PrePrint - University of Illinois at Chicagotrl.lab.uic.edu/1.OnlineMaterials/Mixtures/00...In the above equations, the partial molar volume, V; and 8µ;/8x1 are related to the direct

Chapter 10, Pages 359-380,

Equations of State for Fluids and Fluid Mixtures Professor Jan V. Sengers, et al. (Editors)

International Union of Pure and Applied Chemistry 2000

10 MIXTURES OF DISSIMILAR MOLECULES

(The Fluctuation Theory Prediction Approach)

Enrico Matteoli, Esam Z. Hamad, G.Ali Mansoori University of Illinois at Chicago

(M/C 063) Chicago, IL 60607-7052, USA

Page# List of content 10.1 Introduction 10.2 Background 10.3 Grand Canonical Ensemble 10.4 Analytic Theory of Dissimilar Mixtures

10.4.1 Analytic Solutions for Low to Moderate Pressures 10.5 Binary Mixtures 10.6 Test of Cu Closure

10.6.1 Hard-Sphere Mixtures 10.6.2 Lennard-Jones Mixtures 10.6.3 Real Mixtures

10.7 Vapor-Liquid Equilibria 10.8 Liquid-Liquid Equilibria 10.9 Summary References

Email addresses of authors:E. Matteoli ([email protected]),E.Z. Hamad ([email protected]; [email protected] ), GA Mansoori ([email protected]; [email protected]).

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DOI: http://dx.doi.org/10.1016/S1874-5644(00)80021-1 ISBN: 9780444503848

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10.1 INTRODUCTION

The behavior of mixtures of dissimilar molecules is modeled using a statistical- mechanical solution theory based on the grand canonical ensemble. The strength of this solution theory is its applicability to dissimilar mixtures consisting of species for which, generally, intermolecular potential-energy functions are not accurately known and conventional statistical-mechanical theories of mixtures have failed to predict their behavior. Other models of mixtures of dissimilar molecules are dealt with in other chapters of this book.

The solution theory reported in this chapter is based upon the exact relations between the direct correlation-function integrals of mixtures. These relations indicate that, for example, for a binary mixture, only one of the three direct correlation-function integrals is independent. These relations are combined with an appropriate closure expression for the cross-direct correlation-function integrals to derive analytic expressions for partial and total molar properties, activity coefficients, and equation of state of dissimilar mixtures. The results reported here indicate the possibility of introduction of simple closure expressions relating unlike and like-interaction direct correlation integrals. This approach is shown to be applicable for vapor-liquid and liquid-liquid equilibria calculations of dissimilar mixtures.

One major advantage of this solution theory is the fact that it makes it possible to produce analytic expressions for total and partial molar properties of dissimilar mixtures at finite concentrations. Also, because of independence of this solution theory from the nature of intermolecular-interaction potentials in solutions, it is applicable to mixtures consisting of complex molecules with or without intermolecular-polar and association forces.

10.2 BACKGROUND

Accuracy in prediction of properties of mixtures is one of the major concems in scientific research and engineering calculations (1-3). The traditional molecular theories of mixtures usually start with the consideration of intermolecular potential-energy functions and their applications in a certain statistical-mechanical ensemble (4-8). In the case when the intermolecular potential-energy functions of species of the mixture are not well defined, and this is almost always the case for real fluids, we have to resort to other means of prediction.

One of the interesting problems in statistical mechanics is the prediction of mixture properties from those of the pure components comprising the mixture. There exists a number of techniques to do this. The conformal-solution theory (9-12) has offered a practical and rather accurate theoretical approach to this problem when the intermolecular potential-energy functions of species of a mixture conform to a certain mathematical form. For mixtures of dissimilar molecules for which the intermolecular potential-energy functions are not conformal, the conformal-solution theory may be quite successful. Another statistical- mechanical route to predict behavior of mixtures is the statistical-mechanical solution theory presented here.

The origin of the present statistical-mechanical theory of mixtures goes back to the equation of state derived from the grand canonical ensemble (13-18). The theory introduced here provides us with a strong tool for developing analytic expressions for total and partial thermodynamic properties of multi-component mixtures at finite concentrations as well as solutions at infinite dilution (18-21). It has also been applied to phase equilibria (vapor-liquid and liquid-liquid equilibria) of mixtures with success. In this chapter the details of this theory will be presented and it will be applied to property prediction of a number of dissimilar mixtures of practical interest.

E Matteoli, EZ Hamad, GA MansooriMixtures of Dissimilar Molecules

In "Equations of State for Fluids and Fluids Mixtures" J.V. Sengers, et al. (Ed’s), Chapter 10, pp.359-380, IUPAC (2000).

