Fluid Phase Equilibria, 24 (1985) 87-114 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

87

N R T 1: AH EMPIRICAL EQUATION DR AN IWSPIRIWG MDDEL FOR FLUIDS MIXTURES

PROPERTIES?

Henri RENON

Ecole Nationale Superieure des Mines de Paris

CNRS UA 878

60 Boulevard Saint-Michel

75006 PARIS (FRANCE)

and

MASSACHUSSETTS INSTITUTE OF TECHNOLOGY

Department of Chemical Engineering

CAMBRIDGE, MA 02139 (USA)

(Received and accepted July 26, 1985)

ABSTRACT

The NRTL and other local composition models give good empirical repre-

sentation of phase equilibria. They are currently extended to electrolyte

solutions, and high pressure phase equilibria and transformed into group

contribution models. They inspire theoretical work to find out about real

randomness i.e. distribution of nearest neighbors, in fluids, potentially

leading to improved understanding of fluid state and better empirical models.

INTRODUCTION

The NRTL model was originally developped and presented (RENON, 1966;

RENON and PRAUSNITZ, 1968 a, 1968 b, 1969) as an empirical model which could

improve over the WILSON equation for the simultaneous representation of

Vapor-Liquid Equilibria (VLE), Liquid-Liquid Equilibria (LLE), heat of

0378~3812/85/$03.30 0 1985 Elsevier Science Publishers B.V.

mixing, limiting activity coefficients in binary and multicomponent

mixtures. Many experimental data were considered and a systematic compari-

son with other models for gE was presented for the deviations between

experimental and calculated values of properties of binary mixtures and

multicomponent systems.

In this work, the empirical advantages and shortcomings of the NRTL and

other local composition models will be reviewed. The industry in accepting

these models widely recognized their practical advantages.

Since 1966, the academic research has taken two main directions:

l- extending the usefulness of the local composition models by reducing the

number of their parameters, transforming them into group contribution

models, introducing them into an equation of state approach or applying them

to electrolyte solutions

2 - questioning the reasons of the success and the limitations of these

model. Presently the criticism of the hypotheses which were presented ori-

ginally to explain the model, are turning very precise and based on the most

advanced methods of theoretical study of liquid and liquid mixtures. These

efforts will be critically reviewed and some suggestions for future develop-

ment will be made.

EMPIRICAL ADVANTAGES AND LIUITATIONS OF THE LOCAL COMPOSITIONS MODELS

The NRTL model can be considered as an equation for the excess GIBBS

energy gE of multicomponent mixtures with adjustable parameters :

C

CJE/RT = if, xi3$ 'j 'ji Gji

k?I 'k Gki

G ij = exp (- ~1.. T..)

Jl Jl

(1)

where Xi is the mole fraction of component i, R the gas constant, T the

absolute temperature and 'ijs 'jfs T ij = 'ji

temperature

dependent parameters.

The important practical advantages of the model are in order of decrea-

sing importance :

- It represents fairly well - within one or two per cent the mole frac-

89

tions in absolute value and the pressure in relative value - the VLE and LLE

of binary mixtures by adjusting three parameters at ane temperature.

- It predicts VLE of ternary and multicomponent mixtures from binary

parameters obtained by fitting binary data.

- It represents multicomponent LLE if one adjusts the binary parameters

to ternary data. There is usually not enough information about binary data

to obtain three parameters per binary. But one may adjust nine parameters to

data for a ternary system at one temperature.

- It gives a consistent representation of VLE, LLE and heat of mixing

if one accepts to make all parameters temperature dependent .

The main problems (disadvantages) encountered .in the application of the

model are:

- the two parameters TIP and 721 are strongly correlated, when

the maximun value of gE is relatively low.

- There are problems of multiple solutions : it is possible to find

several sets of parameters which will represent equally well given sets of

data. There are also sometimes many solutions for mutual solubilities of

two components for a given set of parameters. In that case JACQ and

ASSELINEAU (19831 did provide a method to select the single physical

solution.

- The model cannot be used to assess the quality of experimental data

using the maximum likelihood principle to obtain the error in measurements

from very precise measurements. In this case, polynomial expression of the

excess GIBBS energy, like REDLICH-KISTER (19481 equation, should be used as

recommended by PENELOUX and coworkers (PEEIELOIJX et al., 19761.

- The third parameter o12 which appears in the exponential cannot be

fitted for low values of the maximum of the excess GIBBS energy of a binary

mixture (gEFIAX < 400 J/mole) because it has no influence in that case.

However, a12 is very useful for strongly nonideal mixture especially in

case of phase splitting and should be fitted (figure Il.

- A very precise simultaneous representation of hE and gE is not

always possible. Even if the general trend of variation of gE and hE

versus temperature can be taken care of by the temperature dependence of the

parameters, the shape of both gE and hE versus composition cannot neces-

sarily be represented by an empirical equation with only three temperature

dependent parameters.

