Group contribution methods for molecular mixtures. I. Interaction site models

Fluid Phase Equifibria, 43 (1988) 231-261 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

231

G R O U P CONTRIBUTION M E T H O D S FOR MOLECULAR MIXTURES.

I. I N T E R A C T I O N

SITE MODELS

LLOYD L. LEE

School of Chemical Engineering and Materials Science, University of Oklahoma, Norman, OK 73019 (U.S.A.) J.M. HAILE

Department of Chemical Engineering, Clemson University, Clemson, SC 29634 (U.S.A.) (Received May 5, 1987; accepted in final form April 18, 1988)

ABSTRACT Lee, L.L. and Haile, J.M., 1988. Group contribution methods for molecular mixtures. I. Interaction site models. Fluid Phase Equilibria, 43: 231-261. This paper uses interaction site models from statistical mechanics to obtain microscopic interpretations of group contribution expressions for thermodynamic properties of mixtures. Conventional developments of group contribution methods first divide properties into combinatorial and residual terms and we provide a microscopic interpretation of this division as a separation of properties into hard- and soft-core contributions. For the general property of an interaction site model, we find that only a few additional assumptions are needed to attain rigorous group contribution forms for the residual portion of the property. For some properties, including the internal energy, free energy, and chemical potential, the general group contribution forms reduce to integrals over sets of new pair distribution functions--group-group distribution functions. We define these new pair functions and show how they are calculable from the usual site-site distribution functions. All properties may not be simplified in this way, however. In particular, the pressure in interaction site models can be expressed in the general group contribution form, but it cannot be simplified to an integral over just the group-group distribution functions: the pressure depends explicitly on local orientational structure.

INTRODUCTION Group contribution (GC) methods are now well established (Wilson and D e a l , 1962; D e r r a n d D e a l , 1969; F r e d e n s l u n d et al., 1975; N i t t a e t al., 1977) a n d h e a v i l y u s e d p r o c e d u r e s f o r e s t i m a t i n g t h e r m o d y n a m i c p r o p e r t i e s of fluid mixtures. These methods are based on the premise that thermodynamic properties can be decomposed into contributions from structural 0378-3812/88/$03.50

© 1988 Elsevier Science Publishers B.V.

232 groups of atoms, rather than from molecules or individual atoms. Moreover, a particular group's contribution to a given property is presumed to be the same, independent of how the groups are arranged into molecules. The utility of a GC method is that while a mixture may contain hundreds of different molecular species, it may nevertheless contain only a few structurally distinct chemical groups. The methods therefore offer the possibility of (a) providing considerable generality in correlating thermodynamic properties of known multicomponent mixtures and (b) providing simple tools for predicting the properties of new molecular mixtures composed of known structural groups. The ideas embodied in GC methods can be traced to Langmuir's (1925) study of fluid interfaces, with the first experimental tests apparently performed by Smyth and Engle (1929a, b). However, in spite of their age and popularity, the theoretical foundations of group contribution methods have only recently begun to receive attention. Thus, Naumann and Lippert (1981) have proposed a statistical mechanical theory which treats molecular fluids as mixtures of formally independent molecular interaction sites (IMIS model), while Currier and O'Connell (1987) have presented a way of connecting statistical mechanical quantities to GC formulations for activity coefficients. Our objective is a more complete development of relations between GC expressions and microscopic concepts. We identify three steps in the development of GC methods: (1) division of properties into group and non-group terms; (2) development of GC expressions for particular mixtures; (3) assumption of universality.

Step 1 In conventional approaches, GC ideas are actually applied not to the full thermodynamic property F, but only to the part of F that is directly attributable to group interactions. Thus, F is first divided into a combinatorial term Fcom and residual term F r F = Fcom d- F r (1) The combinatorial term accounts for packing molecules of various sizes and shapes; it makes a purely entropic (non-energetic) contribution to F. In contrast, the residual term includes entropic contributions plus energetic effects arising from intermolecular forces. GC premises are applied only to the residual term Fr.

Step 2 To devise GC expressions for particular mixtures, we appeal to interaction site models (ISM) of statistical mechanics. In the statistical mechanics

233

of molecular fluids, interaction site models (ISM) have proven valuable in both theoretical and computer simulation studies (Streett and Gubbins, 1977). Interaction site models treat specific locations on molecules, rather than atoms or groups, as sources of interactions among polyatomic molecules. Thus the interaction between a pair of polyatomic molecules labeled 1 and 2 would be given by an ISM in the form

EEu, (r

(2)

i
where index i(j) runs over all sites on molecule 1(2). The ISM form for intermolecular interactions is popular and has been m u c h used, but it is not necessarily reliable. Basically, eqn. (2) presumes that the interaction between two molecules is a superposition of interactions between spherical charge distributions; multibody effects are either ignored or are roughly included by using "effective" values for the potential parameters. The advantages of such a model are (a) the orientational dependence in eqn. (2) is implicit and (b) molecular shape is defined in a quantitative way, through the relative positions of sites. However, the model does not necessarily account for the distortion of electron clouds when atoms are bonded into molecules. Moreover, highly asymmetric molecules promote significant mulfibody induction interactions (the charge distribution on one molecule distorts the distribution of another molecule) which cannot be reliably modeled as a sum of pair interactions. These issues have been discussed in a cogent fashion by Price (1988). The ISM idea in statistical mechanics is similar to the group contribution idea in molecular thermodynamics; however, the two ideas are not the same. Two of the more prominent distinctions are the following: (1) The locations of sites on a molecule do not necessarily conform to positions of atoms or of centers of mass of groups. For example, interaction sites may be located so as to reproduce effecvs of lone-pair electrons, whose interactions make important contributions to the properties of m a n y substances. Another example is the arrangement of sites in such a way so as to reproduce known values of multipole moments of a molecule (Tildesley, 1984). In neither of these examples would the locations of sites be necessarily synonymous with locations of groups. (2) A second distinction is that ISMs make an assumption only about the form of a microscopic quantity, the intermolecular potential function. In contrast, G C ideas make more far-reaching assumptions about the constitution of macroscopic quantities, namely thermodynamic properties. Currier and O'Connell (1987) have shown that invoking an ISM does not necessarily lead to GC expressions for thermodynamic properties. One objective of this paper is to develop a general basis for determining what assumptions must

234 be added to interaction site models to obtain group contribution expressions for macroscopic properties.

