATURATED LIQUID PROPERTIES FROM THE
MIE (n,6) POTENTIAL


Introduction


The ability of the perturbation theory of liquids developed by

Barker and Henderson 4 to reproduce liquid properties calculated by

means of Monte Carlo or molecular dynamics makes this theory an excellent

tool for studying pair potential energy functions for liquids. Hanley

and Klein '2 have recently shown that various three-parameter potentials

(Kihara, Mie (n,6), exponential-6 and Morse potentials) are equivalent

with respect to their ability to reproduce experimental second virial

coefficients and transport properties of gases. However, the Mie (n,6)

potential is of special interest for mixture property calculations

because simple mixture rules have been found (Chapter 2) for the three

parameters (c/k, a, n) which accurately reproduce cross-term second

virial coefficients for a wide variety of gaseous mixtures. To our

knowledge the only Mie (n,6) potentials that have been studied by

Monte Carlo methods to any extent for the liquid are the (12,6) and

(18,6) potentials for liquid argon. 14'15 The Lennard-Jones (12,6)

potential with the parameters c/k and a determined by Michels et al.16

performs remarkably well (better than that of Hanley and Klein) for

liquid argon as demonstrated by Monte Carlo15 and perturbation theory14

calculations. The effect of the source of potential parameters is

demonstrated for argon and methane by using two empirical sets of

(12,6) potential parameters for each of these liquids in perturbation

theory .see Table 12). Corrections for nonadditivity of pair potentials










have not been included; thus to the extent that many-body interactions

are important the (n,6) potentials should be regarded as effective

pair potentials.

The Mie (n,6) potential is of the form

[ [o n J6
4,(r) = < c (42)
r r

where 1
nn n-6
*(43)
n-6 6{6


Barker-Henderson Perturbation Theory

The expression for the residual free energy is derived by

Barker and Henderson and is given by


A /flkT = A /NtkT + A + A, + A (44)
o 1 3

where
( 2
A = 27p. go(r)q.(r)r dr (45)


A= E 1 g (r).2 (r)r2dr (46)
-o
0


A = -o (r)r dr (47)
0

r
and A g (r) and (?p/;P) are the residual Helmholtz free energy,

radial distribution function and compressibility, respectively, of a

hard-sphere reference fluid with diameter d defined by
1
d = d/o = [1 exp(-,.,,)]d(r/o) (48)

0











where a is the distance parameter in the pair potential and
-l
S = (kT)-.

The second order term (A2 + A3) is derived from the local

compressibility approximation of Barker and Henderson. This is the

approximation adopted for this study.

The residual internal energy, U is obtained by numerical

differentiation of the residual Helmholtz free energy according to the

following equation


U r/NkT = -T 5(Ar/NkT)/;T (49)


The residual entropy is calculated from the relation


SrlN = (Ur/NkT Ar /NkT) (50)


The numerical integration were performed using a Gaussian

integration routine. Percus-Yevick hard-sphere radial distribution

functions were used for g (r). These functions were chosen because

they yield accurate values for the first order term in the Helmholtz

free energy when compared with Monte Carlo calculations. Carnahan
18
and Starling's expressions were used for hard-sphere pressure,

compressibility and free energy. Analytical expressions were utilized
17 ,19
to generate the radial distribution function 19 and its density

derivative (Appendix B).

McDonald and Singer15 have computed internal energies and

pressures for the (18,6) pair potential using the Monte Carlo method

at three different state points. At these state points pressures and










internal energies were recalculated using the (12,6) and the (18,6)

pair potentials in the Barker-Henderson perturbation theory. The

results are compared in Table 10. In the case of the (12,6) potential

the agreement for energies and pressures is excellent. For the (18,6)

potential the energies compare well with Monte Carlo values. The

pressures do not compare as well as do those for the (12,6) potential.

The Monte Carlo and the perturbation theory pressures are both negative

values. However, it is important to note that while the state points

were the same for both pair potentials, the reduced temperatures for

the (18,6) potential were considerably lower than those for the (12,6)

potential. The differences in Monte Carlo and perturbation theory

pressures may well be due to slow convergence of perturbation theory

at low reduced temperatures.


Liquid Properties from Best Virial Coefficient (n,6) potential


Iie (n,6) potential functions for this study were selected from

the tables of Klein. For each substance the (n,6) potential parameters

n, c/k and o which best fit the experimental second virial coefficient

were chosen. Klein has determined parameters for CF4 only for

potentials with n up to 40. An optimal set of parameters for CF, with
4
n = 136.3 was determined in Chapter 2. The best parameters for each

substance studied are given in Table 11 along with the (12,6) parameters

determined by Klein. In general the properties (residual energies and

entropies) along the saturation curve (Table 12) calculated using the

best gas potentials in Table 11 agree better with experimental data than










Table 10. Comparison between Monte Carlo
calculations and perturbation
theory.


V
T 3
(OK) (cm /mole)


(12,6) Potentiald


(18,6) Potentiale


a b c


-Ur (cal/mole)


97.0 26.90

108.0 28.48

136.0 32.52


1480 1424 1420

1387 1352 1351

1192 1186 1189


1566

1457

1233


1501 1517

1410 1427

1214 1233


P (atm)


97.0 26.90

108.0 28.48

136.0 32.52


609 663 660

443 499 498

289 351 351


-519

-415

-113


-278 -273

-225 -224

- 53 58


alionte Carlo calculations.15
bMacroscopic compressibility approximation and perturbation theory.

CLocal compressibility approximation and perturbation theory.
dparameters from Michels et al.16 /k = 119.8 K, a = 3.405 8 .

eParameters from Dymond et al. c/k = 160.3 K, a = 3.277 A .


b c










Table 11. (n,6) Potential energy parameters
from second virial coefficients.


Molecule n c/k (K) a (A)
Argonb 12 119.8 3.405
Argon 12 115.06 3.515
ArgonC 13 123.99 3.458
Argon 16 147.50 3.315

Nitrogen 12 94.77 3.804
Nitrogen 16 118.12 3.650
Nitrogen 24 148.59 3.491

Methane 12 143.25 4.056
Methaned 12 148.63 3.775
Methane 17 189.17 3.779
Methane 27 244.71 3.519

Perfluoromethane 12 151.90 4.742
Perfluoromethane 30 260.33 4.378
Perfluoromethanee 136.3 373.31 4.186



a
Except where otherwise specified potential parameters are those of
1,2
Hanley and Klein.1,2
bMichels' parameters.16
Parameters determined for this work.
parameters obtained from richels' argon parameters as follows
1/3
c/k' = 119.8 (T /T ) and aCH = 3.405 (VB /V )/3, where
4 CH, Ar 4 CH, Ar
Z4 4
TB and VB are Boyle temperature and Boyle volume, respectively, taken
from D. R. Douslin, R. H. Harrison, and R. T. Moore, J. Phys. Chem. 71,
3477 (1967).
parameters from Chapter 4.











Table 12. Saturated liquid properties.


Argon


Potential

(12,6)c


T (K) Experimental

Residual Energy (-U r/NkT)


83.81
87.29
90.0
95.0
100.0
110.0
120.0
130.0
140.0
150.86


8.52
8.05
7.70
7.08
6.53
5.52
4.68
3.93
3.22
1.91


(12,6)b


8.34
7.87
7.53
6.97
6.46
5.58
4.76
4.01
3.29
1.76


8.49
8.03
7.70
7.14
6.63
5.72
4.92
4.16
3.43
1.84


(13,6) (16,6)'


8.64
8.16
7.82
7.24
6.71
5.78
4.95
4.17
3.42
1.83


8.77
8.26
8.00
7.29
6.73
5.76
4.90
4.11
3.35
1.77


Residual Entropy (-S r/Nk)


83.81
87.29
90.0
95.0
100.0
110.0
120.0
130.0
140.0
150.86
Pressures


T (OK)


3.59
3.45
3.34
3.14
2.95
2.59
2.29
1.97
1.65
0.97
pos.