- 2 0.6

> “- 0.4

x

2 0.2

-0.2 -I 0 I 2 4 5 6

m ACTIVITY COEFFICIENT FOR SYMMETRIC EPUIMOLAR MIXTURE AS

A FUNCTION OF ACTIVITY COEFFICIENT AT INFINITE DILUTION

- There are limitations of the applicability of the model in the

critical region. Especially, the very flat maximum of the binary miscibility

curve (T vs x1, x2) cannot be represented (NICOLAIDES and ECKERT, 1978).

Although a minority of authors addressed directly the empirical pro-

blems of application of the local composition models and the majority looked

more towards theoretical developments to be reviewed later, there are a num-

ber of models developed using the local composition concept which have limi- tations similar to the NRTL equation. But it is necessary to explain now the common feature to all these models which originate in WILSON (1974) idea to

take into account empirically the local order in solutions according to the

equation.

X.. x. exp (-g.i/RT) J1= J (3) 'ii xi exp (-gii/RT)

with a normalization relation on the local mole fraction which is for a

binary mixture

‘3 + ‘ii = 1 (4)

91

All the family of local compositions equations are obtained by introdu-

cing these local compositiorainto excess functions expressions. It is known

that molecules first neighbours are not randomly distributed but that there

are influences of sizes, shapes and energies of interaction on the distribu-

tion of species around a given molecule. Equation (3) can be considered as a

quantitative assumption on this effect. It has two conspicuous features: non

randomness depend on the difference of energies Sjf-gii only and

not on size, shape, or energy gjj.

The way authors introduce these "local mole fractions" into an

expression of gE is often arbitrary and can be considered

hypothetical. WILSON (1964) introduced them in FLORY's equation.

gE/RT = CXi In (@ii/Xi)

where the local volume fraction oii is given by :

Q.. = vi xii

11 vi xji +v i 'ii

very much

(5)

(61

Without turning yet to theoretical arguments, let us consider what are

the relative empirical advantages of some local composition equations.

The original WILSON equation had two parameters only fgji - gii) and

(Sij-Sjj)' it was unable to represent phase splitting. The three

parameter WILSON equation (SCATCHARD and WILSON, 19641 for a binary

mixture :

gE/RT = C (XI ln(@II/xl) + x2 ln (@&x2) ) (7)

has a third parameter C and is able to represent partially miscible

mixtures. It was shown IRENON and PRAUSNITZ, 19691 that equation (7) can

be obtained by integration with respect to temperature of the NRTL equa-

tion used to represent the excess enthalpy with temperature independent

energy parameters (gji - gii) = ~~~ x RT and C = 1 / o... 1J

It was not originally proposed to extend it to multicomponent mixtures.

The UNIOUAC equation (ABRAklS and PRAUSNITZ, 19751 is another extension

of the original WILSON equation where phase splitting can be obtained by

adding a "combinatorial", size and shape dependent term to a modified three

parameter "residual" WILSON term.

g E- E

- g combinatorial E

+ g residual

with

(8)

gE @i 8.

= c qi xi In - + f C qi xi In $ combinatorial i 'i i

gE residual = i - r. qi xi In (C ej ~~~~ (8.b)

(8.4

The advantage / disadvantage of the UNIQUAC equation is that it has

only two binary parameters T . . and T.. but it requires informa-

tion (not fitted) on size and shapgof molecu es. '1' It has the same other

advantages / disadvantages as NRTL equation.

Many other equations were proposed, some on theoretical arguments (they

will be reviewed later), other mainly an empirical grounds. The idea to add

a NRTL term and a WILSON term was many times visited. HEIL and PRAUSNITZ

(1966) applied such a model to polymer solution even before the NRTL equa-

tion was invented. WANG and CHAO (1983) proposed recently a very similar sum

of terms. ROWLEY and BATTLER (1984) discuss it in relation with simultaneous

representation of hE and gE. My suggestion (a little different from

theirs) is to use NRTL with temperature dependent parameters to represent

hE as a function of temperature and add an empirical "athermal" term to

the proper integral of hE with respect to T.

hE(T,X) dT

T* (3”

This problem was encountered in the development of the UNIOUAC equation

where the combinatorial term is independent of temperature and was

considered by SAYEGH and VERA (1980).

HIRANUMA (1974 - 1981) described a modified form of the three parameter

WILSON equation for multicomponent mixtures.

N gE / RT. = - CCi Xi In (J~I Afj Xj)

He modified it again in 1983 (HIRANUMA, 1983).

To conclude this rapid and incomplete review, it should be emphasized

that the empirical value of all suggestions should be proven by extensive

application to all properties (hE, gE, LLE, VLE) of binary and multicom-

93

ponent mixtures that the of multiple and correlations

parameters should checked for new modification.

is doubtful the' various will succeed improve

significantly the most used local equations WILSON,

and UNIQUAC if they the correlated parameters

T21 ’ ~12 and do not add other parameters.