Step 3 By universality we mean that numerical values for group a - g r o u p /3 portions of properties found in Step 2 for a particular mixture are assumed to apply to any mixture containing groups a and /3. In this paper we accomplish the following: (a) The conventional division of properties performed in Step I has not been given a clear interpretation on a molecular scale, so we attempt such as interpretation by identifying combinatorial and residual terms as hard- and soft-core contributions, respectively. (b) We then show that under a few well-posed assumptions the objectives of Step 2 can be realized in rigorous ways; i.e. within the ISM framework, GC expressions for residual thermodynamic properties can be rigorously correct. (c) In addition, for certain properties, including the free energies and activity coefficients, the correct GC expressions can be simplified by introducing g r o u p - g r o u p pair distribution functions. (d) We define these g r o u p - g r o u p distribution functions and show how they are related to the more familiar site-site distribution functions. Forces acting between polyatomic molecules depend on molecular orientations as well as molecular separations. One of the advantages offered by ISM is that the orientational dependenc, e of the interaction is implicit within the site-site distances r 0. For the properties cited in item (c) above, this orientational dependence remains implicit in the statistical mechanical averages that yield those properties; howew~r, the pressure is an exception. That is, the ISM expression for the pressure contains both implicit and explicit dependence on local orientational structure. Nevertheless, the pressure can be made to adhere to a GC form. (e) We make some observations on the importance of the explicit orientation contribution to the pressure and its possible impact on development of GC equations of state. The assumption in Step 3 seems at best an approximation. We will address Step 3 in a subsequent paper, in which universality will be tested using molecular simulation. Before embarking on this program, we must establish a notational convention. NOMENCLATURE AND NOTATION In what follows, we formally distinguish between (1) molecules, (2) species, (3) mers, (4) groups, and (5) sites. (1) A molecule is any stable, self-contained structural arrangement of

235

a t o m s that retains the characteristics of an e l e m e n t or c o m p o u n d . T h e total n u m b e r of molecules in a system is N. (2) A species is a p a r t i c u l a r k i n d of molecule. W e use u p p e r c a s e R o m a n letters A = C, D, etc. to indicate species, so that a m u l t i c o m p o n e n t m i x t u r e c o n t a i n s N A molecules o f c o m p o n e n t A. T h e total n u m b e r of molecules in the m i x t u r e is t h e n

U = ~.,U A

(3)

A

( F o r letters used in superscripts a n d subscripts, we reserve the first two c h a r a c t e r s of the R o m a n a n d G r e e k a l p h a b e t s for use as d u m m y indices, so t h a t a s u m m a t i o n starts f r o m the third c h a r a c t e r in the case a n d a l p h a b e t of the d u m m y index.) (3) A m e t , a t e r m a d o p t e d f r o m p o l y m e r chemistry, is a n y structural a r r a n g e m e n t of a t o m s that f o r m s a subdivision of a molecule. Mers o f the s a m e kind m a y a p p e a r repetitively o n o n e molecule and o n molecules of d i f f e r e n t species; e.g. a n e t h a n e molecule is c o m p o s e d of t w o - C H 3 mers. O n e m o l e c u l e of species A c o n t a i n s n A mers, so that in a m i x t u r e the total n u m b e r o f mers, n, is

n= Y'NAn A

(4)

A

(4) A g r o u p is a p a r t i c u l a r kind of mer. T h e c o n c e p t of " g r o u p " is related to that o f " m e r " in the same way that " s p e c i e s " is related to " m o l e c u l e " . T h u s , an e t h a n e m o l e c u l e c o n t a i n s two mers of o n e group: - C H 3. A m o l e c u l e o f species A c o n t a i n s uA g r o u p s and a m i x t u r e c o n t a i n s a total of 1, groups. H o w e v e r , in general l, 4: y ' u A A

b e c a u s e molecules o f d i f f e r e n t species m a y c o n t a i n some of the same groups. T h u s , a m i x t u r e of A - - p r o p a n o l (i,A = 3) a n d B = b u t a n o l (~'B = 3) has p =

3 =~ ~'A -I- P B -

W e use lowercase G r e e k letters o~ = y, ~, e, etc. to indicate groups, so that o n e m o l e c u l e o f species A c o n t a i n s n~ mers of g r o u p a. T h e total n u m b e r of m e r s o n o n e m o l e c u l e of species A is then l,A

,A=En

(5)

Ot

U s i n g eqn. (5) in eqn. (4) allows the total n u m b e r of mers in the m i x t u r e to b e w r i t t e n as UA

n= ENAEnA A

a

(6)

236

Let n" be the total number of mers of group a in the mixture, so that

n" = ~_, NAn~A

(7)

A

then, in addition to eqn. (6), the total number of mers can be written as P

.=E,°

(8)

O~

(5) A site is a particular geometric position on the structure of a molecule. We use lowercase roman letters a = c, d, etc. to indicate sites. The notation used for counting sites is the same as for mers; the distinction between counting mers or sites is apparent from context. Thus, a molecule of species A contains a total of n A sites. We have noted in the Introduction that in ISM formulations sites need not correspond to mers. The distinction between mers and sites originates in a difference in interpretation of molecules: mers mimic a collection of atomic nuclei and are based on an interpretation of molecules as geometric structures, while sites are attempts to localize the primary sources of intermolecular interactions--electron clouds. In G C methods only mers are involved in intermolecular interactions, so that on one molecule of species B each mer occupies one site. Thus in G C methods the number of sites on a molecule equals the number of mers. For example, 2-pentanol and 3-pentanol molecules are each composed of the same number and kinds of mers. Interaction site models would be able to distinguish between these two molecules because the - O H mers are attached to different sites in the two structures; conventional G C methods, however, could not distinguish between these two molecules. This nomenclature is illustrated in Table 1 for a mixture of ethanol and methylethylketone. Symbols for properties whose values depend either on particular sites, mers, groups, or molecules or on some combination of all four will carry subscripts for molecular species and superscripts for group, mer, and site designations. Super- and subscripts formed from the lowercase italic letters i, j, k, or l are d u m m y indices that may run over molecular species, sites, mers, or groups, depending on context. For a generic property F: F ~ Bb depends on a mer of group a ( f l ) occupying site a(b) on a molecule of species A(B); F~ depends on a mer of group a ( B ) occupying any site on a molecule of species A(B); F~b depends on a site of type a(b) appearing on a molecule of species A(B); F ab depends on a site of type a(b) appearing on molecules of any species; F "~ depends on a mer of group a(/3) occupying any site on a molecule of any species.

237 TABLE 1 N o t a t i o n a d o p t e d in this paper applied to a sample mixture of ethanol (A) and methylethylketone (B) Entity

Species Molecules Mers Groups Mers of each group o~= C H 2 fl = C H , y = OH 3=OC

species A = ethanol

Number per molecule of species A

Number per molecule of species B

Total n u m b e r in mixture

-

2

1 nA= 3 uA = 3

1 riB=4 VB=3

N=NA+N B n = 3NA + 4 N B v=4

v~ = 1 v~ = 1 v~=l v~=O

v~=l

n ~= NA + NB n B NA + 2 N B nV=NA n8 = N B

v~ = 2 v~=O v~ = 1

=

spedes B = methylethylketone

C O N V E N T I O N A L DIVISION OF PROPERTIES

In a conventional group contribution approach, a generic t h e r m o d y n a m i c property F is divided into combinatorial Fcom plus residual F r terms, as displayed in eqn. (1). This division of F is equivalent to a statistical mechanical division into hard-core plus soft-core contributions F = Fhc + Fsc

(9)

The hard-core term is identified with the high temperature limit of F, since that limiting behavior is dominated by short-range repulsions characteristic of the size and shape of the molecules; i.e. the entropic contributions that are intended to be included in combinatorial terms are those resulting from the packing together of objects of various sizes and shapes. The residual term is primarily the energetic consequence of longer-range interactions among the molecules. The residual term does not account for all intermolecular interactions, because the combinatorial term includes the effects of hard-object interactions that occur on contact. Nor does the combinatorial term account for all repulsive forces, because the residual term generally includes effects from short-range "soft" repulsions. Moreover, while the hard-core term is purely entropic, the residual term encompasses both entropic and energetic effects.