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10.3 GRAND CANONICAL ENSEMBLE

In the grand canonical ensemble, where the temperature T, the volume V, and the chemical potentialµ, of each component in the system are fixed, the partition function, 3 is (22-24):

S(T,V,µ) = I; ... I;Q(T,V,N)eµ·NlkT (10.l) N1 Ne

where c indicates the number of components, Q is the canonical partition function, µ=µ1 ,µ2, ••• ,µc, N; is the number of molecules of component i, N Ni, N2, . . . , Ne,

C

and the scalar product µ·N= L µkNk . k=l

The radial-pair distribution function, giJ, in the grand canonical ensemble is defined as

gy(r)={[V2 /NiNj ) L L ... f 1tNm!)eµ-N/kTJ ••• Je-<ll/kTdr3 ••• drN}/8T,V,µ) (10.2)N/::.j m=l

where <Dis the interaction potential, r is the vector of coordinates, k is Boltzmann's constant and 7t denotes the product over all components.

A relation between the isothermal compressibility, Kr -(IIV)(oV/oP)r, and g(r) can be derived from the grand-canonical ensemble (22, 24)

C C

pkTKr =IBl/pI; I: x1xj IBlij i=1j=1 (10.3)

where p is the number density and IBliJ denotes the cofactor of the element By in the c x c

matrix Band IBI is the determinant ofB. The elements, By, of the matrix are:

(10.4)

where Sy is the Kronecker delta, and Gy is called the radial-distribution-function integral, or fluctuation integral, or the Kirkwood-Buff integral. It is defined by the following expression, where r indicates the distance between molecules i and j:

(10.5)

In addition to the mixture isothermal compressibility, the Kirkwood-Buff solution theory gives the partial molar volumes and the derivatives of the chemical potential with respect to mole fractions in terms of the fluctuation integrals (24-26):

kT[o < N- > loµ ·]vr. = < N- ><N. > G .JV O·· < N- > I J , .µii) I J If I] l (10.6)

361



where <N;> is the average number of particles of type i in the grand canonical ensemble. Provided information is available about the Gij integrals, Equation (10.6) can be used to

calculate chemical potentials of components of a mixture. Also the information about Gij can provide us with the means of calculating other properties of a mixture such as the isothermal compressibility Kr, as given by Equation (10.3), and the partial molar volumes Vz· as given by the following expression:

C C C

PVi = I X j I B lij I I I X jXk I B ljk j=l j=lk=l

where B, IBlif and IBl;k have been defined previously.

(10.7)

The above equations have been studied extensively for the case of infinitely dilute solutions (27 -31). The major difficulty in utilizing the fluctuation theory in mixture-property calculations at finite concentrations is the lack of knowledge about the fluctuation integrals, Gy's. For solutions with finite compositions, expansions of the Gy's in powers of concentration are available (25,26). However, the coefficients in these expansions are given in terms of third, fourth and higher order correlation functions. Since little is known about correlation functions of order higher than two, the expansions are currently oflimited use.

Theoretical calculation of Gij integrals requires knowledge about the radial-distribution functions, gij. The radial-distribution functions can generally be calculated using the theory of intermolecular potential-energy functions in the context of a partition function. However, for complex mixtures, such as mixtures consisting of asymmetric, highly polar, and associating molecules the intermolecular potential-energy functions are not very well known. Also the existing techniques of calculating radial-distribution functions from the knowledge of intermolecular potential-energy functions require extensive numerical calculations ( 4-6 ,23).

Pearson and Rushbrooke (32) reformulated the fluctuation theory expressions, Equations (10.3), (10.6) and (10.7), in terms of direct correlation-function integrals, Cif, as defined by

Cij

= f cij(r)dr . (10.8)

In this equation cy(r) is the direct correlation function, which rs related to the radialdistribution function by the Omstein-Zemike relation

C

cij(r12)=hij

('l.2)- I Pkfcik(r13)hjk(r23)dr3; hij(r) = gij(r)-1; Pk =pxk k=l

(10.9)

where hij is the total correlation function and Xk is the mole fraction of component k. In terms of the direct correlation-function integrals, the isothermal compressibility, the partial molar volume, and the chemical potential become (32 ,33)

(10.10)



362

C C C

pV; = (1-p I x1Cu)/(l-p I I xjxk cjk)j=l j=lk=l

(10.11)

(10.12)

According to Equations (10.10)-(10.12), the fluctuation theory expressions are reformulated with respect to another molecular function, ciJ rather than giJ. It should be pointed out that the following relations hold between the fluctuation integrals, G/s, and the direct correlationfunction integrals, Cy's, (32,33)

G=C+GXC (10.13)

where G is a c x c matrix with elements pCiJ and X is a diagonal matrix with elements XiJ

=<N;IN>. In solving the above equations for isothermal compressibility, partial molar volumes and

chemical potentials in a mixture, knowledge about the CiJ's and their interrelationships is necessary. In the theory described below, a method is introduced by which one can derive analytic expressions for the Cy's.

