THERMODYNAMIC PROPERTIES OF MULTICOMPONENT MIXTURES FROM
THE SOLUTION OF GROUPS APPROACH TO DIRECT
CORRELATION FUNCTION SOLUTION THEORY
By
JCILH CHARLES TELOTTE
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR
THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1985
To Bonnie, for her support and encouragement
and patience and love
ACKNOWLEDGEMENTS
The author would like to thank Dr. John P. O'Connell
for his guidance and understanding throughout the years.
Thanks also go to Dr. Randy Perry for many helpful discus
sions and the members of the supervisory committee.
TABLE OF CONTENTS
PAGE
ACKNOWLEDGEMENTS................................ iii
LIST OF SYMBOLS ................................. vi
ABSTRACT........................................ x
CHAPTER
1 INTRODUCTION ............................... 1
2 THERMODYNAMIC THEORY................ ...... .. 4
Thermodynamic Properties of Interest....... 5
Calculation of Liquid Volumes.............. 6
Chemical Potentials and Fugacities......... 7
Fluctuation Solution Theory................ 9
Summary .................................... 13
3 GROUP CONTRIBUTIONS ........ ............... 14
"Reactive" System Theory ................... 18
RISM Theory. ................................ 19
Differences between RISM Theory and
"Reactive" Solution Theory............... 20
Thermodynamic Properties from Group
Functions. ................................ 21
Summary................... ................... 24
4 FORMULATION OF MODELS FOR DIRECT
CORRELATION FUNCTIONS ...................... 25
Thermodynamic Consistency .................. 25
Basis for Model Development................ 28
Analysis of the Model....................... 30
Property Changes Using the Proposed Model.. 32
5 MODEL PARAMETERIZATION..................... 34
Corresponding States Theory................ 35
Choice of a Reference Component............ 39
Method of Data Analysis..................... 40
Analysis of Argon Data..................... 44
Analysis of Methane Data................... 49
Use of the Correlations to Calculate
Pressure Changes. .................. ...... 53
6 USE OF THE GROUP CONTRIBUTION MODEL FOR
PURE FLUIDS ................................ 59
Extension of the Model to Multigroup
Systems.................................. 59
Designation of Groups ...................... 61
Pressure Change Calculations............... 61
Summary .................................... 68
7 DISCUSSION ................................. 73
Fluctuation Solution Theory................ 74
Approximations ............................. 74
Comparison Calculation.......... ............. 82
Summary .................................... 83
8 CONCLUSIONS ................................ 84
APPENDICES
1 FLUCTUATION DERIVATIVES IN TERMS OF DIRECT
CORRELATION FUNCTION INTEGRALS ............. 86
2 PERTURBATION THEORY FOR DIRECT CORRELATION
FUNCTION INTEGRALS USING THE RISM THEORY... 92
3 THREE BODY DIRECTION CORRELATION FUNCTION
INTEGRALS .................................... 96
4 NONSPHERICITY EFFECTS...................... 98
5 HARD SPHERE PROPERTIES..................... 110
6 COMPUTER PROGRAMS........ .................. 119
7 COMPRESSIBILITY THEOREM FROM RISM THEORY... 156
REFERENCES...................................... 163
BIOGRAPHICAL SKETCH ............................. 167
LIST OF SYMBOLS
an,aB nth order expansion coefficient in perturbation
function for pair aB
C matrix of direct correlation function integrals
CO matrix of shortranged group direct correlation
function integrals
C matrix of purely intermolecular group direct
correlation function integrals
C.. molecular direct correlation function integrals
C05 group direct correlation function integral
c.. molecular direct correlation function
cas group direct correlation function
Cijk three body direct correlation function integral
c.. group direct correlation function
f temperature dependent function
g temperature dependent function
gij pair correlation function
H matrix of total correlation function integrals
h supermatrix of group total correlation functions
h.. total correlation function
1J
h group total correlation function
ij3
HE excess enthalpy
I, I identity matrix and supermatrix
i molecule
K matrix of differences between direct correlation
function and its angle average
k wave vector
k Boltzmann's constant
M general property
M general partial molar property
N total number of moles, number of components
N. number of moles of species i
1
O lowest order
P pressure
R gas constant
Position vector
r distance
r position vector
T temperature
V, V volume, molar volume
V. partial molar volume
1
W matrix of intramolecular correlation functions
W intramolecular correlation function integral
X matrix of mole fractions
X independent set of mole fractions
X. mole fraction of molecule i
1
Z compressibility factor
a generalized hard sphere parameter
= 1/k T, dimensionless inverse temperature
Yi activity coefficient
5(.) Dirac delta function
vii
6.. Kroniker delta
E energy
n packing fraction
i. chemical potential
v matrix of stoichiometric coefficients
5X th order packing fraction
p total density
p vector of densities
P matrix of densities
pipa density of species i or group a
0 supermatrix of intramolecular correlation functions
o. diameter
1
pa8 perturbation function
2 orientation normalization factor
Q matrix of correlation functions
2 correlation function integral
_,] matrix or supermatrix of intramolecular
correlation functions
Subscripts
i,j,... species quantity
n nth order term in series expansion
a,B,... group quantity
Superscripts
E excess property
HS hard sphere property
viii
mix mixture property
o reference state property
PYC, PYV using PercusYevick compressibility or virial
forms
r residual property
ref reference
T transpose of a matrix or vector
Special Symbols
vector
matrix
supermatrix
partial molar property
< > ensemble average or angle average
fourier transform
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of
Doctor of Philosophy
THERMODYNAMIC PROPERTIES OF MULTICOMPONENT MIXTURES
FROM THE SOLUTION OF GROUPS APPROACH TO DIRECT
CORRELATION FUNCTION SOLUTION THEORY
By
John Charles Telotte
May 1985
Chairman: John P. O'Connell
Major Department: Chemical Engineering
A solution of groups technique was developed for use
with fluctuation solution theory. The general expressions
for calculation of pressure and chemical potential changes
from some fixed reference states have been shown. A new
corresponding states theory correlation for direct correla
tion function integrals was proposed and used with the
group contribution technique for calculation of pressure
changes during compression for several nalkanes and
methanol.
This work gives a detailed analysis of the RISM theory
of liquids. Shown are new results for perturbation theory
and a generalized compressibility theorem for RISM fluids.
The use of the RISM theory for calculation of thermodynamic
properties of real fluids also is given.
The use of hard sphere reference fluids for development
of equations of state has been explored. A generalized
hard sphere equation of state was developed. It was shown
that the most accurate hard sphere equation of state is
not the best reference system for construction of a liquid
phase equation of state of the van der Waals form.
CHAPTER 1
INTRODUCTION
The rational design of chemical process equipment
requires knowledge of the thermodynamic properties of the
substances involved. More specifically, volumetric
properties are needed to size a piece of equipment,
enthalpies are needed to determine heat duties, and fugaci
ties are used to decide feasibility of reactions and
separations.
The general problem in physical property correlation
and estimation is then to determine a procedure for calcula
tion of volumes, enthalpies, and fugacities, for multi
component, multiphase mixtures. This has been accomplished
for some pure components using an equation of state with
a large number of parameters. Here, a more modest problem
is addressed. A formalism is developed that allows the
volumetric properties and chemical potentials for liquid
mixtures to be expressed in terms of a single group of
parameterized functions.
The approach used here tries to recognize the essential
differences between liquid and vapor phase properties.
A thermodynamically consistent formulation for the volumetric
properties and chemical potentials is developed using an
equation of state. Here, unlike many other schemes, the
1
equation of state uses a liquid reference state. This
is all made possible by utilization of the KirkwoodBuff
(1951) or fluctuation solution theory.
In an effort to allow for extrapolation of present
results to state conditions not examined, the correlations
are put into a corresponding states form. This can be
justified from analysis of the microscopic theory that
is the basis for this macroscopic approach.
Finally, to allow for actual physical property estima
tion, the theory is cast in a "groupcontribution" form.
This is easily accomplished formally because the fluctuation
solution theory relates molecular physical properties to
microscopic correlations. These quantities are well defined
for groups, or sites, as they will be often termed.
The next chapter in this work goes into detail in
discussing the thermodynamic aspects of the problem and
the possible approaches that are available to express the
required physical properties. Then, the flucutation solution
theory is outlined.
Chapter 3 is devoted to the discussion of group con
tribution modeling. Two possible means of linking the
group contribution ideas to the fluctuation theory are
presented and their differences expounded upon. This chapter
concludes with the formulation of the physical property
relationships from group correlation function integrals.
Approaches to modeling correlation functions are the
emphasis of Chapter 4. General forms, as suggested by
microscopic perturbation theories, are outlined and
discussed. The thermodynamic consistency requirements
for model formulation are given along with the final expres.
sions for the thermodynamic property changes from the chosen
form.
Because the correlation function integrals contain
unknown functions, expressed in corresponding states form,
some experimental data are required for the model parameteri
zation. In Chapter 5 possible data for this parameterization
are discussed critically, and two sets of general correla
tions are developed. One of these correlations is then
decided upon based on the problem of interest and accuracy.
Chapter 6 then presents calculations for several
nalkane molecules' compressions. Both correlations and
predictions are shown. Chapter 7 is a general discussion
of all of the previously reported work, and Chapter 8 offers
several conclusions and recommendations for future work.
There are also several appendices which give greater
detail on developments presented in the body and listings
of several useful computer programs.
CHAPTER 2
THERMODYNAMIC THEORY
Thermodynamics is a science that has its basis on
certain empirical observations that have become known as
laws. The simplest of these observations, and the ones
most often used in practice, are that mass and energy are
conserved quantities. These statements allow for the con
struction of useful mass and energy balance equations.