Experimental


3.47
3.34
3.24
3.09
2.93
2.63
2.32
2.00
1.64
0.82
pos.
Methane


4.05
3.89
3.77
3.58
3.39
3.02
2.65
2.25
1.86
0.90
pos.

Potential


(12,6)


Residual Energy (-Ur/NkT)


90.66
95.0
100.0
105.0
110.0
111.66
120.0
130.0


10.61
9.96
9.32
8.73
8.18
7.99
7.19
6.34


3.82
3.67
3.57
3.40
3.21
2.88
2.57
2.16
1.78
0.88
pos.


(17,6)


3.33
3.21
3.12
2.97
2.81
2.53
2.24
1.94
1.62
0.82
neg.


(27,6)a


10.29
9.65
9.01
8.44
7.92
7.76
7.00
6.21


10.59
10.05
9.47
8.94
8.45
8.30
7.58
6.81


11.89
11.13
10.38
9.72
9.10
8.91
8.01
7.09


11.67
10.86
10.08
9.38
8.75
8.55
7.64
6.71


(12,6)d









Table 12 (Continued)


Experimental


Residual Entropy (-Sr/Mk)


100.0
105.0
110.0
111.66
120.0
130.0
Pressures


3.75
3.58
3.44
3.40
3.16
2.86
pos.


3.80
3.65
3.51
3.46
3.23
2.97
pos.


Nitrogen
Potential


Experimental


(12,6)


Residual Energy (-U r /NkT)


10.62
10.21
9.11
8.18
7.81
7.41
6.11
5.06
4.12


9.86
9.52
8.66
7.90
7.58
7.25
6.10
5.10
4.22


Residual Entropy (-S r/Nk)


63.18
65.0
70.0
75.0
77.35
80.0
90.0
100.0
110.0


4.30
4.19
3.90
3.63
3.50
3.37
2.97
2.63
2.20


5.07
4.94
4.60
4.27
4.15
4.01
3.46
2.92
2.43


4.38
4.28
4.00
3.74
3.64
3.51
3.06
2.63
2.20


Pressures pos.


T (OK)


(12,6)


(17,6)


(27,6)a


5.77
5.48
5.22
5.15
4.74
4.28
pos.


4.24
4.08
3.90
3.85
3.58
3.29
pos.


T (OK)


3.41
3.26
3.13
3.07
2.87
2.64
neg.


(16,6)


(24,6)a


63.18
65.0
70.0
75.0
77.35
80.0
90.0
100.0
110.0


10.61
10.22
9.23
8.36
8.00
7.61
6.32
5.23
4.26


10.93
10.49
9.42
8.48
8.11
7.70
6.33
5.18
4.19


3.86
3.77
3.49
3.29
3.20
3.10
2.72
2.35
2.00

neg.


(12,6)d


pos. pos.










Table 12 (Continued)


Experimental


Perfluoromethane
Potential
(12,6) (30,6)


(136.3,6)a


Residual Energy, (U Ur 16.49)/NkT


116.49 0 0
127.60 0.39 -0.15
144.27 1.01 -0.23
166.49 1.88 -0.21
194.27 2.56 0.03
224.83 3.62 0.93
227.66 4.44 1.80

Residual Entropy, (Sr Sr )/Nk
116.490


116.49
127.60
144.27
166.49
194.27
224.83
227.66

Pressures


0
0.37
1.03
2.00
2.92
4.00
4.65

pos.


0
0.81
1.85
3.02
4.34
6.10
6.92

pos.


aBest gas
bMichels'


potential.
parameters.


Hanley and Klein's parameters.
parameters estimated from Michels' argon parameters and corresponding
states (see Table 11).


T (OK)


0
0.38
0.86
1.39
2.03
3.24
4.20



0
0.44
1.04
1.73
2.61
3.89
4.56

pos.


0
0.57
1.22
1.89
2.60
3.82
4.77



0
0.40
0.89
1.47
2.17
3.23
3.82

neg.











do those calculated using the (12,6) potential. However, the pressures

predicted by the best gas potentials were all negative along the

saturation curve. Experimental densities were used in the calculations

at all temperatures. Results for CF, only are given in Figure 1.


(n,6) Potentials for Liquids


The effect on the calculated properties of varying the value of

n was studied. For each n the second virial coefficient potential
1
parameters of Klein were used except for the (13,6) potential

parameters for argon. Klein has not reported parameters for the (13,6)

potential. These parameters have been determined for this work using

the same second virial coefficient data used by Klein. The residual

energies and residual entropies were calculated using these potentials

in order to find which one of them best predicts these liquid pro-

perties. For all molecules examined it was found that one of the

values of n between 12 and the best n for virial coefficients gave the

best agreement between predicted and experimental liquid properties.

The results are tabulated in Table 12. Experimental densities were

again used at all temperatures. Results for CF, are given in Figure 1.

For these intermediate n-values all pressures calculated along the

saturation curve were positive. In fact in every case it was found that

the best n for the prediction of liquid residual energy and entropy was

the highest value of n which still predicted positive pressures. This

observation suggests that an excellent estimate for n to be used for

the liquid could be obtained by merely choosing the (n,6) potential which

correctly predicted some experimental liquid density using c/k and o

obtained from a fit of the second virial coefficient.










O (12,6) Potential

A (30,6) Potential


0 (136.3,6) Potential


0
t0
-4









ca
E-

'-I
CI)
di


^^A




A



Experimental 0



0


SExperimental




Experimental


170


190


210


Temperature, T(0K)


Residual properties of liquid CF4.


4.0



3.0



2.0


1.0 -



0.0


%z.









0
I-









ta
C1
-41


1-1


0 0


6.0



5.0



4.0



3.0



2.0



1.0



0.0


130


Figure 1.











Conclusions


The Barker-Henderson perturbation theory using the (18,6)

potential function agrees with Monte Carlo calculations of McDonald

and Singer using the same potential. Barker and Henderson have

previously shown the excellent agreement between perturbation theory

and Monte Carlo calculations for the (12,6) potential. It is reasonable

to assume that other (n,6) potentials can be utilized in perturbation

theory to yield thermodynamic properties of pairwise additive fluids.

The best (n,6) potentials obtained by Klein for second virial

coefficients yield residual properties that agree better with experi-

mental data than do those calculated with Klein's (12,6) potential.

For the systems studied (argon, nitrogen, methane, and per-

fluoromethane) (n,6) potentials were found which give better computed

values for the residual properties along the saturated liquid curve

when compared to the properties computed with virial coefficient

potentials. The results suggest that it may be possible to obtain

a suitable value of n for the liquid using only one experimental P,V,T

point for the liquid in addition to the gas phase second virial

coefficient data.










CHAPTER 6

EXCESS PROPERTIES OF THE MET'ANE-PERFLUOROIETHANE SYSTEM
FROM THE ONE-FLUID VAN DER WAALS PRESCRIPTION
IN PERTURBATION THEORY


Introduction


Theories of liquid mixtures are usually based either on two-

parameter corresponding states with one of the mixture components taken
91 ,22
as the reference substance, or on theories of the liquid state
23 24
which employ a two-parameter pair potential. 24 Common to most of

these theories is an adjustable parameter 4 which takes into account

the deviation of the unlike-pair energy parameter c.. from the geometric

mean of the two like-pair energy parameters. For molecules which

differ greatly in character the factor 4 is usually significantly

less than 1.0, e.g., 4 = 0.909 for the CH4 + CF4 interaction, when the

same pair potential or reference fluid is used for both molecules in

a binary mixture. It has been shown (Chapter 5) that methane and

perfluoromethane obey different (n,6) pair potentials and these poten-

tials are not the same ones for gas and liquid phases. It is the

purpose of this chapter to demonstrate the importance of taking into

account the different pair potentials of CH4 and CF4 in predicting

the thermodynamic properties of gas and liquid mixtures of these

molecules.