EXTENSIONS OF THE LOCAL COMPOSITION APPROACH

Before reaching the "theory" underlying the local composition equation,

it is good to consider the extensions of this type of equation to equation

of state, electrolyte solution and group contributions.

Equations of state

The realization that equations of state can be used to represent devia-

tions from ideality in both vapor and liquid phase is not new. (It can be

attributed to VAN OER WAALS at the end of the 19th century). These

methods became popular among engineers with the article of SOAVE (19721

using the REDLICH-KWONG (1949) equation of state for both vapor and liquid

phase. We understood that the equation of state can be used mainly to repre-

sent vapor pressure of pure components by letting the energy parameter a

vary with reduced temperature and adjusting it on saturation pressure of

pure components, actually exactly as wanted. Using the one liquid VAN DER

WAALS model, one represents fairly well the vapor-liquid equilibria of

multicomponent mixture of hydrocarbons and not too polar molecules by

adjusting one or two parameters of interaction between unlike molecules

aij or bij ( SOAVE (19721, PENG and ROBINSON (1976), MATHIAS

(1983), EVELEIN and MOOR (1979) 1. VIDAL (19781 and HURON and VIDAL (1979)

went a step further by introducing expressions of excess GIBBS energy into

the mixing rules. They opened the so-called "symmetric" method of calcula-

tion of VLE to local composition models by demonstrating that, under

limiting conditions, the excess GIBBS energy of the fluid at infinite

pressure is related to the'mixing rule for a , assuming that b is given by

a linear mixing rule for a class of cubic equations of state :

p=% - (-1 (10)

including the REDLICH-KWONG (19491, PENG-ROBINSON (19761, where $, X2 are

non adjustable constants. Recently, SOAVE (1984) introduced a NRTL excess

Gibbs energy at infinite pressure into the mixing rule of a VAN DER WAALS

equation with temperature dependent of parameters. He applied it success-

94

fully to equilibria with polar components.

The more general relation, without taking the limit of infinite

pressure:

gE = RT [ In Q (T, 1

T, PI 1 (111

is not so easy to use because the equation of state is pressure explicit

and the fugacity coefficient are analytic functions of volume, not pressure.

Their contribution has some of the limitations of earlier work by BEHAR

(1970) and CHAUDRON (1973) using an infinite pressure excess GIBBS energy in

the liquid phase. The cubic equations of state do not represent well the

liquid volume, except if a volume translation is applied as described by

RAUZY (1982). Yet it is not known if they can represent well enough the

excess volume, the role of which is important to obtain the contribution of

non ideality at operating pressure from the same at infinite pressure.

MOLLERUP (1981), WHITING and PRAUSNITZ (1982 a.1 introduced density

dependent mixing rules into the attractive part of cubic equation of state

in order to eliminate the shortcoming they found in the rules introduced by

VIDAL and HURON that the second virial coefficient of a mixture is not a

quadratic function of mole fraction when using local composition or any non-

quadratic mixing rule. The P,V,T properties of the gas phase are not too

well represented by cubic equation of state and even the fugacity coeffi-

cients at saturation may be in error when the pressure is high enough,

making the representation of VLE an empirical achievement. It seems that

the use of density dependent NRTL or UNIOUAC mixing rule was not too

successful (MOLLERUP, 1983) with VAN DER WAALS type equations and a nex

formulation is offered by MOLLERUP in the same article: the random-non random

mixture equation. The current orientation seems to be towards either

simpler density dependent local composition (MATHIAS and COPEMAN, 1983),or

equation based on results of computer simulation to be reviewed hereafter.

Group contributions

Another major empirical development of local composition use was the

group contribution models. The principle of these models is very simple, the

excess GIBBS energy is divided into two parts, one (Sl depends on size and

shape of components (configurational), the other (G) (residual) is a group

contribution. One can write for the activity coefficient i:

1nYi = lny: + lnyf (12)

the group contribution

95

term is expressed as :

- Itirk')) (13)

where VKi is the number of structural group K in component i, and 1nI'K

respectively the group contribution calculated in the solution

and in pure liquid ', (,)

calculated with the same equation :

1nCK = f (Xi' T)

where Xi is the fraction of groups, depending of the composition when cal-

culated for the solution.

The ASOG (Analytical Solution of Groups) equation, introduced by

WILSON and DEAL (1962) is based on WILSON equation for lnry . It was

developed by DERR and DEAL (1969), TOCHIGI and KOJIMA (1976). A monograph

was published by the same authors (KOJIMA and TOCHIGI (1979)).