10.4 ANALYTIC THEORY OF DISSIMILAR MIXTURES

To derive the relations among the CiJ integrals, we use the mathematical fact that the mixed second partial derivatives of a function with respect to two variables are equal at all points where the derivatives are continuous. Let us consider the mole fractions x;, total pressure P, and the absolute temperature T as the independent variables in a mixture in one phase. Then, for the chemical potential, µ; = µ;(T,P,x1,x2, •• , Xc-i), of component i in a mixture one has (19):

8[ 8µ/8x j ]/8P = 8[ 8µ/8P]8x j = [ 8V/8x j ]p, T i,j = 1, .. , c (10.14)

(10.15)

In the above equations, the partial molar volume, V; and 8µ;/8x1 are related to the direct correlation-function integrals by Equations (10.11) and (10.12). By inserting Equations (10.11) and (10.12) into Equations (10.14) and (10.15), we shall derive a number of expressions between the Cy's of the mixture. In a c-component mixture there exist c( c+ 1 )/2 direct correlation-function integrals which are considered as unknowns in Equations (10.14) and (10.15). One should note that there are c(c-1) relations of the type of Equation (10.14) and c(cl )(c-2)/2 relations of the type of Equation (10.15). Of all these equations only [c(c+l)/2)-1 of them are independent. Since there are c( c+ 1 )/2 unknowns, one additional relation will be needed to determine all the unknowns regardless of the number of components involved. The



363

resulting equations for the direct correlation-function integrals consist of a set of non-linear partial differential equations . The general analytic solution of the above set of equations is not presently available. In what follows we report the analytic solution of these equations for mixtures at low to moderate pressures .

10.4.1 Analytic Solutions for Low to Moderate Pressures

For liquids at low to moderate pressures, it may be assumed that the activity coefficient is not dependent on pressure . In mathematical form, indicating with µ;0 the standard chemical potential and using the definition of activity coefficient,/;, of component i in a mixture,

this means

[oln•-!oP]. r = [o(µ--µ· )/oP]. rlkT=(V-v·)/kT=OYI x,, l lO X1, l l

If�= vi, Equation (10.11) can be written in the following form:

C C C

p L xjcij = 1-vip(l-p L L XjXk Cjdj=l j=lk=l

Given that

Equation (10.17) then becomes

i=l, ... ,c

i =1, ... ,c

where the mixture isothermal compressibility is now given by

(10.16)

(10.17)

(10.18)

(10.19)

(10.20)

The set of Equations (10.15) does not simplify upon making the assumption of low pressure. To overcome this difficulty and to provide the one additional relation needed, the following closure relation is assumed to be valid for all the cross-direct-correlation-function integrals ( CiJ,#j)(l 9,20):

iJ = 1, ... ,c and i * j (10.21 )



364

where aJi and au are binary interaction parameters ( aJi*au which, at low to moderate pressures, are functions of temperature only. In earlier publications (18, 19) we demonstrated that this form of the Cu closure expression is quite satisfactory in representing the cross-directcorrelation-function integrals of various non-polar, polar, and hydrogen-bonding binary mixtures. Substituting Equation (10.21) in Equation (10.19) gives the following set of linear equations:

C

( I: xp11)PC11 +xza12PC22 + ••• +xcalcPCcc =l-v1f(kTK r)j=I

C

x1a21PC1 1 +( I: xpj2)PC22 + ... +xca2cPCcc =1-vzl(kTK r) j=I

C

x1aC1PC1 I+ xzac2PC22 + ... + ( L x p Jc )pCcc = 1- vc l(kTK r) j=I

(10.2 2)

where a;,=l . The above relations represent a set of c linear questions for the unknowns pC;;. To solve for pC;;, it is more convenient to use matrix notation:

A.C=b (10.23)

where A is a matrix with elements A;; = I:xp1;, Au = xpu and b and C are vectors with elements b,=1-v;lkTKTand C,=pC;;, respectively. The solution of Equation (10.23) is:

C

pCkk = L [Alik b/[A I i=I

where [A[;k is the cofactor of the element A;k in the determinant [A[.

(10.24)

By using the direct correlation-function integrals calculated above one can derive the expression for the activity coefficient of a species in a multi-component mixture in the following form

(10.25)

As an application, this theory, which we repeat is valid only at low to moderate pressures when Equation (10.21) is valid, will be used for derivation of activity coefficients and vaporliquid and liquid-liquid equilibria calculations of binary mixtures.