One-Fluid Perturbation Theory of Mixtures


The theory chosen for the present study is the one-fluid theory

of mixtures with van der Waals prescription for the mixture potential










energy parameters.

3 2 33 + 2 3 (51)
c o = xclo + 2x1x2c1 o12 + x2c22 (51)
m m 1 1 1 1 2 12 12 2 2 2

[2 3 3 2 31
m = xlxl + 2x1X2a12 + x2 1/3 (52)



The perturbation theory of Barker and Henderson14 was used to calculate

the properties of the pure liquids and liquid mixtures reported herein

(see Appendix C) using the respective potential energy functions and

parameters. Except where otherwise noted calculations were made with

first order perturbation theory. Carrying out the calculations to

second order does not significantly change the values of the predicted

excess properties.

Leonard et al.23 have presented a comparison of the one-fluid

theory using van der Waals prescription with both (12,6) Monte Carlo

calculations and the multicomponent version of perturbation theory.

Their results show the van der Waals predictions to be as good as or

better than those obtained with the multicomponent perturbation theory.

As noted by Leonard et al. the van der Waals results are not based on

the (12,6) potential as are the Monte Carlo and perturbation results.

Calculations have been made with the one-fluid van der Waals prescription

using the (12,6) potential in perturbation theory. These calculations,

reported in Table 13 for one temperature and composition, give a better

comparison of the van der Waals theory with other (12,6) theories of
23
mixtures than results reported by Leonard et al. The one-fluid

van der Waals prescription in perturbation theory predicts an excess










Table 13.


Comparison of one-fluid van der Waals
model with Monte Carlo and multicomponent
perturbation theory calculations.


T = 115.80K, P = 0 and x, = x2 = 0.5, E /k = 119.80K,

E2/k = 167.0K, o1 = 3.405 a, 02 = 3.633 12 = (E1 )/2

012 = (ao + 02)/2.


Theory

Monte Carloa


Multicomp. Pert.b

vdW c

vdWd

vdWe


GE (J/mole)

34 10


32.7

42.2

38.1

36.7


VE (cm 3/mole)

-0.54 .20


-0.73

-0.79

-0.70

-0.63


a cDonald's Monte Carlo calculations reported by Leonard et al.23
bTaken from ref. (23).
c 23
CReported by Leonard et al. 23 Not based on (12,6) potential.

Calculated from vdW prescription in second order perturbation theory.
Macroscopic compressibility approximation to second order contribution
to free energy was used.14
Calculated from vdW prescription in first order perturbation theory.











volume and excess free energy almost the same as multicomponent

perturbation theory and within the estimated uncertainty of the Monte

Carlo results.


The Methane-Perfluoromethane System


Earlier calculations for liquid mixtures in this system have

been made under the assumption that all pair interactions follow the

same pair potential model, usually the (12,6) potential. In all such

calculations it is found that it is necessary to employ an empirical

factor ( multiplying the geometric mean of the like-pair energy

parameters. Heretofore has been determined from binary mixture data.

For the methane + perfluoromethane system the value of required to

fit either cross-term second virial coefficient data or excess properties

of liquid mixtures is significantly less than 1.0 (approximately 0.91)

for the 12,6 potential.

It has been demonstrated (Chapter 2) for cross-term second

virial coefficients that when the two like-pair interactions in binary

systems are allowed to obey different (n,6) potentials, the unlike-

pair potential parameters are all the simple geometric mean of the like-

pair parameters. This observation eliminates the need for binary data

in gas mixture calculations. In order to determine whether these

findings are useful for predicting liquid mixture properties, the gas

phase potential energy functions have been used to predict a value for

(. This t is then used to calculate liquid mixture properties with a

theory which requires the use of the same pair potential for all

intermolecular interactions. Such a theory is the one-fluid theory with











van der Waals prescription for the mixture potential parameters. The

unlike-pair parameters E.. and a.. are calculated for any one reference

potential chosen to represent all intermolecular interactions.

The method used to estimate these parameters makes use of

properties of the second virial coefficient at the Boyle temperature.

The Boyle temperature TB is defined as the temperature at which the

second virial coefficient B(T) is equal to zero. The Boyle volume VB

is defined in the usual way, namely,


VB = TB (dB(T)/dT) (53)


When two species (i and j) obey the same pair potential energy function

their potential parameters are related as follows:


(c/k) /(E/k) = T /TB. (54)
i j


i./c. = (V /V ) 1/3 (55)
i *B. (55)


Eqs. (54) and (55) are not exact if the two molecules do not obey the

same pair potential. Eqs. (54) and (55) provide a means of estimating

the parameters of one molecule from those of another when both are to

be represented by the same potential energy function, e.g., the (12,6)

potential.
25
Douslin et al. have shown that the second virial coefficients

of pure CH4 and CF4 as well as the cross-term second virial coefficients

of the CH4 + CF4 pair all fall very nearly on one reduced curve provided

the reducing parameter for the temperature is TB and that for the second

virial coefficient is VB. This correlation is followed in spite of










the fact that all three sets of virial coefficient data obey different

pair potentials as shown in Chapter 2. This near coincidence suggests

that Eqs. (54) and (55) will be good estimates for the parameter ratios.

Eqs. (54) and (55) are commonly employed with the experimental like- and

unlike-pair Boyle temperatures and Boyle volumes to predict the (12,6)

parameters for CF4 + CF, and CH, + CF4 from those of CH4 + CH Methane

is usually taken as the reference fluid for these mixture calculations.

In part of this work the experimental Boyle properties have not

been used. Instead the (n,6) gas phase potential energy functions and

the proposed mixture rules of Chapter 2 have been used with the series

expression for the (n,6) second virial coefficient to calculate what

are presumably good estimates of the experimental Boyle properties.

The gas phase potential energy parameters for the (n,6) potential are

listed in Table 14 for the like and unlike pair. Each of the unlike-

pair parameters is the geometric mean of the respective like-pair para-

meters. This set of parameters fits the experimental cross-term second

virial coefficient for the CH + CF system in the temperature range 273.16
4 4 .

to 623.160K with a root-mean-square deviation of 1.20 cm 3/mole.

Using the Boyle temperatures and volumes estimated by the

potentials in Table 14 with CH4 as the reference fluid, the ratios f

and g in Table 15 were calculated from Eqs. (54) and (55). These ratios

are independent of the pair potential chosen for the reference liquid.

Parameters obtained from these ratios are for the same potential as

that of the reference liquid.

It can be seen from Table 15 that the predicted c = c../(cj)/2

is 0.91688 and results entirely from forcing all of the molecular



















Table 14. Gas phase potential parameters.


Molecular Pair

CH4 + CH

CF4 + CF4

CH4 + CF
4 4


c/k (K)

218.00

373.31

285.27


0 (R)

3.568

4.186

3.865


n

21.00

136.30

53.50


Reference

3

Chapter 2













Table 15. Predicted and experimental potential
parameter ratios.


Molecular
Pair

CH4 + CH

CF4 + CF

CH4 + CF
4 4


f. = E./C -
l i ret


= T /TB
B. e .
i ret


1.000 (1.000)

1.0127 (1.017)

0.92266 (0.917)


gi = c /o = (VB /V )1/3
S 2 .ref B. B .
i ret


1.000 (1.000)

1.245 (1.242)

1.130 (1.132)


aValues in parentheses were determined from the experimental Boyle
properties given in reference (25) and used for calculations in
references (21) and (23).











interactions to obey the same potential (as yet unspecified), since

this value was predicted on the basis of C = 1.0 for the gas phase

potentials. Parameters obtained from both the predicted and experimental

ratios (in parentheses) in Table 15 have been used in the one-fluid

perturbation theory with van der Waals prescription for the mixture

parameters (hereafter referred to as the vdW perturbation model) to

predict the excess properties of the equimolar CH4 + CF4 mixture at 1110K.

