
Citation 
 Permanent Link:
 https://ufdc.ufl.edu/UF00097604/00001
Material Information
 Title:
 Intermolecular pair potentials in the theoretical description of fluids and fluid mixtures
 Creator:
 Calvin, Donald William, 1945
 Place of Publication:
 Gainesville, Fla.
 Publisher:
 University of Florida
 Publication Date:
 1972
 Copyright Date:
 1972
 Language:
 English
 Physical Description:
 xiii, 143 leaves. : illus. ; 28 cm.
Subjects
 Subjects / Keywords:
 Entropy ( jstor )
Fluids ( jstor ) Geometric mean ( jstor ) Liquids ( jstor ) Molecules ( jstor ) Perturbation theory ( jstor ) Trucks ( jstor ) Vapor phases ( jstor ) Virial coefficients ( jstor ) Volume ( jstor ) Chemical Engineering thesis Ph. D Dissertations, Academic  Chemical Engineering  UF Fluid mechanics ( lcsh ) Statistical mechanics ( lcsh ) City of Gainesville ( local )
 Genre:
 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Thesis:
 Thesis  University of Florida.
 Bibliography:
 Bibliography: leaves 140142.
 Additional Physical Form:
 Also available on World Wide Web
 General Note:
 Typescript.
 General Note:
 Vita.
Record Information
 Source Institution:
 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 022659417 ( AlephBibNum )
13943639 ( OCLC ) ADA4947 ( NOTIS )

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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 PercusYevick hardsphere 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 DYMONDALDER PAIR POTENTIAL.
Introduction ..................................
UnlikePair Potential .........................
Semitheoretical Mixture Rules .................
Empirical Mixture Rules .......................
The DymondAlder Potential ....................
Conclusions ...................................
3. A MIXTURE RULE FOR THE EXPONENTIAL6 POTENTIAL.....
Introduction ..................................
Mixture Rules .................................
Conclusions ...................................
4. THE RELATIONSHIP BETWEEN THE MIE (n,6) POTENTIAL
AND EXPONENTIfAL6 POTENTIAL ........................
Introduction ..................................
Equivalence of Potential Parameters...........
UnlikePair 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
BarkerHenderson 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 METHANEPERFLUOROMETHANE
SYSTEM FROM THE ONEFLUID VAN DER WAALS PRESCRIP
TION IN PERTURBATION THEORY......................... 44
Introduction................................... 44
OneFluid Perturbation Theory of Mixtures ..... 44
The MethanePerfluoromethane System........... 47
Potential Parameters Independent of Choice
of Reference Fluid ............................ 58
Averaged Excess Properties .................... 61
Conclusions................................... 64
7. CORRESPONDING STATES FOR FLUID MIXTURESNEW
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
MoleFraction Averaged Excess Properties. 82
TABLE OF CONTENTS (Continued)
Page
ThreeParameter OneFluid Theory ......... 84
The Virial Coefficient Least Squares (vcls)
Prescription................................... 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 Unlikepair parameters............................. 22
6 Cross virial coefficient B_2....................... 23
7 Second virial coefficients of pure gases predicted
with (n,6) potential using exponencial6 parameters 26
8 Exponential6 potential parameters................. 28
9 Crossterm 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 onefluid 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 onefluid and twofluid 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 onefluid 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 threeparameter vcB
prescription....................................... 91
30 Comparison of the onefluid 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 Likepair and unlikepair (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 Likepair and unlikepair (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 Bestfit parameters for equation (E1)............. 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 likepair
potential energy (mutual energy of a pair of molecules of the same
species) and the unlikepair 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 unlikepair parameters in terms of the like
pair parameters for the Mie (n,6) potential energy model are examined
for their abilities to predict crossterm 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 crossterm second virial coefficient. The geometric mean rule for
the energy and distance parameters is also shown to perform well for
predicting the crossterm 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 crossterm second virial coefficients for the
exponential6 potential. Relationships developed between parameters
for the Mie (n,6) potential and the exponential6 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
unlikepair parameters for the exponential6 potential should follow
the total geometric mean rule. This set of rules predicts good values
for the crossterm 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 onefluid or twofluid
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 onefluid theory. The method
used obviates the empirical determination of the unlikepair potential
energy parameters. A method is developed for obtaining both like
pair and unlikepair 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 exponeritial6 potential and the Dymond
Alder potential. Likepair 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 unlikepair potential parameters which characterize the interaction
between a pair of molecules of different species from the likepair
parameters of the respective molecules. The resulting unlikepair
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 unlikepair parameters in the gas phase can be
used indirectly to estimate unlikepair 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 compositiondependent potential
parameters for use in the onefluid and twofluid 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 unlikepair
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 DYMOINDALDER 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 threeparameter 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 n6 EO6
A = (21
66 n6
and
B n n6 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
n6
and
6c r
B n6 (5)
The most familiar form of the Mie (n,6) potential is the LennardJones
potential in which n is 12.
UnlikePair Potential
From the leading term in the London theory of dispersion forces
the unlikepair attraction coefficient A.. (Note that the double
subscript "ij" will refer to the unlikepair intermolecular interaction
and the single subscript "i" or "j" will refer to the likepair
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 unlikepair 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 ThomasFermiDirac 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 [(ni6)(nj6)]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 [(ni6) (nj6)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) 6n. .
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 crossterm 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 unlikepair energy and distance parameters
should be the simple geometric mean of the respective likepair
parameters. Such simplicity while appealing is hardly a basis for
choosing these forms. However, results of a least squares fit of
the crossterm second virial coefficient for the CH, + CF, system
shown in Table 2 indicate that the bestfit 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
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 ) 1 r
3o > E
41 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 
11
cu
Cl 1 ,
o e
Q ifl
Q~i 0
Cl 
TI Cu C
C1 Cu Cu
') U) U)
: i 2
Table 2. Unlike potential parameters.
System .
CH4 + Ar
CH + C2H6
CH4 + C(CH3 )4
4 i4
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 crossterm 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 crossterm 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 rootmeansquare 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 rootmeansquare deviation of only 1.06 cm3/mole. 4
a rootmeansquare 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 DvrmondAlder 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 unlikepair 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 LennardJones (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 likepair 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 twoparameter potential for the likepair 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
crossterm 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 LennardJones (12,6) potential and
the Sutherland (=,6) potential. Where the third virial coefficient
of a pure system has been calculated with both the LennardJones 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 LennardJones and Sutherland potentials.
CHAPTER 3
A MIXTURE RULE FOR THE EXPONENTIAL6 POTENTIAL
Introduction
It was shown in the previous chapter that for the Mie (n,6)
intermolecular pair potential the unlikepair parameters (c. .,o.,n..)
ii ij 12
are the geometric mean of the respective likepair parameters for second
virial coefficients. It is shown below that these mixture rules can be
extended to define mixture rules for the exponential6 potential,
(r) = (6/) exp[a(l r/r )] (r /r) (21)
(16/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(lo/r ) (22)
m m
Mixture Rules
Since c and a in the exponential6 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 = Brn (23)
where B is constant. From (23) we have
d rn 4 ep.
d nr
For the exponential6 potential the form is
,(r)rep = Kebr
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_ /dnr)(dn rep/d nr)) 1/2
which for the eponentia potential can be written as
which for the exponential6 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 unlikepair parameters for three
binary mixtures for which exponential6 likepair 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 likepair parameters
12
were taken from Sherwood and Prausnitz. Results of the prediction
of the crossterm second virial coefficient with these parameters
are given in Table 6.
Conclusions
In two of the three cases the rootmeansquare deviations for
the exponential6 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 crossterm 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
exponential6 potential.
Table 5. Unlikepair 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
EXPONENTIAL6 POTENTIAL
Introduction
Hanley2 and Klein2,1 have recently shown that five of the
common threeparameter 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)
(n6) m m
and the exponential6 potential
4(r) = (6) {exp[a(lr/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 exponential6 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 exponential6 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 (exp6) (35)
r = r (36)
(n,6) (exp6)
n (n,6) = (exp6) (37)
This suggests that where sets of the threepotential 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,
exponential6 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 exponential6 parameters.
Number of
Temperature experimental m Data
Gas range (K) points dev (cm /mole) ref.
CF, 273.16623.16 15 0.11 b
C(CH ), 303.16548.16 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 273.16623.16 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;;ponential6
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 exponential6 potentials.
UnlikePair Parameters
The implied equivalence of the potential energy functions
suggests that the mixture rules for the exponential6 potential
parameters should be even simpler than those suggested in Chapter 3.
The exponential6 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 exponential6
potential and the parameters in Table 8 to predict the crossterm
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 exponential6 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 exponential6
potential as with the (n,6) potential, but they are much better than
Table 8. Exponential6 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 In6
0 = r 
m n
for the exponential6 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. Crossterm second virial coefficient.a
System
CH4 + Ar
CH4 + C2H6
CH4 + CF
CH4 + C(CH3)4
N2 + Ar
N2 + C2 6
rms dev
(exp6)
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
aCrossterm 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(exp6). 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 exponential6 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) = (exp6) (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
exponential6 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 exponential6 potentials are sufficiently alike
with respect to the prediction of second virial coefficients that
31
sets of the three exponential6 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 exponential6 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 threeparameter potentials
(Kihara, Mie (n,6), exponential6 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 crossterm 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 LennardJones (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 manybody 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 n6
*(43)
n6 6{6
BarkerHenderson 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
hardsphere 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. PercusYevick hardsphere 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 hardsphere 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 BarkerHenderson 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 nvalues 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
11
0 0
6.0
5.0
4.0
3.0
2.0
1.0
0.0
130
Figure 1.
Conclusions
The BarkerHenderson 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'ANEPERFLUOROIETHANE SYSTEM
FROM THE ONEFLUID 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 twoparameter pair potential. 24 Common to most of
these theories is an adjustable parameter 4 which takes into account
the deviation of the unlikepair energy parameter c.. from the geometric
mean of the two likepair 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.
OneFluid Perturbation Theory of Mixtures
The theory chosen for the present study is the onefluid 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 onefluid
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 onefluid 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 onefluid
van der Waals prescription in perturbation theory predicts an excess
Table 13.
Comparison of onefluid 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 MethanePerfluoromethane 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 likepair energy
parameters. Heretofore has been determined from binary mixture data.
For the methane + perfluoromethane system the value of required to
fit either crossterm 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 crossterm second
virial coefficients that when the two likepair 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 onefluid theory with
van der Waals prescription for the mixture potential parameters. The
unlikepair 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 crossterm 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
unlikepair 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 likepair para
meters. This set of parameters fits the experimental crossterm second
virial coefficient for the CH + CF system in the temperature range 273.16
4 4 .
to 623.160K with a rootmeansquare 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 onefluid
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 unlikepair hardsphere diameter is the arithmetic
mean of the two likepair 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 unlikepair hardsphere 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 PercusYevick hardsphere 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 twoparameter
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 threeparameter 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 twoparameter potential for all intermolecular interactions in the
mixture. The value of P. obtained empirically to fit the crossterm
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
threeparameter potential. The crossterm second virial coefficient
for this system has been successfully predicted assuming j = 1.0 with
the threeparameter (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
unlikepair 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 usingthe 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 crossterm 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
44 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
u1 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) Cs
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 likepair gas phase potentials
with good results. The unlikepair potential parameters estimated in
this way predict the crossterm second virial coefficient for the
CH4 + CF4 pair with a rootmeansquare 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 onefluid 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 onefluid 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 molefraction
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 molefraction 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
likepair and unlikepair 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 onefluid 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 likepair and unlikepair 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 unlikepair 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 singlecomponent gas
phase potentials with the mixture rules proposed in Chapter 2 allows
the prediction of both likepair and unlikepair 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 likepair and unlikepair 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 MIXTURESNEW PRESCRIPTIONS
Introduction
The most accurate theories of fluid mixtures proposed to date
are the onefluid and twofluid van der Waals theories21'22'30 and the
23
Leonard, Henderson, Barker multicomponent perturbation theory. Limited
results for the onefluid 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 onefluid and
twofluid van der Waals theories with Monte Carlo calculations for
both hardsphere and (12,6) mixtures has been made by Henderson and
Leonard in references 30 and 31. Results show that the onefluid
van der Waals theory is superior to the twofluid van der Waals theory
and the threefluid theory. In the previous chapter it was shown
(Table 16) that the onefluid van der Waals (vdW) theory was superior
to the multicomponent perturbation theory for the methane + perfluoro
methane system.
The onefluid and twofluid 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
compositiondependent potential energy parameters for the imaginary
fluids. Leland, Rowlinson and Sather21 have examined the thermodynamic
consequences of the onefluid van der Waals prescription for mixtures
of soft spheres and find it superior to other onefluid theories.
In the present chapter new prescriptions are presented for
calculating potential parameters for the one or two imaginary fluids
in either the onefluid or the twofluid 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 onefluid 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 onefluid 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 onefluid 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 twofluid 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 twofluid 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 onefluid 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 onefluid
or twofluid vdW theories one is testing a combination of the onefluid
or twofluid 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
LennardJones (12,6) potential. The properties of dilute LennardJones
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 onefluid and twofluid 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 2224).
As has already been shown30,31 for the MdW prescription, the
onefluid vcB prescription is superior to the twofluid vcB prescription.