365

10.5 BINARY MIXTURES

For a two-component mixture, Equations (10.14) and (10.15) reduce to

(10.26)

(10.27)

Then; by substituting Equations (10.11) and (10.12) in Equations (10.26) and (10.27), we get the following two relations between Cu, C22, and C 12 :

kTo{[l-x1PC11 -x2pC22 + x1x2 p2 (C11C22 -c1/)J/[1- IIxixjpC!i ]}loP

=x18{[l -X1PC11 -xzpCu]/[p(l-LLXjX jPCij )]}lox1

and

(10.28)

(10.29)

These two equations constitute two independent expressions relating Cu, C22 and C 12 in a binary mixture. In principle, Equations (10.28) and (10.29), together with Equation (10.21) relating C 1 2 to Cu and C22, can be used to solve for the three quantities C 11, C2 2 and C 12 with the following results:

(10.30)

(10.31)

(10.32)

where

(10.33)

and where K7i and Kr2 are the isothermal compressibilities of the pure components 1 and 2,respectively. With the availability of pure-fluid thermodynamic data and binary interaction parameters a21 and a12, Equations (10.30) - (10.33) can be used for calculation of mixture direct-correlation-function integrals.



366

Given the direct-correlation-function integrals, activity coefficients can be obtained by substituting the C;/s in Equation (I 0.25), which for binary mixtures can also be written as

lnfi = -ft2 (pC12 -W -l )d.xzlx1 (10.34)

Substituting for pC12 from Equation (10.32) and performing the integration, an analytical expression for activity coefficient results

lnfi = -Lftln[x1 + LzlL1 )x2] + (1/2)(L11:- l)ln[a/a12x? + ai1 -2)xz + I] + (1/2)[(2a12Lz - L1)1: + 1-2,!2]/qln[( l + xzlr-)/(1 + xzlr +)]

with

(10.35)

(10.36)

The expression for ln.fi can be obtained by interchanging subscripts I and 2 in Equation (10.35) and in the expression for r±. The quantities a, q and , remain unchanged under this operation.

To test the accuracy of the closure given by Equation (10.21), we apply it to various binary liquid-mixture calculations. Then the activity-coefficient expression, Equation (10.35), is used for vapor-liquid and liquid-liquid equilibria calculations of a number of binary dissimilar mixtures.

10.6 TEST OF Cu CLOSURE

In order to demonstrate the accuracy of the CiJ closure, given by Equation (10.21), we apply it first to two model-fluid mixtures (hard-sphere and Lennard-Jones mixtures). Then it will be applied to a variety of real-fluid binary mixtures consisting of dissimilar and complex molecules.

10.6.1 Hard-Sphere Mixtures

The validity of Equation (10.21) can be tested by comparing the fluctuation integrals or direct correlation-function integrals calculated by the present theory with their exact values. For this purpose we first consider a model-fluid mixture (hard-sphere mixture) for which accurate direct correlation-function integrals are already available. The Percus-Yevick theory (34) gives quite accurate expressions for the direct correlation-function integrals of a hardsphere mixture. In Table 10.1 the C12/C11 values calculated by the Percus-Yevick theory arecompared with the result of geometric mean closure, C12

=(C1 1C22t, and the arithmetic mean (a12

=a21=1/2) closure, Equation (10.21), for a binary equimolar mixture of hard spheres with

cr11/cr22=1.5 .. Table 10.1 shows clearly that both geometric- and arithmetic-mean closures are accurate enough for a hard-sphere mixture. However, the geometric-mean closure is more



367

accurate at high densities, while the weighted arithmetic-mean closure is more accurate at low densities.

Table 10.1 Values of C 1z/Cn for an equimolar hard-sphere mixture (crn/cr22=1.5) calculated according to different closure equations and compared with the Percus-Yevick theory (19).

0.05 0.5265 0.5047 4.1 0.5243 0.4 0.10 0.4806 0.4677 2.7 0.4772 0.7 0.20 0.4066 0.4034 0.8 0.4039 0.7 0.30 0.3530 0.3526 0.1 0.3537 -0.20.40 0.3141 0.3142 0.02 0.3202 -1.90.50 0.2844 0.2844 0.0 0.2969 -4.4

10.6.2 Lennard-Jones Mixtures

The hard-sphere intermolecular potential-energy function is purely repulsive. to test the proposed closure models for mixtures of molecules with intermolecular potentials which possess both attractive and repulsive parts, the Lennard-Jones-potential model is considered here. The chemical potential of a mixture of Lennard-Jones fluids at infinite dilution is predicted here and it is compared with available computer-simulation results. The chemicalpotential predictions are compared with the infinite-dilution simulation data (35) in Table 10.2 According to this table for molecules with the same intermolecular energy parameters 812/822=1, the weighted arithmetic-mean closure shows better agreement with the simulation data at cr121<:s22> 1. Geometric-mean closure shows better agreement with the simulation data at cr12/cr22<l.