238

To illustrate how the hard-core portions of a t h e r m o d y n a m i c property can be obtained from the property's limiting value at high temperature, consider a multicomponent mixture of N molecules in a closed container of volume V at temperature T. The Helmholtz free energy for such a mixture can be written as the sum of kinetic and configurational terms (Reed and Gubbins, 1973, p. 169) 131 = - I n Q K - In Z

(10)

where fl = 1/kT, QK is the kinetic partition function, and Z is the configurational integral. The QK term depends on temperature, but is independent of density, while the configurational integral depends on both temperature and density

z= f ... fexp[-13U(r N, ,oN)]

dr u duN

(11)

In eqn. (11) the vectors r i locate molecular centers of mass and vectors u i are molecular orientations. The notation x N represents the set {x 1, x 2 , . . . , x N }, while dx N represents (dx 1, d x 2 , . . . , dx N }. We now separate the intermolecular potential U into hard- and soft-core terms U - - Uhc + Usc

(12)

put eqn. (12) into eqn. (11), and replace exp(-13Usc) with its power series (McQuarrie, 1976, p. 302)

z= f... fexp[-13Uhc(r u,

uu)] ( 1 - 13Usc+ O(132)) dr u d*0u

(13)

The high temperature limit of eqn. (13) (13 = ( 1 / k T ) ~ 0) is then lim Z = Z h c = B--'0

f..

fexp[--flUhc(r N, uN)] dr u d u N

(14)

Note that in eqn. (14) (13Uhc) does not really change with 13 because Uhc is either zero or infinity. Thus the high temperature limit of (13A) gives the hard-core portion of 13A lim (13A) = 13Ahc

(15)

fl~0

Now the soft-core portion of A is expressible as the difference between the full property and the hard-core portion

flAsc = 13A - 13Ah~

(16)

239

and the Gibbs-Helmholtz equation can be used to obtain Asc in terms of the configurational internal energy Uc; thus 0/~

Nv

= g-

gidealgas = g c

(17)

Separating variables in eqn. (17), integrating, at fixed N and V, from the high temperature limit to the temperature of the system, and using eqn. (15) gives f l A - ( f l A ) i d e a l g a s -- f l A h c q- ( f l A ) i d e a l gas =

foflUc dfi'

(18)

An equivalent procedure for obtaining eqn. (18) has been given by Maurer and Prausnitz (1978). Combining eqns. (16) and (18) yields

BAse= f0BUc dfl'

(19)

In the Appendix we show that, under the assumption of a pairwise additive site-site potential, the configurational internal energy Uc can be written in terms of the site-site pair distribution functions gAB(r, ab p, T, {x}) as HA n B

Uc = ~.,~.,NANB~_,Vj-" fuab(r) gAB(r, ab A

B

a

P, T, {x}) dr

(20)

b

Here we have emphasized that the site-site distribution function depends not only on site-site separation r, but also on the mixture density p, temperature T, and composition { x}. Combination of eqns. (19) and (20) gives r/A nB

flA~c= flA~= 21----ZENAN B~a Y'.fdruab(r VA B b --

ab )fo__/~dfl ,gAB(r, p, T, (x})

(21) Equations (20) and (21) can be expressed for a general residual property F r as r/A g/B

• E f f[uab(r), gAB(r, p, T, (x})] dr fr-~ -2VEY'~NANB ab A B a" b

(22)

where f depends on both the site-site pair potential and the site-site pair distribution function. Besides the internal energy and free energy, other properties that conform to eqn. (22) include the residual chemical potential and the residual pressure.

240 GROUP CONTRIBUTION EXPRESSION FOR GENERIC PROPERTIES

In this section we start from the general ISM expression (22) and derive the group-contribution expression for a residual generic thermodynamic property. The ISM formalism assumes that interactions among polyatomic molecules can be computed from pairwise additive sums over interaction sites, as in eqn. (2). G C methods, however, do not always make clear whether g r o u p - g r o u p contributions are restricted to intermolecular interactions or whether they also include intramolecular forces. Typically intramolecular forces are implied, even when not explicitly included, to ensure molecular integrity. Moreover, when numerical values for g r o u p - g r o u p contributions are extracted from experimental thermodynamic data, intramolecular contributions are implicit in the data. Large molecules (i.e. those containing many groups) will have intramolecular forces than can be modeled as pairwise additive g r o u p - g r o u p interactions, analogous to intermolecular forces. However, intramolecular forces m a y also involve contributions (such as b o n d rotations) that are not susceptible to an interpretation in terms of simple pairwise additive terms. Here we consider G C ideas applied only to intermolecular portions of properties. Consider a multicomponent mixture of N molecules in a volume V at temperature T. The system is uniform and isotropic, so that potential functions and spatial distribution functions depend only on scalar distances r -- ]r I, and not on vector separations r. For the assumed site-site potential (eqn. (2)), a residual generic thermodynamic property F r can be written from eqn. (22) as t/A t/B

Fr= ~, Y'~NANBY'~ Y'~F~,b A

B

a

(23)

b

where

FAab- ~__~ff[uab(r). , gA,(r, ab p, T, (x i})]

dr

(24)

In writing eqn. (24) we have emphasized that the site-site pair distribution function not only depends on the distance between site a on a molecule of species A and site b on a molecule of species B, but also depends on the fluid density, temperature, and composition. N o t e that eqn. (23) comprises a total of n2 terms, where n is the total number of sites in the mixture. At this point eqn. (23) is rigorous under the assumed site-site form of the pair potential. We now want to carry eqn. (23) into a G C expression. To do so, we must have an identifiable structural mer occupying each interaction site, so that the site-site pair' potential u a b ( r ) is equally interpretable as a g r o u p - g r o u p potential, u"~(r). That is, the pair potential between

241

a mer of group a and a mer of group fl is independent of the sites occupied by those mers. If this condition is satisfied, then the sums over sites in eqn. (23) are actually sums over mers. We then divide each sum over mers in eqn. (23) into two parts: a sum over all v different groups and a sum over all mers of the same group. Thus eqn. (23) becomes Fr= E A

ENANBE E E EFAkg B

a

/3

k

(25)

/

Rearranging the order of summations and integrations in eqn. (25) we can write 1 ~ ~ fdr~ Fr=2v A -

-

~-~NANnY'. Y'~f[uab(r) B

k

l

kl , gAB(r)]

(26)

The inner two double sums in eqn. (26) contain a total of (n~n~) terms, where n ~ is the total number of mers of group a in the mixture. We now define the g r o u p - g r o u p average F ~/3 of each integral in eqn. (26) by

n"nBF~l~- 2 ~ f d r ~ - " Y' NANBY" y" f[u~/3(r), A

B

k

gAk~(r)]

(27)

1

where we have interpreted the site-site pair potential in its equivalent g r o u p - g r o u p form. Combining eqns. (26) and (27) gives the g r o u p - c o n t r i bution expression p

l*

Fr= ~., Zn"nBF "B

(28)

We emphasize that in passing from eqn. (23) to eqn. (28) no assumptions have been made about either the site-site distribution functions or the integrals in eqn. (27); eqn. (28) is merely a rearrangement of the order in which terms in eqn. (23) are accumulated plus a utilization of definition (27) for the average integral. The above derivation of the G C expression (eqn. (28)) is rigorous under three assumptions: (1) The intermolecular potential is of a pairwise additive site-site form (eqn. (2)). (2) The residual thermodynamic property F r depends on the site-site pair potential and site-site pair distribution function as given in eqn. (24). Most properties of interest satisfy this requirement. Moreover, properties adhering to the form of eqn. (24) divide into at least two subclasses: (a) those whose f can be rewritten in terms of a g r o u p - g r o u p pair distribution function, g~(r) (defined in the next section), and (b) those whose dependence on local structure cannot be expressed in terms of just a g r o u p - g r o u p distribu-

242

tion function. Members of subclass (a) include the internal energy, Helmholtz free energy, and chemical potential. The group-group distribution functions are defined in the following section, and the general GC expressions for these thermodynamic properties are developed in the section headed "Group-Group Pair Distribution Functions" (pp. 245-246). The prominent member of subclass (b) is the pressure, which is discussed in the section "Comments on the Pressure" (pp. 251-255). (3) A chemical mer must be identifiable with each interaction site; i.e. mers must coincide with sites. This enables site-site pair potentials to be interpreted in equivalent group-group forms, thereby making the group-group averages defined in eqn. (27) meaningful.