As a general rule the excess properties predicted by the onefluid 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 onefluid
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 likepair and unlikepair (12,6) parameters the
virial coefficient of an equimolar mixture can be calculated as a
function of temperature. Then the onefluid 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 onefluid vdW theory
or onefluid vcB theory as outlined above is the requirement that all
molecules and the mixture itself obey the same twoparameter
corresponding states or twoparameter pair potential. It was demonstrated
in Chapter 6 using the onefluid vdW prescription that in order to make
Table 21. Comparison of onefluid and twofluid
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
onefluid
HE (J/mole)
34 i 40
34
VE (cm 3/mole)
0.54 .20
0.50
0.32
twofluid
vdWc
onefluid
twofluid
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 = 133.5K,
3.596. GE is in J/mole.
011/012
I I
1/12 = .810
GE (MC)
GE (vcB)
onefluid
twofluid
GE (vdW)
onefluid
twofluid
c11/E12 = .900
GE (MC)
GE (vcB)
onefluid
twofluid
GE (vdW)
onefluid
twofluid
E /C 12 = 1.000
GE (MC)
GE (vcB)
onefluid
twofluid
CE (vdW)
onefluid
twofluid
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)
onefluid 31 3 69 130
twofluid 29 12 23 56
GE (vdW)
onefluid 37 3 64 125
twofluid 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)
onefluid
twofluid
HE (vdW)
onefluid
twofluid
SI/12 = .900
HE (MC)
HE (vcB)
onefluid
twofluid
HE (vdW)
onefluid
twofluid
E11/cE2 = 1.000
HE (MC)
HE (vcB)
onefluid
E twofluid
H (vdW)
onefluid
twofluid
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)
onefluid
twofluid
HE (vdW)
onefluid
twofluid
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)
onefluid
twofluid
VE (vdW)
onefluid
twofluid
11/E12 = 0.900
VE (MC)
VE (vcB)
onefluid
twofluid
VE (vdW)
onefluid
twofluid
c11/c12 = 1.000
VE (MC)
VE (vcB)
onefluid
twofluid
VE (vdW)
onefluid
twofluid
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)
onef luid
twofluid
VE (vdW)
onefluid
twofluid
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 onefluid 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.
MoleFraction 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 unlikepair potentials. To further illustrate the use of the
vcB prescription only likepair and unlikepair 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 CarnahanStarling hardsphere
equation of state and free energy has been used with PercusYevick
hardsphere 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 likepair and unlikepair 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 onefluid (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 molefraction
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 molefraction averaged
excess properties are given in Table 28. Results are also given in
Figure 2 for excess free energy. The molefraction 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.
ThreeParameter OneFluid 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 unlikepair 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
alfolefraction 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 Threeparameter
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
11
0
E
X
U
x
w
0 Averaged

Full Text 
PAGE 1
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 IN PARTIAL FULFILLMENT OF THE REQUIREMENTS OF THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1972
PAGE 2
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. W. Dufty, Dr. J. P. O'Connell and Dr. W. 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 BMN for evaluating the second virial coefficient for the Mie (n,6) potential, to Dr. A. W. Westerberg for supplying the subroutine RMINSQ, a nonlinear least squares routine, and to Dr. K. Rajagopal for supplying the subroutine PYGX for evaluating the PercusYevick hardsphere 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. W. 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. ii
PAGE 3
TABLE OF CONTENTS Page ACKNOWLEDGMENTS ii LIST OF TABLES vii LIST OF FIGURES x ABSTRACT xi CHAPTERS : 1. INTRODUCTION 1 2. MIXTURE RULES FOR THE MIE (n,6) INTER^IOLECULAR PAIR POTENTIAL AND THE DYMONDALDER PAIR POTENTIAL. 4 Introduction 4 UnlikePair Potential 5 Semi theoretical Mixture Rules 6 Empirical Mixture Rules 8 The DymondAlder Potential 15 Conclusions 15 3. A MIXTURE RULE FOR THE EXPONENTIAL6 POTENTIAL 19 Introduction 19 Mixture Rules 19 Conclusions 21 4. THE RELATIONSHIP BETWEEN THE MIE (n,6) POTENTIAL AND EXPONENTIAL6 POTENTIAL 24 Introduction 24 Equivalence of Potential Parameters 25 UnlikePair Parameters 27 Conclusions 30 ill
PAGE 4
TABLE OF CONTENTS (Continued) Page 5. SATURATED LIQUID PROPERTIES FROM THE MIE (n,6) POTENTIAL 32 Introduction 32 BarkerHenderson Perturbation Theory 33 Liquid Properties from Best Virial Coefficient (n,6) Potential 35 (n,6) Potentials for Liquids 41 Conclusions A3 6. EXCESS PROPERTIES OF THE METHANEPERFLUOROMETHANE SYSTEM FROM THE ONEFLUID VAN DER WAALS PRESCRIPTION IN PERTURBATION THEORY 44 Introduction 44 OneFluid Perturbation Theory of Mixtures 44 The MethanePerf luoromethane System 47 Potential Parameters Independent of Choice of Reference Fluid. 58 Averaged Excess Properties 61 Conclusions 64 7. CORRESPONDING STATES FOR FLUID MIXTURESÂ— 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 MoleFraction Averaged Excess Properties. 82 iv
PAGE 5
TABLE OF CONTENTS (Continued) Page ThreeParameter OneFluid Theory 84 The Virial Coefficient Least Squares (vcls) Prescription 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 APPENDICES : 117 A. DETERMINATION OF (n,6) POTENTIALS FROM THE SECOND VIRIAL COEFFICIENT 118 B . RESIDUAL THERMODYNAMIC PROPERTIES 120 C. CALCULATION OF EXCESS FREE ENERGY 122 D. EXPERIMENTAL PROPERTIES OF THE CH. + CF, SYSTEM 123 4 4 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
PAGE 6
TABLE OF CONTENTS (Continued) Page Conclusions 138 LIST OF REFERENCES 140 BIOGRAPHICAL SKETCH 143 vi
PAGE 7
LIST OF TABLES Table Page 1 Pure component parameters 9 2 Unlike potential parameters 11 3 Cross virial coefficient B^Â„ with the (n,6) Mie potential 12 4 Cross virial coefficient B^ _ with the Dymond and Alder potential 16 5 Unlikepair parameters 22 6 Cross virial coefficient B _ 23 7 Second virial coefficients of pure gases predicted with (n,6) potential using exponential5 parameters 26 8 Exponential6 potential parameters 28 9 Crossterm 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 onefluid 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 lll^K, P = 0, X = x^ = 0.5, with (12,6) potential 53 17 Excess properties of the CH, + CF, mixture at 111Â°K, P = 0, x^ = x= 0.5 with various potentials and reference liquids 55 vii
PAGE 8
LIST OF TABLES (Continued) Table Page 18 Calculated properties of liquids at 111Â°K, 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 onefluid and twofluid prescriptions with Monte Carlo calculations 73 E 22 Comparison of excess free energy (G ) from vcB and vdW prescriptions with Monte Carlo (MC) calculations 74 23 Comparison of excess enthalpy (H ) from vcB and vdW prescriptions with Monte Carlo (MC) calculations 76 E 24 Comparison of excess volume (V ) from vcB and vdW prescriptions with Monte Carlo (MC) calculations... 78 25 Comparison of the onefluid vcB and vdW prescriptions 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 threeparameter vcB prescription 91 30 Comparison of the onefluid 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
PAGE 9
LIST OF TABLES (Continued) Table . _ _ Page 33 Gas phase potential parameters for various mole cules 103 34 Likepair and unlikepair (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 Likepair and unlikepair (12,6) potential parameters estimated from gas phase (n,6) potentials Ill 37 Estimated excess properties using estimated (12,6) parameters from Table 36 112 38 Comparison of predicted and experimental viscosities 134 39 Bestfit parameters for equation (E1) 136 ix
PAGE 10
LIST OF FIGURES Figure Page 1 Residual properties of liquid CF, ^2 2 Excess free energy of the CH, + CF, system at 111Â°K, P = 87 3 Molar volume of the CH, + CF, system at lll^K, P = t ^ !.... 92
PAGE 11
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 likepair potential energy (mutual energy of a pair of molecules of the same species) and the unlikepair potential energy (mutual energy of a pair of molecules of different species) have been studied. The pairwise additivity approximation for conf igurational energy is assumed throughout this work. Formulae for the unlikepair parameters in terms of the likepair parameters for the Mie (n,6) potential energy model are examined for their abilities to predict crossterm second virial coefficients. The London dispersion formula and an assumption of geometric mean repulsion energies are shown to apply only for cases wherein the xi
PAGE 12
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 crossterm second virial coefficient. The geometric mean rule for the energy and distance parameters is also shown to perform well for predicting the crossterm 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 crossterm second virial coefficients for the exponential6 potential. Relationships developed between parameters for the Mie (n,6) potential and the exponential6 potential show that the three parameters in the latter (c.r ,a) can reasonably be equated to three parameters (e,r ,n) in the (n,6) potential. Furthermore, the m . implied equivalence of these potential parameters suggests that the unlikepair parameters for the exponential6 potential should follow the total geometric mean rule. This set of rules predicts good values for the crossterm 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 onefluid or twofluid xii
PAGE 13
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 onefluid theory. The method used obviates the empirical determination of the unlikepair potential energy parameters. A method is developed for obtaining both likepair and unlikepair 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
PAGE 14
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 development 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 intermolecular 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 exponeritial6 potential and the DymondAlder potential. Likepair 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 unlikepair potential parameters which characterize the interaction between a pair of molecules of different species from the likepair parameters of the respective molecules. The resulting unlikepair potential parameters are used to calculate accurate values of crossterm second virial coefficients in gas mixtures.
PAGE 15
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 unlikepair parameters in the gas phase can be used indirectly to estimate unlikepair 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 + perfluorome thane system. A simple method is proposed for accurately estimating the excess properties of such mixtures. It is further shown (Chapter 7) that statistical mechanics provides relationships for calculating compositiondependent potential parameters for use in the onefluid and twofluid theories of liquid mixtures. The new prescriptions called the virial coefficient prescriptions are shown to give accurate estimates of the properties of
PAGE 16
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 unlikepair energy parameter e.. from the geometric mean of the respective likepair parameters which is observed in mixture calculations where all molecules are assumed to obey the same pair potential.
PAGE 17
CHAPTER 2 MIXTURE RULES FOR THE MIE (n,6) INTERMOLECULAR PAIR POTENTIAL AND THE DYMONDALDER PAIR POTENTIAL Introduction The Mie (n,6) model for the intermolecular pair potential function is of the form (r) = B (1) It has been studied in some detail for argon and nitrogen by Klein and Hanley ' and for methane by Ahlert, Biguria and Gaston. The repulsionterm exponent n as a third adjustable parameter gives this model a 2 . flexibility equivalent to that of other threeparameter models, in a simple analytical form. The coefficients A and B in Eq. (1) may be written in terms of the parameters e and a, the depth of the potential minimum and the intermolecular separation at which (t)(r) = 0, respectively. A = and B = Â—T n n
PAGE 18
6e r B ^ . (5) n6 The most familiar form of the Mie (n,6) potential is the LennardJones potential in which n is 12. UnlikePair Potential 4 From the leading term in the London theory of dispersion forces the unlikepalr attraction coefficient A. . (Note that the double subscript "ij" will refer to the unlikepair intermolecular interaction and the single subscript "i" or "j" will refer to the likepair intermolecular interaction.) may be written as ^ij = (Vj^^^^ ^i Â• ^^^ where 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 Harkness based on molecular beam scattering results. Mason and coworkers use a purely repulsive potential of the form ^^ = B^r""i (8) to represent the intermolecular interaction at small separations. For the unlikepair repulsion interaction they suggest that rÂ£E = ( ^ <,,ieÂ£ ^1/2 (g^
PAGE 19
or B n. . n. n, X 1 r r Â•* * 1/2 (10) With this assumption dimensional considerations require that n. . = (n. + n.)/2 (11) and therefore. B. . = (B.B.) 1/2 (12) Abrahamson has made theoretical calculations of the interatomic repulsion interaction of both like and unlike inert gas atoms. These calculations were based on the ThomasFermiDirac statistical model of the atom and showEq. (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) n. . 6z.. r ^J IT m. . B.. =Â— iÂ— ii = xj n. . 6 6e . r X m. or e.. = (E.e.) 1/2 r r m. m, , 1/2 n, 6e. r ^ n. 6 J ^"i1 '' 1/2 (13) "ij m. [(n.6)(nj6)] 1/2 (14) ij Similarly with Eq. (6) and Eq. (4)
PAGE 20
n. ,Â£ . .r ii = or iJ n, . 6 :..=(Â£.Â£.) f r, ij 1 J n I 6 c .n. r 1 1 m. 1 n. 6 1 '"ii '' 6 e .n.r J J m n. 6 J 1/2 [(n.6)(n^6)] 1/2 r r m. m. 1/2 "ij (15) (16) where Â£ =iVLll 1/2 2(n^) 1/2 n n ij (n, f n.) (17) Elimination of e.. between Eqs. (14) and (16) yields after algebraic manipulation
PAGE 21
n.. and r given in Eqs. (11), (16), (18) has been tested in seven J ij binary gas mixtures. Pure component parameters (Table 1) were determined from a fit of the second virial coefficient (see Appendix A). The crossterm second virial coefficient was calculated using the semitheoretical values of e.., 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 + N 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 ^^ ~ n, and I^ = I the unlikepair energy and distance parameters should be the simple geometric mean of the respective likepair parameters. Such simplicity while appealing is hardly a basis for choosing these forms. However, results of a least squares fit of the crossterm second virial coefficient for the CH, + CF, system 4 4 shown in Table 2 indicate that the bestfit results are reasonably reproduced by the geometric mean e 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
PAGE 23
10 m in (U 3 C Â•H 4J Ci o u CM
PAGE 24
11 .