The effect on predicted properties of the choice of reference

fluid, source of potential energy parameters, and choice of the potential

energy function of the reference fluid has been studied. Results for

the (12,6) potential with CH4 as the reference fluid are given in Table

16 along with the results obtained by Leland et al.21 and Leonard
23
et al.23 for the same system with their respective theories and

experimental ratios f and g from Table 15. Also included are the one-

fluid perturbation results for E = 1.0 and a.. = (o. + o.)/2 with the

(12,6) potential.

The excess properties predicted by the vdW perturbation model

(Table 16) using the experimental f and g from Table 15 and the (12,6)

potential are nearly the same as the predictions of the Leland, Rowlinson

and Sather theory with experimental f and g. The predictions of the vdW

perturbation model are far superior to the Leonard, Barker and Henderson

multicomponent perturbation theory predictions for the CH4 + CF4

system. The apparent failure of the multicomponent perturbation theory

for this system may be due to the usual assumption in perturbation theory

calculations that the unlike-pair hard-sphere diameter is the arithmetic

mean of the two like-pair diameters. It is significant that this










Table 16. Excess properties of the CH, + CF4
mixture at ll1K, P = 0, xI = x2 = 0.5,
with (12,6) potential.


Experimental Dataa

vdW Perturbation Model
(predicted f and g)

M. Klein CH4 parameters
(1st order theory)
b
1. Klein CH4 parameters
(2nd order theory)

Sherwood and Prausnitz CH
parameters

Leland, Rowlinson and Satherd
with experimental f and g

Leonard, Barker and Hendersone
with experimental f and g

vdW Perturbation Model
(experimental f and g)

Sherwood and Prausnitz CHR
parameters

vdW Perturbation Model
(; = 1.0, 0.. = (o. + o.)/2)

Sherwood and Prausnitz CH4
parameters


V (cm 3/mole)

0.845






0.89



0.97


0.64


0.90


-0.97


1.05


CE (J/mole)

360






209



213


224


279


209





296


-0.79


aTaken from reference 27.
parameters taken from reference 1.
Parameters taken from reference 12.

Taken from reference 21.
Taken from reference 23.











assumption makes the unlike-pair hard-sphere radial distribution

function independent of the mixture rules for the potential energy

parameters.

The use of the predicted f and g in the vdW perturbation model

with the (12,6) potential yields estimates of the excess properties

comparable to the results obtained with experimental f and g although

the excess free energy in the former case is somewhat lower. When

values calculated with the predicted f and g are compared to those

obtained under the assumption ( = 1.0 and 12 = (o1 + 02)/2 in the

liquid mixture it is seen that most of the required deviation of & from

1.0 is accounted for by the predicted values. The success of these

values obtained from the Boyle point correlation on the basis of

C = 1.0 in the gas phase demonstrates that much of the deviation of

C from 1.0 usually observed in gas and liquid mixture calculations for

this system results from the artificial requirement that all pair

interactions in the mixture obey the same pair potential model.

The dependence of the predictions on the source of potential

parameters is seen from Table 16 by comparing the predicted excess

properties using Klein (12,6) CH4 parameters with those using Sherwood

and Prausnitz CH4 parameters. Table 17 illustrates the effect of

varying the reference fluid and the potential energy function on the

predictions of the vdW perturbation model. With the experimentally

determined f and g excess volumes vary from 0.50 to 1.05 cm 3/mole

and excess free energies vary from 296 to 746 3/mole. The highest value

for the excess free energy is the least reliable as it occurs with

the (136.3,6) potential for which reduced liquid densities are so large











Excess properties of the CH4 + CF4 mixture
at 1110K, P = 0, x- = x 2 = 0.5 with various
potentials and reference liquids.


Ref.
Potential Liquid

Predicted f and g

12,6 CH4

12,6 CH 4

12,6 CF4

12,6 CF4


Source for
Parameters



1

12

1

12


%VE (cm /mole)



0.89

0.64

0.63

0.59


GE (J/mole)



209

224

227

226


Experimental

12,6

30,6

21,6

136.3,6


S= 1.0, 02
' 12


f and g

CH4

CF4

CH4

CF4


12

1

3

Chapter 2


= (oI + 02)/2


12 -0.79


Table 17.


1.05

0.50

0.59

0.61


12,6 CH4











that the Percus-Yevick hard-sphere radial distribution functions used

in the calculations are not accurate.

The dependence of the results on the potential energy function

and the reference fluid limits the ability to discriminate between

possible mixture rules which might be proposed for parameters. In

fact, when one examines the wide variations in the predictions for the

actual volumes and residual free energies of a particular pure component

or of the mixture using the various potentials and reference systems,

it is remarkable that the excess properties are as insensitive as they

are to the choice of pair potentials. Some examples of the properties

predicted for the liquids are presented in Table 18.

It would appear from the above results that while two-parameter

theories of liquid mixtures may hold promise in predicting excess

properties of liquid mixtures there is little hope of predicting the

actual magnitude of the properties of mixtures and pure liquids with

a single pair potential. There is consequently considerable incentive

for development of useful theories, such as the Leonard, Barker,
23
Henderson multicomponent perturbation theory, which allow the use

of different pair potentials for the constituent molecules. Rogers

and Prausnitz24 have recently used the Leonard, Barker, Henderson

theory with the three-parameter Kihara potential with considerable

success to predict the magnitudes of both pure and mixed liquid

properties for the argon + neopentane and methane + neopentane systems

with an empirically adjusted C. The values obtained for are 0.994

and 0.988,respectively, when the Kihara potential is used for the liquid

state. .These values of C, which are not far from 1.0, further support










Table 18. Calculated properties of liquids at 111K, P = 0.


All results are for first order perturbation theory with experimental
f and g. Volumes (V) are in cm 3/mole, and residual Gibbs free energy
(Gr) is in J/mole.


Potent jala


(12,6)13

(30,6)12

(136.3,6)2

Exptl. Datab


Ref.
Liouid


CH4


CF4


V Gr V Gr


Pote n tia-


CH 4

CF4

CF4


39.74

27.32

19.22

37.70


-3839

-5914

-9716


75.58

52.08

36.63

49.47


- 3958

- 6095

-10027


Equimolar
mixture


V I r


58.71

40.20

28.23

44.43


-3589

-5534

-9068


aSuperscripts indicate references for potential energy function parameters.
bCalculated from data of Croll and Scott.27










the conclusion reached in this work that much of the deviation of the

observed C from 1.0 that is usually observed results from the use of the

same two-parameter potential for all intermolecular interactions in the

mixture. The value of P. obtained empirically to fit the cross-term

second virial coefficient in the methane + neopentane system with the

(12,6) potential is approximately 0.93.28 This value is much lower

than that found by Rogers and Prausnitz for the liquid state using a

three-parameter potential. The cross-term second virial coefficient

for this system has been successfully predicted assuming j = 1.0 with

the three-parameter (n,6) potential in Chapter 2.