These balances, when augmented with constitutive relations
such as the conditions of phase or reaction equilibria,
can be used for design ofchemical process units. However,
the power of these balance expressions cannot be fully
utilized unless the physical properties that appear in
the expressions are available.
In the following discussion one other empirical obser
vation is required. This is not known as a law but is
often listed as a postulate of thermodynamics (Modell and
Reid, 1974). This result may be stated in many ways, but
the simplest formulation is as follows: For a simple,
homogeneous system of N components, with only PV work
and thermal interactions with its surroundings, that N+l
independent, intensive variables are required to specify
the state of the system. The simplest use of this observa
tion is the existence of equations of state, such as
P = f(T,V,X)
Thermodynamic Properties of Interest
As stated previously, this work is concerned with
the volumetric properties and chemical potentials of the
components in liquid mixtures. The chemical potentials
are of interest for phase or chemical equilibrium calcula
tions because of the constraints that are imposed. For
equilibrium between phases a and B the constraints are
vi = 1. V. in both a and B (22)
1 1 1
and the chemical equilibrium constraints are
I v. v. = 0 V reaction j (23)
iJ 1
These equilibrium conditions are used to determine both
feasible operating conditions for chemical processes and
minimum work requirements for some processes.
To size a piece of process equipment one must have
knowledge of volumetric properties of the fluids involved.
The molar volume of a mixture is found from knowledge of
the component partial molar volumes by
V = V x. V. (24)
 i 1
(21)
An important aspect that must be considered in con
struction of thermodynamic models is the relationship between
chemical potentials and partial molar volumes
( ) = (25)
P T,X
When both the chemical potentials and partial molar volumes
are determined from the same equation of state, then equation
(25) will always be satisfied. However, if different
correlations are used for calculating these two different
properties, then a thermodynamic inconsistency exists and
can cause computational difficulties.
Calculation of Liquid Volumes
Reid, Prausnitz, and Sherwood (1977) give an extensive
review of techniques for correlating liquid molar volumes.
Most effort has been placed on calculation of saturated
liquid volumes because of the insensitivity of liquid volumes
to pressure. Most of the correlations are in corresponding
states form.
Calculations of compressed (subcooled) liquid volumes
often are based on a knowledge of the saturated liquid
volume at the system temperature. In general, the volumes
are found from expressions of the form
V = V(T,P)
(26a)
Vmix = V(T,P,X) (26b)
These are calculationally convenient forms. If a complete
equation of state is used, the volumes are found from solu
tion of the implicit relation
P = P(V,T,X)
and care is often necessary to ensure that the proper volume
is calculated. Equation 21 can be satisfied for several
values of the volume at a fixed temperature and composition.
Partial molar volumes are found from the definition
V = (27)
1 T,P,Njfi
For most correlations this is evaluated by using one of
the pure component correlations with a set of mixing rules
applied to the parameters.
Chemical Potentials and Fugacities
Chemical potentials are used for solution of phase
or reaction equilibria problems. However, the chemical
potentials themselves are not often used because the
equilibrium expressions can be written in terms of the
fugacities, defined by
S= 1 + RT Zn (fi/f ) (28)
ii
where the superscript o refers to a given reference state
of a specified pressure, composition, and phase at the
system temperature, T. The fugacity can be found from
two general approaches based on different reference states.
For the ideal gas reference state a complete equation of
state is required, one that can reasonably predict the
system volume. Another, more common, approach is to use
a liquid reference state and an expression for the excess
Gibbs free energy. In this formulation the fugacity is
written as
f. = X. Y f (29)
1 1 1 1
The Y. term, known as the activity coefficient, is found
1
from the free energy model and is used to correct for com
position nonideality in the liquid solution.
Mathias and O'Connell (1981) have proposed a slight
variant on this liquid reference state scheme. Their
approach is based on using the temperature and the component
densities, 0i = N./V as the independent variables to describe
the state of the system. Then, at constant temperature
the following relation holds:
3 Zn f.
d Zn f = ) dp (210)
1 3 j
J T 'Pk/j
Fugacity ratios can be calculated based on any reference
state if the partial derivatives in equation 210 are known.
Expressions for these derivatives in terms of integrals
of microscopic correlation functions are presented in the
next section.
Fluctuation Solution Theory
Fluctuation solution theory (Kirkwood and Buff, 1951;
O'Connell, 1971, 1982) is a bridge that connects the thermo
dynamic derivatives to statistical mechanical correlation
function integrals. The basic relation of fluctuation
theory is
3
IL = (
j T,V,P j 1 1 (211)
where the brackets denote an average over an equilibrium
grand canonical ensemble and B = 1/k T. These averages
are related to correlation function integrals by
6.. = (i ) f dld2gij(1,2) (212)
1 3 13 1 Q2 ]
where fdl is an integration over all phase space coordinates
required for molecule 1 and Q is the normalization constant
for the orientation dependence. The function g.j(1,2)
is known as a pair correlation function and is directly
related to a two molecule conditional probability density.
It is often more convenient to work with the total correla
tion function, h. (1,2), defined by
h..(1,2) = gi. (1,2) 1 (213)
Combination of equations 212 and 213 leads to
3
! = .. + H. (214)
STV,,kj 13 1 1 3 13
where we have defined
H.i = 1 I dld2 hi (1,2) (215)
Now because of the translational invariance of an equilibrium
ensemble not subject to external fields
hij(l,2) h (R1,R2, ) = h ((RR2, 2) (216)
and thus we can also write
1 dRd dh (, (217)
ij V I dRd2 2 1 2
To simplify the further analysis we rewrite equation 214
using matrix notation as
=1 ) = [N + NHN]i (218)
j T,V,Ik j
where the elements of the N and H matrices are
(N) = .ij (219a)
(H)i = H. (219b)
To calculate changes in chemical potentials, one is
interested in the inverse of equation 218
j) = {[N + NHNI j (220)
j T,V, 13
kpj
This is most easily expressed in terms of integrals
of the direct correlation functions, c.i(1,2), introduced
by Ornstein and Zernike (1914). These direct correlation
functions are defined by
h. (1,2) = c. (1,2) + V J d3cik(l,3)hkk(3,2) (221)
3 k
Because this integral is not of full convolution form,
it may seem that it is not possible to relate the fluctuation
derivatives to integrals of the direct correlation functions
but that is incorrect. Appendix 1 gives the details of
the relationship required. Using equations 220 and Al23
we find
3Bu. 1
( i) = [Nh /v]j (222)
3 T,V,
k j
and if the independent variables used are the component
densities the result is
3ui 1
p ) = [e C]i (223)
If equation 223 is combined with equation 210 one
finds that
Snf
) Cij (224)
TPkfj
These relations are most useful because they are
required to derive the differential equation of state.
If the GibbsDuhem equation (at constant T) is written
as
dPS = I pi [ d p. (225)
i j pj T,pkj
then it is shown that the equation of state can be found
through knowledge of the direct correlation function
integrals. Combining equations 225 and 223 yields
dPB = [ [1 C p. C..]d p (226)
A knowledge of the C.. then allows for the calculation
of both pressure changes and fugacity ratios relative to
any chosen reference state. These results can also be
used to determine partial molar volumes of all components
in a mixture.
13
Summary
This chapter has dealt with some of the properties
of interest for process design. The relation, required
by thermodynamic consistency, between the partial molar
volumes and the chemical potentials has been emphasized.
Mention was made of the common forms of correlations for
these quantities, asserting that often liquid reference
state approaches are used for both. The final section
presented the fluctuation solution theory that allows for
calculation of the partial molar volumes and chemical poten
tials, based on any reference state, to be expressed simply
in terms of ope set of functions.
CHAPTER 3
GROUP CONTRIBUTIONS
One of the more powerful tools developed for physical
property estimation has been the group contribution concept.
The term group normally refers to the organic and inorganic
radicals but can be more specific. A molecule of interest
can be described by the number of the different types of
groups of which it is composed.
There are two general methods for using the group
contribution concept. In the first approach some molecular
property is written in terms of the state variables and
a set of parameters, 6,
M = M(T,P; 8) (31)
and the parameters for a given set of substances are found
as sums of group contribution
i a 9 ia (32)
all
groups a
The most common forms of this type of formulations have
been for predicting critical properties (Lydersen, 1955)
and ideal gas specific heats (Verma and Doraiswamy, 1965).
"In some cases this technique can be justified on molecular
grounds.
The second use of group contribution ideas has been
to assume that the groups actually possess thermodynamic
properties and that the molecular properties are then a
sum of these group properties,
M. = M
1 a l a (33)
groups
a in i
This idea is the basis for two popular activity coefficient
correlations, ASOG (Derr and Deal, 1968) and UNIFAC
(Fredendslund et al., 1975). A model for the group proper
ties must be developed for this technique to be useful.
The true utility of the group contribution approach
stems from its predictive ability. Because all of the
molecules in a homologous series are formed of the same
groups, only in different proportions, the data for several
of the elements of the series can be used to establish
the group property correlations. These can then be used
to predict the properties of the other series members.
This has even greater scope for mixtures.
Consider for example mixtures of nalkanes and
nalkanols. They can be considered to be made of only
three groups, OH, CH2, and CH3. Thus, any mixture of
the alkanes and alkanols can be described by the concentra
tion of these three groups. If a viable theory exists
for some property in terms of the group functions, then
the properties of all mixtures of these groups are set.