PAGE 25
12 Table 3. Cross virial coefficient B Â„ with the (n,6) Mie potential.
PAGE 26
13 Table 3 (Continued)
PAGE 27
14 being the arithmetic mean of the two like repulsion exponents (Eq. 11). Once again, however, a glance at the least square mixture parameters for CH + CF, reveals that the value of 59.54 for the unlike repulsion 4 4 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 E = Ce e )^^^ Â£..=(Â£.Â£.) 1/2 V ^l/2 ij = (ij> ^ij = ^Â°i^3^ n^. = (n^n.)^/2 n^. = (n^ 4n. ) /2 The results of calculating B , the crossterm second virial coefficient, 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 3 mixtures tested. The rootmeansquare deviations of 0.6 to 4.2 cm /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 3 a rootmeansquare deviation of only 1.06 cm /mole. However, this
PAGE 28
15 particular set of rules failed to exhibit the general applicability found for the rules given in set A. The DvmondAlder Potential For molecules which obey the same form of the reduced pair * * * * potential energy function = 4)/e and r = r/r , the empirical rules always require the unlikepair energy and distance parameters c and r (or a) to be the simple geometric mean. For the m 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 Gubbins to perform well for the other inert gases and for 0Â„, N and CH, but not for CF,. The potential energy parameters (e, o) for the other molecules are obtained from the argon parameters by corresponding states relationships. Results of predicting the crossterm second virial coefficient for four binary gas mixtures using the geometric mean rule for e and a 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 crossterm second virial coefficient only for those cases where these rules are equivalent to the geometric mean rule for all parameters. The
PAGE 29
16 Table 4. Cross virial coefficient B^^ ^^^'^ the Dymond and Alder potential. 3 B 2 (cm /mole) Mixture
PAGE 30
17 set A of rules, in which each parameter (e, a, 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 likepair intermolecular interaction allow the prediction of binary mixture properties without any knowledge of mixture properties. This is in contrast to the use of a twoparameter potential for the likepair interaction 11 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 crossterm second virial coefficients of mixtures of molecules which obey the Djonond 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 (e, o) determined from least squares fits of the second virial coefficient. These are the LennardJones (12,6) potential and the Sutherland (Â°',6) potential. Where the third virial coefficient of a pure system has been calculated with both the LennardJones and Sutherland potentials, the values predicted by the (12,6) potential are higher than experimental data, while the values predicted by the (<=Â°,6) potential are lower than experimental data. The (n,6) potentials
PAGE 31
18 reported in the present chapter for pure systems have values of n ranging from 17.74 for ethane to 138.68 for sulfur hexaf luoride. 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 LennardJones and Sutherland potentials.
PAGE 32
CHAPTER 3 A MIXTURE RULE FOR THE EXPONENTIAL6 POTENTIAL Introduction It was shown in the previous chapter that for the Mie (n,6) intermolecular pair potential the unlikepair parameters (c . . ,a . . ,n. .) 13 ij ij are the geometric mean of the respective likepair parameters for second virial coefficients. It is shown below that these mixture rules can be extended to define mixture rules for the exponential6 potential. Mr) = (16/a) ^ exp[a(l r/vj] (r /r)^ a mm (21) Here r is the value of r at which 4)(r) = e . The value of r at which m d)(r) = (i.e., r = a) can be determined from r by solving the following m equation numerically. Â£n(a/r ) = (1/6) { Â£n(6/a) + a(la/r )} . (22) m m Mixture Rules Since e and o in the exponential6 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 a for the unlike pair may be obtained in the following way. In the (n,6) potential the repulsive energy is of the form (},(r)^^= Br'" (23) 19
PAGE 33
20 where B is constant. From (23) we have d in .1^ d ilnr = n . (24) For the exponential6 potential the form is , , rep ,, br (j)(r) Â— ^ = Ke (25) and d_Jjn_$^^ = d Â£nr br (26) ij The results of the (n,6) mixture rule study (geometric mean for n^ J give reÂ£ dan^..^^^)/dZnr = { (dJincj) .^^/dJlnr) (d)ln(t) Â— ^/d^Lnr) } X 1 *J .1/2 ij (27) which for the exponential6 potential can be written as 1/2 b..r = (b.b.)"' r . (28) From Eq. (21) we see that b = ct/r , which implies m ii_ = m. iJ a. a. . ^ J r r m. m. ' 1 J ' 1/2 or (29) a.. = y(a.a.) and r = Y(r r ) (30) where y is to be determined.
PAGE 34
21 Substitution of Eq. (30) into Eq. (22) along with 1/2 a.. = (a .a .) 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 m. 1 X Resulting values for y are near one (Table 5) and thus a. . is nearly the simple geometric mean. Values of unlikepair parameters for three binary mixtures for which exponential6 likepair 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 a. The likepair parameters 12 were taken from Sherwood and Prausnitz. Results of the prediction of the crossterm second virial coefficient with these parameters are given in Table 6. Conclusions In two of the three cases the rootmeansquare deviations for the exponential6 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 + NÂ„ mixture the deviation of 1.8 cm /mole is greater than 3 0.6 cm /mole found with the Mie (n,6) potential. However, either potential model fits the crossterm 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 exponential6 potential.
PAGE 35
22 Table 5. Unlikepair parameters. e../k ('K) a., d) V ^ ^ a System i1 U U Â— iJÂ— ^ Â— CH, + Ar 185.14 3.431 3.752 20.77 0.9996 4 Ar+N2 156.05 3.373 3.665 23.21 0.9990 CH, + CF, 301.68 3.848 3.983 83.77 0.9872 4 4 _
PAGE 36
23 Table 6. Cross virial coefficient B 12'
PAGE 37
CHAPTER 4 THE RELATIONSHIP BETWEEN THE MIE (n,6) POTENTIAL AND EXPONENT IAL6 POTENTIAL Introduction 9 9 1 Hanley and Klein ' have recently shown that five of the common threeparameter 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 and the exponential6 potential Hr) = (fl^ {exp[a(lr/r^)] (a/6) (r^/r)^} ; (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 &n(t>^^^ ^ _^ (33) d Inv For the exponential6 potential lJi^*=^ = (a/r )r . (34) d )lnr m' As n and a approach infinity both potentials become identical to the Sutherland (",6) potential. Thus in this limit the parameters e and r^ for the two potentials would be the same when found from fitting the 24
PAGE 38
25 same set of second virial coefficient data. This suggests that for large values of n and a the parameters e, 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 a/r in the exponential6 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) m are the same value, (b) that (})(r ) by Eq. (31) is equal to <)(r ) by Eq. (32) and (c) that dÂ£n()/dJ?.nr at r by Eq. (33) is equal to that by m Eq. (34) then the following equivalence of parameters is obtained: ^(n,6) = '(exp6) ^^^^ r^ = r^ (36) (n,6) (exp6) (n,6) (exp6) This suggests that where sets of the threepotential 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, exponential6 parameters for six pure gases, determined by either 12 1 Sherwood and Prausnitz or Klein from fitting second virial coefficients, 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
PAGE 39
26 Table 7. Second virial coefficients of pure gases predicted ^ with (n,6) potential using exponential6 parameters. Number of ^^^ Temperature experimental o Data Gas range (Â°K) points dev (cm /mole) ref . CF, 273.16623.16 15 0.11 b C(CH ) 303,16548.16 16 2.2 c,d,e C^H, 220.0 500.0 11 2.4 f 2 6 N 400.0 700.0 4 1.4 f CH, 273.16623.16 15 1.61 b Ar 81.0 600.0 14 7.7 f ^he 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 those used by others for the determination of the exponential6 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. ^D. R. Douslin, R. H. Harrison, and R. T. Moore, J. Phys . Chem. Tl, 3477 (1967). '^J. 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. ]_, 1 (1954). S. D. Hamann, J. A. Lambert, and R. B. Thomas, Australian J. Chem. 8^, 149 (1955). ^J. H. Dymond and E. B. Smith, The Virial Coefficients of Gases (Clarendon, Oxford, 1969). e
PAGE 40
27 experimental uncertainties. As might be expected the worst results are obtained for argon which has the lowest value of a 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 exponential6 potentials. UnlikePair Parameters The implied equivalence of the potential energy functions suggests that the mixture rules for the exponential6 potential parameters should be even simpler than those suggested in Chapter 3. The exponential6 mixture rules could be taken to be the same as those for the (n,6) potential in Chapter 2. .,. = (e,e^).^/2 (38) a.. = (a.o.)^/^ (39) ij 1 J Â• a.. = (a.a.)^^^ . (40) These mixture rules have been used with the exponential6 potential and the parameters in Table 8 to predict the crossterm 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 exponential6 potential are similar to those with the (n,6) potential in Chapter 2. The predictions for the CÂ„H, + NÂ„ and CH, + C(CH.), systems are not as good with the exponential6 / o 2 4 J 4 potential as with the (n,5) potential, but they are much better than
PAGE 41
28 Table 8. Exponential6 potential parameters, Component CF, C(CH3)^ CH, Ar e/k (
PAGE 42
29 Table 9. Crossterm second virial coefficient. 3 rms dev (cm /mole) System
PAGE 43
30 the predictions of the (12,6) potential with any of the eleven sets g of mixture rules tested by Good and Hope. It should be pointed out that for a given set of e, r , m n (or a) the value of a, ,. does not equal to a, ,,. A given (n,6; (exp6) ^ mixture is characterized by e . , e . , r , r and n. , n. (or equivalently 1 J m. m. 11 ^ Â•' a^ and a ). Using Eq. (39) for the exponential6 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, ^(n,6) = ^(exp6) (^^^ 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 CH, + C(CH^), system is" 4 3 4^ 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 Eqs. (38) to (40) would predict r = 4.787 A. If taken to be m. . ^J o exponential6 parameters, the same rules predict r = 4.782 A. This m. . ij difference would lead to a difference of only 0.3% in the predicted virial coefficients. Conclusions The (n,6) and exponential6 potentials are sufficiently alike with respect to the prediction of second virial coefficients that
PAGE 44
31 sets of the three exponentlal6 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 exponential6 potential.
PAGE 45
CHAPTER 5 SATURATED LIQUID PROPERTIES FROM THE MIE (n,6) POTENTIAL Introduction The ability of the perturbation theory of liquids developed by 14 Barker and Henderson 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 1 2 and Klein ' have recently shown that various threeparameter potentials (Kihara, Mie (n,6), exponential6 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 (e/k, o, n) which accurately reproduce crossterm 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 14 15 (18,6) potentials for liquid argon. ' The LennardJones (12,6) potential with the parameters e/k and a determined by Michels et al. performs remarkably well (better than that of Hanley and Klein) for 15 14 liquid argon as demonstrated by Monte Carlo and perturbation theory 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 32
PAGE 46
33 have not been included; thus to the extent that manybody interactions are important the (n,6) potentials should be regarded as effective pair potentials. The Mie (n,6) potential is of the form (})(r) = < e Â•
PAGE 47
34 where a is the distance parameter in the pair potential arid * * The second order term (A_ + AÂ„) is derived from the local 14 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^/NkT = T 8(A'^/NkT)/3T Â•. (49) The residual entropy is calculated from the relation s^/m = (u^/NkT k^/mcr) . (50) The numerical integrations were performed using a Gaussian integration routine. PercusYevick hardsphere 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 hardsphere pressure, compressibility and free energy. Analytical expressions were utilized 17 19 to generate the radial distribution function ' and its density derivative (Appendix B) . McDonald and Singer 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
PAGE 48
35 internal energies were recalculated using the (12,6) and the (18,6) pair potentials in the BarkerHenderson perturbation theory. The results are compared in Table 10. In the case of the (12,5) 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 Mie (n,6) potential functions for this study were selected from the tables of Klein. For each substance the (n,6) potential parameters n, e/k and a which best fit the experimental second virial coefficient were chosen. Klein has determined parameters for CF, only for potentials with n up to 40. An optimal set of parameters for CF, with 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
PAGE 49
36 T Table 10. Comparison between Monte Carlo calculations and perturbation theory. 3, , ^ ,,o ,. T,.._...,d /.o ^^ T,_.Â„_..Â„.e (Â°K) (cm /mole) (12,6) Potential (18,6) Potential a b c a b c U (cal/mole) 97.0 26.90 1480 1424 1420 1566 1501 .1517 108.0 28.48 1387 1352 1351 1457 1410 1427 i36.0 32.52 1192 1186 1189 1233 1214 1233
PAGE 50
37 Table 11. (n,6) Potential energy parameters* from second virial coefficients. Molecule
PAGE 51
38 
PAGE 52
39 Table 12 (Continued) T (Â°K) Experimental Residual Entropy (S^/Nk) (12,6)' (12,6) (17.6) (27.6)' 100.0
PAGE 53
40 Table 12 (Continued) P er f lu or ome thane
PAGE 54
41 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 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 properties. 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 nvalues 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 e/k and a obtained from a fit of the second virial coefficient.
PAGE 55
42 o \o iH I t3 0) 3 13 Â•H CO (U o CO I a, o u 41 0) ce 3 T3 H CO erf O (12,6) Potential A (30,6) Potential D (136.3,6) Potential 4.0 3.0 2.0 1.0 0.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0 h Experimental JL X 130 150 170 190 210 Temperature, T(Â°K) Figure 1. Residual properties of liquid CF,.
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43 Conclusions The BarkerHenderson 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 experimental data than do those calculated with Klein's (12,6) potential. For the systems studied (argon, nitrogen, methane, and perf luoromethane) (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.