Table 10.2 The chemical potential at infinite dilution of Lennard-Jones mixtures (19) for kT/822=1.2, cr2/=0.7.

0.3 0.5 0.75 1.0 1.5 2.0

10.6.3 Real Mixtures

1.0 1.0 1.0 1.0 1.0 1.0

Simulated Data 35

-1.30

-1.63-1.83-1.93-2.00-2.48

WJJ1.,/kT C12=(C11C22t

-1.75

-1.80-1.86-1.99-2.26-2.58

+4.14-1.29-1.92-1.99-1.65-1.52

Data on fluctuation integrals and direct correlation-function-integrals are available for a variety of binary mixtures (20,37). Such data can be used for the test of closure expressions. The experimental Cij integrals for real mixtures are calculated from experimental Gi

j integrals (5,22) by inverting Equation (10.13). In Table 10.3 the results of this fit are reported for 22 different binary mixtures; for eight of these mixtures, the calculated C12 values are compared with the experimental data in Figures 10.l - 10.4.

The validity of the C12 closure, Equation (10.21), can be tested for real mixtures by plotting C12/C11 data versus Cii/C11 data and considering the linearity of the resulting curve.



368

. . . . . . . . . . . . . . . . . . . .-20

-80 e

e

-140 e

V+l'Rlll'ANDL ..

-200

0 .I .2 .3 .4 .S .6 .7 .8 .9

XI

10 . . . . . . . . . . . . . . . . . . .

-40

pCti-90

e e e e e e e e e e e e z

e e ..

..

..

-I 40 I)

.,

V+ACETDNE .,

-190 .,

0 .I .2 .3 .4 .5 .6 .7 . 8 .9

XI

Figure 10.1 The variation of pCij with composition for water(W) (1) + propanol (2) and for water (1)

+ acetone (2) at 25°C. The points represent experimental data (18,20) for pC12 (triangle), pCil (square),

and pC22 (circle). The solid curve represent the pC12 prediction by the present theory using weightedarithmetic-mean C12 closure.

Figures 10.5 and 10.6 show the plot of C12/Cll versus C22/Cll for four different binary mixtures. The linearity of these plots suggests that the weighted arithmetic-mean closure, Equation (10.21), is quite sufficient for real mixtures. As a further test of the validity of the weighted arithmetic-mean closure for other real mixtures the deviations of C12 data from the

C12 value calculated using Equation (10.21) are reported in Table 10.3. The maximum deviation for all mixtures in Table 10.3 is about 22%. This large error happens where the Cu goes to zero and crosses the horizontal axes. Since some of the experimental data have uncertainties up to 30-40% (37), one concludes that the weighted arithmetic-mean closure is a good closure approximation for real mixtures.



369

pC11 -100

-zoo

. . . .. . . .

.

. V+DMSO .

. . . . . .. . .

.,

D .1 .2 .3 .4 .S .6 .7 .8 .9

Xl

l D -�-...-�-...--,--.,.--,--.,.--,----,. . . . . . . . . . . . . . . . . . . . . . . .

-3D

-7D

�IJO

-!SD W+llttltil:JETHRNOL

O .! .2 .3 .4 .s .6 ·' •• • 9

Xl

Figure 10.2 The variation of pCif with composition for water (W) (1) + dimethylsulfoxide (DMSO) (2)

and for water (1) + aminoethanol (2) at 25°C. The points represent experimental data (18,20) for pC12

(triangle), pC11 (square), and pC22 (circle). The solid curve represents the pC12 prediction by the present theory using weighted-arithmetic-mean C12 closure.

pClj

-26

-3D

-34

-31

-42

TCM+TJ1f

0 .! .2 .3 .4 .s .• .7 .8 .9

Xl

e e e e -36

"99 e e 11

-4D

-44

-48

-sz

0 • I

. . . . . . . . . . . .

.? . .l ,4 .5 .6 .7 .6 ,S . l

Xl Figure 10.3 The vanat10n of pCif with composition for tetrachloromethane (TCM) (1) +

tetrahydrofuran (THF) (2) and for TCM (1) + dioxane (2) at 25°C. The points represent experimental

data (18,20) for pC12 (triangle), pC1 1 (square), and pC22 (circle). The solid curve represents the pC12

prediction by the present theory using weighted-arithmetic-mean C12 closure.