Potential Parameters Independent of Choice of Reference Fluid

As demonstrated previously the predictions of the vdW per-

turbation model are dependent on the choice of the reference fluid and

the particular set of potential energy parameters chosen for that

fluid. Both of these arbitrary choices can be avoided when the gas

phase (n,6) potential energy functions are known for the like- and

unlike-pair interactions. This is accomplished by utilizing Eqs. (56)

and (57). Given two different (n,6) potentials representing the same

molecule, say (n1,6) and (n2,6), the following relationship can be

used to estimate the parameters of the (nl,6) potential from those of

the (n2,6) potential:


(a/k) = (c/k) (T ) /(TB) (56)
n n2 B n B n



oa = a {(V ) /(V ) } (57)
nw n B n2 B n

where T = T /(c/k) and V = V /(No ).
B B B B









Thus, if the (n1,6) potential is chosen to represent all

interactions in the liquid mixture, and the known gas phase potential

for a given interaction is the (n2,6) potential, then the (nl,6)

parameters can be estimated. From the gas phase potential parameters

in Table 14 the liquid phase parameters have been calculated by

Eqs. (56) and (57) for all three pair interactions in a given potential.

Of course these parameters will be in the ratios given as the predicted

ratios in Table 15.- Since only one set of parameters results for each

potential model chosen, the choice of a reference system does not

arise. Parameters for the (12,6) and (30,6) potentials have been

determined in this manner and used to predict the excess properties of

the CH4 + CF4 mixture. Results are given in Table 19. The predicted

excess properties with these (12,6) parameters are comparable to those

in Table 16 obtained with CH, as the reference fluid and (12,6)

parameters of Sherwood and Prausnitz using-the predicted values of

f and g.

In order to determine the experimental Boyle properties of
25
the CH + CF system, Douslin et al. originally fit (n,6) potentials

to the pure and cross-term second virial coefficients of CH CF, and

CH4 + CF,. The experimental Boyle properties were then determined

from these potentials and has been done in the present work with the

gas phase potentials in Table 14. Douslin and coworkers chose to

represent the CH CF, and CH, + CF, interactions by the (28,6),

(500,6) and (30,6) potentials, respectively. These gas phase potentials

have also been used to estimate parameters independent of the reference

system for the various interactions with the (12,6) and (30,6)







60












,-- I -I -. s r 0 1 o
-' CD 0 -.r


-4-4 a
w 0 0



) 0 -. ,. .

ca 3 n L Lr i *



0 .



Q- (i

S141
0) iD
o On
0 o Co o< ONT L) r
00 O 1
0) 0 a w
0. Q





0) m.0 0 La





u-1 E- Cl n r >
01'O







o 0









4 .4
0 0
-7 o< C O n

0) r4 C- C7% *-|














ca C (n u CL) c IT
0) 3






















*0 *
H 0) C-s
U O














-14 4 ,- dc'-
C'O 4I 0.. C







a) 0 )a
00 0 0 WJ
i & i- z U Cl.. U
01-










potentials as was done with the gas phase potentials in Table 14.

The parameters and corresponding excess properties are also given in

Table 19. Results are comparable to those in Table 16 with experimental

f and g. It can be seen that when values are available for gas phase

potential parameters Eqs. (56) and (57) can be used to estimate

parameters for any (n,6) potential to be used for the liquid phase,

eliminating the need for some other source of liquid parameters and

the arbitrary choice of reference fluids.

It should be mentioned that the total geometric mean rule of

Chapter 2 can be applied to Douslin's like-pair gas phase potentials

with good results. The unlike-pair potential parameters estimated in

this way predict the cross-term second virial coefficient for the

CH4 + CF4 pair with a root-mean-square deviation of 2.22 cm /mole.


Averaged Excess Properties


In the above scheme for estimating potential parameters

independent of the reference system one choice still remains. One must

choose the single (n,6) potential to be used in the one-fluid theory

to represent all interactions in the liquid. For the CH, + CF4

mixture the predicted excess properties are probably more sensitive

to this choice than to either of the two choices eliminated by the

above scheme. For mixtures of molecules which obey the same (n,6)

potential in the liquid this potential would be the natural choice for

the one-fluid potential. In the case of CH, + CF, system the choice

is complicated by the fact that the two pure liquids require significantly

different (n,6) potentials in perturbation theory as shown in Chapter 5.











One crude method for taking into account the different pair potentials

obeyed by the different molecules is suggested by what has been done

in the past with corresponding states mixture theories which employ

pure fluid experimental data. With such theories it has been common

practice with binary mixtures of molecules which are very different

to calculate two sets of excess properties, one set with one liquid as

reference and the second set with the other liquid as reference. The

resulting two sets of excess properties can then be mole-fraction

averaged to yield one set of values. Such a method may be used for vdW

perturbation theory calculations by performing the calculations separately

with both liquid potentials (parameters may be independent of the

reference fluid) and mole-fraction averaging the resulting excess

properties. Reasonably good choices for the liquid potentials for

CH4 and CF, would be the (12,6) and (30,6) potentials, respectively.

These average excess properties for the equimolar mixture of CH4 + CF4

were computed from values in Table 19 and are shown in Table 20. From

Table 20 it can be seen that even this crude method of taking into

account the presence of molecules with different pair potentials gives

better estimates of the experimental values than those predicted by

either separate pair potential.

The dependence of predicted excess properties on the single

pair potential chosen points out the necessity of somehow accounting

for the different pair potentials obeyed by different molecules even

when experimentally derived estimates are available for f and g of

like-pair and unlike-pair interactions.











Table 20.


Averaged excess propertiesa (parameters
independent of reference fluid).


V (cm 3/mole) GE (J/mole)


Experimental data

Douslin's gas phase potentials
(Exptl. f and g)

Gas phase potentials from Table 14
(Pred. f and g)


0.845


0.76


0.45


360


335


308


second order perturbation theory used with macroscopic compressibility
approximation to the second order term given in reference 29.










Conclusions

The one-fluid van der Waals prescription for mixture potential

energy parameters in the perturbation theory of Barker and Henderson

reproduces well the Monte Carlo calculations for the (12,6) potential.

The vdW perturbation model predicts the excess properties of

the equimolar CH, + CF4 liquid mixture when CH, is taken as the

reference fluid with the (12,6) potential as well as the Leland,

Rowlinson and Sather theory and better than the Leonard, Barker and

Henderson theory using the like-pair and unlike-pair potential energy

parameters in the ratios obtained from the experimental Boyle proper-

ties. Such predictions are shown to be dependent on a number of

arbitrary choices, such as (1) the reference fluid, (2) the particular

single pair potential for all interactions in the liquid, and (3) the

source of potential parameters.

In view of the large variations in magnitudes of mixture and

pure fluid properties predicted by the various potential energy

functions and reference fluids it appears unlikely that it will be

possible to predict the magnitude of both pure fluid and mixture

properties with a single pair potential.

Probably the most important result of this work is the demon-

stration that most of the deviation of the unlike-pair energy parameter

Eij from the geometric mean rule for the CH4 + CF, system arises from

forcing all pair interactions to obey one form of the pair potential.

It is further shown that knowledge of the single-component gas

phase potentials with the mixture rules proposed in Chapter 2 allows

the prediction of both like-pair and unlike-pair parameters for any











pair potential chosen to represent all interactions in the liquid.

These parameters are independent of the choice of reference fluid, but

they will depend on the particular liquid potential used. The same

procedure for estimating liquid potential parameters is recommended when

both like-pair and unlike-pair gas phase parameters are 'nown as in

the case of the CH4 + CF4 mixture.

Finally, a crude method is illustrated for taking into account

the different pair potentials of the constituent molecules in the

liquid mixture. The method predicts extremely good estimates of the

excess properties of the equimolar CH4 + CF4 mixture.