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CHAPTER 6 EXCESS PROPERTIES OF THE METHANEPERFLUOROMETHANE SYSTEM FROM THE ONEFLUID VAN DER WAALS PRESCRIPTION IN PERTURBATION THEORY Introduction Theories of liquid mixtures are usually based either on twoparameter corresponding states with one of the mixture components taken 21 22 as the reference substance, ' or on theories of the liquid state '5 / which employ a twoparameter pair potential. ' Common to most of these theories is an adjustable parameter 5 which takes into account the deviation of the unlikepair energy parameter e^ from the geometric mean of the two likepair energy parameters. For molecules which differ greatly in character the factor E, is usually significantly less than 1.0, e.g., E, = 0.909 for the CH^ + CF^ 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 potentials 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 CH^ and CF^ in predicting the thermodynamic properties of gas and liquid mixtures of these molecules. OneFluid Perturbation Theory of Mixtures The theory chosen for the present study is the onefluid theory of mixtures with van der Waals prescription for the mixture potential 44
PAGE 58
45 energy parameters. 3 e a = m m 23^3^23 x^e^a^ + 2x^X2e^2Â°^2 + ^2'2''2 m 23^. 3^23 ^1Â°1 + 2x^X20^2 + '^2''2 1/3 (51) (52) lA The perturbation theory of Barker and Henderson 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. 23 Leonard et al. have presented a comparison of the onefluid 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 onefluid 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 onefluid van der Waals prescription in perturbation theory predicts an excess
PAGE 59
46 Table 13. Comparison of onefluid van der Waals model with Monte Carlo and multicomponent perturbation theory calculations. T = 115. 8Â°K, P = and x^ = X2 = 0.5, e^/k = 119. 8Â°K, E^/k = 167. 0Â°K, a^ = 3.405 X, o^ = 3.633 X, z^^ = (e^c^)^^^, ^12 "^ ^^1 "*" ^2^^^' Theory Monte Carlo Multicomp. Pert. vdW*^ vdW^ vdWÂ® G^ (J/mole)
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47 volume and excess free energy almost the same as multicomponent perturbation theory and within the estimated uncertainty of the Monte Carlo results. The MethanePerfluorome thane 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 5 multiplying the geometric mean of the likepair energy parameters. Heretofore E, has been determined from binary mixture data. For the methane + perfluorome thane system the value of B, required to fit either crossterm 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 crossterm second virial coefficients that when the two likepair interactions in binary systems are allowed to obey different (n,6) potentials, the unlikepair potential parameters are all the simple geometric mean of the likepair 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 5. This E, 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 onefluid theory with
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48 van der Waals prescription for the mixture potential parameters. The unlikepair 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 T is defined as the temperature at which the second virial coefficient B(T) is equal to zero. The Boyle volume V^ is defined in the usual way, namely, Vg = Tg(dB(T)/dT)^ . (53) . 6 When two species (i and j) obey the same pair potential energy function their potential parameters are related as follows: (e/k) /(e/k) = Tg /Tg (54) Â•' i j o.lo. = (V^ /V_ )'^'^ . (55) 1 J B. B. 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 CH, and CF, as well as the crossterm second virial coefficients 4 4 of the CH, + CF, pair all fall very nearly on one reduced curve provided 4 4 the reducing parameter for the temperature is T^ and that for the second virial coefficient is V^. This correlation is followed in spite of
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49 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. (5A) and (55) will be good estimates for the parameter ratios. Eqs. (5A) and (55) are commonly employed with the experimental likeand unlikepair Boyle temperatures and Boyle volumes to predict the (12,6) parameters for CF + CF and CH, + CF from those of CH, + 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 26 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 unlikepair parameters is the geometric mean of the respective likepair parameters. This set of parameters fits the experimental crossterm secpnd virial coefficient for the CH, + CF, system in the temperature range 273.16 4 4 3 to 623.16Â°K with a rootmeansquare deviation of 1.20 cm /mole. Using the Boyle temperatures and volumes estimated by the potentials in Table 14 with CH 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. 1/2 It can be seen from Table 15 that the predicted 5 = ^Â±i^^^Â±^A^ is 0.91688 and results entirely from forcing all of the molecular
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50 Table 14. Gas phase potential parameters, Molecular Pair
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51 Table 15. Predicted and experimental potential parameter ratios. Molecular f. = c./e^^^ = T^/T^ g = ojo^^^ = (V /V^ )l/3 j^air 1 ref j ref CH^ + CH^ 1.000 (1.000) 1.000 (1.000) CF^ + CF^ 1.0127 (1.017) 1.245 (1.242) CH^ + CF^ 0.92266 (0.917) 1.130 (1.132) "values in parentheses were determined from the experimental Boyle properties given in reference (25) and used for calculations in references (21) and (23).
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32 interactions to obey the same potential (as yet unspecified) , since this value was predicted on the basis of E. = 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 onefluid 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 CH, + CF, mixture at 111Â°K. 4 4 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 CH, as the reference fluid are given in Table 16 along with the results obtained by Leland et al. and Leonard 23 et al. for the same system with their respective theories and experimental ratios f and g from Table 15. Also included are the onefluid perturbation results for B, = 1.0 and a . . = (a . + a )/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 CH, + CF, 4 4 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 unlikepair hardsphere diameter is the arithmetic mean of the two likepair diameters. It is significant that this
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53 Table 16. Excess properties of the CH + CF, mixture at lll^K, P = with (12,6) potential. mixture at lll^K, P = 0, x = x = 0.5, V^ (cm^/mole) G^ (J/mole) Experimental Data* 0.845 360 vdW Perturbation Model (predicted f and g) b M. Klein CH, parameters (1st order theory) 0.89 209 b M. Klein CH, parameters (2nd order theory) 0.97 213 Sherwood and Prausnitz CH, parameters'0.64 224 Leland, Rowlinson and Sather with experimental f and g 0.90 279 Leonard, Barker and Henderson with experimental f and g 0.97 209 vdW Perturbation Model (experimental f and g) Sherwood and Prausnitz CH, parameters'" 1.05 296 vdW Perturbation Model (5 = 1.0, o. . = (a. + a.)/2) Sherwood and Prausnitz CH, parameters*^ 0.79 5 iaken from reference 27. Parameters taken from reference 1, parameters taken from reference 12. Taken from reference 21. Taken from reference 23.
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54 assumption makes the unlikepair hardsphere 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 C = 1.0 and cj^ ~ (^i "* oÂ„)/2 in the liquid mixture it is seen that most of the required deviation of E, 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 E, = 1,0 in the gas phase demonstrates that much of the deviation of 5 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) CH, parameters with those using Sherwood and Prausnitz CH, 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 3 determined f and g excess volumes vary from 0.50 to 1.05 cm /mole and excess free energies vary from 296 to 746 J/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
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55 Table 17. Excess properties of the CH, + CF, mixture at 111Â°K, P = 0, X = X = 0.5 with various potentials and reference liquids. V^ (cm^/mole) G^ (J/mole) 0.89 209 0.64 . 224 0.63 227 0.59 226
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56 that the PercusYevick hardsphere 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 twoparameter 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 24 and Prausnitz have recently used the Leonard, Barker, Henderson theory with the threeparameter 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 C are 0.994 and 0.988, respectively, when the Kihara potential is used for the liquid state. ..These values of 5, which are not far from 1.0, further support
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57 Table 18. Calculated properties of liquids at 111Â°K, P = 0. All results are for first order perturbation theory with experimental 3 f and g. Volumes (V) are in cm /mole, and residual Gibbs free energy (G ) is in J/mole.
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58 the conclusion reached in this work that much of the deviation of the observed 5 from 1.0 that is usually observed results from the use of the same twoparameter potential for all intermolecular interactions in the mixture. The value of E, obtained empirically to fit the crossterm second virial coefficient in the methane + neopentane system with the 28 (12,6) potential is approximately 0.93. This value is much lower than that found by Rogers and Prausnitz for the liquid state using a threeparameter potential. The crossterm second virial coefficient for this system has been successfully predicted assuming 5 = 1.0 with the threeparameter (n,6) potential in Chapter 2, Potential Parameters Independent of Choice of Reference Fluid As demonstrated previously the predictions of the vdW perturbation 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 likeand unlikepair interactions. This is accomplished by utilizing Eqs. (56) and (57). Given two different (n,6) potentials representing the same molecule, say (n,6) and (n2,6), the following relationship can be used to estimate the parameters of the (n^,6) potential from those of the (n2,6) potential: (e/k) = (E/k) (T*) /(T*) (56) n. n2 is nB n^ * * 1/3 a = a {(V ) /(V ) } (57) n^ n_ i5 n_ Jts n. where T^ = Tg/(e/k) and V* = V^/ (No^) .
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59 Thus, if the (n ,6) potential is chosen to represent all interactions in the liquid mixture, and the known gas phase potential for a given interaction is the (n_,6) potential, then the (n^ ,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 CH, + CF, mixture. Results are given in Table 19. The predicted 4 4 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 crossterm second virial coefficients of CH, , CF, and CH, + 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)
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60
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61 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 likepair gas phase potentials with good results. The unlikepair potential parameters estimated in this way predict the crossterm second virial coefficient for the 3 CH, + CF, pair with a rootmeansquare 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 onefluid theory to represent all interactions in the liquid. For the CH, + CF, 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 onefluid 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.
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62 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 molefraction 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 molefraction averaging the resulting excess properties. Reasonably good choices for the liquid potentials for CH, and CF, would be the (12,6) and (30,6) potentials, respectively. These average excess properties for the equimolar mixture of CH, + CF, 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 likepair and unlikepair interactions.
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63 Table 20. Averaged excess properties (parameters independent of reference fluid) . v'^ (cm^/mole) G^ (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
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64 Conclusions The onefluid 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, + CF, 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 likepair and unlikepair potential energy parameters in the ratios obtained from the experimental Boyle properties. 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 demonstration that most of the deviation of the unlikepair energy parameter e, , from the geometric mean rule for the CH, + CF, system arises from forcing all pair interactions to obey one form of the pair potential. It is further shown that knowledge of the singlecomponent gas phase potentials with the mixture rules proposed in Chapter 2 allows the prediction of both likepair and unlikepair parameters for any
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65 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 likepair and unlikepair gas phase parameters are known as in the case of the CH, + CF, 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 CH, + CF, mixture.
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CHAPTER 7 CORRESPONDING STATES FOR FLUID MIXTURESÂ— NEW PRESCRIPTIONS Introduction The most accurate theories of fluid mixtures proposed to date 21 22 30 are the onefluid and twofluid van der Waals theories ' ' and the 23 Leonard, Henderson, Barker multicomponent perturbation theory. Limited results for the onefluid 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 onefluid and twofluid van der Waals theories with Monte Carlo calculations for both hardsphere and (12,6) mixtures has been made by Henderson and Leonard in references 30 and 31. Results show that the onefluid van der Waals theory is superior to the twofluid van der Waals theory and the threefluid theory. In the previous chapter it was shown (Table 16) that the onefluid van der Waals (vdW) theory was superior to the multicomponent perturbation theory for the methane + perfluorome thane system. The onefluid and twofluid 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 compositiondependent potential energy parameters for the imaginary 21 fluids. Leland, Rowlinson and Sather have examined the thermodynamic consequences of the onefluid van der Waals prescription for mixtures of soft spheres and find it superior to other onefluid theories. 66
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67 In the present chapter new prescriptions are presented for calculating potential parameters for the one or two imaginary fluids in either the onefluid or the twofluid 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 B^(T) = x^B^(T) + Zx^x^B^^^T^ + X2B2(T) . (58) The Boyle temperature of the mixture (T ) is defined analogously m to that of a pure component. \^h ) = Â° = 4^(^B > ^ 2x^Vl2(^B ^ + 4^2^h ^ ' ^'^^ m m mm The Boyle volume of the mixture is given by Vg = T^ (^V^^^T = ^B {lOB^/aT) + 2.^.^0B^^/^T) m m B m B B m m m + x^OB^/ST)^ } . (60) B m
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68 Under the usual onefluid 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 narameters e and a of the one fluid which will represent the mixture ^ mm in terms of those of some reference fluid (R) . m R a /a_ = (V^ /V_ )^/^ . . (62) For any (n,6) potential T and V^ as well as T^ and V^ are readily mm 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 onefluid mixture it is unnecessary to consider a particular reference fluid. One can simply use the following relations. (^/^)m(n.6) = ^B /^b/ ', ' ^"^ ^ ' m (n,6) and where and ' m (n,o; (n,6) (n,6; B, ,>. B. ,. (n,b; (n,6) (n,6; Thus, the onefluid Boyle (vcB) prescription is contained in either Eqs. (61) and (62) or Eqs . (63) and (64). The prescription yields
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69 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 twofluid vcB prescription is readily derived by writing Eq. (58) as B^(T) = x^[x^B^(T) + x^B^^(T)] + X2[x2B2(T) + x^B^2(T)^ Â• (67) or B (T) = xB'(T) + xBid) . (68) m 11 ^ z Here the mixture virial coefficient is given as that of an ideal mixture of two imaginary fluids with virial coefficients BMT) and B'(T). The Boyle properties of the two fluids determine the potential parameters of these fluids to be used in twofluid 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 TÂ„ = a/Rb , (70)
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70 and the Boyle volume is given by V = b . (71) B For a onefluid mixture of van der Waals gases m and B mm V, = b . (73) B m m Writing Eq. (59) for a van der Waals fluid, = x^(b^ a^/RTg ) + 2x^X2 (b^2 " ^U^^^B ^ "^ ^2 ^^2 " ^2^^^^ * (74) Rearranging we have 9 7 2 Tg = (x^a^/R + 2x^X23^2^^ + x^a^/R) / (^^h^ + 2x^X2b^2 + ^2^2^ * m (75) Using Eqs. (70) and (71) we obtain \ ^4\\ ^ '^1^2^B^2'^2 " ^2^B2^B2>/^^1^B, ^ 2V2^B^2 ' '''^2' * (76) \ = ^B <4^l/^^B " 212^2/^^B + 4^2/^^B > Â• ^''^ m m m m ^ From Eqs. (76) and (77) we have ^B = (4V ^ ^x^x^V + x^^V^ Â• ^''^ ml iz it Eqs. (76) and (78), when written in terms of e/k and 0, are immediately recognized as the van der Waals mixture rules given in Eqs. (51) and (52) of' Chapter 6.