The UNIFAC (UNIQUAC Functional Group Activity Coefficients) equation

published by FREDENSLUND, JONES and PRAIJSNITZ (1975) uses the UNIQUAC equa-

tion to calculate both the shape and group contributions to ln,! . It is

described in a monograph by FREDENSLUND, GMEHLING and RASMUSSEN (1977). The

group of the Institute of Technical Chemistry of the Technical University of

Denmark and of the Technical University of Dortmund publishes updates of the

method with list of parameters for application to VLE (SKJOLD - JORGENSEN

et al (19791, GMEHLING et al (1982), MACEDO et al (1983)), and LLF:

(MAGNUSSEN (1981)).

The sets of parameters are different for VLE and LLE. Empirical

improvements and extensions are constantly taking place (SKJOLD, JORGENSEN

et al, 1980 - 1982, KIKIC et al, 1980, JENSEN et al, 1982), (SANDER et al,

1983, BEN YAIR, 1983).

In both the UNIFAC and ASOG method the configurational term is indepen-

dent of temperature.

VERA applied a"group contribution method derived from ASOG for heat of

mixing, calling it AGSM (Analytical Group Solution Method) (LA1 et al,

1978). Parameters are different from ASOG, he combined both methods in a

Simplified Group Method Analysis (SIGMA) able to represent both types of

data (ASHRAF and VERA, 1980) without combinatorial term lnv; . But the

method was not fully developed and extensively applied (VERA and VIDAL,

19841.

One of the latest development is the use of a simple group contribution

rule based on the NRTL equation in an equation of state (SKJOLO - JORGENSEN,

1984). The second virial coefficient has the correct form because the

BYLTZMAN exponential in the definition of local fraction contain the factor

V . This equation is used to calculate gas solubilities of gases including H2S and CO2 in various solvents. The accuracy of the K values is 15 per cent except when large amounts of water are present.

Electrolytes

The NRTL model was extended to electrolytes by CRUZ and RENON (1978)

and CHEN et al (1982). BALL et al (1985) made a comparison of these models,

using two parameters for representing the osmotic coefficient of solutions

of simple strong electrolytes in water in the range of molality 0 - 6 M.

The results are similar to those obtained with the simpler PITZER equation. The prediction for mixed salts is not quite as good. But the advantage of

the NRTL models is the possibility of treatment of mixtures of electrolytes and non electrolytes (non aqueous solvents, weak electrolytes). Other

extensions are u&er way (SANDER et al 1985).

THEDRETICAL DEVELDPWEMTS

A theoretical framework 'using radial distribution functions comes

naturally to mind in order to explore the role of order in the theory of

fluids. This is even more true when simulation methods (Molecular Dynamics

and Monte-Carlo) or theoretical method offer the possibility to obtain radial distribution functions and thermodynamic properties of mixtures of

molecules interacting through a given intermolecular potential. There are

however difficulties to define local composition and to derive the residual thermodynamic properties from them. Then, the older cell-lattice theories

look a little obsolete in this context although they may still provide use-

ful methods of deriving empirical models if the assumptions usually taken

for granted in building models are modified.

Radial Distribution Function

The number of molecule i in a spherical shell of radius L centered at

the center of a molecule j is :

n.. =- '3

r2 gij(r) dr (15)

97

If Lij is suitably chosen after the first peak of the distribution

function, ni j can be interpreted as the number of molecules i around

central molecule j, the coordination number of j is :

Zj = ; nij 1

The number of interactions ij counted from central molecule i or j

should be equal

Nj nij = Ni nji

this condition is fullfilled if Lij = Lji because gij(r) = gji(r).

The local mole fraction of i in the volume r< Lij around molecule j

is:

n..

'ij = 1J

+n "ij jj

(16)

The configurational energy of a mixture at given volume and temperature

is:

UCoNF (N1,...NC, V, T)

where N = i N is1 i

4,r2uij(r) gij(N1,...NC, V, T, c) dr

(17)

Uij is the potential of interaction between molecules i and j.

The radial distribution functions gij are related to the potentials

ufj(r), they are functions of density, composition and temperature and

cannot be obtained in any simple way -in the general case.

Simulation methods for the interactions of molecules exist. They

provide distribution functions for given Uij(r), composition, density

and temperature, and also the thermodynamic properties. They are very useful

to physicists to find which approximation can be used to solve the equations

of the statistical mechanics relating the radial distribution functions to

the potentials uij. The problem is essentially solved for fluids of hard

sphere, and some rigid hard bodies. Therefore, we can consider that a fair

approximation of the thermodynamic properties of molecular systems at very

high temperature is available and the configurational Helmholtz energy

A~O"F (N l,...NC,V,T) from which all other can be derived is

available by integration of U CONF:

98

TT

AcoNF(NI, . ..NC. V,T)= ACoNF(NI,...NC, V,T) -T /

UCoNF(N,,...NC.V,T)

T=m T* dt (18)

where

ACoNF(N1,...NC, V, T) = A (N1,...NC, V, T) - AIG(N1,...NC, V, T) (19)

The equation of state is readily obtained from ACoNF by :

p (N, v, Tl =t$?,q..NC + y

The excess properties are obtained through :

GE (N1,...NC, p, T)= AE(Nl,.+ v, T) = A coNF(Nl,...Nc, V, T)

- ‘$ ACoNF (ipir, Vxi, Ni) - RT Xxi In xi 1

(20)

(21)

There is no basic difference between the equation of state approach and

the excess properties approach.