Group Symbols: • = C H 3 (group c~)

O =CH 2 (group ~ )

[] = O H (group ?)

• = OC (group 5)

Molecular Species Symbols:

= ethanol

~

(speciesA)

= methylethylketone (speciesB) "

Sample Diagram for One Integral in the Residual Internal Energy: A bond between two group symbols indicates the pair of mers over which the integration is being done. (Here site "a" of species A holds a CH3-mer.)

aa

FAA

.-aa dr = - 1 ~ uaa(r)~~AAfr~ 2V

=

Residual Internal Energy in the Site-site form of Eq. (20):

Uc =

n A nA 2 ab NAZ Z FAA +

a b 2 U c -- N a UAA + 2 N A N B Uha

a b 2 + N B UBB

a b

(a )

Fig. 1. (a) D e f i n i t i o n o f diagrammatic notation for comparing the site-site and group-group expressions for the residual internal energy in Uc in mixtures of ethanol and M E K .

243

UAA =

UBB

-

+z

+z

+2

LDOj

+2

z

+2

+

+

Cb) Fig. 1. (b) Diagrammatic representation of the site-site form of Uc given in Fig. 1 (a).

In Fig. 1 we compare the site-site (eqn. (23)) and g r o u p - g r o u p (eqn. (28)) forms for the residual internal energy in a sample mixture of ethanol and methylethylketone. All integrals in the two forms are written explicitly using a diagrammatic notation defined in Fig. la. Assuming ethanol to be composed of three mers and M E K four, the site-site form for Uc (eqn. (23)) contains 28 different integrals; these are displayed in Fig. lb. These 28 integrals reduce to just ten integrals when U~ is resummed into the g r o u p - g r o u p form of eqn. (28). These ten integrals are shown in Fig. l c along with their definitions in terms of the site-site integrals. Two primary points are to be appreciated from the figure. (a) The numerical value for Uc will be the same both from the site-site expressions of Fig. la,b and from the g r o u p - g r o u p expression of Fig lc. So if the site-site distribution functions are already known, the g r o u p - g r o u p form offers no advantage in

244

I

t~

"~

÷

÷

'~=

~

~ ol

"i-

~

÷

x

H

~

It

II

Z

&

e,l

Z • q-

Z

÷

,'°°,,, ~:z~~

II

,,

•

÷

'4'-

° ,,,..,

+

'44-

"b,,q ~

o

245

determining a property value. (b) If the site-site distribution functions are not known, then the g r o u p - g r o u p form m a y be advantageous since a smaller n u m b e r of integrals need be evaluated and those integrals are independent of molecular species. We emphasize this independence by writing the g r o u p - g r o u p equation for Uc in Fig. l c using diagrams containing just two groups, suppressing the species on which those group reside. Of course, the g r o u p - g r o u p integrals are independent of species only for particular molecular components at a specified composition. If the composition is changed, then, in general, the site-site distribution functions will change, and hence the ten g r o u p - g r o u p integrals will each change in value. GROUP-GROUP PAIR DISTRIBUTION FUNCTIONS

For certain thermodynamic properties the general GC expressions (27) and (28) may be simplified to expressions involving integrals over g r o u p - g r o u p pair distribution functions. In this section we define the g r o u p - g r o u p pair distribution functions and show how they are related to the site-site functions. We consider a multicomponent mixture of polyatomic molecules at constant N, V, and T, with intermolecular forces prescribed by an ISM expression, as in eqn. (2). The site-site distribution function gAB(r) ab is proportional to the probability of site a on a molecule of species A being found at a distance r from site b on a molecule', of species B. This function is defined by Gray and Gubbins (1984, p. 178) as gAB(r ) _

V

NANB

r

i~j

Here, NA (NB) is the total number of molecules of species A (B) present in the mixture. The indices i and j extend over all molecules of species A and B, respectively, and the stipulation i 4=j avoids sampling intramolecular contributions to g~b(r) when A = B. The scalar ri~b is the distance between site a on the ith molecule of species A and site b on the j t h molecule of species B. The symbol 3 in eqn. (29) is the Dirac delta function: it vanishes unless its argument is zero. The angle brackets represent an ensemble average, i.e. (using r i and l ~ i as in eqn. (11))

((...)) = (1/z)f(...)

exp( -

U/kT)

dr, drj dr u-2 do~ dtoj dto u-2

(30)

in which all integrations are done with the particular site-site distance ri~b = Iri~ - rb[ held fixed. In eqn. (30) N = I ~ N A is the total n u m b e r of molecules. If molecules A and B are of the same species (A = B), each

246 molecule containing n A sites, then the pair has at most nA(n A + 1) site-site pair functions gAA(r). ab If the molecules are of different species, containing n A and n u sites, respectively, then the pair has at most nAn B pair functions ab gAB(r). In both cases fewer distinct pair functions m a y actually occur because of symmetry among kinds and distributions of sites on the molecules. N o w if the ISM expression for the intermolecular potential applies to the mixture and further if a structurally distinct mer can be identified with each site, then we can introduce a g r o u p - g r o u p pair distribution function gaB(r). This function is proportional to the probability of finding a met of group a a distance r from a mer of group fl, regardless of which sites or molecules contain the mers. Formally the definition is

gaB(r)_

V n a?lB

y,~_~8(rk, k

r)

(31)

l

where n ~, defined in eqn. (7), is the total number of mers of group a present in the mixture. In expression (31) index k(l) runs over all mers of group Definition (31) for the g r o u p - g r o u p distribution function is unambiguous and physically meaningful. It simply defines a measure of the probability of finding two mers (one of group a, another of group fl) a distance r apart. In some cases, g~/~(r) can be extracted from diffraction experiments. For example, a mixture of alkanes, ignoring for the m o m e n t any distinction between - C H 2 and - C H 3 groups, can be considered to be c o m p o s e d of b u t a single kind of mer. One X-ray diffraction experiment on such a mixture would provide the g r o u p - g r o u p function defined in expression (31). (One requirement for such an experiment to yield g~(r) is that the mer should provide a single (or dominant) scattering center. This requirement is satisfied in our example because the single electron in hydrogen scatters little of the incident X-ray radiation compared to that scattered b y the electrons in carbon.) We now show how the g~(r) function is related to the site-site functions defined in eqn. (29). To do so, we divide the sum over k in expression (31) into three parts: a sum over all molecular species, a sum over all molecules of each species, and a sum over all sites containing a mer of group a on each molecule. A similar division is made for the sum over l in expression (31). The result, which is rigorously equivalent to expression (31), is

V ?l a ?l fl

E EE B

i~j

(32) b

247

Moving the first and third pairs of sums in eqn. (32) outside the ensemble average and then using the definition of the site-site functions (29), eqn. (32) becomes gaB(r )

1

-

~

g/a/,/B A

Y'NANBY"~EgAB( ab r ) B

a

(33)

b

That is, the g r o u p - g r o u p function is an appropriately weighted average of all the site-site distribution functions involving the two groups. Since g~n(r)=gn~(r), a mixture containing 1, groups will have v ( u + l ) / 2 g r o u p - g r o u p distribution functions. We emphasize that the gaB(r) values depend on density, temperature, composition, and the structure of the constituent molecules.