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71 Thus it is seen that the vcB and vdW 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 vcB Prescription for (12,6) Systems It should be pointed out that when evaluating the onefluid or twofluid vdW theories one is testing a combination of the onefluid or twofluid corresponding states assumption and the particular van der Waals 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 vcB prescription is exact within the assumption of corresponding states. The most commonly used model for the intermolecular pair potential in real systems is the LennardJones (12,5) potential. The properties of dilute LennardJones 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. 8Â°K for potential parameters characteristic of the argon + krypton 23 mixture have been reported by Leonard, Henderson and Barker. Singer 33 and Singer have made similar calculations for other mixtures at 97Â°K. 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 onefluid and twofluid mixtures using the vcB and vdW prescriptions. The resulting excess properties are compared to
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11 the Monte Carlo estimates of both McDonald (Table 21) and Singer and Singer (Tables 2224). 30 31 As has already been shown ' for the vdW prescription, the onefluid vcB prescription is superior to the twofluid vcB prescription. As a general rule the excess properties predicted by the onefluid 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 onefluid prescriptions agree remarkably well with the Monte Carlo calculation. Another comparison of the vcB and vdW prescriptions can be made for gas phase mixtures using the second virial coefficient. For a given set of likepair and unlikepair (12,6) parameters the virial coefficient of an equimolar mixture can be calculated as a function of temperature. Then the onefluid 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 vcB Prescription for Mixtures of Molecules with Different (n,6) Potentials One of the most serious limitations of the onefluid vdW theory or onefluid vcB theory as outlined above is the requirement that all molecules and the mixture itself obey the same twoparameter corresponding states or twoparameter pair potential. It was demonstrated in Chapter 6 using the onefluid vdW prescription that in order to make
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73 Table 21. Comparison of onefluid and twofluid prescriptions with. Monte Carlo calculations. T = 115. 8Â°K, P = and X = x^ = 0.5, e /k = 119.8*'K, 1/2 z^l^l. = 167. 0Â°K, o^ = 3. 405 X, a^ = 3.633 X, z^^ = (e^e^^ Â• ^12 " ^''l "^ Â°2^'''^
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74 Table 22. Comparison of excess free energy (G ) from vcB and vdW prescriptions with Monte Carlo (MC) calculations. 1/2 T = 97Â°K, P = and X = x^ = .5, z^^ = (Â£^^2^ " 133. 5Â°K, '12 = (a + o )/2 =3.596. G is in J/mole. ^11^^12
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75 Table 22 (Continued) ^11/^12
PAGE 89
76 Table 23. Comparison of excess enthalpy (H ) from vcB and vdW prescriptions with Monte Carlo (MC) calculations.^ 1/2 T = 97Â°K, P = and x^ = x^ = 0.5, e^^ = (Â£^^2^ " 133. 5'K, a = (a + aJ/2 = 3.596. H^ is in J/mole. ^11^^12 1.00 e,,/e,^ = .810 11' ^12 H^ (MC) H^ (vcB) onefluid twofluid H^ (vdW) onefluid twofluid ^11/^12 = ^QQ H^ (MC) H^ (vcB) onefluid twofluid H^ (vdW) onefluid twofluid ^11/^12 = ^QQQ H^ (MC) H^ (vcB) onef luid twofluid H (vdW) onefluid twofluid 124 Â± 34 78 95 1.03 111 111 29 Â± 20 20 24 28 28 1 Â± 10 (163) 174 142 206 159 60 Â± 7 67 47 75 51 5 Â± 12 1.08 1.12 (336) 330 238 362 111 167 Â± 27 145 87 153 91 54 Â± 20 (500) 453 286 484 301 263 Â± 35 206 118 214 122 101 Â± 24
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77 Table 23 (Continued) ''11/^12
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78 Table 24. Comparison of excess volume (V ) from vcB and vdW prescriptions with Monte Carlo (MC) calculations. 1/2 T = 97Â°K, P = and x^ = x^ = 0.5, e^^ = (Â£^^2^ " 133. S^K, 12 = ^'^l ^ Â°2^ E 3 a,Â„ = (a, + aÂ„)/2 = 3.596. V is in cm /mole. ^11^^12
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79 Table 24 (Continued) Â°ll/^12
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80 Table 25. Comparison of the onefluid vcB and vdW prescriptions for equimolar mixtures of (12,6) gases. o o e /k = 119. 8Â°K, a^ = 3.405 A, E^/k = 167.00Â°K, a^ = 3.633 A, 1/2 12 " ^^1^2' ' "12 e_ = (e Â£ )'^% o = (o^ + 02)72 Mixture Second Virial Coefficient (cm^/mole) tCk)
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81 Table 25 (Continued) ^c /k = 135.547Â°K a = 3.6075 1 m m e /k = 135.665Â°K a = 3.6080 A, obtained from least squares fit of m m exact values.
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82 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. MoleFraction 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 unlikepair potentials. To further illustrate the use of the vcB prescription only likepair and unlikepair Boyle temperatures and Boyle volumes are used. As in Chapter 6 the CH^ + CF^ mixture is used for demonstration purposes. For the pure liquids CH, and CF, (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 111Â°K. 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 CarnahanStarling hardsphere equation of state and free energy has been used with PercusYevick hardsphere radial distribution functions. Potentials found are given in Table 26 along with calculated molar volume and residual free energy.
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83 Table 26. Liquid phase potentials for CH, and CF,. T = 111Â°K, P = 0, experimental densities taken from Table 18. Â„/ 3, , . Calculated V(cm /mole; Liquid _n Â£/k(Â°K) a (2) Exptl Calcd G^(J/mole) CH, 13.2 161.105 3.7172 37.70 37.70 4226.2 4 CF, 34.2 273.106 4.3141 49.47 49.47 6688.3
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84 Using both likepair and unlikepair gas phase potentials given by Douslin the Boyle temperature and Boyle volume of the CH, + CF. 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 onefluid (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 lll^K. The (13.2,6) potential parameters of pure CF, are also estimated and used to calculate the pure liquid CF, 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 molefraction 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 molefraction averaged excess properties are given in Table 28. Results are also given in Figure 2 for excess free energy. The molefraction 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. ThreeParameter OneFluid 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
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83 Table 27. Mixture properties and excess properties for the CH^ + CF^ system with the (13.2,6) and (34.2,6) potentials. T = 111Â°K, P = 0, vcB b 3 , prescription, volumes in cm /mole, free energies in J/mole. (13.2,6) Potential (34.2.6) Potential '^^.
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86 Table 28. Averaged excess properties from vcB prescription. T = lll^K, P =
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87 A (13.2,6) Potential ^ Averaged O (3A.2,6) Potential O Threeparameter 0) o B M to CO
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88 possible with the present onefluid theory to predict the actual mixture properties and both sets of pure fluid properties with a single (n,6) potential. At present the only theory with sufficient flexibility to allow the use of different pair potentials for different molecular pairs is the multicomponent perturbation theory of which the Leonard, Henderson, Barker theory" is one version. Unfortunately calculations made with this theory are very expensive and of questionable accuracy for such systems as' the CH^ + CF^ system (Chapter 6, Table 16). As mentioned before the onefluid theory in its present form does not have sufficient flexibility to allow the use of more than a single pair potential for all interactions. The second method to be discussed in this chapter for taking into account the different pair potentials obeyed by different molecular pairs in a mixture is actually an attempt to extend the onefluid theory to threeparameter potentials. This would give the onefluid theory a flexibility equivalent to that of multicomponent perturbation theory.. A binary mixture of species 1 and 2 at zero pressure and fixed temperature is characterized by the following set of variables: e^. a^, n^, z^, a^. n^, e^^' 'u^ "i2 ^""^ ^1' ^^^ threeparameter onefluid theory assumes that such a mixture can be represented by an imaginary fluid with composition dependent potential parameters E o , n . It should be emphasized that Eq. (58) on which the Boyle m' m m prescription is based does not require that the virial coefficients B , B and B obey the same pair potential. As a result the Boyle properties calculated from Eqs. (59) and (60) are characteristic of a mixture composed of intermolecular interactions represented by
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89 B,, B and B^. If some means were available for estimating n as a function of n , n , n Â„ and composition, then the Boyle prescription could be used to estimate e and a . For the present investigation mm Â° the following arbitrary relation has been assumed for n . m For mixtures in which all interactions obey the same (n,6) potential the threeparameter theory reduces to the twoparameter onefluid vcB prescription. Values of n^Â„ and n^Â„ for the liquid are 13.2 and 4 4 34.2, respectively. Unfortunately, no accurate means of estimating n. for the CH, + CF, pair in the liquid is available, and the resulting n and predicted liquid mixture properties are quite sensitive to the value of n^2* ^^ order to determine n. Â» for use in Eq. (79) one piece of liquid experimental mixture data is required. The experimental molar volume of the equimolar liquid mixture at 111Â°K and zero pressure is used in this work. As was done for the pure fluids, a value of n m was found to correctly predict the mixture volume. Values of e. and a m m were estimated from the Boyle temperature and Boyle volume of the gas phase equimolar mixture according to the vcB prescription. The equimolar mixture was found to obey the (21.65,6) potential. Using Eq. (79) a value of 19.60 was found for n _. This value was then used to calculate n at other compositions. For each composition T and m V are calculated using Douslin's likepair and unlikepair gas phase m potentials. These Boyle properties are used with the respective values of n to determine the corresponding values of e and o . The composition m TO m m '^
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90 dependent e , a and n are used to estimate the actual mixture proper*^ m m m r e ties as well as excess properties at 111Â°K and zero pressure. In this case the ideal mixture properties are calculated using the (13.2,6) and (34.2,6) potentials for liquid CH, and CF,, respectively. Results of the threeparameter onefluid theory calculations are given in Table 29. Results are also given in Figure 2 for excess free energy and in Figure 3 for mixture volume. Both the predicted mixture properties and excess properties compare favorably with experimental properties. This success suggests that the threeparameter onefluid theory holds promise for treating mixtures of molecules with different pair potentials. The Virial Coefficient Least Squares (vcls) Prescription Use of the vcB prescription forces the second virial coefficient and its first temperature derivative at one temperature calculated from the onefluid theory to equal that of the mixture in question. Since the corresponding states assumption on which the onefluid theory is based is only approximately correct, parameters calculated from the vcB prescription do not give exactly correct mixture second virial coefficients at all temperatures. While the vcB prescription gives excellent results for both the mixture second virial coefficients and liquid mixture excess properties, Eq. (58) can serve as the basis for another onefluid prescription. For a given mixture composition Eq. (58) can be used to generate B (T) over a range of temperature. m The resulting points may simply be fit by least squares to determine the onefluid potential parameters which represent the mixture behavior at a given composition.. Such a least squares fit is not expensive and will result in the best possible set of onefluid parameters at least
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91 Table 29. Calculated mixture and excess properties for the CH, + CF, system with the threeparameter vcB prescription. T = lll^K, P = Mixture Properties Excess Properties
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92 0) O 6 e a i O > 3 u X A (13.2,6) Potential Q Threeparameter O (34.2,6) Potential 70.0 60.0 50.0 40.0 30.0 0.5 1.0 Mole fraction CF, , X 4 CF, Figure 3. Molar volume of the CH, + CF, system at 111Â°K, P = 0.
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93 for the dilute gas mixtures. Calculations have been made for the properties of both gas phase and liquid phase mixtures of (12,6) molecules. These calculations were made for the two sets of potential parameters used in Table 25. The least squares program was written to minimize the sum of squares of the fractional deviations. The least squares (vcls) prescription, of course, gives the best reproduction of the gas phase mixture virial coefficients. These results are the bestfit values in Table 25. The liquid mixture excess properties are nearly the same as those predicted by the vcB prescription. Results are reported in Table 30. ' Another calculation has been made for mixtures of (12,6) molecules. The second set of exact mixture virial coefficients in Table 25 has been used for a threeparameter least squares fit. This was done in order to determine the best e , a and n to represent the mm m mixture of (12,6) molecules. If the onefluid corresponding states assumption were completely correct one would expect the best n found for the mixture to be 12. The actual value of n found was 12.1, m indicating that the mixture virial coefficients do nearly correspond to the (12,6) potential. The vcls prescription holds little or no advantage over the vcB prescription for mixtures of molecules all of which obey the same twoparameter pair potential. The advantage of the vcls prescription may lie in its possible extension to a threeparameter onefluid theory to treat mixtures of molecules which obey different pair potentials. 2 Hanley and Klein have pointed out the insensitivity of the second virial coefficient to the parameter n in the (n,6) potential over certain
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94 a Table 30. Comparison of the onefluid vols prescription and Monte Carlo calculations. Â£j^/k 119. 8Â°K, a^ 3.405 A, e^"' " "7.0UK,
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95 ranges of temperature. Assuming that the same is true for the mixture second virial coefficient the problem of lack of sensitivity could be avoided, since Eq. (58) can be used to generate exact values of B (T) m over any temperature range desired. Conclusions Two new prescriptions have been presented for use with either the onefluid or twofluid theory of mixtures. As was shown earlier for the vdW prescription the onefluid vcB prescription appears to be somewhat superior to the twofluid vcB prescription. Both the onefluid vcB and vcls prescriptions yield accurate estimates of mixture second virial coefficients and liquid excess properties for mixtures of (12,6) molecules. The vcB prescription is shown to reduce to the vdW prescription for mixtures of molecules which obey the vdW equation of state. When used to calculate mole fraction averaged excess properties the vcB prescription yields accurate estimates of the excess properties of the CH, + CF, system at 111Â°K over the entire range of composition. A method is illustrated for extending the vcB prescription to provide a threeparameter onefluid theory. Good estimates of both actual mixture properties and excess properties are possible. Finally, it is proposed that the vcls prescription be extended to three parameters in order to treat mixtures of molecules which obey different pair potentials.