Using this formalism, it is clear that properties can be separated into

two terms: altemperature independent term (sometimes called "free volume")

which is 'v_b in VAN DER WAALS or REDLICH-KWONG equations of state and gE(comb) in UNIOUAC , and a temperature dependent term which depends on

attractive forces, the --$$ term in the REDLICH-KWONG-SOAVE equation of state.

Perhaps because the temperature independent term is known only at infi-

nite temperature, errors in this term are not too important empirically (ASHRAF and VERA, 1980), except in the extreme case of solutions of polymers

where molecules differ strongly in size, shape and conformation (the last

property can in this case be temperature dependent and the approach may fail).

RESULTS OF SIWIATIONS

NAKANISHI and his coworkers performed Molecular Dynamics (MD) and Monte

Carlo (MC) calculations for liquid mixtures of molecules interacting

according to the LENNARD-JONES potential. They investigated strong size and

energy differences between molecules.

The first MD results (TOUKUBO and NAKANISHI, 19761 clearly show that

the number of solvent molecules around a solute molecule vary with the

solute solvent attractive energy parameter E 12 (simulation of 108 sol-

99

vent molecule around 1 solute molecule). The first peak of the distribution

function is higher and positioned further away if e 12 is larger. The results are for solute and solvent molecules of same size, a number density

of 0.75 and kT/cll=l(typical liquid State). They extended this work to molecules differing in size (TOUKUBO and and NAKANISHI, 1977 a.) and in

size and energy (NAKANISHI et al, 1978). The results are given in the following table 1.

"12 ?2

Position of maximum Height of Coordination

of first peak first peak number at

(r/u) r = 1.41 o

?l 0.5 El1 1.07 2.27 9.2

o11 %l 1.09 2.52 9.6

0.5 ;; 2 ?l

1.13 3.36 10.4

?l 0.83 2.84 2

Q El1 1.57 2.26 0.5 “11 0.5 El1 0.81 2.31 2

Tl 2 ?l 1.65 2.72

Table 1: Results of MD calculations of NAKANISHI et al.

The radial distribution is shown in figure 2. The number of solvent mole- cules within the distance r of the solute is also plotted, calculated accor-

ding to equation (15). It depends strongly on distance r. This coordination

number calculated for r >1.136 increases with increasing energy ~~~~ It

does not change any more with ~12 if r>1.706.

100

Energy has therefore an influence on close-packing of molecules, but it is

not very strong in the liquid state. He also compared M.D simulations of

equimolar mixtures of molecules with c22 = 2 EII and EI~ equal to EII ,

-and?2 which he call respectively associated (unlike interaction is

weak), standard (Lorentz - Berthelot rules) and solvated (unlike interaction

is strong) in approximately the same conditions as before (TOUKUBO and

NAKANISHI, 1977 b., GIERYCZ et al, 1984). The difference in mixing rule has

a large influence on the nearest neighbor shell, especially on the local

mole fraction and not so much on the coordination number (total number of nearest neighbors). A major difficulty is to decide at which distance L the

upper limit of the integral for the calculation of nij by equation (15)

should be set. The local mole fraction varies with this upper limit of inte-

gration. The above mentioned authors set it in the region of the maximum of

the first peak g(r), assuming that the molecules in that region make most of

the contribution to the internal energy U. They also assume that -E.. in

the LENNARD-JONES potential are equivalent to the e.. or gi. in WILS N 'd or NRTL local composition definitions.The result (NAKAN\iHI and TdUKUBO, 19791,

is that WILSON hypothesis is poor. A factor oij = 0.4 as in the NRTL

model improves the situation only in special cases. The ratio of local mole fractions around molecule i g depends not only on the difference between

eji and E ii but also on ejj contrary to the WILSON-NRTL assumption.

NAKANISHI et al (19821 conclude that the WILSON or NRTL assumptions are ina-

dequate for liquids.

101

The excess Gibbs energies obtained have the features of the experimen-

tal values for strongly non ideal mixture : it goes from large negative

(solvationl to large positive (association) with phase splitting in that

last case. For spherical molecules of equal size the excess GIBBS energy is

always nearly symmetrical (NAKANISHI et al, 19821. This is in contradiction

with NRTL, WILSON and other local composition models which attribute

dissymmetry of excess functions to differences in energy parameters.

Increase in size ratio between molecules makes the excess energy more endo-

thermic and dissymmetrfc, assuming the geometric rule for E~~(GIERYCZ et al,

1984).