Example To illustrate eqn. (33) consider an equimolar binary mixture in which species A is composed of m o n o m e r s and species B is composed of dimers. All mers are the same on both species. The total n u m b e r of site-site aa ab distribution functions for such a mixture is nine: one gAA(r), tWO gAB(r) (because gAB ab = gaB), ba tWO g~bA(r), and four gaB(r). bb Because of symmetry aa gAB(r), and gbb(r). Since all the only three of these are distinct: gAA(r), ab" mers are the same (i.e. group a), only one g r o u p - g r o u p distribution function occurs and eqn. (33) yields

g'~'~(r)

=

+ 4 gbb(r)] ~[gAA(r) + 4 gAB(r) ab

(34)

aa

GROUP CONTRIBUTION

EXPRESSIONS FOR ENERGETIC

PROPERTIES

In this section we show that for certain t h e r m o d y n a m i c properties the G C expressions (27) and (28) reduce to terms in which the site-site pair can be replaced by g r o u p - g r o u p pair distribudistribution functions gAB(r) ab tion functions g~n(r). For this replacement to be valid, the residual property must have its f(u, g) in eqn. (24) expressible as a product of fl, which depends on the site-site pair potential, and f2, which depends on the site-site pair distribution function, i.e.

f[uab(r), g~(r)] =fl[uab(r)lfE[g~(r)]

(35)

Using eqn. (35) in expression (27) for the g r o u p - g r o u p portion of Fr, we find F~ B

1 2Vn,~n/~ f

drfl[u°¢(r)]2

A

2 NANBY'~ Y~.f2 [ g ~ ( B

k

l

r)]

(36)

248

Now for those properties that obey eqn. (35), f2 is typically linear in the site-site distribution functions, so that

E ENAN.E Ef2[g A

B

k

ENAN.EEg (r)

(r)] =f2

l

B

k

(37)

1

Introducing the group-group distribution function from eqn. (33) into the right-hand side of eqn. (37) yields lhs(37) =

n~nBf2 [ g"•( r )]

(38)

Where lhs stands for left-hand side. Now combining eqns. (37) and (38) with eqn. (36) gives the group contribution expression for F r in terms of the site-site (group-group) pair potential and the g r o u p - g r o u p distribution function 1

= 2-F fil

[uo (r)l j2 [ go.(r)l

dr

(39)

This result (eqn. (39)) is a special case of the more general result (eqn. (27)). Note that eqn. (39) together with eqn. (28) reduces to the correct expression for the configurational internal energy in the special case of a pure fluid composed of spherically symmetric monomers (see Reed and Gubbins, 1973, p. 179, or eqn. (42) below). More generally, the GC expression (39) used with eqn. (28) is rigorously correct under the three assumptions cited on pp. 241-242, with assumption 2 for f replaced by the more specialized expressions in eqns. (35) and (37). Properties that conform to eqns. (35) and (37) include the internal energy, for which fz is merely the pair distribution function itself, and the Helmholtz free energy, for which f2 is the integration over the reciprocal temperature shown in eqn. (21). An exception to eqn. (35) is the pressure, for which structural information beyond the site-site distribution functions is needed. Note that the development of the GC expression (39) does not make any assumption about site-site distribution functions. None of the gAB(r) ab need be equal nor must the sites be distributed on the molecules in any symmetric way. For eqn. (39) to be correct, we need only average over the site-site functions g~b(r) to obtain the group-group distribution functions g~/J(r). The important limitation implicit in eqn. (39) is that the g°¢(r), and hence the group contributions to properties F ~B, depend on density, temperature, composition, and molecular structure. At this point we have no reason to expect that values for F °~ obtained from one mixture will apply to the same groups (a and /3) either in other mixtures or even in the same mixture at other state conditions. That is, the realization of Step 3 in devising GC methods requires additional assumptions beyond those leading to eqn. (39).

249

Examp~ To illustrate ~the interpretation of the group contribution expressions (28) and (39), we consider a pure fluid composed of flexible, straight-chain tetramers; thus, the model is an approximate representation of n-butane. In the model the - C H 2 and - C H 3 mers are modeled as one group: identical L e n n a r d - J o n e s spheres. We have performed molecular dynamics simulations on such a fluid and the molecular model is described in more detail elsewhere (Haile, 1986). The model has 16 site-site pair distribution functions; however, because of symmetry only three of these are distinguishable: the distribution of end site-end site distances, gEE(r); the distribution of inner site-inner site distances, glI(r); and the distribution of end site-inner site distances, gEl(r). We suppress subscripts that indicate different molecules since in a pure fluid all molecules are identical. Because all mers are the same here, this model substance contains only one g r o u p - g r o u p distribution function, g~(r). Equation (33) gives this g r o u p - g r o u p function in terms of the site-site functions as

g ~ ( r ) = (a/4)[gEE(r) + 2 gEI(r) + glI(r)]

(40)

Figure 2 shows the simulation results for the three site-site functions, as well as the g r o u p - g r o u p function computed from eqn. (40). The three site-site functions are clearly different; for example, the end sites on adjacent molecules can pack closer together (at the simulated state condition) than can inner sites. We now test the G C expressions for this model fluid by computing the intermolecular internal energy. At 282 K and 605 kg cm-3, the value of the intermolecular contribution to the configurational internal energy obtained directly from the simulation is Uc/N= 12.0 + 0.1 kJ g - i mol-1. (The uncertainty quoted is an estimate of statistical precision.) The GC expression for this property is given by eqn. (28) as

U J N = 16 U ~

(41)

with U ~ given by eqn. (39) as

U ~ = 2~rpj° u ~ ( r ) g ~ ( r ) r 2 dr

(42)

where p----N/V is the molecular number density. In the simulation the site-site (i.e. group-group) pair potential was a L e n n a r d - J o n e s model (eqns. (6) and (9)) having { = 419 J g - i m o l - i and o = 3.5 ,~. U ~ a g the g r o u p - g r o u p pair distribution function from Fig. 2, performing the integration in eqn. (42) numerically and substituting the result for U ~ into eqn. (41) we obtain the GC result Uc/N = 11.9 kJ g-1 mol-1. (The uncertainty in

250 1.4 gii gei

i.2

~-

gee

.N~

. f - ~

/.~

"/;/,J

i

group-group

~ ' / "

LL

,-- 0.B

-g

~"

l//

7' 0.6 O3 r'~

.~

0.4

110 El_

/

0.2

t

i

02

~

3

I 4

I 5

I 6

I 7

I 8

r (Angstroms) Fig. 2. M o l e c u l a r d y n a m i c s s i m u l a t i o n results for the s i t e - s i t e a n d g r o u p - g r o u p p a i r d i s t r i b u t i o n f u n c t i o n s in a pure fluid o f flexible t e t r a m e r molecules, a p p r o x i m a t e l y m o d e l i n g

n-butane. Broken lines are the site-site functions: ( . . . . . ) = gena-ena, ( . . . . . ) = gena-in; ( ..... ) = gin-in" ( ), group-group function, computed from eqn. (40). Each mer was the same Lennard-Jones (6,9) group, with c = 419 J tool -1 and a = 3.5 A. Using these parameters the simulated state condition was 282K and 605 kg c m -3. Molecular flexibility included modes for bond vibration, bond-angle bending, and rotation about the central bond.