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CHAPTER 8 ESTIMATION OF EXCESS PROPERTIES FOR VARIOUS SYSTEMS USING THE TOTAL GEOMETRIC MEAN RULE IN THE GAS PHASE Introduction In the previous chapters methods have been developed for (1) estimating gas phase unlikepair potential energy parameters from gas phase likepair parameters (Chapter 2) , (2) estimating likepair and unlikepair liquid phase parameters from known gas phase parameters (Chapter 6) and (3) estimating onefluid mixture potential energy parameters using the vcB and vcls prescriptions (Chapter 7) . In the present chapter these ideas are combined to estimate excess properties of several binary liquid mixtures. No attempt is made to account for the presence of molecules with different liquid phase (n,6) potentials. Instead all calculations of pure liquid and mixed liquid properties are made with the (12,6) potential using the Monte Carlo interpolation formulae of Singer and Singer. As a result only excess properties of a mixture are compared with experiment since pure component and mixture properties may not be predicted accurately by the (12,6) potential. The primary purpose of the present chapter is to further examine the validity of the explanation offered in Chapter 6 for the deviation of E from (e c )^^'^ usually observed in liquid mixture calculations. Allowing for the different gas phase (n,6) potentials obeyed by CH^ and CF through use of the total geometric mean rule (Set A of Chapter 2) did account for most of the deviation of e^^ from the geometric mean observed in the liquid when all interactions were assumed to obey 96
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97 the same (n,6) potential. The success of the total geometric mean mixture rule for several binary gas phase mixtures in Chapter 2 suggests that it may be possible for many mixtures to estimate both likepair and unlikepair potential parameters to be used in the liquid phase. Selection of Gas Phase (n,6) Potentials 12 3 Hanley and Klein ' and Ahlert, Biguria and Gaston have studied the selection of (n,6) potentials from gas phase data. Appendix A discusses the determination of the (n,6) potentials used in Chapter 2. In general the (n,6) potential found to represent a given molecule depends on the particular set of experimental gas data used to determine the parameters. The potential found will depend on the temperature range covered by the experimental data as well as the accuracy of the data. As a result for a given molecule there is often available a wide range of values of n found from fitting different sets of data. The choice of the value of n to. represent a molecule is then somewhat arbitrary. Instead of making this choice for each molecule in question another approach has been taken for the purpose of demonstration. A value of n for CH, has been taken equal to 21 as was done in Chapter 6. For other molecules the gas phase n is adjusted so that the excess free energy of an equimolar liquid mixture with CH, predicted by the onefluid vcB theory agrees with the experimental value. The total geometric mean rule is used for the gas phase potentials, and the (12,6) potential is used for the liquid phase. The parameters e and o for the (n,6) potentials used are obtained by least squares from second virial coefficient data. For
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98 mixtures not involving CH^ the value of n for one molecule is held at that found from the mixture of that molecule with CH^. The value of n for the other molecule is varied to obtain agreement between predicted and experimental excess free energy of the mixture in question. The procedure is analogous to the usual procedure of varying 5 discussed in Chapter 5 to obtain agreement with experimental excess free energy. The purpose is to demonstrate that the resulting values of n found for the gas phase from fitting the liquid mixture properties are reasonable values for the gas phase n. Since these values of n were obtained from the assumption of the total geometric mean rule in the gas phase, this supports the explanation proposed in Chapter 6 for the deviation of e in the liquid from the geometric mean. The procedure followed is outlined below for the Ar + CH^ mixture. (1) CH, is assumed to obey the (21,6) potential in the gas 4 phase. (2) A gas phase value of n for Ar is assumed, and parameters E and a for this value of n are found by least squares from the second virial coefficient. (3) Unlike gas phase parameters (h^^ , e^^ and o^^ are calculated for the Ar + CH, pair using the total geometric mean rule of Chapter 2. (4) Using Eqs. (56) and (57) from Chapter 6 likepair and unlikepair parameters are estimated for the (12,6) potential from the gas phase potentials. (5) Using the vcB prescription of Chapter 7 onefluid mixture
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99 (12,6) parameters are calculated for the equimolar mixture. (6) With parameters from steps (4) and (5) the properties of the pure fluids and the onefluid mixture are calculated at 91Â°K from the Monte Carlo interpolation formulae of Singer and Singer. (7) The excess properties are then calculated and the excess free energy compared to the experimental value. (8) If the calculated and experimental excess free energies do not agree, a new value of n for argon is guessed, and the procedure is repeated until the excess free energies agree. The resulting gas phase values of n for various molecules in binary mixtures with CH, are given in Table 31 along with various values of n for the same molecules found from gas phase second virial coefficient data. The values of n found from fitting the liquid mixture data appear to be reasonable values for use in the gas phase. For mixtures which do not contain CH, the value of n for one 4 of the components is fixed at the value found for this component in the presence of CH, . The n for the other component is varied to obtain 4 agreement between calculated and experimental equimolar excess free energy. The resulting values for n in these mixtures are given in Table 32. Again the yalues are reasonable values of n for the gas phase. There are some inconsistencies in the results of Table 31 and Table 32. For example the apparent value of n =15.0 when in the presence of N (n^ = 16.0), while n = 11.25 in the presence of CH^(n^^ =21.0).
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100 Table 31. Gas phase values of n for various molecules found from liquid mixtures with CH, . Potential for CH, chosen as (21,6). 4
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101 Table 32. Gas phase values of n for various molecules found from liquid mixtures not containing CH, . Binary Compc 1
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102 Similarly, n = 17.0 in the presence of CH, (n = 21.0), while 4 nÂ„, = 13.5 when in the presence of Ar(n = 11.25). For other systems CO Ar which have components in common the results are consistent. This is seen in that n^ =17.0 when in the presence of both CH, (n = 21.0) 4 and Ar(n. = 11.25). Similarly, n. = 16.0 when in the presence of Ar ^ ' 0NÂ„(n^ = 16.0), and n is nearly the same value when in the presence of Ar(n = 11.25). Parameters e and a for gas phase potentials used Ar in this chapter are given in Table 33. Some of the inconsistency may be due to the approximate nature of the (n,6) potential especially for systems in which angular dependent forces are present. Part of the inconsistency may be due to use of the (12,6) potential for all liquid interactions and to the particular choice of parameters e and a for the potentials used in the gas phase. The latter two problems were dealt with in Chapter 6. The remainder of the inconsistency can probably be attributed to the approximate validity of the total geometric mean rule. The use of the total geometric mean rule in the gas phase will always predict 5 <_ 1.0 for the liquid potential. The value of the predicted E, is equal to 1.0 only when the two molecules obey the same (n,6) potential in the gas phase. The values of E, predicted by the total geometric mean rule and the gas phase (n,6) potentials are given in Table 34 along with estimated likepair and unlikepair (12,6) parameters for the various binary mixtures. Calculated and experimental excess properties are given in Table 35Two sets of (12,6) parameters and calculated excess properties are given for the N^ + Ar mixture.
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103 Table 33. Gas phase potential parameters for various molecules. Molecule
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104 Table 34. Binary Mixture CH, + Ar 4 U = .9706) CH, + Kr 4 U = .9974) U = .9955) CH, + CO 4 U = .9974) CH, + CF, 4 4 U = .8869) Ar + Kr U = .9853) N^ + Ar U = 1.0000) ^2 + Ar (5 = 1.0000) Likepair and unlikepair (12,6) potential parameters estimated from gas phase (n,6) potentials. Molecular Pair CH, + CH, 4 4 Ar + CH, 4 Ar + Ar CH, + CH, 4 4 CH, + Kr 4 Kr + Kr CH, + CH, 4 4 CH^ + N^ CH, + CH, 4 4 CH, + CO 4 CO + CO CH, + CH, 4 4 CH, + CF, 4 4 CF, + CF, 4 4 Ar + Ar Ar + Kr Kr + Kr N^ + N^ N^ + Ar Ar + Ar N2 + N2 N2 + Ar Ar + Ar (12,6) :
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105 Table 34 (Continued) Binary Mixture Ar + O2 a = .9871) (C = 1.0000) Ar + CO a = .9967) Molecular Pair Ar + Ar Ar + 0^ O2+O2 N2+N2 N2 + O2 Â°2 + Â°2 Ar + Ar Ar + CO CO + CO (12,6) Parameters e/k 120.00
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106 Table 35. Estimated excess properties using (12,6) parameters from Table 34. Binary Mixture CH + Ar (91Â°K) CH + Kr (116Â°K) CH, + N^ (91Â°K) CH, + CO (91Â°K) CH + CF, (lllK) Ar + Kr (lie^K) N^ + Ar*^ (84Â°K) N^ + Ar^ (84Â°K) Excess Property
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107 Table 35 (Continued) Binary Mixture Excess Property Calculated Experimental Ar + (84'K) G^ 38 37 60 0.14 N + (78Â°K) g" 50 43 46 0.21 Ar + CO (84''K) g" 58 57 + 0.10 Excess Proper
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108 Comparison of the two sets of excess properties demonstrates the sensitivity of the results to the source of parameters. Just as e, estimated for the (12,6) potential will always be less than or equal 1/2 to (e.e.) , a., estimated for the (12,6) potential will always be 1/2 greater than or equal to (a. a.) . When the gas phase potentials for a pair of molecules differ significantly as in the case of Ar + CH,, the estimated a is even larger than (a + a.)/2. When this is the case ij _ 1 J calculated excess volumes are more positive than those calculated from the assumption of a.. = (a. + o.)/2 in the liquid. This is the reason for the success of the method used for the mixtures Ar + CH, and Ar + 0. The usual assumption of an arithmetic mean a. . does not predict sufficiently large excess volumes for these two mixtures. For some other mixtures, however, the arithmetic mean a., would be an improvement over the a., predicted from the gas phase potentials. The Ar + Kr mixture is one such case where a slight improvement would be seen if a. . were taken to be the arithmetic mean in the liquid. Mention should be made of the rather large negative excess volumes observed for the mixtures Ar + CO, CH, + CO and CH, + NÂ„ . 4 4 2 In the latter two cases use of the arithmetic mean a. . in the liquid would result in even more negative values. In the first case there would be only a slight improvement if the arithmetic mean a. . used in preference to the value predicted on the basis of the total geometric mean rule in the gas phase. As a result another explanation for the large negative value must be sought. Calculations were repeated using the onefluid vdW prescription. Similar results were obtained indicating that the problem does not arise from use of the were
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109 vcB prescription. Therefore, it must be concluded that the large negative values resulted from either the use of the particular set potential parameters or the use of the (12,6) potential for all interactions in the liquid. Mixtures of Molecules with Very Different (n,6) Potentials The systems discussed to this point except for the CH, + CF, system were all composed of molecules with similar (n,6) potentials with n ranging from 11.25 to 21.0. In these systems E, does not differ significantly from the value 1.0. The systems for which a method for estimating 5 is most desirable are those which contain molecules with very different (n,6) potentials. It is for these systems that Â£, must be significantly less than 1.0. To demonstrate the usefulness of the total geometric mean rule in the gas phase for estimating unlikepair liquid phase potential parameters in systems which exhibit large deviations of E, from 1.0, calculations are made for three such systems. These systems are the CH, + CF, system treated earlier, the CÂ„H + CF, system and the Kr + CF, system. For these systems n for pure components is not found by fitting the liquid mixture excess properties as was done in the previous section. Instead, for each pure component n is selected somewhat arbitrarily from values available from least squares fits of second virial coefficient data. For CH, the (21,6) potential is used. For CF, and C^H, the (136.3,6) and (17.74,6) potentials, respectively, are used. For Kr the (18,6) potential found by Klein is used. The
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110 parameters for these potentials are given in Table 33. Estimated like, pair and unlikepair (12,6) parameters for the various interactions are given in Table 36. The calculated and experimental excess properties are given in Table 37. The results are quite good considering the fact that the (12,6) potential is not a good potential for some of the molecules involved and the fact that no experimental binary mixture data were used in the determination of unlikepair parameters. Conclusions The total geometric mean rule for gas phase (n,6) potentials can be useful for estimating unlikepair potential parameters for use in liquid mixture calculations. The method used is most applicable to systems containing molecules with very different (n,6) potentials.
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Ill Table 36. Likepair and unlikepair (12,6) potential parameters estimated from gas phase (n,6) potentials. (Gas phase potentials were obtained from least squares fit of second virial coefficients.) (12,6) Parameters Binary Mixture Molecular Pair e/k CH4 + CF^
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112 Table 37. Estimated excess properties using estimated (12,6) parameters from Table 36. Binary Mixture
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CHAPTER 9 CONCLUSIONS 1. A geometric mean rule for each of the three parameters (e, a and n) in the Mie (n,6) potential predicts accurate estimates of the crossterm second virial coefficients for many unlikepairs of molecules. This set of geometric mean rules is called the total geometric mean rule, 2. The total geometric mean rule for the Mie (n,6) potential suggests that when molecules obey the same form of the pair potential the mixture rules for the energy and distance parameters should be the geometric mean. This is supported by the agreement between calculated and experimental crossterm second virial coefficients of unlikepairs of molecules which obey the Dymond and Alder numerical potential. 3. The (n,6) and exponential6 potentials are sufficiently alike with respect to the prediction of second virial coefficients that sets of three exponential6 parameters can be used for the three parameters in the (n,6) potential. The total geometric mean rule for the (n,6) potential parameters can also be used for the exponential6 potential parameters. A. The best (n,6) potentials obtained by Hanley and Klein for second virial coefficients yield better estimates in perturbation theory of residual properties of liquids than those calculated with Hanley and Klein (12,6) potentials. 5. The (n,6) potentials found to give the best estimates of liquid properties in perturbation theory have values of n between the best value of n for the gas phase and the value 12. 113
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114 6. In every case studied the best value of n found for use in the liquid phase is the largest value of n which predicts positive pressures for saturated liquid densities. This suggests that (n,6) potentials for liquids may be found by choosing n so that some experimental liquid density is correctly predicted by perturbation theory. 7. The use of the onefluid van der Waals prescription for mixture potential energy parameters in perturbation theory reproduces Monte Carlo (12,6) calculations for excess properties of liquid mixtures as well as the multicomponent perturbation theory of Leonard, Henderson and Barker. The onefluid van der Waals perturbation theory yields better estimates of the experimental excess properties of the CH^ + CF^ system than the multicomponent perturbation theory. The failure of the multicomponent perturbation theory for the CH^ + CF^ system is probably due to the treatment of the unlikepair hardsphere diameter in this theory. 8. Predictions of the onefluid theory are dependent on the choice of (1) the reference fluid, (2) the single twoparameter pair potential for all interactions in the liquid, and (3) the source of potential parameters. 9. It is not possible to predict the correct magnitude of both pure liquid and liquid mixture properties with a single twoparameter pair potential when the pure components obey different (n,6) potentials in the liquid phase. 10. Most of the deviation of the unlikepair energy parameter e . . from the geometric mean for the CH^ + CF^ pair which is usually observed in mixture calculations results from the artificial requirement that both molecules obey the same twoparameter pair potential. By
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115 allowing different molecules to obey different twoparameter pair potentials in the gas phase e,, can be set equal to the geometric mean of the two likepair parameters according to the total geometric mean rule. 11. Using the likepair threeparameter gas phase potentials and the unlikepair threeparameter gas phase potentials obtained from either the total geometric mean rule or some other source, parameters e and a can be estimated for both likepair and unlikepair interactions in any single twoparameter pair potential (e.g., the (12,6) potential) to be used for all interactions in either gas or liquid mixtures. 12. Using relations involving the likepair and unlikepair Boyle properties of molecules it is possible to estimate both likepair and unlikepair potential parameters which are independent of the choice of reference fluids. 13. It is possible with the onefluid theory of mixtures to account for the presence of molecules with different pair potentials by molefraction averaging sets of excess properties predicted with each of the pure component pair potentials. 14. Statistical mechanics provides relations for calculating onefluid or twofluid mixture potential energy parameters. The two new virial coefficients prescriptions presented are shown to give accurate estimates of the properties for both gas and liquid mixtures of (12,6) molecules. The van der Waals prescription is a special case of one of the virial coefficient prescriptions. 15. Using the onefluid virial coefficient Boyle prescription, molefraction averaged excess properties are accurate estimates of the
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116 experimental excess properties for the CH, + CF, system at all compositions. 16. A threeparameter onefluid virial coefficient prescription allows accurate estimation of both mixture properties and excess properties for systems of molecules with different (n,6) potentials. 17. The total geometric mean rule for gas phase (n,6) potentials is useful for estimating unlikepair potential parameters for use in liquid mixture calculations. The method used is most applicable to systems containing molecules with very different gas phase (n,6) potentials and obviates the need for empirical determination of unlikepair parameters.