Figure 31 shows an application of these ideas for
calculation of the excess enthalpy of alkanealkanol mix
tures. The figure shows the surface of the excess enthalpy
for all mixtures of the hydroxyl, methyl, and methylene
groups calculated using the UNIFAC equation (Skjold
Jirgensen et al., 1979). Any compound made of the three
groupsis represented by a point in the base plane. For
example, the point (XCH = 1/2, XCH = 0, XOH = 1/2) is
that for methanol. The possible group compositions for
any mixture are found along the line connecting two molecular
points. In the figure these lines are drawn in for methanol
pentane, ethanolpentane, and pentanolpentane. The predic
tion of the excess enthalpy is then found by the intersection
of a vertical plane along the composition path and the
property surface. To obtain the enthalpy prediction of
the molecular system, the ideal solution value must be
subtracted from the group estimate. The ideal solution
line simply connects the property surface at the points
of the pair molecules. The enthalpy prediction for a
molecular system of methanol and pentane and at (XME =
M
2/3, XE = 1/3) is shown as the value Ho in the figure.
The power here is that this one diagram can be used to
find excess enthalpies for all alkanealkanol mixtures.
17
\ LEX ", 
 L
jOLrz ;O
00
Z
  
systems at 298 K calculated using UNIFAC
theory.
"Reactive" System Theory
Equation 33 can also be written as
Mi = v ui Ma (34)
where via the stoichiometric coefficient, represents the
number of groups of type a in molecule i. This expression
is completely analogous to that found for a system of groups
"reacting" to form the molecule
I Via" i (35)
With this idea the group contribution expressions have
a physical interpretation and the thermodynamics of
"reactive" systems (Perry et al., 1981) can be applied
to obtain many results.
The motivation here has been to use the fluctuation
solution theory in terms of the direct correlation function
integrals. It is then desirable to determine the relation
ship between the molecular and group direct correlation
functions. This has been accomplished in a very general
fashion by Perry (1980). The development is lengthy, but
the salient features are presented below.
The most important aspect of this approach is that
even if the groups do not have a thermodynamics, there
are still welldefined correlations between groups. This
allows for the development of the KirkwoodBuff theory
19
in terms of group fluctuation, with the constraints offered
by equation 35. The result is
i T 1
j ~ = [MT (e' c' ]6)
jp T kj 1 (36)
where p' is the matrix of group densities and C' is a matrix
of group direct correlation function integrals.
While this theory is formally exact for the "reactive"
system, it can be difficult to apply. The correlation
function integrals contain both inter and intramolecular
contributions. For the total correlation functions these
effects can be separated, but no known analogous result
exists for the direct correlation functions (Lowden and
Chandler, 1979). This is a problem that becomes of great
importance in attempting to model the correlation function
integrals.
RISM Theory
Chandler and Andersen (1973) have formulated a molecular
theory for hard sphere molecules known as RISM. The mole
cules are assumed to be composed of overlapping hard spheres
or groups. They show how the molecular OrnsteinZernike
equation 218 can be reduced to a group form with explicit
separation of the intermolecular and intramolecular correla
tions. This allows them to define group direct correlation
function integrals that have only intermolecular
20
contributions. The development is detailed but the essential
result is
cij(1,2) = i Vje C B,(ri r (37)
Note here that the molecular correlation function has orien
tation dependence even though the group functions,
(a 6
caB(ri,rj), are written for spherically symmetric inter
actions. Chandler and Andersen discuss many attributes
of these group functions but put no emphasis on the thermo
dynamic ramifications of these findings.
Differences between RISM Theory and
"Reactive" Solution Theory
The major difference between the RISM theory and the
"reactive" solution theory is the nature of the direct
correlation functions. In Perry's formulation we have
4
c B(r = direct correlation function (38)
between group a and group B
and the RISM theory uses
c "(ri,r) = direct correlation function (39)
Between group a on molecule i
and group B on molecule j
This shows that the RISM functions contain less information
but may be easier to model because they seer. analogous
21
to molecular functions for which models have been developed.
Chandler and Andersen have also presented a variational
theory which can be used to obtain the direct correlation
functions for hard body systems.
Figure 32 shows the results obtained using data from
Lowden and Chandler (1973) for a system of hard diatomic
molecules. The diatomic molecules are treated as overlapping
spheres of diameter o separated by a distance L. The figure
shows that the calculations agree to a reasonable degree
with the Monte Carlo calculations for this type of system.
Another important aspect of the ChandlerAndersen
theory is that they discuss how the c.B functions could
be written for real molecules. In general, they show that
perturbation theories that would be valid for molecular
systems would also apply to RISM group system. The RISM
formulation will be employed in the present work.
Thermodynamic Properties from Group Functions
Equations 223 and 226 can be combined with equation
37 to express the molecular thermodynamic differentials
in terms of group correlation function integrals. The
first quantity required is
C.. = i(y ) dld2 cij(1,2) (310)
1: s 1
Using equation 37 this is
45
30
15
I I
0.0 0.3 0.6 0.9
3
Figure 3.2. Comparison of RISM () theory prediction and Monte
Carlo (e* ) for a hard, homonuclear diatomic molecule
with separation to diameter ratio of 0.6.
c. f dld2 c Cr., r.)
C'3 = r r Vi vjs ( 12 2) I dld2 ce6 (+a :
(311)
To evaluate this quantity, a coordinate transformation
is required.
dld2 = dR1dR2dSi, d 2 drl, dr2d~id 2 (312)
Here, i., represents the set of angles needed to specify
the orientation of molecule i with respect to a fixed
coordinate system. The above transformation is canonical,
and the angular integration can be performed to yield
C.. = I v, ( ) dr dr c (?, r.)
ii] ]B i 2 i j aO 1 ]
(313)
Now, the systems under
that
consideration are homogeneous so
c (ri?, rt) = c a(ri ) c t(r)
06 i ] j B 16
Then
Cj = .i vj6 Ce
where
CaB = ( d cB()
This leads to
(314)
(315)
(316)
(3BiJ_5 6.
i Ti n Uia B. 3 C 3 (317a)
j T,Pk7j i a B
or in differential form
do.
d. i v (j C dp ) (317b)
i p aB
and the corresponding expression for the pressure variation
is
dPB = dP D CaF dpg (318)
a b
Summary
Group contribution approaches are valuable for predict
ing thermodynamic properties. In this chapter two general
approaches for incorporating group contributions into fluc
tuation theory have been presented and analyzed. Finally,
the thermodynamic property differentials for molecular
systems have been expressed in terms of group direct correla
tion function integrals. Actual property changes can be
formulated once models are expressed for the integrals.
This is the subject of the next chapter.
CHEAPER 4
FORMULATION OF MODELS FOR
DIRECT CORRELATION FUNCTIONS
In previous chapters it has been shown that changes
in thermodynamic properties of molecular systems can be
expressed in terms of integrals of site direct correlation
function. Expressions for these integrals are required
before the calculations can be performed.
The purpose of this chapter is twofold. First, thermo
dynamic requirements on models for the direct correlation
functions will be presented and examined. With these
restrictions on model form established, a feasible model
for the correlation function integrals will be presented.
Thermodynamic Consistency
One aspect that has been emphasized throughout this
work is the necessity of the formalism to meet thermodynamic
consistency requirements. Because our proposed calculational
scheme employs direct correlation function integrals, the
models that are developed for these quantities are subject
to consistency tests. The easiest check that can be employed
is one of equality of cross partial derivatives of the
dimensionlesss) chemical potentials.
2 2
( ) = ( )a ] V i,j,k (41)
aPjSpk p kj
This can be expressed in terms of the direct correlation
functions integrals because
) = C. (42)
j T,Pkj
Combination of equations 41 and 42 leads to
aC. aC
DC3 ik) (43)
r___ = (aplk
Pk T,pZ/k j Tr,/j
It must be noted that this is equivalent to the equality
of threebody direct correlation function integrals (Brelvi,
1973)
C.. = C. (44)
ijk = Cikj (44
where the subscripts can take on all values associated
with the species in the mixture. The case of interest
here is that in which the direct correlation function
integrals for the molecular species are written in terms
of group quantities
Cij = ( I vi vj CB (45)
a 8
Then equation 43 takes the form
[ k B B T, ik
nTT,p p
S a [K r Vio VkB CB ) (46)
j a B T,p j
This requirement can be expressed in terms of group prop
erties if the chain rule is used for the derivatives
_P i YaP (47)
yV 1 P y y
Combination of equations 46 and 47 leads to
C
ac
aBy iy kB y Bp nny
i Y UkVji (a ] (48)
This relation will only be satisifed in general if
a3C 3C
0y) T6,p (49)
T' Y P Z
Thus, the consistency requirements for group direct correla
tion function integrals are equivalent to those of the
molecular quantities. This constraint will be employed
in formulation of the working models for the group direct
correlation function integrals.
Basis for Model Development
It has already been shown that the thermodynamic con
sistency tests for a group formulation and a molecular
formulation are equivalent when written in terms of direct
correlation function integrals based on the RISM theory.
This should not be surprising considering that stability
conditions for fluids were shown to be equivalent for the
two approaches by Perry (1980). Thus, in developing models
for group direct correlation function integrals, the same
considerations should apply as those used by Mathias (1979)
in developing molecular models.
The general philosophy that will be employed is to
use as much theoretical information as possible to develop
these models. This requires use of some concepts of statis
tical mechanical perturbation theory to obtain approximate
forms for the correlation function integrals. The analysis
of Chandler and Andersen (1972) has shown that for RISM
theory direct correlation function results of molecular
perturbation theory are easily extended to the group func
tions. Appendix 2 contains a complete development of an
exact perturbation theory for the direct correlation func
tions based on the RISM theory. In this chapter we shall
only deal with the important ramifications of these results.
It is always possible to write
B ref + a ref (410)
C =a6 + (Cc a (410)
where the superscript ref refers to some reference system.