CHAPTER 7

CORRESPONDING STATES FOR FLUID MIXTURES--NEW PRESCRIPTIONS


Introduction


The most accurate theories of fluid mixtures proposed to date

are the one-fluid and two-fluid van der Waals theories21'22'30 and the
23
Leonard, Henderson, Barker multicomponent perturbation theory. Limited

results for the one-fluid van der Waals theory and the multicomponent

perturbation theory were presented in Chapter 6 (Table 13) and compared

with Monte Carlo results. Extensive comparison of the one-fluid and

two-fluid van der Waals theories with Monte Carlo calculations for

both hard-sphere and (12,6) mixtures has been made by Henderson and

Leonard in references 30 and 31. Results show that the one-fluid

van der Waals theory is superior to the two-fluid van der Waals theory

and the three-fluid theory. In the previous chapter it was shown

(Table 16) that the one-fluid van der Waals (vdW) theory was superior

to the multicomponent perturbation theory for the methane + perfluoro-

methane system.

The one-fluid and two-fluid theories are corresponding states

models in which the thermodynamic properties of a mixture are related

to the properties of one or more imaginary fluids, respectively. The

van der Waals prescription is merely a prescription for calculating

composition-dependent potential energy parameters for the imaginary

fluids. Leland, Rowlinson and Sather21 have examined the thermodynamic

consequences of the one-fluid van der Waals prescription for mixtures

of soft spheres and find it superior to other one-fluid theories.










In the present chapter new prescriptions are presented for

calculating potential parameters for the one or two imaginary fluids

in either the one-fluid or the two-fluid theory. The prescriptions

arise from exact statistical mechanical expressions for gas mixtures.

The new prescriptions will be referred to as the virial coefficient

(vc) prescriptions. The first to be discussed is called the Boyle

prescription (vcB); it reduces to the van der Waals prescription for

fluids which obey the van der Waals equation of state. The second

is called the least squares prescription (vcls) and is the most

general of the vc prescriptions.


The Boyle Prescription (vcB)


Statistical mechanics provides the following expression for

the second virial coefficient B (T) of a binary gas mixture.
m

2 2
B (T) = xlBB(T) + 2x:x 12B(T) + xB2 (T) (58)


The Boyle temperature of the mixture (TB ) is defined analogously
m
to that of a pure component.



B (T )= = xB ) + 2x B 12(TB ) + xB2(TB ) (59)
m B 11 B 2 B
m m m m

The Boyle volume of the mixture is given by


V TB (B /;T) = TB {x2(3B /T) + 2xlx2(CBl2/3T)T
m m B m B B


+ x2(B2/3T) } (60)
m










Under the usual one-fluid assumption that the mixture at a

given composition obeys two parameter corresponding states with some

reference fluid, we have the following relations for the potential

parameters c and a of the one fluid which will represent the mixture
m m

in terms of those of some reference fluid (R).


c /c = T /T (61)
m 1


a /a = (V /V )1/3 (62)
mm RR

For any (n,6) potential TB and V as well as TB and V are readily
m m R R
evaluated using the series expansion for B(T) mentioned in Appendix A

and its temperature derivative. In fact, for any (n,6) potential

chosen to represent the one-fluid mixture it is unnecessary to consider

a particular reference fluid. One can simply use the following

relations.
*
(c/k) m = T /T (63)
m(n,6) B B
m (n,6)

and

S= (V /V* )1/3 (64)
(n,6) B (n,6)

where
*
TB = TB /(/k)(n6) (65)
(n,6) (n,6)

and
V = V /(No3) (66)
B(n,6) n,6) (n,6)


Thus, the one-fluid Boyle (vcB) prescription is contained in

either Eqs. (61) and (62) or Eqs. (63) and (64). The prescription yields










potential parameters for some fluid which, according to the corresponding

states assumption, will have the same thermodynamic properties as the

mixture of given composition at all temperatures and pressures. Unlike

the parameters from the vdW prescription the parameters from the vcB

prescription are exact within the corresponding states assumption.

The two-fluid vcB prescription is readily derived by writing

Eq. (58) as


B (T) = x[1Bx1 (T) + x2B 2(T)] + x2[x2B2(T) + xlB 2(T)]

(67)
or

B (T) = xB'(T) + x B'(T) (68)
m 1 22

Here the mixture virial coefficient is given as that of an ideal mixture

of two imaginary fluids with virial coefficients B'(T) and B2(T). The

Boyle properties of the two fluids determine the potential parameters

of these fluids to be used in two-fluid theory calculations.


Relation of the vcB Prescription to the vdW Prescription

32
For a single component van der Waals gas the second virial

coefficient is given by


B(T) = b a/RT (69)


where a and b are the usual van der Waals constants.

The Boyle temperature is given by


TB = a/Rb ,


(70)










and the Boyle volume is given by


VB = b .


For a one-fluid mixture of van der Waals gases


T = a /Rb
B m m
m


V = b "
B m
m

Writing Eq. (59) for a van der Waals fluid,


0 = x1(b1 al/RTB )
m


+ 2x1x2(b12 al2/RTB ) +
m


2


- a2/RTB )


Rearranging we have

2 2 2 2
T = (x2al/R + 2x x2a /R + x2a2/R)/(x b1 + 2x 2b b + x2b2)
m


(75)


Using Eqs. (70) and (71) we obtain


2
TB = (XlTB VB
m 1 1


+ 2x TB VB
12 12


2 2
2 B 2 B x1 B
2 2 1


+ 2x x2 VB
-1 2 BI


2 2
VB = T (xlal/RTB
m m m


2
+ 2x x2al2/RTB
m


(76)
(77)


+ x2a2/RTB ) .
m


From Eqs. (76) and (77) we have


VB = (Xl BI
m 1B


+ 2xx V + 2
12B2 2B2


Eqs. (76) and (78), when written in terms of E/k and o, are immediately

recognized as the van der Waals mixture rules given in Eqs. (51) and


(52) of Chapter 6.


(71)


(72)


(73)


(74)


9
2 B2


(78)











Thus it is seen that the vcE and vdVJ prescriptions are identical

for a van der Waals fluid, and the vcB prescription is for real

systems the analog of the vdW prescription for van der Waals systems.


The vcE Prescription for (12,6) Systems


It should be pointed out that when evaluating the one-fluid

or two-fluid vdW theories one is testing a combination of the one-fluid

or two-fluid corresponding states assumption and the particular van der

WJaals prescription, since the van der Waals prescription would not be

exact even if the corresponding states assumption were correct. With

the vcB prescription, however, one is testing independently the

corresponding states assumption since the vcE prescription is exact

within the assumption of corresponding states. The most commonly

used model for the intermolecular pair potential in real systems is the

Lennard-Jones (12,6) potential. The properties of dilute Lennard-Jones

gases in both pure and mixed states can be calculated from the series

expansion for the (12,6) second virial coefficient. For pure and mixed

dense liquid properties Monte Carlo computer simulation results are

available for zero pressure. Monte Carlo calculations of McDonald at

115.80K for potential parameters characteristic of the argon + krypton
73
mixture have been reported by Leonard, Henderson and Earker. Singer
33
and Singer have made similar calculations for other mixtures at

970K. In the present work the Monte Carlo interpolation formulae of
33
Singer and Singer have been used to calculate the properties of pure

(12,6) fluids as well as one-fluid and two-fluid mixtures using the vcB

and vdW prescriptions. The resulting excess properties are compared to










the Monte Carlo estimates of both McDonald (Table 21) and Singer and

Singer (Tables 22-24).

As has already been shown30,31 for the MdW prescription, the

one-fluid vcB prescription is superior to the two-fluid vcB prescription.

As a general rule the excess properties predicted by the one-fluid vcB

prescription are either equal to or slightly more negative than those

predicted by the vdW prescription. In almost every case, however, there

is little difference between the vcB and vdW results. Both one-fluid

prescriptions agree remarkably well with the Monte Carlo calculation.

Another comparison of the vcE and vdW prescriptions can be

made for gas phase mixtures using the second virial coefficient.