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APPENDICES
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APPENDIX A DETERMINATION OF (n,6) POTENTIALS FROM THE SECOND VIRIAL COEFFICIENT The values of e, a, and n for the pure systems studied were determined from a least squares fit of the experimental second virial coefficient using a least squares program based on the method of M. J. D. Powell. The common series expression was used to represent the second virial coefficient for the Mie (n,6) potential. 2 12 Hanley and Klein ' have recommended that for determining potential energy parameters emphasis be placed on the use of data in the reduced temperature range below T^ Â„ , = 2.0, where the reducing parameter for the temperature is the value of e/k for the LennardJones (12,6) potential. Unfortunately for many of the binary systems for which data exist the experimental values for the pure components are not available in this low reduced temperature range. Thus even though the present investigation employed data over as wide a temperature range as possible, more than one set of potential parameters e, a and n exist for at least some of the pure systems which can be identified with local minima in the sumof squares objective function. The parameters are listed in Table 1 that reproduce the experimental second virial coefficients for the singlecomponent systems in the temperature ranges and with the rootmeansquare deviations shown. Where more than one set of parameters was found, the set which gave the lowest value for the best sum of squares of the deviations was used. One further comment should be made. The rootmeansquare 3 deviation for CF, (0.12 cm /mole) given in Table 1 is greater than the 118
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119 3 12 value of 0.05 cm /mole for CF found by Sherwood and Prausnitz with the best set of parameters for the Sutherland (Â°Â°,6) potential using the same smoothed second virial coefficient data. This suggests that the Mie (136.30,6) potential used in this work, while giving an excellent fit, is not the "best" Mie (n,6) potential for the CF, data. However, the value of n = 136.30 was retained on the basis of the value of 138.68 found for the repulsion exponent of SF, , a similar molecule.
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APPENDIX B RESIDUAL THEEMODYNAMIC PROPERTIES Properties reported in Chapter 5 are residual properties. These are defined by Y^ y1^*^^^*^ Â„ideal gas f^,. where X is any thermodynamic property. For internal energies the residual properties are the same as configurational properties. 35 Rowlinson gives values for the experimental configurational energy argon, methane and nitrogen in the saturated liquid state. Entropies for liquid argon were taken from reference 36. The ideal gas values at the same T and V were calculated from the monatomic ideal gas partition function and subtracted from values of entropy in reference 36 to obtain experimental residual entropies. 37 Din gives entropies of liquid methane and nitrogen along the saturation curve. Interpolation between state points was carried out using the following equation AS = C Â£n(T./T) (B2) all where C is the heat capacity of a liquid which is maintained at all temperatures in equilibrium with an infinitesimal amount of vapor. This quantity is tabulated by Rowlinson. . Din gives values of the entropy of methane in the ideal gas state at one atmosphere. These were corrected for the effect of pressure using the ideal gas partition function to yield entropies of the ideal 120
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121 gas at saturated liquid temperatures and volumes. Ideal gas entropies of nitrogen at saturated liquid conditions were calculated from the partition function for an ideal diatomic gas and the physical constants given on page 431 of reference 38. It should be pointed out that the value given in this reference for the mass of a nitrogen molecule is in error. In the case of perf luoromethane ideal gas energies and entropies at one atmosphere are tabulated in reference 39. Entropies were again corrected for the effect of pressure using the ideal gas partition function. The experimental saturated liquid energies and entropies were taken from reference 40. The values reported for CF, are relative to 116.49Â°K, taken arbitrarily as the zero for energy and entropy. Calculated values reported for these properties for a given (n,6) potential were obtained by subtracting predicted quantitites at 116.49*'K from energies and entropies at other state points. The program for evaluating the hardsphere radial distribution 41 function and its density derivative was written by K. Rajagopal.
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APPENDIX C CALCULATION OF EXCESS FREE ENERGY Perturbation theory provides a means of calculating the residual properties of fluids. These properties are defined in Appendix B. For mixing at constant pressure and temperature the excess thermal properties of a mixture are not merely the differences between the residual properties of the mixture and the molefraction averages of the respective residual properties of the pure components. This is the result of the fact that all of the fluids (the mixture and the pure components) are not at the same volume when pressure and temperature are fixed. The following expression was used to calculate the excess Gibbs free energy (G^) at zero pressure for a binary mixture from the residual free energy of the mixture (G^) and that of the two pure components (Gand G^). X X ^^ = ^m " ^""l^l "^ ''2^2^ "^ ^^ '^"^^l^ ^2^''^m^ ^^"^^ All calculations of liquid properties in Chapter 6 have been 14 made with the perturbation theory of Barker and Henderson. Second order calculations employed the macroscopic compressibility approximation. PercusYevick hardsphere radial distribution functions and free f 18 energies have been used. CarnahanStarling hardsphere pressures and compressibilities have been used. Volumes for pure liquids and liquid mixtures at zero pressure were determined by numerical solution of the analytical expression for pressure in perturbation theory. 122
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V at 106. 7Â°K
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124 lll^K has been used with Eq. (D1) to determine a different value of the constant A to be used at 111Â°K. The value of A found to give the E 3 correct equimolar V (0.845 cm /mole) at 111Â°K is 3.38. This value of A is used with Eq. (D1) to estimate V^^o^^ at other compositions. J.J.1 K The volume of the CH, + CF, mixture at lll'K and at compositions 4 4 different from x^ = x is estimated by adding the calculated value E of V, ,^oÂ„ to the volume of an ideal mixture of CH. + CF. at 111 K. Ill K 4 4 Excess free energy at 111Â°K is calculated as a function of 27 composition from the following relation given by Thorp and Scott. G^/RT = x^x^El.Se + 0.36(x^ x^)] , (D2) where subscript 1 again refers to CH,.
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APPENDIX E A NEW APPROACH TO THE REFERENCE STATE FOR LIQUID TRANSPORT PROPERTIES Introduction A simple approach to the reference state for liquid transport properties explains the relation between the free volume theory of 42 43 Doolittle or Cohen and Turnbull and the conf igurational entropy 44 45 46 theory of J. H. Gibbs. ' ' It is suggested that both theories are based on the same reference state. In the new approach this state is the perfectly ordered liquid with a specific volume equal to that of the crystalline solid with a nearest neighbor separation equal to the radius of the first peak in the experimental liquid radial distribution function. As an approximation to this volume we follow the lead of Eyring's significant structure theory in using the volume of the solid at the normal melting point. This reference state satisfies the requirement of infinite viscosity, zero conf igurational entropy, and zero free volume. It also reduces by one the number of adjustable parameters in the free volume equation by fixing the reference volume V as the volume of the solid at the melting point. The theory is supported by empirical findings of Hogenboom, 48 Webb, and Dixon for liquid hydrocarbon viscosities, by the success of Eyring's liquid theory, and by results of correlating liquid viscosities for argon, benzene, and molten NaCl reported herein. The reference state predicted by the criterion of zero conf igurational entropy performs well in the free volume equation for viscosity supporting the unified view of free volume and conf igurational entropy. 125
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126 Two Current Theories In the past there has been considerable discussion of the glass transition in both simple liquids and polymer melts. A recent view 44 45 46 proposed first by Gibbs and DlMarzio ' ' Is that the observed glass transition temperature is determined by relaxation phenomena; however, there is a lower limit to the observed glass transition that would be observed in an experiment of infinite duration provided crystallization could be prevented. The glass at this temperature and density is called by Gibbs the ground state for the amorphous liquid. Gibbs maintains that this ground state is the point at which the 43 configurational entropy vanishes, while Cohen and Turnbull identify this state as the point of disappearance of the socalled free volume (V^) which appears in the free volume transport equations. The theories can be formulated in terms of either temperature or molar volume. In this work the latter will be used. In its 42 simplest form the free volume equation originally proposed by Doolittle for viscosity (n) is given as follows Â£n n = A + B/V^ (E1) where V^ = V V f o Here A and B are empirical constants; V is the volume per mole of the liquid, and V is the volume per mole of the liquid in the ground o state. Both of the above mentioned views assign to the ground state the thermodynamic significance of a fundamental reference state for
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127 liquid transport properties. In the following development it is suggested that the two theories are based on essentially the same reference state, one which can be predicted from solid properties. Hole Theory of the Liquid Gibbs and DiMarzio developed their theory for prediction of the glass transition on the basis that the glass transition was a true second order transition. They felt that the glass transition inter49 vened to prevent a paradox pointed out by Kauzmann. One aspect of the paradox arose when one extrapolated liquid entropies at constant pressure to temperatures well below the freezing point. The apparent entropy of the liquid eventually became lower than that of the corresponding crystalline phase. At the same time the apparent molar volume of the liquid became less than that of the crystalline phase. Gibbs employed the canonical configurational partition function for the liquid to predict the point of the second order transition. This point was characterized by a temperature TÂ„ (or equivalently VÂ„) . While Gibbs wrote the partition function for a polymer melt, this work will deal only with that for a simple liquid. The following assumptions for the simple liquid are analogous to those of Gibbs for the polymer melt: 1. The liquid can be accurately represented by a hole theory, consisting of molecules and empty spaces or of occupied and unoccupied lattice sites. 2. Vibrational contributions are not included in the definition of configurational entropy, as they are assumed to be the same in solid, liquid, and glass states.
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128 3. The ground state for the liquid is defined as the state of zero conf igurational entropy. For a simple liquid the conf igurational partition function Q can be expressed as follows: (n + n ) ! nE /2kT " n! n Â° ^ Â° 'f Â• '^Â» o Here n and n are the number of occupied and unoccupied lattice sites, respectively. The v' term, not to be confused with V in Eq. (E1) , arises from vibrations of molecules about their lattice sites. The E represents the potential energy of a molecule located on its lattice site, while T is the absolute temperature, and k is Boltzmann's constant. Gibbs leaves out the vibrational factor vl according to assumption 2, saying that it is unaffected by a phase transition. Thus, with Eq. (E2) and the usual definition of conf igurational entropy (S ) S^ = k Â£n Q^ + kT^ 9T c (E3) n,V we obtain neglecting the v' term s'^ = kÂ£n{(n + n )I/(n!n !)} , (E4) where we assume for simplicity that n and E are functions of volume only. In his treatment of polymer melts Gibbs chose to neglect the combinatorial factor (n + n )!/n!n ! when equating the conf igurational o o ^ <= o entropy to zero on the assumption that it was near to 1.0 relative to other terms in his expression. In the case of simple liquids, however.
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129 it can be seen from Eq. (E4) that when vibrational terms are neglected the only contribution to the conf igurational entropy is due to the combinatorial factor. By defining V as the state at which s'^ = it is seen that this is the point at which n =0. Thus if we define the free volume as the volume due to the presence of vacancies in the lattice, we have the point of zero conf igurational entropy coincident with the point of zero free volume. We can also see that in this state since n^ = we have a perfectly crystalline phase, not an amorphous glass as suggested by Gibbs and DiMarzio. If the dependence of n o on volume were known then the value of V (and corresponding T ) could be determined as the apparent volume of the state obtained by extrapolation to n = 0. o The Real Liquid It is seen above that in the simple hole theory under assumptions analogous to those of Gibbs the state of zero free volume coincides with the state of zero conf igurational entropy. A case can be made for the applicability of such a reference state in real liquids. One must first consider the full significance of zero conf igurational entropy as defined herein. First it must be said that in a real system the vibrational contribution to configurational entropy would not be expected to be exactly the same in the three condensed states, except at CK neglecting differences in zero point energies. Only here would the Gibbs definition of configurational entropy neglecting vibrational terms be strictly correct. For the present purposes the configurational entropy of a simple liquid other than that arising from vibrations will be referred to as the structural entropy since it depends only
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130 on the arrangement of molecules throughout the system in question. It is this structural entropy that is given in Eq. (E4) . In fact, it can be said that the only state of either the model or a real simple liquid with zero structural entropy (or conf igurational entropy in the Gibbs sense) would be a perfectly ordered crystalline state with no vacancies. Note that the temperature of this hypothetical state need not be CK. It is only required that the state is a crystalline one, since the Gibbs definition of s'^ includes only the structural contribution to the entropy of a simple liquid. Thus any crystalline solid free of vacancies and structural defects would satisfy this criterion at any temperature. The question that remains is whether there is reason to believe that such an ordered state, as opposed to the amorphous glass, is a reasonable choice for the ground state of the liquid. Experimental work of Mikolaj and Pings^Â°'^^ on the radial distribution function for liquid argon along with the success of the Eyring significant structure 52 theory and the recent theory of liquid structure by Bhatia support such a view. The experimental results of Mikolaj and Pings show that the position of the first peak in the radial distribution function for liquid argon is very nearly constant over the range of liquid conditions studied. The position of the first peak can be interpreted as the location of the shell of nearest neighbor molecules around a given molecule. For liquid argon this peak is located at an average distance of 3.81 % with a median of 3.84 + .10 1. This is near to the value of 3.885 1 corresponding to the nearest neighbor separation in solid argon at the melting point. It is also near to the value of 3.822 X,
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131 the location of the minimum in the LennardJones potential for argon with Michels' parameter. This potential has been shown by Monte Carlo and molecular dynamics calculations to be a good effective pair potential for liquid argon. IVhile the position of the first peak in the liquid radial distribution function is nearly constant, the coordination number (N ) or number of nearest neighbors, calculated from the area under 2 the first peak of the function ATrpr g(r), increases with increasing density, where p is the average number density and g(r) is the radial 52 distribution function. In fact Bhatia has shown that linear extrapolation of the coordination number from a value of zero at zero density yields a value of 12 in the supercooled liquid at a density near that of the solid at the melting point. At the density and temperature at which the apparent value of N becomes 12 naturally N , the number of i o vacant nearest neighbor lattice sites, becomes zero. Bhatia has assumed such a linear dependence of N in his theory, but the actual value of N depends on which of several methods are used to calculate it. Such evidence supports the view that the solid, originally with N^ = 12 in the crystalline state, expands in total volume on melting by reducing the number of nearest neighbors while keeping the nearest neighbor separation essentially constant. In the hole model discussed previously this is analogous to the introduction of vacant lattice sites in the lattice structure. From the variation of N with density contraction of the liquid on cooling appears to occur by a cooperative rearrangement of the molecules to expel the excess volume that was acquired on melting. This is accomplished by more efficient packing
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132 increasing the coordination number but still maintaining nearly the same nearest neighbor separation that existed in the solid at the melting point. This suggests that if it were possible to supercool the simple liquid to the density of the solid at the melting point without forming the glass, the liquid structure would approach smoothly the structure of the solid at the melting point at least with respect to the first coordination shell of nearest neighbor molecules. At this hypothetical point the coordination number would have increased to near 12 and the nearest neighbor separation would still be almost the same in the liquid as in the solid at the melting point. In this case the reference state or ground state of the liquid might well be taken as the perfectly ordered state with a density near that of the solid at the melting point. A more correct reference volume (V ) would be that of the crystalline state with a nearest neighbor separation equal to the radius of the first peak in the experimental liquid radial distribution function. To the extent that this is the same as that in the solid at the melting point, the density of the solid at melting should be a good approximation to the reference state density. For argon V would be a slightly smaller volume than V the volume of the solid at the melting point. sm With the proposed reference state the criterion of zero structural entropy would be met by virtue of the crystalline structure. Also it is the crystalline state with which we naturally associate infinite viscosity. At the same time with the free volume defined as that volume in excess of the volume of the solid at the melting point the free volume also vanishes in the reference state.