The purpose of perturbation theory is to determine a refer
ence system so that an approximate form for the perturbation
(second on the righthand side) term can be made that yields
useful results. Because this work is concerned with dense
fluids, we require a reference system that can adequately
represent the behavior at high densities. In liquids the
optimal choice of a reference system seems to be one with
only repulsive forces (Weeks, Chandler, and Andersen, 1971).
Values of Ca3 are not available for this type of model
system. However, the system with purely repulsive forces
can be well represented by a system of hard spheres if
the hard sphere diameters are chosen as functions of tempera
ture (Barker and Henderson, 1967). This approach will
be followed in this work. Even with this choice of reference
system the perturbation term cannot be exactly identified.
A further approximation used is that the zero density limit
of this term is adequate. Thus, the model employed here
is
HS lim HS
C CB + lim (CaB C) (411)
Appendix 2 shows the evaluation of the required limit which
gives the working relation
Ca C H + (a) /Tn (412)
n=o aB
30
The constants in equation 412 are dependent on the inter
molecular potential (written as a sum of group of interac
tions). No explicit calculation of these terms is attempted
here because we choose to determine the expressions on
the basis of experimental data. This model form is com
pletely analogous to the model used by Mathias (1979).
Analysis of the Model
Several interesting features of the direct correlation
function model seem to merit mention. Equation 412 is
highly similar to the RISM form of the mean spherical
approximation (Lebowitz and Percus, 1966; Chandler and
Andersen, 1972) and is essentially equivalent if the hard
sphere diameters are chosen as functions of temperature
only. The assumptions involved in obtaining the present
approximant and the mean spherical form are different,
but it seems that for our purposes this difference is
immaterial.
Of greater interest here is the relationship suggested
by the present model for the threebody direct correlation
function integrals. These can be found from
3C.
C = () (413)
ijk 8k T,p /k
The form for the C.. proposed in equation 412 then suggests
that
HS
C CHS (414)
ijk ijk
This is probably not a very accurate approximation, but
the flexibility in the model form inherent due to the
determination of the parameters from experimental data
may make this adequate. A better interpretation of this
analysis is that the density dependence of the threebody
direct correlation function is assumed to be approximated
by that of the hard sphere quantity. This analysis also
suggests an extension of the RISM theory to threebody
correlation functions, a derivation of which can be found
in Appendix 3.
Cjk i ji jky CaBy (415)
Finally, thermodynamic consistency of the proposed model
must be considered. If equation 412 is rewritten in
explicit form
HS
C (pT) = C (p,T) + B (T) (416)
then the consistency requirement is met if
ECHS aCHS
( "B ( y)
py Tp T,pp (417)
HS
In this work, the Cae are determined from the generalized
hard sphere equation of state, which is consistent, then
this must be true. Note that if terms of higher order
in density had been included in the perturbation term,
they also would have to have been chosen to be thermodynami
cally consistent.
Also of interest here is the functional dependence
of the hard sphere diameters. If they are chosen to be
functions of density, as is normally done in the Week
ChandlerAndersen perturbation scheme, then this dependence
would have to be such that equation 417 was satisfied.
This is an aspect not always appreciated.
Property Changes Using the Proposed Model
This chapter has presented and analyzed a feasible
model for group direct correlation function integrals.
The utility of this model can only be tested by its use
to calculate changes in molecular thermodynamic properties.
The required formulas are developed below.
Chemical Potential
Equation 317b expresses the differential of a molecular
chemical potential in terms of the group direct correlation
function integrals. This is conveniently rewritten in
terms of the residual chemical potential as
dBf = i vj CaB dPg (418)
Insertion of equation 416 and integration leads to
Ari = I )i [A HS + r A Ap] (419)
aa a
where ',HS is the residual chemical potential calculated
from the hard sphere equations for group a in a solution
of groups.
Pressure Change
To calculate pressure changes, it is again easiest
to work with residual properties. The required differential
form is
dP = p CB dp (420)
ca a
Insertion of the model and integration leads to
Apr = APr,HS + (1) Y A(p p ) (421)
6 5 2 a a aS
where A r,HS is the change in the residual pressure (divided
by k T) calculated from the hard sphere equation for the
solution of groups.
It should be noted that the ideal gas state used for
the residual property changes on either side of equation
421 are different. This should be expected for the group
equation of state is written in terms of the group densities
which sum to a larger value than the molecular densities.
However, because this form is not used as a complete equa
tion of state, this should not present a problem.
CHAPTER 5
MODEL PARAMETERIZATION
This work has been concerned with formulating the
thermodynamic properties of molecular systems in terms
of integrals of group direct correlation functions. In
the last chapter a form for these terms was developed
C = CRE + (51)
and further, the reference state was identified as that
of a hard sphere system. The full dependence on state
variables is written as
C (p,T) = CHS (P,T; ) + 0a (1/T) (52)
where U is the vector of hard sphere diameters for the
groups in the system. The purpose of this chapter is to
develop empirical forms for the functions U'aS and to estab
lish the dependence of the hard sphere diameters on state
parameters (Weeks,Chandler, and Andersen, 1971).
Corresponding States Theory
The utility of the form proposed in equation 52 can
be enhanced if the perturbation functions and hard sphere
diameters could be written in corresponding states form.
This is a twostep process:
1. Determine if a corresponding states principle
exists for the direct correlation function
integrals for liquids.
2. Determine if the functional form proposed in
equation 52 can represent the observed
correspondence.
To answer these questions simple molecules will be
considered first. The molecules considered here can be
considered to consist of only one group. The relation
to thermodynamics used in the analysis is
( ) = 1/(lpC) (53)
T
Figure 51 shows a corresponding states correlation for
the quantity on the lefthand side of equation 53 for
simple molecules, using only one parameter. Further, Brelvi
and O'Connell (1972) showed that this behavior can be found
for polyatomic molecules also, if the density is large
enough (p > 2Pc). These results suggest that for large
densities a oneparameter corresponding states formulation
may be valid for the direct correlation function integrals.
2
in o
CCl4
3 4
0 CH
m CO2
N
0.7 0.8 0.9
3
p3
Figure 51. Bulk modulus versus reduced ensity for simple
fluids. The line is argon data and o is
LennardJones potential parameter.
Note that the temperature dependence seems to have little
effect. Inclusion of temperature dependence in these cor
relations should be able to enhance the accuracy.
Figure 52 shows another test of this correspondence.
This is in an integrated form of equation 53.
PB = PBef + S (1pC) dP (54)
Pref
Here, a characteristic volume for each molecule, V*, and
a characteristic temperature, T*, have been found to yield
the correspondence as
P = Pref' f(, T; ref) (55)
where
P = PV*/RT
= pV*
T = T/T*
The function, f, in equation 55 is a corresponding
states correlation for the density integral of the direct
correlation function. This is what is desired. The data
presented seem to suggest that a corresponding states formu
lation can be found for pure component direct correlation
function integrals for dense fluids. A detailed discussion
100
argon
50 methanol
V acetic acid
2 nCl6H34
nC16 H 34
A 25 38
20
10
T = T/T*
P = PV*
P =+C
RT
2
T=0.92 T=0.60 T=0.50
T=0.80
Figure 52. Corresponding states correlation for liquids.
Figure 52. Corresponding states correlation for liquids.
39
of the origins of this correlation is presented in Appendix
4.
Mathias (1979) has shown that these concepts can be
extended to mixturedirect correlation function integrals
with the proper choice of mixing rules. He used a form
similar to equation 52, but the group contribution concept
was not employed.
The applicability of the twoparameter corresponding
states principle seems valid for the direct correlation
function integrals. The question of the use of the func
tional form must now be addressed. This is again most
easily accomplished by analysis of a set of experimental
data for a simple molecule. After the noble gases are
used for this type of testing, however, the major interest
here is for hydrocarbons so that a hydrocarbon reference
system may prove to be more useful.
Choice of a Reference Component
Mathias (1979) had great success in developing a
corresponding states correlation for direct correlation
function integrals, using a form similar to equation 52,
for molecular systems, using argon as a reference component
to determine model parameters. This work is concerned
with hydrocarbon properties so that it seems reasonable
to examine the difference in direct correlation function
integrals between argon and a simple hydrocarbon, methane.
The comparison is performed here using the dimensionless
quantity 1pC. Table 51 shows this comparison for two
isotherms, in the dense fluid region.
The behavior is similar on both isotherms. It is
seen that the fluids act quite similarly at lower densities
but behave differently as the density becomes larger.
Methane is increasingly more compressible as the density
increases. This can be explained by the polyatomicity
of methane. Both molecules are essentially spherical,
but at the higher densities the methane seems to allow
for interlocking of the hydrogens.
These effects can be important for the development
of the group contribution correlations. If the groups
were chosen as monatomic, then argon would probably be
the better choice of reference substance due to physical
similarity. However, if the groups are chosen as the common
organic radicals, then methane may be the better choice
of reference substance.
The actual choice of a reference substance will be
made after it has been determined whether equation 52
is a viable functional form for the direct correlation
function integrals.