For a given set of like-pair and unlike-pair (12,6) parameters the

virial coefficient of an equimolar mixture can be calculated as a

function of temperature. Then the one-fluid vcB and vdW prescription

can be used to estimate the mixture virial coefficient. Results for

two sets of parameters are given in Table 25. The vcB prescription

is in these two cases seen to be superior to the vdW prescription,

but again it is seen that the difference between the two prescriptions

is not great.


The vcE Prescription for Mixtures of Molecules
with Different (n,6) Potentials


One of the most serious limitations of the one-fluid vdW theory

or one-fluid vcB theory as outlined above is the requirement that all

molecules and the mixture itself obey the same two-parameter

corresponding states or two-parameter pair potential. It was demonstrated

in Chapter 6 using the one-fluid vdW prescription that in order to make











Table 21. Comparison of one-fluid and two-fluid
prescriptions with Monte Carlo calculations.



T = 115.8K, P = 0 and xI = >:2 = 0.5, cl/k = 119.80K,

c2/k = 167.00K, o1 = 3.405 t, 02 = 3.633 1, c12 = (cLc2)2/2

.12 = (a1 + 02)/2


Theory


Monte Carloa


vcBb


GE (J/mole)


34 i 10


one-fluid


HE (J/mole)


-34 i 40


-34


VE (cm 3/mole)


-0.54 .20


-0.50


-0.32


two-fluid

vdWc
one-fluid

two-fluid

Multicomp. Pert.


-0.30

-0.73


alIcDonald's Monte Carlo calculations reported by Leonard et al.23
bCalculated from vcE prescription using Singer's33 Monte Carlo inter-
polation formulae for properties of (12,6) fluids at P = 0.
cCalculated from vdW prescription using Singer's33 interpolation
formulae.
dTaken from reference 23 for multicomponent perturbation theory.










Table 22. Comparison of excess free energy (G E) from
vcB and vdW prescriptions with Monte Carlo
(MC) calculations.a


T = 970K, P = 0 and

12 = (a0 + 02)/2 =


xI = x2 = .5, F2 = (E12) 1/2 = 133.5-K,

3.596. GE is in J/mole.


011/012


I I


1/12 = .810

GE (MC)
GE (vcB)
one-fluid
two-fluid

GE (vdW)
one-fluid
two-fluid

c11/E12 = .900

GE (MC)

GE (vcB)
one-fluid
two-fluid

GE (vdW)
one-fluid
two-fluid
E /C 12 = 1.000


GE (MC)
GE (vcB)
one-fluid
two-fluid
CE (vdW)
one-fluid
two-fluid


1.00


1.03


1.08


130 14


123
118


35 7


1 1


184 14


188
149



209
160




60 7



61
44



66
47



-2 2


-2
-1


-2
-1


1.12


(345)


342
225



362
236




122 10



121
71



127
74



-21 2


-35
-22


-35
-22


(273)


280
205



301
107




97 8



100
62



105
65



-12 + 2


-15
-10


-15
-10












Table 22 (Continued)



S11/o12 1.00 1.03 1.08 1.12


E 11/E2 = .111

GE (MC) 35 7 3 7 -50 t 10 -95 10

GE (vcB)
one-fluid 31 -3 -69 -130
two-fluid 29 12 -23 -56

GE (vdW)
one-fluid 37 3 -64 -125
two-fluid 32 15 -20 -53



aMC calculations of J. V. L. Singer and K. Singer.33 Values in
parentheses were estimated in reference 33 where no MC calculations
were available. Properties of all (12,6) fluids for evaluation
of vcB and vdW prescriptions were estimated using the MC interpolation
formulae in reference 33.











Table 23. Comparison of excess enthalpy (H E) from
vcB and vdW prescriptions with Monce Carlo
(MC) calculations.a


T = 970K, P = 0 and x1 = x2 = 0.5, 12 = (c1E2)1/2

012 = (al + 02)/2 = 3.596. HE is in J/mole.


= 133.50K,


011 /12


S/11 E2 = .810

HE (MC)

HE (vcB)
one-fluid
two-fluid

HE (vdW)
one-fluid
two-fluid

SI/12 = .900

HE (MC)

HE (vcB)
one-fluid
two-fluid

HE (vdW)
one-fluid
two-fluid
E11/cE2 = 1.000

HE (MC)
HE (vcB)
one-fluid
E two-fluid
H (vdW)
one-fluid
two-fluid


1.00


124 34


111
111




29 20


28
28



1 10


0
0

0
0


1.03


1.08


4 4 4


(163)



174
142



206
159




60 7



67
47



75
51



5 12


0
0

0
0


(336)



330
238



362
111




167 27



145
87



153
91



54 20


0
0

0
0


1.12


(500)



453
286



484
301




263 35



206
118



214
122



101 24


0
0

0
0





77





Table 23 (Continued)


011/a12


rl1/rl2 = 1.111


HE (MC)

HE (vcB)
one-fluid
two-fluid

HE (vdW)
one-fluid
two-fluid


1.00


34 i 20


1.03


1.08


4 4 4


-12 20



-28
0



-20
4


-48 t 27



-106
-39



-98
-35


aMC calculations of J. V. L. Singer and K. Singer.33 Values in
parentheses were estimated in reference 33 where no MC calculations
were available. Properties of all (12,6) fluids for evaluation of
vcB and vdW prescriptions were estimated using the MC interpolation
formulae in reference 33.


1.12


-40 t 35



-167
-70



-159
-66









Table 24. Comparison of excess volume (V ) from vcB
and vdW prescriptions with Monte Carlo (MC)
calculations. a


T = 970K, P = 0 and xI = x = 0.5, c12 = (c2)
2 1


012 = (ao + 02)/2 = 3.596.


= 133.50K,


VE is in cm 3/mole.


11


/12 = 0.810


VE (MC)

VE (vcB)
one-fluid
two-fluid


VE (vdW)
one-fluid
two-fluid


11/E12 = 0.900


VE (MC)

VE (vcB)
one-fluid
two-fluid

VE (vdW)
one-fluid
two-fluid

c11/c12 = 1.000


VE (MC)

VE (vcB)
one-fluid
two-fluid

VE (vdW)
one-fluid
two-fluid


1.00


-0.61 t 0.19


-0.76
-0.52



-0.70
-0.49


-0.15 t 0.09


-0.19
-0.13



-0.17
-0.12


0 0.05


1.03


-0.87 0.19


-0.94
-0.71



-0.89
-0.68


-0.25 0.09


-0.30
-0.24



-0.28
-0.23


-0.01 0.05


-0.05
-0.05



-0.05
-0.05


1.08


(-1.29)


-1.46
-1.21



-1.40
-1.18


-0.43 0.11


-0.68
-0.62



-0.67
-0.62


-0.02 t 0.06


-0.32
-0.32



-0.32
-0.32


1.12


-1.61)



-2.05
-1.80



-1.99
-1.77


-0.59 0.17


-1.18
-1.11



-1.16
-1.11


-0.02 0.12


-0.72
-0.72



-0.72
-0.72










Table 24 (Continued)


a11/ 12


11 i12 1.111

VE (MC)


1.00


-0.13 t 0.09


VE (vcB)


one-f luid
two-fluid


VE (vdW)


one-fluid
two-fluid


-0.19
-0.13



-0.18
-0.12


1.03


-0.09 0.09



-0.17
-0.11



-0.15
-0.10


1.08


0.00 0.11



-0.33
-0.28



-0.32
-0.27


1.12


0.09 0.17



-0.65
-0.59



-0.64
-0.59


.*IC calculations of J. V. L. Singer and K. Singer.33 Values in
parentheses were estimated in reference 33 where no MC calculations
were available. Properties of all (12,6) fluids for evaluation of
vcB and vdW prescriptions were estimated using the MC interpolation
formulae in reference 33.