370

-22

-30

-34

-3•

-40

-so

-60

CT-HEX•ETHTLETHER

o .1 .z ,3 .4 .s .s .7 .e .s

Xl

. . . . . . . . . . . . . . . . . .

e e e • . . . . . . .

. . . . .

CT-HEX•f'ROf"Y'LETHER e

11 11

0 .1 .2 .3 .4 .s .6 .7 .6 .9

Xl

Figure 10.4 The variation of pCu with composition for cyclohexane (CY-HEX) (1) + ethylether (2) and for CY-HEX (1) + propylether (2) at 25°C. The points represent experimental data (18,20) for pC12

(triangle), pC11 (square), and pC22 (circle). The solid curve represents the pC12 prediction by the present theory using weighted-arithmetic-mean C12 closure.

-:r-----------------------�

0

. ,...

-

--------------�• • •

0 ,.._ __ ...._ __ --'---..l----l..---L----....1---....J-0.4 .-0.2 0.0 0.2 . 0.4

C22/C 11

0.6 o.e 1.0

Figure 10.5 The test of the linearity of C12/C11 versus C22/C11 for water (I)+ methanol (2) (circles) and water (I)+ amionoethanol (2) (squares) binary mixtures. The curves represent the weighted arithmeticmean closure, Equation (10.21), compared with the experimental data at different compositions (19,20).



371

0

,.:..--------------------,

0 ,;

..;.

"

��

. ..,,

•

__ ,,,,,.,,'■

•

i;:__ __ ...__ __ _._ __ _,__ __ __, ___ ,..._ _ __,

4.0 8.0 12.0 16.0 20.0 24.0 28.0

C22/C11

Figure 10.6 The test of the linearity of C12/C11 versus C2z/C11 for tetrachloromethane (1) + ethanol (2)(circles) and tetrachloromethane (1) + n-butanol (2) (squares) binary mixtures. The curves represent theweighted arithmetic-mean closure, Equation (10.21), compared with the experimental data at differentcompositions (19,20).10.7 VAPOR-LIQUID EQUILIBRIA

To test the possibility of vapor-liquid-equilibrium (VLE) prediction by the activitycoefficient expression, Equation (10.35), a number of complex systems are chosen at low pressures. The vapor phase is assumed to be an ideal gas and the vapor pressure of the pure component, P;0 is represented by the Antoine equation. Two kinds of pure-liquid property data are needed in the activity-coefficient equation: they are the specific volume and the isothermal-compressibility values. The VLE data are fitted to the activity-coefficient expression, by minimizing the following objective. function

(10.37)

where P is the calculated total vapor pressure, y is the calculated vapor molar fraction of one component, and S

p and S

y are the values of the sums of squares of P and y deviations,

respectively, that result when "i,(P-Pexp)2 and "i,(y--Yexi are minimized individually. Equation

(10.37) allows optimization of Equation (10.35) simultaneously with respect to both experimental vapor-pressure, Pexp, and vapor-phase mole fraction, Yexp

, data. The results of the calculations are shown in Table 10.4.



372

Table 10.3 Deviations ofreal-fluid direct correlation inteS!:als from the relation C12=a12C 1 1+a21C22-

Mixture Temperature s Max Dev Xr at [C o] xlOO [%] Max Dev.

water+methanol 0 2.1 7.7 0.0 25 1.9 7.5 0.0 60 2.2 -2.6 0.4

water+ethanol 25 14.5 22 0.0 50 3.2 4.1 0.1 90 4.1 6.7 0.1

water+propan-1-ol 25 2.7 7.4 0.0 water+tert-butanol 25 1.5 -4.6 1.0 water+acetonitrile 30 2.3 -11 1.0 water+acetone 25 1.9 -4.4 1.0 water+dimethylsulfoxide 25 5.9 -8.7 0.0 water+tetrahydrofuran 25 4.3 -8.8 1.0 water+ 1,4-dioxane 25 4.0 -5.9 1.0 water+2-aminoethanol 25 6.9 -6.2 1.0 CC14-methanol 25 0.32 -2.4 0.0 CC l4+ethanol 25 0.28 -2.1 0.0 CC l4+propanol 25 0.12 -1.3 0.0 CC14+n-butanol 25 0.12 -1.2 0.0 CC14+tetrahydrofuran 25 0.0004 .005 0.9 CCl4+ 1,4-dioxane 25 0.00003 0.003 0.0 CY-hexane+diethylether 25 0.014 -0.4 1.0 CY-hexane+dipropylether 25 0.030 -0.6 0.0 CY-hexane+dibutylether 25 0.002 0.2 0.0 CY-hexane+ethylbuty !ether 25 0.020 -0.5 0.0 CY-hexane+dimethoxymethane 25 0.24 -2.1 1.0 CY-hexane+diethoxymethane 25 0.0005 -0.02 1.0 CY-hexane+diethoxyethane 25 0.19 -1.8 0.0 CY-hexane+ 25 0.16 1.0 1.0

diethyleneglycol dimethylether

In this table S=L{(C12,pred - C12,exp} 2

and %Dev.=100{ C12,pred - C12,exp)/C12,exp}.