Method of Data Analysis
This section addresses the question of whether the
form proposed in equation 52 can be used to fit the direct
COMPARISON
TABLE 51
OF DIRECT CORRELATION FUNCTION INTEGRALS
FOR ARGON AND METHANE
T/T = 0.76 T/TC = 2.0
pVc (1pC)argon ( 1pC)methane pVc (1pC)argon (1pC)methane
2.26 6.44 6.35 2.1 7.23 7.14
2.31 7.97 7.82 2.2 8.50 8.35
2.37 9.68 9.44 2.3 9.96 9.73
2.43 11.51 11.25 2.4 11.65 11.29
2.48 13.57 13.24 2.5 13.58 13.07
2.54 15.83 15.43 2.6 15.79 15.06
2.59 18.30 17.83 2.7 18.29 17.30
2.65 21.05 20.46 2.8 21.13 19.91
2.70 23.88 23.33 2.9 24.33 22.59
2.76 27.10 26.45 3.0 27.94 25.68
42
correlation function integrals of real fluids. The analysis
will be performed in terms of the dimensionless quantity,
C, definedby
PBS
C pC = 1 ( (56)
The thermodynamic derivative is evaluated from correlated
experimental data. From equation (52) this is
C CHS (p, T; o) + p6 (57)
For this work the hard sphere diameter is treated as a
function only of temperature. Thus, along an isotherm,
if the proposed functional form is proper, for the proper
choice of hard sphere diameter, the value of 6 should be
constant given by
S= (CHS C)/p (58)
for any value of p. The suitability of the proposed model
is determined by how well this is obeyed. The hard sphere
properties are determined using the generalized hard sphere
expressions as presented in Appendix 5. It should be noted
that all of the hard sphere equations are generated by
the same microscopic form of the direct correlation
function. Here the hard sphere direct correlation function
43
integrals are found using equation 56 with the corresponding
form of the hard sphere equation of state. For these pure
component cases we have
HS (1+a)n4 4(1+a)n3 + 2 8
C 4 (59)
(1n)
where n = V/6.po
and in corresponding states form
n = [T/6 o3/V*][pV*] (510a)
= X pV* = Xp (510b)
Here, the characteristic volume, V*, has been introduced
as a reducing parameter for both the hard sphere volume
and the density. On each isotherm there are actually two
parameters that can be used to vary the model, a and X.
The analysis here has been done sequentially, a value of
a is chosen and then the optimum X has been found for that
value of a. The characteristic volume has also been used
to reduce the perturbation term 0 as
u = (t,/V*)V* = V*f(T) (511)
The model can then be written as
C(p,T) = CHS(n; T,X,a) + pf (512)
Figure 53 gives a simplified flowchart of how the calcula
tions are performed for each isotherm for a fixed value
of a. The subscript i denotes the number of data points
on the isotherm, NP. The optimization routine, RQUADD,
can be found in Appendix 6.
Analysis of Argon Data
3P8
The values of ( needed for this analysis were generated
from the equation of state of Twu et al. (1980). These
authors claim to be able to reproduce dense fluid pressures
to within the experimental accuracy. Eleven temperatures,
evenly spaced between the triple and critical points, were
chosen to generate liquid phase data. Each isotherm con
sisted of 16 densitycorrelation function integral pairs
evenly spaced between the vaporizing and freezing densities.
Supercritical data were also used. The isotherms were
for values of T/T from 1.1 to 3.0. The densities used
were in 0.1 increments of P/P from 1.5 to 3.0, or the
freezing density, whichever was lower. A listing of the
program used to generate the data is given in Appendix
6.
Figure 53. Flowchart for calculation of X at a fixed a.
For the initial analysis values of a of 0, 1, 2,
3, 4, and 5 were used. Table 52 shows the average
absolute deviation in the direct correlation function
integrals found for the optimum value of x at each isotherm.
A noticeable minimum exists in the range of a = 4. Note
that the CarnobanStarling equation (a = 1) does not yield
the best predictions, even though it is the "best" hard
sphere equation of state.
Several isotherms were chosen to examine the sensitivity
of the results to the value of a, on a finer scale. These
results are given in Table 53. For the liquid phase iso
therms a value of a = 4.3 seems to be optimal. The super
critical data are best fit with a of 4.0 or 4.1. It
was decided to use a constant value of a = 4.2 to fit
all of the argon data.
With a set at 4.2 the value of X and f could then
be determined. The optimum values of X found were correlated
as a polynomial in inverse temperature. For this analysis
the characteristic parameters were chosen to be equal to
the critical parameters. The form that fit X best was
X = 0.12342 + 0.15069 _0.18545 + 0.15947
T 2 3
T T
0.070261 0.012111 (5
+ (513)
4 5
T T
TABLE 52
REGRESSION ANALYSIS OF ARGON DATA
T Average Absolute Deviation in C for = a
0.5966
0.6297
0.6629
0.6960
0.7292
0.7623
0.7954
0.8286
0.8949
0.9280
0.8617
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.2
2.4
2.6
2.8
3.0
0.0101
0.0392
0.0785
0.1225
0.1651
0.2062
0.2445
0.2758
0.3104
0.3053
0.2994
0.2840
0.2413
0.2175
0.1990
0.1837
0.1707
0.1611
0.1530
0.1498
0.1403
0.1310
0.1242
0.1192
0.1152
0.1126
0.0102
0.0395
0.0789
0.1228
0.1651
0.2059
0.2436
0.2736
0.3056
0.2990
0.2963
0.2764
0.2344
0.2088
0.1899
0.1740
0.1611
0.1504
0.1411
0.1344
0.1286
0.1196
0.1131
0.1084
0.1047
0.1023
0.0094
0.0361
0.0714
0.1100
0.1461
0.1798
0.2094
0.2304
0.2408
0.2374
0.2430
0.2142
0.1835
0.1580
0.1383
0.1227
0.1101
0.1003
0.0920
0.0854
0.0798
0.0713
0.0651
0.0610
0.0582
0.0566
0.0058
0.0219
0.0427
0.0642
0.0830
0.0983
0.1085
0.1104
0.1122
0.1280
0.1071
0.1260
0.1058
0.0912
0.0801
0.0713
0.0643
0.0536
0.0495
0.0425
0.0371
0.0324
0.0290
0.0261
0.7266
1.3274
1.7500
2.0472
2.2372
2.3746
2.3629
2.1992
2.0497
2.2885
1.4495
1.1049
0.8986
0.6364
0.5516
0.4845
0.3864
0.3505
0.2948
0.2536
0.2214
0.1960
0.1749
48
TABLE 53
EFFECT OF HS EQUATION ON FITTING ARGON DATA
Average Absolute Deviation in C for = a
3.8 3.9 4.0 4.1 4.2 4.2 4.4
.7954 .1435 .1277 .1085 .0848 .0547 .0214 .0401
.8286 .1516 .1329 .1104 .0831 .0535 .0263 .0595
.8617 .1507 .1297 .1071 .0829 .0592 .0399 .0866
.8949 .1477 .1294 .1122 .0916 .0732 .0667 .1264
.9280 .1581 .1434 .1280 .1144 .1049 .1152 .1851
1.1 .1484 .1368 .1309 .1235 .1258
1.2 .1344 .1303 .1260 .1263 .1364
1.3 .1126 .1092 .1058 .1064 .1065
0590 .0585 .0618 .0698
1.8 .0607
The maximum error in calculated x values from equation
513 was 0.105% with an average error 0.022%. With x values
from equation 513 and a set at 4.2 the optimal values
of f were then found. These values were also fit to a
polynomial as
f = 14.70 23.237 +25.221 11.636
f = 1.4098 + +
2 ~3 ~4
T T T T
2.0463
+ 2 (514)
T5
This polynomial form reproduced the f values with an average
error of 0.15% for the isotherms analyzed. The temperature
dependence of X and f is shown in Figure 54. Table 54
gives a summary of the values of X and f for argon along
with the average error in the prediction of the direct
correlation function integrals. In general, the correlation
function integrals were correlated to within the experimental
accuracy. The only range of temperatures for which the
fit was not excellent was near the critical point, which
is to be expected.
Analysis of Methane Data
There is less high quality PVT data available for
dense methane than for argon. In this work the equation
9.0
l/V*
5.0
Figure 54.
.0.20
/I 3
 /V*
6
0.18
0.16
1.0 2.0
T/T*
Dimensionless hard sphere diameter and
perturbation function based on argon
reference.
TABLE 54
CORRELATION OF ARGON DCFI USING a = 4.2
Average Error
T X f V inC
c
0.5966
0.6297
0.6629
0.6960
0.7292
0.7623
0.7954
0.8286
0.8617
0.8949
0.9280
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1. 9
2.0
2.2
2.4
2.6
2.8
3.0
0.21159
0.20916
0.20692
0.20482
0.20283
0.20094
0.19915
0.19745
0.19583
0.19430
0.19286
0.18649
0.18348
0.18086
0.17852
0.17643
0.17453
0.17280
0.17120
0.16971
0.16833
0.16582
0.16359
0.16159
0.15978
0.15814
890.79
820.09
759.79
707.47
661.13
620.05
583.39
550.40
520.61
493.31
467.98
373.74
335.43
306.28
282.27
262.19
245.04
230.17
217.10
205.48
195.05
177.04
161.94
149.05
137.88
128.10
.0035
.0128
.0525
.0372
.0466
.0532
.0562
.0547
.0596
.0732
.1076
.1264
.1377
.1166
.1037
.0936
.0844
.0764
.0700
.0648
.0605
.0527
.0458
.0393
.0343
0308
52
of state of Mollerup (1980) was used to generate nine sub
critical isotherms and one supercritical isotherm, as was
done for argon. These supercritical isotherms measured
by Robertson and Babb (1969) were also analyzed. The pro
cedure used for the analysis of the methane data was the
same as for argon, except some positive values of a were
also investigated.
For the liquid phase isotherms a value of a = 4.2
was clearly the best. For the supercritical data values
of a from 3.0 to 4.3 could produce almost equivalent
results, but a = 4.0 was the optimum. It should be noted
that for the supercritical isotherm values of a = 9 could
also offer a reasonable representation of the data, but
not as good as 4.0. Thus, a value of 4.2 was chosen
to represent the methane data.