Unfortunately the authors did not give the complete radial distribution

data and it is difficult i!iassEss the accuracy of their conclusion (espe-

cially that the product V . 4 does not vary with bulk composition)

because there is an arbitrary and perhaps inconsistent (as we shall see

later) choice of L in the calculation of number of neighbors.

HOHEISEL and KOHLER (1984) extended the work of NAKANISHI and coworkers

by aiming at better accuracy and studying composition dependence (T = 120 K,

liquid density). Local composition are calculated with the same value of L

IL/d = 1.35 after the decay of the first peak of g). They find that the

ratios g , 2 and g / '3 for molecules of same size are (see figure 3) ,2-,i'

bulk composition dependent 'and very different from the WILSON equation but

close to the quasi-chemical modeirto be described later. From the results

for molecules of different sizes, the local composition is harder to define

because the same value of L is chosen beyond all the first peaks. The local

composition x11 from MD for an equimolar mixture is, in accordance with

perturbation theory, getting alternatively lower and higher than 0.5 as r/o

varies. HOHEISEL and KOHLER conclude that local composition are caused

mainly by packing effects and that WILSON type formulas cannot be used at

all.

.*

102

APPLICATION OF RADIAL DISTRIBUTIDN FUNCTION

LEE, CHUNG and STARLING (1983) remarked that the energies of interac-

tion to be used in WILSON definition of local mole fraction should not be

directly the ~~~ of the LENNARD-JONES potential or any average of uij(r)

but the average of the potential of mean force Wij defined from the

radial distribution function by :

/ Lij

/

L

4 TI r2gij(r) dr = ij

0 0 4 TI r2 exp (- DWij) dr = Vij exp (- BWij)(2B)

with

V ij

= 5 L3 3 ij

But g and therefore W are composition dependent (and temperature and

density dependent as well). This confirms the work of HOHEISEL and KOHLER

(1984) in its conclusion that the ratios :

xij / 2 = > exp [ - fl(iiij - Pi,) ] 'ii 'i ii

(23)

are composition dependent. There remains the problem of the choice of Lij

which should be equal to Lji in order to keep consistency between the

nji as shown above. Another problem is the calculation of thermodynamic

properties : the exact value of the internal energy is generally not

obtained by integration of the product uij gij over the range of r ’ limited the region of the first peak of g, because the term arising from the

second or third peak may be important.

Nevertheless, LEE et al (1983) found that by adjusting parameters in

WILSON equation on results of AE from M.D. simulation by NAKANISHI, the

values of local composition in one case are qualitatively in agreement with

the values obtained from the radial distribution function resulting from the

simulation.

SANDLER (1985) suggests the overcome the problem of integration of the

energy equation by using a square-well potential and integrating over the

range of the potential well. Equation (17) becomes :

UCoNF (N1, . ..NC. V, T) = -+ z ij Nj clj "ij

103

(25)

L ij is selected as the upper limit of the range of the square well

potential. SANDLER explores the low density limit in this case.

gij (Nl,...NC,V ~ m, T, r) = exp(- uij(r)/kT) = exp(eij/kT) (26)

and

N. n.. = - 1J

Vi +o:j (R3 _ 1) eeij'kT (27)

The local mole fractions are in this case given nearly exactly by the

expression proposed by WILSON :

N ij =

xi v.. - N Jj

Yj * exp .-% )

(28)

The internal energy is :

“CONF = _ E. .N.N 1Jlj V

ij 2v ij exp (cij/kT) (29)

which would lead for the attractive term in VAN DER WAALS or REDLICH equa-

tion to :

amixture = C C Xi Xj aij(T) 1 J

(30)

aij(T) = RTNAV Vij (e oij/kT

- 1) /2

This shows that non randomness

(31)

is not negligible at low density and

this is not in contradiction with the quadratic form of the second virial

coefficient. The exact result is valid only for the square well potential

which is perhaps not the best suited potential to calculate low density

properties. In the case of other potentials approximations are needed like

truncation of the integration of Uij in equation (17) and taking average

of the potential over the range of integration.

Another important observation is that the coordination

selected range is composition dependent. For a binary mixture

Z2 q n12 + n22 = r N2

v12 exp(s12/kT) +-Q- V22 eXp(s22/kT) N1

number in the

(32)

104

It is the assumption of causes the violation of cient with non density

of state models.

constant z in the usual local composition model that the quadratic character of the second virial coeffi- dependent local composition mixing rule in equation

'1'

SANDLER (1985) transposes the local composition concept into a more

precise framework, but many problems remain to be solved : validity of

calculation of energy from first neighbor interaction through the truncated

energy equation, choice of the upper limit of integration of g(r) to obtain

number of neighbors. It is clear that more should be learned on the beha-

vior of radial distribution functions by simulation, or application of

theories.