373

In this table comparisons are also made with the results of the Wilson correlation (36) which is selected because it gave the best over-all fit to the large number of mixtures studied in VLE calculations. Table 10.4 also includes parameters a21 and a12, and optimized values of Sp

and Sy. Figures 10.7 - 10.9 show the vapor-liquid-equilibrium composition (x-y)- diagrams for the systems methanol + water, 1-propanol+water and ethanol+benzene.

Table 10.4 Parameters of the present activity coefficient model, Equation (10.35), and comparisons with the Wilson correlation (18,19}.

Equation (10.35) Wilson Mixture T/°C U21 U12 10-2s

p 10

2S

y 10-2s

e 10

2S

y

Methanol+Water 60 0.3869 0.3562 2.0 0.10 1.0 0.o7Ethanol+Water 40 0.2900 0.3561 1.4 0.09 1.1 0.06Ethanol+Water 55 0.2920 0.3799 0.7 0.14 0.4 0.13Ethanol+ Water 70 0.2942 0.4048 1.6 0.11 0.6 0.13Propan-1-ol+Water 30 0.2392 0.2722 0.o7 0.46 0.09 0.64Ethyl Acetate+Ethanol 40 0.5105 0.2989 0.03 0.04 0.02 0.04Ethanol+Benzene 25 0.1028 0.6393 0.009 0.008 0.004 0.01Ethanol+Benzene 40 0.09741 0.6458 0.2 0.o7 0.1 0.05Ethanol+Benzene 55 0.09688 0.6498 0.9 0.13 0.5 0.13Acetone+Chloroform 25 0.2907 0.6404 0.04 0.24 0.04 0.24

According to Table 10.4 and Figures 10.7 - 10.9, it can be concluded that the present theory can predict the vapor-liquid equilibria of the systems studied as well as the Wilson equation (36).

0

0

0

0 L:::....-'----''----'---'--......1.----'---'---'----'-----'

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

x i

Figure 10.7 The vapor-liquid equilibrium composition (x-y) diagram for methanol (1) + water (2) at 60°C. The squares are the experimental data, the solid curve is the present theory and the dashed curveis the Wilson correlation; diagonal is Raoult's-law line.



374

"---------------------�

"'

0

o-

0

0

□

0 "--�-�-�--'----'----'----'----''----'---'

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 LO

x,

Figure 10.8 The vapor-liquid equilibrium composition (x-y) diagram for 1-propanol (1) + water (2) at

30°C. The squares are the experimental data, the solid curve is the present theory and the dashed curve is the Wilson correlation; diagonal is Raoult's-law line.

"----------------------71

0

"'

0

0

0

L::.----'--..J.....----'---�--'---'----'--'----'--;---'

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 O.B 0.9 1.0

x,

Figure 10.9 The vapor-liquid equilibrium composition (x-y) diagram for ethanol (1) + benzene (2) at

55°C. The squares are the experimental data, the solid curve is the present theory and the dashed curve is the Wilson correlation; diagonal is Raoult's-law line.



375

10.8 LIQUID-LIQUID EQUILIBRIA

Phase splitting in liquid mixtures may occur when two phases have a lower total Gibbs energy than one phase at constant temperature and pressure. A necessary condition for stability of a two-component homogeneous mixture is (2,3,38,39):

i =l,2 (10.38)

In terms of the activity coefficients, Equation (10.38) becomes:

(10.39)

Substituting for Inf; from Equation (10.35) gives

(10.40)

provided parameters a21 and a12 are positive. For liquid-liquid equilibria (LLE) the left-hand side of the inequality given by Equation (10.40) should be negative. Considering the magnitudes of the different quantities one concludes that the present activity-coefficient model is capable of predicting LLE. The Wilson correlation (36), on the other hand, is known to be unable to predict LLE, because it always satisfies inequality (10.40).

The present activity-coefficient theory is tested versus experimental LLE data. For the variation of the isothermal compressibilities with temperature, which is needed in this calculation, a three-parameter equation is used (40). Temperature and pure-component volume dependence of parameters a21 and a12 for a number of organic binary systems exhibiting LLE are represented by the following expressions (3,4)

a21 = (v2/v1)exp(a021/RT) (10.41)

(10.42)

where a021 and a012 are constants (independent of temperature and composition). Their functional form was suggested by examining the behavior of the a21 and a12 values obtained for all mixtures shown in Table 10.3.