Proceeding as was done before for the argon analysis,
the values of X and f were fit to polynomials in inverse
temperature. The resultant forms were
0.026446 0.080054 0.089994
X = 0.13832 + + 2 3
T T T
S0.039085 0.0060984
4 ~5 (515a)
T T
23.21 50.827 40.662 15.818
f = 5.0851 + 2 3 +
T T T T
2.3173
5 (515b)
T
where the critical properties were used as the reducing
parameters. These functions are plotted in Figure 55.
Table 55 gives the summary of results for the methane
correlation. The proposed model was able to reproduce
the data to within the experimental uncertainty.
Use of the Correlations to Calculate Pressure Changes
The development of the DCFI models for the pure com
ponents was first tested by performing equation of state
calculations for the base substances, argon and methane.
The method of calculation was to choose the lowest density
point on each isotherm as a reference and then to calculate
pressure changes for all other points on the isotherm from
that reference. There are thus two criteria for goodness
of fit, the average percent error in pressure changes,
defined by
ND LPcalc Aexp
aP = [ ] / ND (516)
i=1 APexp
and the sum squared percent error, defined by
ND APcalc Aexp 2
SSEAP = [ ] (517)
i=l APexp
where in both cases ND is the total number of data points
for a compound. When parameters were found to best "fit"
a set of data, the quantity SSEAP was minimized using a
TABLE 55
CORRELATION OF METHANE DCFI USING a = 4.2
Average Error
T X f V in C
c
.4986 .22182 1545.26 .0015
.5511 .21596 1309.43 .0155
.6036 .21095 1131.07 .0188
.6561 .20671 996.35 .0157
.7085 .20302 890.93 .0140
.7610 .19974 805.32 .0335
.8135 .19675 733.53 .0612
.8660 .19400 671.78 .0946
.9185 .19145 617.18 .1413
1.1 .18381 473.60 .1484
1.6173 .16916 260.80 .2262
1.9585 .16316 204.86 .1437
2.4833 .15703 168.74 .0637
13.0
9.0
)/V*
5.0
3.0
1.0 2.0
T/T*
Figure 5.5.
Dimensionless hard sphere diameter and
perturbation functions based on methane
reference.
1.20
113
3/V*
. 18
. 16
56
nonlinear regression routine and the program GROUPFIT listed
in Appendix 6. However, the data comparisons will normally
be made in terms of the quantity AP. This procedure is
common and is used to eliminate bias from the error mini
mization.
The first test performed was to calculate AP for argon
using the characteristic parameters as the critical
parameters and the functions given by equations 513 and
514. The average error was found to be 1.42%. This is
larger than expected, but if only the liquid phase data
reconsidered, the error is just 0.629%, and if the isotherm
that is within 10% of the critical temperature is neglected,
the error for the liquid isotherms is only 0.34%. This
is within the accuracy of the equation of state for this
region of the phase diagram. On the whole the largest
errors are found in the low density data near the critical
temperature. The equation is able to reproduce the highest
temperature isotherms (T/Tc = 2.8 and 3.0) with an average
error of 0.47%. So the fit of the argon data seems satis
factory for the desired usage.
In an attempt to improve the argon representation,
a regression was performed to find the set of characteristic
parameters that minimized the sum square error in the pres
sure changes as per equation 517. The parameters found
were T* = 144.45 K and V* = 74.31 cm3/gmole as compared
to T = 150.86 K and V = 74.57 cm /gmole. The average
c c
57
error for all of the data increased to 1.53%. This worsening
of the percentage error occurred because the fit was made
more even; that is, SSEAP was decreased from 4.549 x 103
to 3.034 x 10 but on the average the error was larger.
This is due to the larger localization of error in the
critical region.
A calculation was also performed on the methane data
using the argon functions and methane's critical parameters
as the characteristic parameters; the error was 3.13%.
And when a regression was performed on the methane data,
the characteristic parameters were found to be T* =
190.54 K and V* = 98.40 cm3/gmole (compare T = 190.53
K and V = 98.52 cm /gmole) with an error of 2.98%. This
shows that the trends found for the methane DCFI as compared
to argon carries over to the pressure change calculations.
Similar calculations to those described above were
performed using the hard sphere diameter and perturbation
term functions as given by equations 515. The average
error in pressure changes for all of the methane data was
0.57% with the largest error from the isotherm at 175 K
(within 9% of the critical temperature). Even at this
temperature the high density data are well described.
In the reduced temperature range of 0.5 to 0.89 the average
error was only 0.18%, well within the accuracy of the data.
As for argon, when the GROUPFIT program was employed to
find optimal characteristic parameters for methane, the
2 2
SSEAP was reduced from 4.13 x 10 to 2.44 x 10 but the
average error was increased to 0.727%. Again .the localiza
tion of the error causes this phenomenon.
A calculation of the pressure changes for the argon
data using the methane functions gave an error of 4.65%,
and when optimized, the argon characteristic temperature
only changed 1.3% and the error reduction was only to 4.56%.
This again shows that there are noticeable differences
in the compressive behavior of argon and methane at high
densities.
In summary, the temperature dependence of the hard
sphere diameter and perturbation term in the DCFI model
are seen to adequately represent the data from which they
were developed. The difference in DCFI noticed for the
atomic species argon and the polyatomic methane are notice
able in calculations of compressions.
The rest of this work will deal with calculations
of hydrocarbon compressions using the group contribution
formulation. The groups of interest are methyl (CH3)
and methylene (CH2). These are physically more similar
to methane than to argon. For this reason all of the group
contribution calculations will be performed using the methane
reference functions.
CHAPTER 6
USE OF THE GROUP CONTRIBUTION MODEL FOR
PURE FLUIDS
The development thus far has been limited to the formal
expressions for calculating property changes of mixtures
based on the group contribution model of direct correlation
function integrals and the parameterization of the model
using data for argon and methane. In both cases the mole
cules were assumed to consist of only one type of group.
This chapter presents the calculations of pressure changes
for pure fluids composed of different types of groups.
The first section discusses the extension of the cor
relations required for systems of several groups. The
remainder of the chapter presents and discusses the results
of the calculations performed for several nalkanes and
one nalkanol.
Extension of the Model to Multigroup Systems
The expressions developed in Chapter 4 for calculation
of pressure changes involve two contributions.
APB APBS + APPERTURBATION (61)
60
The hard sphere contribution is calculated for the solution
of groups using the results shown in Appendix 5. To cal
culate the perturbation contributions for multigroup systems,
the form of the 'P functions must be established for the
terms with a 3 B. In Chapter 5 a corresponding states
expression was developed
a = V* f (T/T ) (62)
where f is a universal function of the reduced temperature.
This result is extended to the unlike terms as
S= V* f (T/T* ) (63)
and the characteristic parameters are found from
1 1/3 1/3)
V = (V* )1/3 + (V )3) (64a)
aB 8 at B6
T* = (T* T )1/2 (k) (64b)
where k is an empirically determined binary interaction
parameter for groups a and 3. It is important to note
that this binary parameter can still be determined from
analysis of pure component volumetric data.
Designation of Groups
Before the group contribution model can be used for
calculation of property changes, the decision as to which
collections of atoms in a molecule comprise a group must
be made. The simplest choice would be to consider each
atom as a group, and thus the properties of all molecules
could be described with minimal information. However,
that is an overly ambitious approach. Even for the molecules
considered here that contain only carbon, hydrogen, and
oxygen atoms, this is unworkable. Typically, the common
organic radicals are designated as the groups.
For this work it must be decided if the nalkanes
are composed of only one type of group, or two. If they
are considered as being comprised of only one type of group,
then the model could never predict excess volumes for
nalkane mixtures. This is reasonable at low pressures,
but for highly compressed systems, noticeable excess volumes
do exist (Snyder, Benson, Huang, and Winnick, 1974). Thus,
the nalkanes will be considered to be composed of two
groups, methyl (CH3) and methylene (CH2). To analyze
methanol the hydroxyl moiety (OH) will also be considered
as a separate group.
Pressure Change Calculations
All of the molecules analyzed in this work will be
considered to be composed of only two types of groups.
The equations required for the calculation of the pressure
changes are
P6 (P)ref = PHS (pHS)ref
PB (PB) PB (PB
1 p2_ 2 ref) 2 (65)
2 111+2 v212222) (65)
where here, v. is the number of groups of type i in the
molecule. The individual terms are found using
PBHS 1 1 3 11 + 3623
1 (153) (163)
4.2 2 3 } (66a)
(13)
with
1 = P[vlo + 2c2 ] (66b)
and the oi terms are found using
3
o /V* = f(T/Tf) (66c)
6 i l 1
where the function of reduced temperature required in equation
66c is given in Chapter 5. The perturbation contributions
are found using equations 63 and 64 with the required
function of reduced temperature again given in Chapter 5.
These functional forms are dependent only on the reference
component chosen for their development, either argon or
methane. For the two group molecules there are then only
five independent parameters, V*, V*, T*, T*, and kl2.
Analysis of nAlkane Compressions
In this section we consider the ability of the proposed
formulation to calculate changes in pressure during compres
sion of four nalkane molecules: ethane, propane,
npentadecane, and noctadecane. Calculation of pressure
changes using the equations given above were performed
for both the argon and methanebased reference functions
as derived in Chapter 5.