The exclusive dependence of local composition on energy parameters is probably limited to low density. When the structure of the fluid depends

mainly of the shape and volume of interacting particles as it is the case in

the dense fluids and liquids, the local compositions defined by WILSON are

probably the least applicable. Even for molecule of similar size, the effect of energy differences is small in the liquid state, much smaller

than predicted by WILSON if the energy parameter is the potential energy, and in better agreement with the first approximation of GUGGENHEIM (quasi-

chemical theory) which involves exchange energies

Many other authors refer to NAKANISHI simulation results and improve

empirical formulation of local composition concept and its utili>zation in

equations of state and excess Gibbs energy equations. Among the,!latest

published articles, HU, LUDECKE and PRAUSNITZ (1984) present a new empirical

model taking into account the observations reported before : the local composition term in the case of high densities takes into account primarily

the molecular geometry (packing effect) and weakly the energy differences.

The coordination number of molecule i varies with composition,, and local mole fraction are optimized through GUGGENHEIM's method which wai‘ found to

give weaker effect of local order more in agreement with simulation results

at high density. The local mole fraction tends to the value given'by WILSON equation at low density.

A completely new formulation of local composition models will probably

result from the observations made through simulation and theory. The role of the structure of the fluid resulting from size and shape of molecules

will have to be taken into account, without neglecting the effects of density dependence. The more rigourous approach of ,model development using equations (17) and (18) can be used, once the hypothesis on u.. and

1J gij are chosen.

105

CELL OR LATTICE THEORIES

A very large amount of work was devoted to development of cell or

lattice models of solutions using the concept of local composition. First

there was criticism of the equations proposed by WILSON and attempts to

rationalize the derivation of empirical models. The lattice theory was used

to obtain the combinatorial part of gE. Finally, authors used the approach

of GUGGENHEIM to apply or develop "quasi-chemical" models.

All this work will be very shortly and incompletely reviewed here, just

citing a few typical publications.

The most often cited criticism of local composition models (WILSON,

NRTL, UNIDUAC) was written by FLEMR (1976) and MC DERMOTT and ASHTON (1977).

It is based on the balance of interactions 12 counted from cells centered

around molecules 1 and 2 respectively (assuming equal coordination number).

X1 91 = x2 52 (33)

This equation makes inconsistent the WILSON or NRTL definition of local

mole fraction :

x12 _ x1 --- x22 x2

w12 (341

taking into account the definitions :

x12 + x22 = Xl1 + x21 = 1

if W12 is constant it should

models cannot be justified as

(35)

be 1 (random solution) and therefore the

one fluid, lattice or cell models. Or W12

should be composition dependent, (what is confirmed by simulation results),

models can be developed in this way for binary systems as proposed by

PANAYIOTOU and VERA (1980). The way to get out of this difficulty is to assume different numbers of nearest neighbors around molecules 1 and 2, and

these numbers depending on the bulk composition as suggested by radial

distribution function theory.

Derivations of the UNIQUAC model using a lattice model was given by

MAURER and PRAUSNITZ (19781 and discussed by KEMENY and RASMUSSEN (19811,

VERA (1982) and FISCHER (19831,based on the two fluid model where the above

condition (equation 331 is not fulfilled. In connection with the UNIGUAC

model, there were efforts to develop lattice models to obtain the combina-

106

torial entropy of liquid mixtures as reviewed by SAYEGH and VERA (1980).

But in view of the theory developed above and the results of comparison

between simulation results with various model, the quasi-chemical method of

GUGGENHEIM (1952) seems theoretically the most promising. It is based on

the writing of the partition function for a model based on specified

assumptions and constraints. The local compositions are obtained by minimi-

zation. The disadvantage of that method is that it requires solutions of

many implicit equations in the case of multicomponent systems. This advan-

tage isthatit uses only one binary parameter on exchange energy for a binary

mixture.

Modifications of the quasi-chemical models were proposed by RENON

(1972) NAGATA and NAGASHIMA (19761, PANAYIOTOU and VERA (1980, 1981), KNOX,

VAN NESS and HOLLINGER (1984). The approach of GUGGENHEIM can be applied in

a variety of manners depending on the assumption on number of neighbors,

constraints taken into account. It would be interesting to develop such a

model for dense fluids with choices of assumptions making the results of

simulation likely to be obtained : coordination numbers varying with compo-

sition, good description of the effect of structure inspired by considera-

tions on geometrical number of nearest neighbors as discussed by DEITERS

(1982, 1983), SANDLER (1983), EDULJEE (1983). The advantage would be to

take into account the fact that the local composition are very much depen-

ding on the relative size of molecules in the case of dense fluid. The

correction term depending on energies of interaction is provided by the

quasi-chemical theory is small. DEITERS (1982) develops such a term in his

semi-empirical equation of state for fluids.