this n u m b e r is difficult to estimate reliably, because the s i m u l a t i o n provides the site-site distribution functions, f r o m which Uc is o b t a i n e d by n u m e r i c a l integration. It is likely to be at least _+ 0.2 kJ g-1 mol-1.) N o t e that eqn. (42) has exactly the form for the configurational internal energy of a pure fluid of spherically symmetric molecules (i.e. spherical m o n o m e r s ) . T h e appearance of eqn. (42) in the f o r m of that for a solution of m o n o m e r s is the essence of a conventional G C method. In eqn. (42) the distribution f u n c t i o n has ostensibly the same physical i n t e r p r e t a t i o n as for a p u r e m o n o mer; however, numerically in eqn. (42) is n o t equal to the radial distribution function in a m o n o m e r i c fluid because the distribution f u n c t i o n in eqn. (42) implicitly contains the effects of molecular structure t h r o u g h the averaging over the site-site distribution functions, as can be seen in Fig. 2.

g~(r)

g~(r)

251 COMMENTS ON THE PRESSURE

The extension of GC ideas to pressure-explicit equations of state is a problem of considerable interest. U n d e r the assumptions of pp. 240-241 the residual pressure for polyatomic fluids can be expressed in the general G C form (eqns. (27) and (28)); however, the functional for the pressure does not simplify to the form of eqn. (35). That is, within ISM, the integrand for the pressure depends upon more than just a functional of the site-site pair potential times the site-site pair distribution function (Bearman, 1977; Nezbeda, 1977; Aviram and Tildesley, 1978). The pressure integrand also depends on a partial measure of local orientational structure, so that, in ISM, the pressure is a more complicated property than the internal energy (eqn. (20)) or the free energy (eqn. (21)) because all the orientational dependence of those properties is contained in the site-site pair distribution functions. Tildesley et al. (1979) have shown that the orientational structure of rigid ISM fluids can be conveniently characterized by the angular site-site distribution function gab(r, oa1, ~02) expanded !in spherical harmonics of the molecular orientations. The function gab(r, ~1, ~°2) is proportional to the probability of finding site a on molecule 1 at a distance r from site b on molecule 2, when the molecules are in orientations ~a and ~02, respectively. Rather than the full gab(r, ~01, 0 2 2 ) , only the zeroth (gab(r)) and first-order (g~bo(r)) spherical harmonic coefficients contribute to the pressure. Thus the ISM expression for the pressure can be written as (Tildesley et al., 1979)

P=pkT

F/A F/B ~

1 ~-"~NAN~-"~ J

6V2 A B

a

+ I _ Z ~ N A N , y,y,n. 6V2 A B

a

b

duab(r)

dr

ab'

'

gA:B(r)rdr

f d/gab(r)dr{lcagmoAB(r) ab ba +/cbgl00AB(r)} dr (43)

b

Here /ca is the distance between the center of the molecule and site a on the same molecule. Physically, the spherical harmonic coefficient g lab 0 0 ( r ) is proportional to the ensemble average of cos(O), where 0 is the angle between the site-site vector r and the molecular centers-of-mass vector R12 = r + lea -lcb. Thus the third term in eqn. (43) depends explicitly on the relative orientations of the molecules, as well as on the distances between molecules. Now the second term on the right-hand side of eqn. (43) is in a form analogous to the general expression (26), with the integrand in eqn. (43) having the form given in eqn. (35). Thus, making the same assumptions as on pp. 240-241, carrying out the resummation and introducing the g r o u p - g r o u p pair function, as in the previous section, the second term on

252

the right-hand side of eqn. (43) will attain the G C forms (28) with (39). The same procedure can be applied to the third term on the right-hand side of eqn. (43), with { lcagloo(r)} playing the role of the site-site function in eqn. (26). Specifically, the GC for the third term will involve g r o u p - g r o u p functions, lc,g~(r), that are related to the corresponding site-site functions by equations analogous to eqn. (33), namely n~

lc~g~(r ) _

1 l'l C~l'l B

E ~.,NANuE A

B

a

ab (r ) Eb lcagl00

(44)

The points we wish to make here are concerned with the third term in eqn. (43). Although the pressure (or the residual pressure, which will involve two terms analogous to the second and third terms in eqn. (43)) in an ISM formalism can, under the same assumptions as on pp. 240-241, be expressed in a GC form for a particular mixture, the GC expression will formally contain explicitly orientation-dependent terms. These kinds of terms are not present in, e.g. the free energy, so we anticipate more difficulty in implementing Step 3 of the GC procedure for the pressure than for energetic properties. Here we consider two characteristics of the third term in eqn. (43): (a) What is its magnitude? That is, is the gaoo(r) ab term as important as t h e gab(r) term? (b) What is the effective range of its integrand? The answer to this may indicate the way forward in formulating reliable G C equations of state. First we ask whether the gl00 term in eqn. (43) makes a significant contribution to the pressure. In general, of course, that term depends on molecular structure, fluid state condition, and the strength and range of intermolecular forces, but a simple molecular fluid should at least offer some guidance. We therefore consider a pure fluid composed of rigid, homonuclear, L e n n a r d - J o n e s diatomic molecules (eqns. (6) and (12)). For such fluids eqn. (43) simplifies to (Tildesley et al., 1979; Nezbeda et al., 1983)

P* = p'T* + G~s+ Gloo

(45)

with

G,,-

G10 0 -

8~rp*2fo~dU*(r*) 3 -~r ~ g~(r*)r .3 d r *

(46)

87tO.2 ~ o ~ d U * ( r * ) ss .)r.2 3VC5 l* drr;- gl°°(r dr*

(47)

and P * = P o 3 / e , p * = 0 o 3, T * = k T / e , u * = u / e , and l * = l / o . A pure homonuclear diatomic has only one site-site pair distribution function gS~(r*) and hence only one g~o(r*) spherical harmonic coefficient. Bohn et al. (1988) have performed molecular dynamics simulations on