Table 25. Comparison of the one-fluid vcB and vdW
prescriptions for equimolar mixtures of
(12,6) gases.

o 0
E 1/k = 119.80K, oI = 3.405 A, E2/k = 167.000K, 02 = 3.633 A,


E12 = (E 2) 1/2


, o12= (o1 + a2)/2


T(K)
100.0
200.0
300.0
400.0
500.0
600.0
700.0
rms % dev


Mixture Second Virial Coeffi-
cient (cm3/mole)
Exact vcBa vdWb
-264.47 -262.41 -261.61
- 76.91 76.75 76.49
- 30.92 30.89 30.74
- 10.56 10.56 10.45
0.74 0.74 0.83
7.83 7.83 7.91
12.64 12.64 12.70
0.30 4.66


Best fit (vcls)c
-262.88
76.89
30.95
10.58
0.74
7.85
12.66
0.27


1 /12 = 1.111, 01/012 = 1.08, 12

S12/k = 133.50K, a = 3.596 A
12 12


= 1 2, c12 = ( + o2)/2


T(K)
100.0
200.0
300.0
400.0
500.0
600.0
700.0
rms % dev


Mixture Second Virial Coeffi-
cient (cm3/mole)
Exact vcBd vdWe
-255.51 -254.82 -254.49
- 73.66 73.61 73.50
- 28.09 28.08 28.50
- 7.82 7.82 7.77
3.45 3.45 3.49
10.52 10.52 10.55
15.31 15.31 15.33
0.14 0.55


Best fit (vcls)f
-254.97
73.66
28.10
7.82
3.45
10.53
15.32
0.09


a /k = 143.8560K


b /k
m


= 143.5700K


c /k = 143.862K
exact values.
d /k = 135.659K
a'


o = 3.5205
m

o = 3.5210
m

o = 3.5225 2, obtained from least squares fit of
m


o = 3.6067 A
m





81





Table 25 (Continued)


eE /k = 135.547K
m

c /k = 135.665K
m
exact values.


o = 3.6075
m

c = 3.6080 A, obtained from least squares fit of
m











accurate estimates of the excess properties of mixtures composed of

molecules with very different (n,6) potentials some method must be

used to account for the presence of different potentials. In this

section two methods are examined which allow for the different

potentials when the vcB prescription is used.


Mole-Fraction Averaged Excess Properties


The first of these methods is analogous to that used in

Chapter 6. The method requires knowledge of liquid phase (n,6)

potentials for pure components along with either gas phase or liquid

phase unlike-pair potentials. To further illustrate the use of the

vcB prescription only like-pair and unlike-pair Boyle temperatures and

Boyle volumes are used. As in Chapter 6 the CH4 + CF, mixture is used

for demonstration purposes.

For the pure liquids CH, and CF4 (n,6) potentials have been

determined using second order perturbation theory as suggested in

Chapter 5 by adjusting n so that predicted pure liquid densities equaled

experimental densities at 1110K. The experimental densities were

assumed to be at zero pressure. Parameters e and a for the (n,6)

potentials were estimated using pure fluid Boyle properties calculated
25
from Douslin's gas phase potentials for CH, and CF The macroscopic

compressibility approximation to the second order term in perturbation

theory is used. In this chapter the Carnahan-Starling hard-sphere

equation of state and free energy has been used with Percus-Yevick

hard-sphere radial distribution functions. Potentials found are given

in Table 26 along with calculated molar volume and residual free energy.




83




Table 26. Liquid phase potentials for CH4 and CF .


T = 111K, P = 0, experimental densities taken from Table 18.


3
V(cm /mole)
Exptl Calcd

37.70 37.70


49.47


49. 47


Calculated
GCr(J/mole)

-4226.2

-6688.3


Liquid

CH4


CF4


n

13.2

34.2


c/k(K)

161.105

273.106


0(R)

3.7172

4.3141










Using both like-pair and unlike-pair gas phase potentials given

by Douslin the Boyle temperature and Boyle volume of the CH4 + CF4
4 *4
mixture are calculated at various compositions. Then all interactions

in the liquid are assumed to obey the (13.2,6) potential found for

liquid methane. Using the mixture Boyle properties one-fluid (13.2,6)

potential parameters for the mixture at various compositions are

estimated. With these parameters the molar volume and residual free

energy of the mixture is calculated for each composition at zero

pressure and 1110K. The (13.2,6) potential parameters of pure CF4 are

also estimated and used to calculate the pure liquid CF4 properties at

the same temperature and pressure. The excess properties are then

calculated. In the same way excess properties are calculated assuming

all interactions obey the (34.2,6) potential found for pure liquid CF .

The resulting two sets of excess properties are then mole-fraction

averaged to obtain one set of excess properties for the various mixture

compositions. The two sets of excess properties for the two respective

(n,6) potentials are given in Table 27. The mole-fraction averaged

excess properties are given in Table 28. Results are also given in

Figure 2 for excess free energy. The mole-fraction averaged excesses

are the best estimates of the experimental values obtained to date with

any theory which requires the use of a single pair potential for all

intermolecular interactions.


Three-Parameter One-Fluid Theory


While it is possible to obtain good estimates of the excess

properties of mixtures composed of molecules with different (n,6)

potentials the preceding results again demonstrate that it is not











Mixture properties and excess propertiesa for
the CH, + CF4 system with the (13.2,6) and
(34.2,6) potentials. T = 1110K, P = 0, vcB
prescription,b volumes in cm /mole, free energies
in J/mole.


(13.2,6) Potential
V G r E
m m V

41.44 -4078.9 0.34
46.74 -3965.1 0.71
53.36 -3942.0 0.92
55.70 -3960.0 0.98
60.05 -4024.1 0.81
66.65 -4188.0 0.54


GE

133.0
241.5
278.7
270.7
232.1
121.4


V
m

28.47
32.05
36.58
38.19
41.22
45.86


(34.2,6) Potential
G r
m V

-6257.5 0.16
-6083.9 0.34
-6049.0 0.47
-6076.3 0.48
-6174.0 0.45
-6422.9 0.27


Pure Fluid Properties


Liquid

CH4

CF4


(13.2,6) Potential
V Gr

37.70 -4226.2


71.73


-4362.0


(34.2,6) Potential
V Gr

25.95 -6481.9


49.47


-6688.3


aExcess properties calculated as discussed in Appendix C.

Required Boyle properties calculated from Douslin's gas phase like-
pair and unlike-pair potentials.25


Table 27.


x
CF4


.100
.245
.432
.500
.630
.835


GE

219.0
399.6
462.5
449.9
386.0
203.0










Table 28. Averaged excess propertiesa from
vcB prescription.


T = 1110K, P = 0


VE (cm 3/mole)


GE (J/mole)


Experimental b


0.37
0.71
0.86
0.845
0.74
0.39


Calculated

0.33
0.62
0.72
0.73
0.58
0.31


Experimentalb

153.5
297.7
364.4
359.9
315.5
167.7


Calculated

141.6
280.2
358.1
360.3
329.1
189.5


alfole-fraction averaged excess properties calculated as follows:


E E
V = x V E
AVG CH, (13.2,6)

E E
AVG C= H (13.2,6)


+ x V
CF4 (34.2,6)

E
+ x G(34.2,6)
CF4 (34.2,6)


bExperimental excess properties at 1110K estimated from data of Croll
and Scott as described in Appendix D.


CF 4
-4

.100
.245
.432
.500
.630
.835
















n (13.2,6) Potential


0 (34.2,6) Potential O Three-parameter


0.0


1.0


0.5

Mole fraction CF XCF4
C4 CF


Figure 2. Excess free energy of the CH, + CF4 system
at 111K, P = 0.


450


350





250





150


1-1
0






E

X





U
x
w-


0 Averaged




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