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133 Verification of the Proposed Reference State 48 The recent work of Hogenboom, Webb, and Dixon has shown that for five hydrocarbon liquids the V in Eq. (E1) which gives the best fit of the experimental viscosity differs from the experimental volume of the solid at the melting point by an average of only 0.96%. It is interesting to note that in four of the five liquids the value of V o was slightly smaller than V (the volume of the solid at the melting sm Â° point). This is in line with the statement made earlier in the case of argon. The above cited authors have stated that the value of V which gives the best fit of viscosity increases slightly with decreasing temperature and might thus equal V in the supercooled liquid region. 53 According to Macedo and Litovitz it is the supercooled region in which Eq. (E1) in its simple form is most applicable. The simple view of the reference state is further supported by correlating the viscosity for three very different liquids using the free volume equation for viscosity given in Eq. (E1). The parameter V which is usually found empirically as a third parameter has been taken as volume per mole of the solid at the melting point. The results of the correlations for saturated liquid argon, saturated liquid benzene, and molten sodium chloride are given in Table 38. The results are encouraging. Values of the parameters A and B which minimize the sum of squares of deviations are given in Table 39 along with the volume of the various solids at the melting point. The simple free volume transport equations have historically performed better for the larger polyatomic molecules than for such
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134 Table 38. Comparison of predicted and experimental viscositiesViscosity (centipoise) Compound
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135 Table 38 (Continued) Viscosities and densities from J. Tiiranermans, PhysicoChemical Constants of Pure Organic Compounds (Elsevier, New York, 1950). 'Viscosities and densities from NSRDSNBS15, Molten Salts: Vol. I Electrical Conductance, Density and Viscosity Data, National Bureau of Standards, Oct. 1968.
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136 Table 39. Bestfit parameters for equation (E1) Compound Argon
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137 simple molecules as argon. The poorer performance of the predicted reference volume in Eq. (E1) for argon may be the result of the failure of the simple form of the equation rather than of the predicted reference state. The correlation for liquid argon would be improved somewhat if instead of equating V and V , we take V equal to the volume of the " o sm' o solid with a nearest neighbor separation of 3.81 A suggested by the radial distribution function data. This lowers the rms % deviation in Table 39 to 5.71. The Glass Transition As mentioned earlier, the mechanism of contraction proposed for the liquid requires a cooperative rearrangement. Since relaxation times in liquids increase with decreasing temperature, it is reasonable to expect that for a given liquid there exists a temperature T. below which such a cooperative rearrangement is extremely unlikely during the duration of an experiment. Below this temperature the liquid would be expected to contract by simply shrinking the existing structure of the liquid as no cooperative phenomena would be involved. In this region the nearest neighbor number N would cease to be a variable, and the previously constant nearest neighbor distance would vary as the cube root of the volume as is the case in crystalline solids. This change from one mode of contraction to the other would be expected to occur not at one particular temperature but over a temperature range. The amorphous character of the liquid that existed at the lower limit of this temperature range T would be expected to be frozen in at all temperatures below T^ . Such is the case with the glass transition.
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138 49 Kauzmann has pointed out that, while there are no discontinuous changes in volume and heat content as the liquid goes through the glass transition, once the glass has formed changes in volume and energy with temperature are similar in magnitude to those of a crystalline solid. 54 Using the free volume equation. Carpenter, Davies and Matheson have had some success in predicting the glass transition point as the 13 point at which n = 10 poise. Such success indicates that attempts to identify the reference state, which in the free volume equations is by definition one of infinite viscosity, with the glass transition are incorrect. Rather, it is suggested that the simple view of the reference state proposed in the present work is more reasonable. It should also be noted that with the present definition of the reference state the Kauzmann paradox is eliminated, since the reference state represents the end to any reasonable extrapolation of liquid properties to high densities. Conclusions A simple view of the reference state for liquid transport properties satisfies the Gibbs requirement of zero configurational entropy and the Doolittle or the Cohen and Turnbull requirement of zero free volume. The reference state is taken to be the perfectly crystalline phase with a nearest neighbor separation equal to that in the liquid. The volume of this phase is taken to be approximately equal to the volume of the solid at the melting point. It has been shown that the proposed approximation to the reference state volume (V ) can be used to correlate liquid viscosities in the simplest form of the free volume
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139 viscosity equation. It should be mentioned at this point that attempts 53 to define V as the closepacked molecular volume contain the assumption that infinite viscosity is associated only with a state in which molecular motion is impossible as molecules are faced with essentially infinite potential energy barriers in all directions. The new approach outlined in this work capitalizes on the fact that this is certainly not a necessary condition, as would be seen in the case of infinite viscosity in the crystalline solids with a volume greater than the closepacked volume. It is suggested that it is the vanishing of the structural contribution to configurational entropy resulting in a perfect crystalline state that is correctly associated with infinite liquid viscosity. The use of the density of the solid at the melting point as the reference density for the liquid is in accordance with the emphasis which Eyring's significant structure theory has placed on this experimental quantity. Eyring's expression for liquid viscosity likewise yields infinite viscosity when the liquid volume equals the volume of the solid at the melting point.
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LIST OF REFERENCES 1. M. Klein, Natl. Bur. Std. (U.S.), AEDCTR6767. 2. H. J. M. Hanley and M. Klein, Natl. Bur. Std, (U.S.), Tech. Note 360 (1967). 3. R. C. Ahlert, G. Biguria, and J. W. Gaston, Jr., J. Phys . Chem. n, 1639 (1970). 4. F. London, Z. Physik 63, 245 (1930). 5. I. Amdur, E. A. Mason, and A. L. Harkness, J. Chem. Phys. 22_, 1071 (1954). 6. A. A. Abrahamson, Phys. Rev. 133, 990A (1964). 7. H. Lehmann, Z. Phys. Chem. (Leipzig) 235_, 179 (1967). 8. R. J. Good and C. J. Hope, J. Chem. Phys. 53_, 540 (1970). 9. J. H. Dymond and B. J. Alder, J. Chem. Phys. 51, 309 (1969). 10. T. M. Reed III and K. E. Gubbins, Applied Sta tistical Mechanics (McGrawHill, New York, to be published). 11. C. A. Eckert, Henri Renon, and J. M. Prausnitz, Ind. Eng. Chem. Fundamentals 6^, 58 (1967). 12. A. E. Sherwood and J. M. Prausnitz, J. Chem. Phys. 41, 429 (1964). 13. A. E. Sherwood and J. M. Prausnitz, J. Chem. Phys., 41, 413 (1964). 14. J. A. Barker and D. Henderson, J. Chem. Phys. 47^, 4714 (1967). 15. I. R. McDonald and K. Singer, J. Chem. Phys. 50, 2308 (1969). 16. A. Michels, H. Wijker, and H. K. Wijker, Physica 15, 627 (1949). 17. W. R. Smith, Ph.D. Thesis, University of Waterloo, Waterloo, Ont., Canada, 1969. 18. N. F. Carnahan and K. E. Starling, J. Chem. Phys. 51, 635 (1969). 19. W. R. Smith and D. Henderson, Mol. Phys. 19_, 411 (1970). 20. J. W. Dymond, M. Rigby and E. B. Smith, Phys. Fluids 9. 1222 (1966). 21. T. W. Leland, J. S. Rowlinson and G. A. Sather, Trans. Faraday Soc. 64, 1447 (1968). 140
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141 22. T. W. Leland, J. S. Rowlinson, G. A. Sather and I. D. Watson, Trans. Faraday Soc. 65, 2034 (1969). 23. P. J. Leonard, J. A. Barker, and D. Henderson, Trans. Faraday Soc. 66^, 2439 (1970). 24. B. L. Rogers and J. M. Prausnitz, "Calculation of High Pressure Vaporliquid Equilibria...," presented at the Los Angeles Meeting of the American Chemical Society, March, 1971. 25. D. R. Douslin, R. H. Harrison, and R. T. Moore, J. Phys. Chem. 71, 3477 (1967). 26. J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids (WiJey, New York, 1967), p. 163. 27. I. M. Croll and R. L. Scott, J. Phys. Chem. 6_2, 957 (1958); N. Thorp and R. L. Scott, J. Phys. Chem. 6Â£, 670, 1441 (1956). 28. D. W. Calvin and T. M. Reed III, unpublished work. 29. J. A. Barker and D. Henderson, J. Chem. Phys. 42, 2856 (1967). 30. D. Henderson and P. J. Leonard, Proc. Natl. Acad. Sci. U.S., 68 , 632 (1971). 31. D. Henderson and P. J. Leonard, Proc. Natl. Acad. Sci. U.S., 67 , 1818 (1970). 32. J. M. Prausnitz, Molecular Thermodynamics of FluidPhase Equilibria (PrenticeHall, New Jersey, 1969), p. 24. 33. J. V. L. Singer and K. Singer, Mol. Phys. 19, 279 (1970). 34. M. J. D. Powell, Computer J. ]_, 303 (1964). 35. J. S. Rowlinson, Liquids and Liquid Mixtures , Second Edition (Plenum, New York, 1969). 36. A. L. Gosman, R. D McCarty, and J. G. Hust, Natl. Bur. Std. (U.S.), NSRDSNBS 27 (1969). 37. F. Din, ed.. Thermodynamic Functions of Gases , Vol. Ill (Butterworth, London, 1961) . 38. E. A. MoelwynHughes, Physical Chemistry , Second Revised Edition (Pergamon, London, 1964). 39. J. H. Simons, ed.. Fluorine Chemistry , Vol. V (Academic, New York, 1964), p. 224.
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142 40. D. L. Flske, Refrigerating Engineering, April (1949), p. 336. 41. K. Rajagopal, Ph.D. dissertation, University of Florida, Gainesville, Fla., 1971. 42. A. K. Doolittle, J. Appl. Phys. 22^, 1471 (1951). 43. M. H, Cohen and D. Turnbull, J. Chem. Phys. 31, 1164 (1959); 34, 129 (1961). 44. J. H. Gibbs, J. Chem. Phys. 25, 185 (1956). 45. J. H. Gibbs and E. A. DiMarzio, J. Chem. Phys. 28, 373 (1958); 28, 807 (1958). 46. J. H. Gibbs, in Modern Aspects of the Vitreous State , edited by J, D. Mackenzie (Butterworth, London, 1960), Chapter 7. 47. H. Eyring and M. S. Jhon, Significant Liquid Structures (Wiley, New York, 1969). 48. D. L. Hogenboom, W. Webb, and J. A. Dixon, J. Chem. Phys. 46_, 2586 (1967). 49. W. Kauzmann, Chem. Revs. 43, 219 (1948). 50. P. G. Mikolaj and C. J. Pings, J. Chem. Phys. 46_, 1401 (1967). 51. C. J. Pings, in Physics of Simple Liquids , edited by Temperley, Rowlinson, and Rushbrooke (North Holland, Amsterdam, 1969), Chapter 10. 52. K. K. Bhatia, Ph.D. dissertation. University of Florida, Gainesville, Fla., 1969. 53. P. B. Macedo and T. A. Litovitz, J. Chem. Phys. 4_2, 245 (1965). 54. M. R. Carpenter, D. B. Davies, and A. J. Matheson, J. Chem. Phys. 46, 2451 (1967).
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BIOGRAPHICAL SKETCH Donald William Calvin was born April 26, 1945, at Princeton, Indiana. At the age of six, he moved with his family to Oak Ridge, Tennessee, where he enjoyed five years in the foothills of the Great Smokey Mountains. His family then moved to Baton Rouge, Louisiana, where the bayou country provided a pleasant setting for the remainder of his education through the baccalaureate level. In May of 1967, he was married to the former Barbara Gene Beaucoudray of Baton Rouge. Upon receipt of a Bachelor of Science Degree in Chemical Engineering at Louisiana State University in January of 1968, he moved to Gainesvillej Florida, to pursue graduate education toward the degree of Doctor of Philosophy in the Department of Chemical Engineering at the University of Florida. Here the pleasant Florida living made the demands of graduate study seem less severe and provided an excellent atmosphere for the early childhood of his daughter, Sandy, born October, 1968. 143
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. iZ /^Â•/^^ T. M. Reed, Chairman Professor of Chemical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. (^ O^^^JSL JU p. O'Connell Associate Professor of Chemical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. W. Wi Menke Associate Professor of Management I certify that I have read this study and that in my opinion it conform.s to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. of Physics This dissertation was submitted to the Dean of the College of Engineering and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of DoctojxjfT^ilosophy , March, 1972 D'5^, College of Engineering Dean, Graduate School
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