The method of BARKER (1952) is a group contribution form of the quasi

chemical theory of GUGGENHEIM. It was presented by KEHIAIAN, GROLIER AND

BENSON (1978) and applied to the representation of excess Gibbs energy,

enthalpy and heat capacities of many organic mixtures. KEHIAIAN and co-

workers published many articles of application of that method (T.O.M.

project). The method relies on considerations of the structure of molecules.

The adjustable parameters are group interchange energy parameter. Often it

is not necessary to take into account the non randomness effect which is

perhaps small enough to be developed analytically and explicitly as sugges-

ted by MICKELEIT and LACMAN (1983). The method has not been sufficiently

validated for the engineering applications, especially for the prediction

of the properties of multicomponent systems or liquid-liquid equilibria.

Another group contribution molecular model for liquids mixtures was

developed by GREENKORN, CHAO and coworkers (NITTA et al (19771, CHIEN et al

(19811, KOUKIOS et al (19841).

co

and

107

NCLUSION

The usefulness of the local composition models (WILSON, NRTL, UNIQUAC)

derived group contribution models (ASOG, IJNIFAC) in engineering applica-

tions has been proved. They can be considered as empirical. The unified

approach of using one equation of state for gas and liquid is gaining ground.

We should look for simple models in spite of the apparent complexity of

interactions. There is a need for critical evaluation of any new equation

in comparison with existing ones, based on treatment of data on many systems

including polar, aqueous, non polar components and also reacting systems and

electrolytes, if possible, with all types of equilibria, vapor-liquid,

liquid-liquid, solid-liquid, heat capacities, volumes in a wide range of

conditions. Test on prediction of multicomponent systems properties should

be performed.

There is a need of data.

- Data obtained by simulation in order to establish the theoretical basis of

the models. It is useful to report the whole radial distribution function

and all properties in a wide range of temperature, pressure and composition,

using simple but meaningful potential, with various combinations of parame-

ters and shapes of hard cores.

- Data in the gas phase are needed (experimental data, especially densities

or thermal properties of mixtures in a wide range of pressure) to evaluate

the fugacities at saturation. Phase equilibria which yield properties needed

to model the liquid phase, heat of mixing, excess heat capacity and compres-

sibility of the liquid, are also needed.

Theoretical development for the representation of the thermodynamic

properties of mixtures will probably use more and more the unified approach

deriving all properties from ACoNF (N,V,T) and decomposition into a

temperature independent "repulsive" part and a temperature dependent-part

resulting from attracting forces. In this approach, it will be necessary to

refine the representation of the properties of pure substances including

improvement for the critical region, the repulsive part, the intermediate

density region. But most of the effort is to be devoted to the understanding

of the mixtures, using the results of simulation to further understand the

role of local order and to check the validity of the approximations needed

in the rigorous statistical mechanics equations relating thermodynamic

properties to intermolecular potentials. The role of energy difference in

the local order is especially important at low density and the role of

shape, size and structure in group from interacting molecules especially

important at high density.

108

LIST OF SYHBOLS

C

Cscf

!tiE

gij

gij

)j

K

L ij

"ij

Ni N av

P

qi r

R

T

TR u.. 1J

u

V

vi V Vij

W ij

W ij

xi

'ji

xi z

a.. 'J

yi

Energy parameter in cubic equations of state

Helmholtz energy

Covolume parameter in cubic equations of state

Number of component in mixture

Parameters in WILSON and HIRANUMA equations

Moiar excess Gibbs energy

Radial distribution of molecule i around j

Parameter in NRTL equation

Parameter in NRTL equatian

Molar excess enthalpy

Distrjhution factor in vapor liquid equilibria, ratio of mole

fraction of Component i in vapor and liquid

Truncation distance in integral of equation (15)

Number of molecules i among first neighbors molecules of j

Total number of nearest neighbors molecules around j

Avogadro number

Pressure

Area parameter in UNIQUAC model

Inte,rmolecular distance

Gas constant

Absolute temperature

Reference temperature

Potential of interaction between molecules i and j

Internal energy

Molar volume of mixture

Molar volume of pure component i

Volume,characteristic volume for interaction ij

Potential of mean force

Average of potential of mean force

Mole fraction of component i

LOCal mole fraction of component j around i

Fraction of group i

Coordination number

Coordination number of species j

Parameter in NRTL equation

Activity coefficient of species i

Contribution of group K to activity coefficient

Contribution of group K to activity coefficient from pure substance

group-composition

109

?j

'i

hl' A2 A ij

"ki

'ij

'ij

Qj

Intermolecular energy parameter

Area fraction

Constants in cubic equation of state (10)

Parameter in WILSON and HIRANUMA equations

Number of structural group K in component i

Intermolecular distance parameter

Parameter in NRTL equation

Local volume fraction

SUPERSCRIPT

CdNF

IG

G

S

Configurational property

Ideal gas property

Residual, group contribution term in group contribution model

Size and shape term in group contribution model

SUBSCRIPT

i Component i

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