The initial plan of attack was to determine the methyl
parameters using ethane data and ethylene and cross parame
ters from the propane data. The model capabilities would
then be examined by using these same parameters for calcula
tions on the long chain molecules. The analysis was per
formed both with and without a binary interaction coeffi
cient. The characteristic parameters for the methyl and
methylene groups as determined by regression analysis are
shown in Table 61 along with the errors in calculated
pressure changes. Several conclusions can be drawn from
these results that will be of future interest. First,
the best fit of all of the data was obtained using the
methanebased correlations with the inclusion of a binary
or 0
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a
interaction coefficient. This binary constant was much
larger than usual and thus gave a large value of the T*
12
parameter. This results from the fact that the model is
much more sensitive to the values of the characteristic
volumes than the characteristic temperatures so that large
adjustments to the binary constant are required to signifi
cantly affect the calculations.
It was also seen that 2V* = 0.9 Ve and
CH C,ethane
2V* + V*C = 0.9 V This suggests an approach
CH 3 CH2 c,propane
for estimating characteristic volumes of the groups from
critical properties. The characteristic temperatures might
also be obtainable using critical values with some sort
of mixing rule applied.
While the results are not as good as the molecular
fit, they could be satisfactory. A more important question
is their ability to predict compression data for other
hydrocarbons. For npentadecane and noctadecane results
were rather poor, the best agreement being an average error
in the pressure changes of 16.6% using parameter set 1.
This is discouraging but not unexpected. The hard sphere
contribution does not behave properly at high densities
as explained in Appendix 5. In addition, the improper
ideal gas limit of the equation of state affects the results.
In an effort to improve the model performance, the
data for the two long chain hydrocarbons were used to deter
mine the methylene and interaction parameters.
The results of interest are shown in Table 62. The
methyl parameters were those found from analysis of the
ethane data. The results are rather encouraging in that
either the four or fiveparameter models could give an
adequate representation of data for both short and long
chain hydrocarbons. However, it should be noted that none
of the parameter sets could give a reasonable representation
of the propane pressure changes.
These results do yield two interesting conclusions.
First, that when the larger hydrocarbons are considered
for the parameter estimation that the binary parameter
becomes very important. Noticeable improvement in the
pressure change calculations occurs when the binary parameter
is included. This suggests that the methyl and methylene
groups should be considered as distinct entities. However,
the magnitude of the binary parameter is much greater than
would be expected. It can also be seen that the methane
based correlations are more apropos for modeling the alkane
compounds because the groups are physically more akin to
methane than to argon.
After studying the aforementioned results, it was
decided to establish a final set of model parameters for
the methyl and methylene groups from an analysis of all
of the nalkane data simultaneously. The methanebased
reference functions were used, and the calculations were
performed both with and without a binary parameter. These
(N
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results are shown in Table 63. It can be seen that use
of a binary parameter does improve the results and that
the order of magnitude of this parameter is reasonable.
Using only five adjustable parameters, the average absolute
error in pressure changes calculated for all four hydro
carbons was 5.9%.
Analysis of Methanol Compressions
A set of compression measurements for methanol were
analyzed to determine if the present model could adequately
calculate pressure changes for molecules other than
nalkanes. Methanol was considered to be composed of one
methyl group and one hydroxyl (OH) group. The methyl group
characteristic parameters used are those listed under
parameter set 11 in Table 63. The regression results
are shown in Table 64. The characteristic volume for
the hydroxyl group appears to be quite reasonable because
V* + VH 0.9 Vmethan which is the same pattern
CH OH c,methanol
seen for the alkanes. The characteristic temperature and
binary parameter found are surprising, but these values
were necessary to obtain a fit of the data about as good
as that for the alkanes.
Summary
While the calculations presented here are based on
recognized assumptions that are believed to be sound, there
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are certain disconcerting aspects. In particular, the
equation of state developed does not properly approach
the ideal gas limit. Even though the calculations are
always performed for pressure differences from a liquid,
or dense fluid, reference state this may be important.
Also, the percentage errors in the pressure change
calculations appear to be rather large. In most process
design calculations the temperature and pressure are known
and one wishes to calculate the volume. It would then
be of interest to know how the present model would perform
in calculating volume changes for given pressure changes.
A simple error analysis shows how this can be estimated.
At constant temperature errors in pressure calculations
and volume calculations are related by
3V
AV = ( ) AP (67)
T
Then average absolute errors in volumes and pressures must
be related by
/ / = PT /P/ (68)
where 8T is the isothermal compressibility. For liquids
the product PBT is almost always less than 0.2 and even
then is only that large at very high pressures. This sug
gests that if the present correlation were used to calculate
72
a volume change for a given pressure change that the average
absolute errors would be less than 20% of the errors reported
here for the pressure changes. This shows that the present
model compares reasonably with existing correlations.
One reassuring result of regression analysis is the
magnitude of the group characteristic volumes that were
calculated. Ratios of the group characteristic volumes
found here are very nearly equal to the ratios of the van
der Waals volumes for these groups. This may allow for
estimation of these group parameters. This type of regu
larity was not seen for the group characteristic tempera
tures. Also, whereas the characteristic volumes for the
different groups did not vary much when determined from
different data sets, the characteristic temperatures did.
As a result, the binary constants did not follow a pattern.
The binary constant was necessary in all cases to obtain
an optimal representation of the data, but the improvement
obtained by introducing this additional parameter was
minimal.
No calculations have been reported for chemical poten
tial changes or for pressure changes for multicomponent
systems. The necessary formulae have been presented in
Chapter 4 and Appendix 5. Pressure change calculations
for multicomponent systems are no more difficult than those
reported here because the molecules considered contained
more than one group.
CHAPTER 7
DISCUSSION
This work has centered on the development of a group
contribution liquid phase equation of state. Fluctuation
solution theory was used to develop the exact relationships
between thermodynamic derivatives and direct correlation
function integrals. Two approximations were made to actually
construct the equation of state. The first approximation
made was the use of the interaction site formalism, or
RISM theory. The last stage in the model formulation was
the choice of an approximate form for the group direct
correlation function integrals. After that development,
the general expressions for calculation of pressure and
chemical potential changes were written and used for
several compounds.
In this chapter, the important aspects of each stage
in the development described above will be examined in
detail. Great emphasis will be placed on the approximations
embodied in the model development as they pertain to the
numerical results.
Fluctuation Solution Theory
The relationship between integrals of molecular correla
tion functions and thermodynamic derivatives have come
to be known as fluctuation solution theory (O'Connell,
1981). Kirkwood and Buff (1951) originally reported these
relationships though they have not been extensively used.
The original results, using total correlation function
integrals, are valid for both spherical and molecular sys
tems. With the introduction of the direct correlation
functions, through the OrnsteinZernike equation, the analy
sis is not as simple. Appendix 1 details a procedure which
surmounts all difficulties encountered. This result is
not new, but the rigorous proof appears to be. This result
could also be derived through the use of the operator tech
nique of Adelman and Deutch (1975).
Approximations
While equations 224 and 225 are exact results for
fluctuation derivatives, one must have values for the direct
correlation function integrals for these to be useful.
Because exact values of the required integrals are not
generally available, some approximations must be used to
obtain these values. The approximations used will determine,
to a large degree, the success of any thermodynamic calcula
tions. In this work, two levels of approximation are used
and they will be discussed separately below.
Group Contributions
One of the major goals of this research was the develop
ment of a group contribution formulation of fluctuation
solution theory. This appears to be a realistic and rational
goal. This is true because fluctuation solution theory
requires knowledge of correlation functions and correlations
among the groups are well defined. The major problem
encountered was the ability to separate the inter and
intramolecular correlations. The RISM theory (Chandler
and Andersen, 1972) purports to affect this separation
and offer an approximate relationship between group direct
correlation functions and molecular correlations. The
calculations reported here are based on this original version
of the RISM theory.
Recently, the direct correlation functions calculated
using the RISM theory have become the objects of considerable
attention (Cummings and Stell, 1981, 1982b; Sullivan and
Gray, 1981). It has been shown that in many situations
for neutral molecules that the sitesite direct correlation
functions are necessarily longranged. This means that
these functions are nonintegrable and thus no compressibility
relation involving these functions may exist. Cummings
and Stell (1982a) have shown how a general compressibility
relation can be derived in terms of a limiting operation,
but this is not a practical solution. However, for the
cases that have been considered in the literature it seem
this problem can be overcome. If the compressibility rela
tion is viewed as the k o limit of a result in Fourier
space, where all of the correlation function transforms
exist, then all of the mathematical manipulations required
may be performed. Then as the long wavelength limit is
taken, we find (Appendix 7) the following relationship:
~PX = lp XiXji iviB C, (o) (71)
DP T,X ij ab
While the individual functions C a(r) may be longranged,
it appears that the collection of terms wcy is well defined
in the k o limit. It is known (Cummings and Stell, 1982a)
that for diatomic molecules this is true, and thus a simple
compressibility theorem applies. Also for triatomics,
as examined by Cummings and Stell (1981), we find that
this summation of terms is nondivergent. As has now become
known, the possible divergencies in the C a(r) functions
are due to intramolecular effects that are not shown
explicitly. It appears that the projection by the v matrix
removes the divergent terms. This is analogous to the
ionic solution case (Perry and O'Connell, 1984) where the
charge neutrality constraint removes the divergent part
of the direct correlations.
Chandler et al. (1982) formulated the proper integral
equation theory for sitesite correlations to offer an
exact formulation involving nondivergent CB(r) functions
that here are labeled C (r). These new functions are
related to the RISM C B(r) functions and are made nondiver
gent by removal of intramolecular correlations involving
only sites a or 6. This requires use of auxiliary functions
SaB(r) that have a complicated density dependence. These
functions do possess the interesting property
eaB(o) = 0 Va,B (72)
It is then possible to derive a compressibility relation
involving only C (r) function integrals and 0
(1P 1 X.X. v. j
ST,X ij a s
J dr C C (r) (73)
In this case, the integrals always exist. However, we
have no knowledge of the functional dependence of 0 on
temperature or density.