253 TABLE 2 Contributions to the pressure in pure fluids of Lennard-Jones homonuclear diatomic molecules

l / (Y

,003

k T/c

Po3/~

oo3 k T/~

G~

Gloo

0.3292 0.505 0.67 0.793

0.64 0.54 0.48 0.44

3.00 2.30 1.90 1.68

4.76 3.03 3.16 1.88

1.92 1.24 0.91 0.74

1.03 0.25 0.36 0.16

1.81 1.54 1.89 0.98

such pure fluids and evaluated the pressure and the site-site pair distribution functions. Thus we have P * directly from simulation and we use the simulation result for gSS(r*) in eqn. (46) to compute G~s numerically. Then eqn. (45) can be solved for Gl0o. (The integration in eqn (46) was done to r * = 3.3 using Simpson's rule; for r * > 3.3, gS~(r*) was set to unity and eqn. (46) was evaluated analytically.) The results of these calculations are shown in Table 1 for four different molecular elongations l*. The results in Table 1 for P * at l * = 0.793 and 0.505 are in agreement with simulation results of Singer et al. (1977), while the result for P * at l* = 0.3292 is in agreement with the previous result of Cheung and Powles (1975). Moreover, the result given in Table 2 for Glo0 at / * = 0.3292 is confirmed directly by integrating eqn. (47) using theoretical estimates for g~oo(r*) made by Nezbeda et al. (1983). The results in Table 2 show that the Gloo term makes a larger contribution to the pressure than the G~ term; in addition, the G~o0 contribution is about the same magnitude as the ideal-gas term, at least at the state conditions studied. Of course, the results in the table only apply to the fluids and state conditions simulated, but they strongly suggest that the G~oo contribution to the pressure cannot be neglected. Next we ask whether the G100 term in eqn. (45) is short-ranged compared to the G~s term. Several points suggest that this might be so: (a) G~oo measures a portion of average orientational structure, which we expect to be most important when molecules are close together. (b) The integrand in eqn. (46) for Gs~ includes a factor of r 3, while that in eqn. (47) for Glo0 involves r 2. (C) At large pair separations, g ~ ( r ) tends to unity, so long-range portions of du/dr can contribute to G~s; however, g~o(r) tends to zero at large r, effectively suppressing long-range contributions to G~o0. To test whether Gloo is in fact short-ranged, we computed the integrands in eqns. (46) and (47) separately for the LJ homonuclear diatomic having l * = 0.3292. For the Gss integrand in eqn. (46), simulation results (Bohn et al., 1988) for gSS(r) were used directly. For the Glo0 integrand in eqn. (47), we used the estimates for g~(r) given by Nezbeda et al. (1983) for the

254

30

20

t0

f_

w

0 o b~ "C3

c

~

-20

-

-30

-

-4°ols

I

t

I

i.~

I

2

2.5

Site-site distance (r/sigma)

Fig. 3. Integrands for the Ga00 and Gss terms occurring in the ISM expression (45) for the pressure in a pure fluid of rigid, monogroup dimer molecules having elongation l/o = 0.3292. The Gss-integrand is from a molecular dynamics simulation [Bohn et al., 1988] at pa 3 = 0.64 and kT/c = 3.0; the Ga0o-integrand is from the theoretical estimate of Nezbeda et al. (1983) (their figure 6(a)) at pa 3 = 0.622 and kT/c = 2.9.

l * -- 0.3292 fluid. The latter are n o t quite at the same state c o n d i t i o n as our simulation, but the two states are sufficiently close to provide a reliable answer as to the relative ranges of the two integrands. Figure 3 clearly illustrates that the i n t e g r a n d for G100 is i n d e e d shortranged c o m p a r e d to that for Gss- Additionally, the figure explains w h y G10o can be larger in m a g n i t u d e t h a n Gss, while s i m u l t a n e o u s l y being shorterranged. Thus, the integral for Gss contains b o t h positive a n d negative areas that tend to cancel w h e n c o m b i n e d in c o m p u t i n g Gss. In contrast, the integral for Glo0 is d o m i n a t e d by an area of a single sign. In general, the range of each i n t e g r a n d will be largely d e t e r m i n e d b y the range of the intermolecular potential u ( r ) ; for example, molecules having strong multipolar, associating, or i n d u c t i o n forces will exhibit longer-range c o n t r i b u t i o n s to the pressure t h r o u g h b o t h Gs~ a n d Glo0. Nevertheless, we expect t h a t in m a n y cases the relative ranges of the two terms in eqn. (45) will be similar to that shown in Fig. 2. W e caution that these conjectures only apply to rigid, axially s y m m e t r i c

255 molecules, because the spherical harmonic expansion of gab(r, ~1, ~2) only applies to such molecules. For classes of more general polyatomic molecules, a more elaborate expansion in orthonormal functions is required and the resulting structural contributions to the pressure could be more complicated. CONCLUSIONS In this paper we have used interaction site models (ISMs) as the basis for exploring the formal development of group-contribution (GC) expressions for thermodynamic properties of fluid mixtures. ISMs provide only moderately reliable approximations to intermolecular forces acting in real molecular fluids (Price, 1988; Gray and Gubbins, 1984, p. 34); nevertheless, they provide a convenient point of departure for the study of GC ideas. We have given a statistical mechanical interpretation of combinatorial and residual contributions to properties in terms of hard and soft-core contributions, respectively. We then found that if the thermodynamic properties for a particular mixture can be expressed via an ISM and if a structural mer can be identified with each interaction site, then the residual properties can be resummed into group contribution (GC) expressions. In the general case the resummation introduces a group-group average that is formed from the site-site contributions of ISM; for some properties the group-group average can be simplified to an average involving group-group pair distribution functions. The GC expression for properties eqn. (28) obtained here is the usual one; however, while Currier and O'Connell (1987) choose to interpret group contributions in terms of site-site distribution functions, analogous to eqn. (27), we believe that the group-group distribution functions will prove more useful in examining the assumptions needed in realizing Step 3 of a GC development. One can arrive at GC expressions for properties by postulating that, not only must the assumptions of pp. 240-241 be made, but in addition the site-site distribution functions must either be the same for all pairs of sites containing mers of the same groups or, at least, certain kinds of symmetries must exist among the site-site functions. A weaker set of hypotheses would require that the site-site integrals F~ub be equal for all pairs of sites containing mers of the same two groups. The analysis in this paper shows that such additional assumptions are not necessary to achieve Step 2: one need only be able to average over the site-site functions to obtain the group-group functions. We suggest that the group-group distribution functions are fully in the spirit of conventional GC approaches to mixture thermodynamics. Moreover, assumptions about the group-group functions, to realize Step 3, appear to be more amenable to physical interpretation than assumptions

256

about the site-site distribution functions or averages over those functions (Currier and O'Connell, 1987). These advantages of the g r o u p - g r o u p functions are exploited in the companion to this paper. We emphasize that the G C expressions for properties devised herein depend on the thermodynamic state condition, the mixture composition, and the kinds of molecular species present in the mixture. In this paper we have not addressed Step 3 in G C developments, in which the values for g r o u p - g r o u p contributions are taken from one mixture and are applied either to the same mixture at other compositions or to other mixtures. Such a procedure is tantamount to assuming that the g r o u p - g r o u p averages (eqn. (27)) or the g r o u p - g r o u p pair distribution functions (31) are independent of composition. Such assumptions are not formally valid, but it m a y be possible to reformulate the G C expressions so as to make Step 3 approximately correct, at least for some properties of some mixtures. One such reformulation is contained in the decision as to how to subdivide molecules into mers. This decision is often crucial to the success G C approaches; for example, G C expressions for methylethylketone are more reliable if the molecule is divided into three mers, ( C H 3 C O ) ( C H 2 ) ( C H 3 ) , rather than into the four shown in Fig. 1 (Benedek and Olti, 1985, p. 499). U n d e r ISM, with a mer coinciding with each site, we have shown that Step 2 is not affected b y this decision. (For example, in the case of M E K , we could subaverage the C O - and C H 3- g r o u p - g r o u p distribution functions and obtain C H 3 C O - g r o u p - g r o u p functions. The G C forms for thermodynamic properties would still be obtained, for a particular mixture.) The decision as to how to divide a molecule impacts the success of Step 3, because these decisions are really ad hoc attempts to take into account how g r o u p - g r o u p pair distribution functions depend on molecular structure. Finally, we have pointed out that the pressure in ISM, while amenable to the general G C form, depends explicitly on local orientational structure. This orientational contribution appears to be non-negligible and m a y confound simple extensions of G C ideas to pressure-explicit equations of state. However, the short-range nature of the orientational terms m a y prove advantageous in formulating useful G C expressions for the pressure. ACKNOWLEDGMENTS

We are pleased to thank J.P. O'Connell and R.P. Currier for a critical reading of an early version of the manuscript. This work was supported in part by a National Science Foundation Presidential Young Investigator Award (1984) to J.M.H.