One further aspect of the group contribution approach
used here needs to be mentioned. This formulation works
on the Ansatz that
cij(l,2) = I jia Ij c Cs B) (741)
jS a
which is only to be considered an approximate relation.
However, the compressibility relation obtained from the
above starting point and that found using the proper integral
equation formalism (as shown in Appendix 7) are the same.
The only difference is that all forms of the RISM theory
assume that each group, as opposed to each type of group,
have separate correlation, i.e., that the set of functions
cij r) should be considered, not just caB(r). This is
definitely true at the level of the correlation functions
themselves. However, it is always possible to define a
)* aS *
set of c C(r) functions that are averages of the c (r)
functions (Adelman and Deutch, 1975) and retain the same
compressibility relations.
Modeling of Direct Correlation Function Integrals (DCFI)
As the DCFI are functions of temperature and density,
it would be possible to model them using polynomial expan
sions for the pure fluids and a solution theory for the
mixture quantities. In this work, we attempted to use
fluctuational forms that have a theoretical basis. This
approach seems to have both merits and demerits. For mole
cules, Mathias (1979) had success in modeling DCFI using
an approach suggested by perturbation theory. In the present
case this is not of the greatest value because analytic
forms are available for group DCFI in only some limited
cases (Morris and Perram,1980; Cummings and Stell, 1982a),
79
and even then, the solutions are not in closed form. Thus,
some reasonable assumptions must be made to proceed with
the modeling.
It is known that the RISM correlation functions can
be related to PercusYevick correlation functions for
molecular species in some limiting cases (Chandler, 1976).
This led to the use of the form for the DCFI presented
in Chapter 5. The important aspect of this approach is
that the groups are dealt with as if they were all indepen
dent. This is not physically the situation. The dependence
is caused by the intramolecular correlations, and they
must be properly accounted for to yield an accurate model.
Originally it was believed that the RISM theory took account
of this behavior, but as was mentioned in the preceding
section, this is not true. Even the proper integral equation
direct correlation functions, equation 73, while short
ranged, still involve intramolecular correlations due to
third body effects. Cummings and Sullivan (1982a) explicitly
show that while the h (r) functions are purely intermolecu
lar quantities that the c (r) are not.
The exact resummation of the cluster expansion for
h o(r) shown by Chandler et al. (1982) suggests a way around
the aforementioned difficulties. It should be possible
to define a set of direct correlation functions, labeled
here as ca (r), that include only intermolecular effects.
Chandler's (1976) analysis can be examined to see that
these diagrams can be isolated. And then, in the same
fashion as was used to eliminate the longranged behavior
of the original RISM direct correlation functions, it would
be possible to write an exact proper integral equation
of the form
S+ Ph = [I Pc ] (75)
where the elements of the W matrix would contain all infor
mation about intramolecular effects. It is then easy to
show the relationship between the different classes of
direct correlation functions, for example
y 1 1 1
= c + ( ) (76)
The elements of the W matrix would be functions of
temperature and density, but the exact functionality could
not be determined in general. It is possible to identify
some of the properties of the new functions, such as
olim W = (77)
and
klm = W V (78)
0 ( a3 a,8
Using equations 75 and 78, in conjunction with the
techniques of Appendix 7, a compressibility relation can
be derived in terms of these new group direct correlation
functions
(3P 1 p .
ST,X ij a i
J dr C (r) (79)
All of the above analyses show that there is no simple
solution to the modeling problem. One may either work
with a simple compressibility relation written in terms
of DCFI whose behavior is not certain, or in terms of well
behaved DCFI but have another unknown quantity present.
The former approach was taken for this work and is discussed
below.
DCFI Model
The rationale behind the development of the DCFI model
has been presented in Chapter 5. It is of interest to
note that the proposed form is a group variation of a van
der Waals, or mean field, model. Appendix 2 details how
higher order correction terms could be appended to the
proposed form.
Because the DCFI model was formulated as a correspond
ing states correlation, the constants in the temperature
dependence of the hard sphere diameters and perturbation
terms had to be determined using experimental data. Volu
metric data for both argon and methane were used to accom
plish this task. The results of the compression calculations
show that the methanebased correlations were definitely
superior for representation of the properties of the larger
molecules.
Comparison Calculation
Chapter 6 presents the results of the group contribution
compression calculations for the nalkane and methanol.
The results seem to indicate that the theoretical basis
is feasible but that the accuracy is not that desired.
It is an accomplishment to be able to perform the compression
calculations for all of the nalkanes using only five
parameters, but the overall accuracy is not high. One
would hope to be able to calculate pressure changes with
an accuracy of about 1% (this allows for density change
calculations accurate to 0.2%), and this cannot be assured.
The theoretical basis behind the equation of state
development is now on firm footing. Our compressibility
relation is exact; one must simply have proper models for
the functions involved. Here, this appears to not be the
case. Because we use the RISM approximation, the c a(r)
functions must contain intramolecular effects that have
no analogy at the molecular level. Thus, our model that
83
is based on an analogy to a previously successful molecular
approach appears inadequate. Note, however, that we seem
to have little recourse. Because if we were to introduce
easily modelable cB8(r)'s, then the aspect of determining
the W function must be addressed, and this is still an
unsolved problem.
Summary
This study has revealed many interesting aspects about
sitesite correlation functions and their relationship
to thermodynamic derivatives. A group contribution liquid
reference state equation of state has been formulated and
tested. The results are encouraging enough to warrant
further investigation but not accurate enough for practical
density calculations. It appears that the major problem
associated with the present approach is the inability to
model the required functions, and suggestions have been
made as to how one can alleviate that problem.
CHAPTER 8
CONCLUSIONS
The goal of this work was the development of a group
contribution technique for calculation of liquid phase
properties of mixtures. The general results are available
but have only been tested on a few pure components. The
calculated pressure changes for four nalkanes and methanol
are in general in error by less than 7%. This is reasonable
but not of high enough accuracy for process engineering
calculations. While the results of this study are not
outstanding, there are several interesting conclusions
that can be drawn.
It has been shown that it is possible to derive thermo
dynamic property derivatives from a RISM theory. A general
ized compressibility theorem was proven, and this was used
along with a model, based on a rigorous perturbation theory,
to develop a van der Waals equation of state.
The liquid phase equation of state was applied to
the representation of compression measurements of argon
and methane. It was shown that proper choice of the
repulsive contribution to the equation of state is
important. Also, that an optimum repulsive contribution
could be determined from a new hard sphere equation of
state.
The use of the group contribution model was limited.
Pressure changes could be calculated for both long and
shortchain alkanes using a fiveparameter model. The
results were of reasonable accuracy for these cases and
when the model was applied to methanol. While the volumetric
parameters in the model appeared to correlate well with
previous molecular results (Mathias, 1979), the temperature
parameters followed no pattern.
An analysis of the present work also shows several
areas that require future work. First, the present model
should be applied for calculation of chemical potential
changes for nalkane mixtures. This, along with mixture
volumetric calculations, would be a strong test of the
model's ability. Secondly, some effort to identify the
intramolecular correlation functions of the form shown
in Chapter 7 may be required to improve the present model.
This appears to be the weakest aspect of the present work.
APPENDIX 1
FLUCTUATION DERIVATIVES IN TERMS OF DIRECT CORRELATION
FUNCTION INTEGRALS
To eliminate the difficulties associated with the
form of the general OZ equation we define an angle averaged
total correlation function, by
= d2 d2 h i(1,2) (All)
and then we implicitly define a new set of direct correlation
functions, the by an OZ type equation
=
+ j k dR3 (Al2)
Now, because the full h.. (1,2) is translationally invariant,
the average function must also be, and thus the
13
function must also possess this property. Then if we
integrate equation Al2 over one coordinate we have
=
l]
+ < Jk ds (Al3)
k ik
And if this is then integrated once more we find
86
= +
k
where we define
= 1 dR
= 1 1 dR
1j V i]
(Al4)
(Al5a)
(Al5b)
and when these terms are collected in matrix form we find
= +
(Al6)
Here, the terminology is the same as that used in Chapter
2 for the definition of the matrix elements. This relation
becomes most useful when it is realized that
= H
(Al7)
as can be shown by direct substitution. And now rearrange
ment leads to
[N + NHN] = N1 ]
(Al8)
which shows that the fluctuation derivatives can be written
in terms of the integrals of the functions. The
1]
relation between cij and is not obvious but can be
derived. By definition
=
dR
k dR3 Cik R1R3>kj 3,R2
and, if the OZ equation is angle averaged, we find
= di d .2 cij(l,2)
+ k f d3 ddd2 c i(1,3)h k(3,2) (Al9)
kv7 1 2 ik kj
k VQ
Now, when these two relations are combined, the link between
the angle averaged c.. and the functions are given
as
= ds d C (1,2)
+ [ 3 dihld cik(l,3)hkj(3,2)
k v3 2
J dR3 (Al10)
And now insert the definition of to find
= d d c (1,2)
+ 3 I d3d~ld 2 cik(1,3)hkj(3,2)
k v1 2
N2 dd2hkj(3,2) (Al11)
k V'2 2 ik 3 kj
and the terms can be grouped as
d d 2 ci (1,2)
1
<= N d3d2hkj (3,2) {1 dn c (1,3)
k VQ
(Al12)
Now, use the fact that
= J d2 (Al13)
to rewrite equation Al12 as
11
1 d {
=  + d3d(2 hkj(3,2) { I dl cik(1,3)
k VQ
} (Al14)
It is now advantageous to define