257 LIST OF SYMBOLS

A

/ F Fcom Fhc

F,~B ab gAB( r )

gS~(r)

g~/3(r)

SS

G100 K l l* /ca n 71A 1,/a

N

NA P p*

Q QK r r* r ab

T

Helmholtz free energy a generic function generic extensive thermodynamic property combinatorial portion of F hard-core portion of F residual portion of F soft-core portion of F contribution to F from interactions between mers of group a and mers of group/3 site-site distribution function; a measure of the probability of finding site a on a molecule of species A a distance r from a site b on a molecule of species B a site-site distribution function in a pure fluid composed of molecules having only one kind of site (e.g. pure nitrogen) g r o u p - g r o u p distribution function; a measure of the probability of finding a mer of group a a distance r from a mer of group/3, independent of the molecules on which the mers are located a coefficient in a spherical harmonic expansion of the angular site-site distribution function integral defined in eqn. (46) integral defined in eqn. (47) Boltzmann's constant distance between interaction sites on a rigid molecule reduced distance, l/o distance between the center of mass and site a on a rigid molecule total number of mers in a system number of mers on one molecule of species A number of mers of group a on one molecule of species A total number of mers of group a in a system total number of molecules in a system number of molecules of species A in a system pressure reduced pressure, Po3/e canonical partition function kinetic portion of Q scalar distance reduced distance, r/o distance between site a and site b distance between site a on molecule i and site b on molecule j absolute temperature

258 T • u Ig*

uab(r)

U

uc Use U ab

V XA

(x} Z Zhc

reduced temperature, kT/c pair potential energy function reduced pair potential, u/c site-site potential energy due to an interaction between site a separated by a distance r from a site b g r o u p - g r o u p potential energy due to an interaction between group a separated by a distance r from a group fl total potential energy of a system configurational portion of U hard-core portion of U soft-core portion of U contribution to U from interactions between mers of group a and mers of group fl system volume mole fraction of species A set of mole fractions configurational portion of Q. hard-core portion of Z.

Greek letters 1/kT 6 E I*

~A P p* o 6~

Dirac delta Kronecker delta pair potential energy parameter total number of groups in a system number of groups on one molecule of species A system number density, N / V reduced density, No 3 / V pair potential distance parameter molecular orientation

APPENDIX

Here we derive eqn. (20) in which the configurational internal energy U~ is expressed in terms of integrals over the site-site distribution functions ab gAB(r). A principal objective here is to clarify the origin of the coefficients NA, etc. appearing on the integrals in eqn. (20), since conversations with colleagues have raised questions as to whether those coefficients are in fact correct. For a multicomponent mixture of N total molecules in volume V at

259

temperature T, the configurational internal energy is given by the ensemble average (Gray and Gubbins, 1984, p. 164)

Uc= ½ f .. fexp(-Bu)u(r

N, .,u) dr u d~U

(A1)

Within the ISM formalism, the intermolecular potential energy U is a pairwise additive sum of site-site potentials //A

/'/B

U( ru, t~U)= ½ E E E E u " b ( r ) A

B

a

(A2)

b

Note that the site-site potential u a b ( r ) only depends on the sites a and b, and not on the molecular species A or B to which the sites are attached. The scalar r in eqn. (A2) is the distance between site a on a molecule of species A and site b on a molecule of species B. Using eqn. (A2) in eqn. (A1) gives nA

} E Y" .

n B

yb' l f ''' fexp(-BU)uab(r)

d' N

(A3)

Now, for a given pair of sites, say a and b, on the right-hand side of eqn. (A3), all integrated terms will have the same numerical value if the same two molecular species are involved. The total number of such equivalent terms is NA(NB -- •AB), where N A is the total number of molecules of species A and dAB is unity if A = B and is zero otherwise. (When sites are distributed on molecules in symmetric ways, additional equalities among terms in eqn. (A3) will occur, but we need not consider such symmetries to arrive at the form of eqn. (20).) Thus, rather than perform all the integrations in eqn. (A3), we need only pick out a representative integral from each term and multiply each by the number of terms that are the same; thus eqn. (A3) becomes /~A /'/B

× ½NA(N.-aA.)f...

fexp(-/3U)dr N-2

d¢o N-2

(A4)

In writing eqn. (A4) we have separated the integrals over the first two molecules of each species from the remaining ( N - 2) integrations. This makes transparent the introduction of the angular pair distribution function gAB(FI2, t~l, ~2), which is defined by (Gray and Gubbins, 1984, p. 164) gAB(r12, t~l, /.02) = NA( N . - 8AB)a2/(pAPBZ) × f . . . f e x p ( - / 3 U ) dr u-2 d ~ u-2

(AS)

260

NA/V

where OA =

and f] = f d ~ . Using eqn. (A5) in eqn. (A4) leaves

n A rl B

Uc= ½Y'~ Y'~ Y'~ ~_~(OAPB/f]z)f ''' f uab(r)gAB( r, A

B

a

t~l, ~2)dr1 dre d6°1 d~2

b

(a6) N o w the site-site pair distribution function is defined in terms of the angular pair distribution function b y ( G r a y and Gubbins, 1984, p. 174)

f fgA.(r,2,,~1, ~2) dt~l dr°2

1

g~b(r) = -~

(A7)

wherein the integrations are done with the site-site distance r = [ r ~ - rbl held fixed. Bearman (1977) and N e z b e d a (1977) have each shown that the Jacobian for a transformation from center of mass c o o r d i n a t e s {dr1 dr2 d ~ l dto2 } to site-site coordinates {dry, dr b dr01 dto 2 } preserves the form of the volume element dradr2do~dto 2. Hence, we apply that transformation to eqn. (A6) and then use the definition (A7) to obtain eqn. (A6) in the form r/A rl B

S, E E EoAo, f ... f A

B

a

uab(r)gXb(r)dr] dr b

(A8)

b

Since r = Ir ~ - r b l , ( d r ~ d r b } = { d r d r b } , f d r b = V , and O A = N A / V , we can write eqn. (AS) as eqn. (20). QED. N o w if symmetries exist among the sites, so that, for example, there are n~ identical sites of type a on molecules of species A, then eqn. (20) can be rigorously written as PA

/~B

Uc= 2-T E ENANBZ E n A an . bf A

B

a

... f

uab

( r ) g Aa bB ( r ) d r

(A9)

b

where uA is the total number of different sites on a molecule of species A. REFERENCES Aviram, I. and Tildesley, D.J., 1978. A Monte Carlo study of mixtures of hard diatomic molecules. Mol. Phys., 35: 365-384. Benedek, P. and Olti, F., 1985. Computer Aided Chemical Thermodynamics of Gases and Liquids. Wiley, New York. Bohn, M., Fischer, J. and Haile, J.M., 1988. Effect of molecular elongation on the quadrupolar free energy in diatomic fluids, accepted for publication, Mol. Phys. Bearman, R.J., 1977. Remark on the classical statistical mechanics of rigid molecules. Mol. Phys., 34: 1687-1693. Cheung, P.S.Y. and Powles, J.G., 1975. The properties of liquid nitrogen. IV. A computer simulation. Mol. Phys., 30: 921-949.

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