The experimental LLE data are fitted to the activity-coefficient equation with Equations (10.41) and (10.42) for a21 and a12, and by taking as objective function, S, the deviation from unity of the ratio of the activities in the two liquid phases:

(10.43)

where superscripts (1) and (2) are for phases 1 and 2, respectively. Table 10.5 shows values of parameters a021 and a012 and optimum S for a number of binary systems. The variation with the compositions of two LLE systems (methanol+heptane and formic acid+benzene) with temperature is shown in Figures 10.10 and 10.11. According to these figures the agreement with the data is satisfactory. The largest deviation of theory from experimental data occurs near the upper critical solution temperature.



376

Overall, the present theory is capable of formulating analytic expressions for activity coefficients in mixtures. for this purpose it is necessary to define closure expressions for the cross-direct-correlation-function integrals. Application of the activity-coefficient expressions resulting from this theory for vapor-liquid and liquid-liquid equilibria calculations has been as successful as the other activity-coefficient expressions available. To make this theory predictive, relations which correlate the aiJ values to temperature, pressure and molecular structure and properties are to be developed; work aiming to this goal is in progress.

Table 10.5 LLE parameters of the present activity-coefficient model for different binary mixtures (18,19). Mixture

Methanol+ Heptane Methanol+Cyclohexane Formic Acid+Benzene Phenol+Octane Methanol+Carbondisulfide Nitrobenzene+Hexane 1,3-Dihydroxybenzene+Benzene

ao21

-1070.9-980.29-748.15-1077.0-797.20-975.97-937.02

In this table, Sis defined by Equation (10.43).

0

ao12

1.0243 1.0191 0.97983 1.0031 1.0186 1.0027 1.0032

0.12 0.21 1.7 7.3 0.45 2.0 5.5

o�--------------------,

0

g

0

N

0

D

� L-.....J-.L-.-'------------------'---'--------.,___�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Xl

Figure 10.10 The liquid composition x versus temperature (T-x) diagram for liquid-liquid equilibri-um of methanol (1) + heptane (2). The squares are the experimental data and the solid curve is the present theory.



377

0

0

0

0 D D r-

": D

0

u

� 0

0

0 "

0

0

0

0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1.0

x,

Figure 10.11 The liquid composition versus temperature (T-x) diagram for liquid-liquid equilibrium of formic acid (1) + benzene (2). The squares are the experimental data and the solid curve is the present theory.

10.9 SUMMARY

Equations (10.14) and (10.15), which are the basis of the present theory, are mathematically exact. The choice of the closure expression, Equation (10.21), relating the cross-direct-correlation integral to the other two integrals in a binary mixture allows us to solve these equations simply and analytically using only the pure-fluid thermodynamic data. It is demonstrated here that simple geometric- and arithmetic-mean closures are needed for simple model fluids. It is also demonstrated that a weighted-arithmetic-mean closure expression is sufficient to represent the relation between the cross-direct-correlation integral to the other two correlation integrals of binary mixtures. Success of this theory in its application to a variety of mixtures comprised of components with polar and associating intermolecular potential energies is indicative of its promise as a strong technique for prediction of properties of complex dissimilar mixtures of practical interest.

The present theory has allowed us to perform calculations of properties and phase equilibria of mixtures. There are several points to be noted about the advantages of the present techniques compared with other existing thermodynamic calculation methods for mixtures: (i) The present theory allows us to perform thermodynamic calculations for the whole range of mixture compositions and not just at the infinite-dilution and high-concentration limits as has been the case for most of the fluctuation-theory techniques. (ii) The present theory is applicable for mixtures consisting of dissimilar species with large differences in molecular size, shape, and energetics. It is specifically useful for polar and associating molecular fluids for which, generally, no accurate intermolecular-potential-energy functions are available. (iii)



378

With the application of the weighted-arithmetic-mean closure for the cross-direct-correlationfunction integrals, Equation (10.21), it has become possible to derive analytic expressions for activity coefficients in complex mixtures consisting the dissimilar molecules.

The resulting activity-coefficient expressions allow us to perform vapor-liquid and liquidliquid equilibria computations for such mixtures. The accuracy of these calculations are is good as the best available phase-equilibria computational techniques for mixtures. (iv) What makes the relations among the direct-correlation-function integrals (or fluctuation integrals) introduced here particularly interesting is the fact that only one closure relation is needed to determine all the binary-mixture properties. Detailed studies are under way in order to develop analytic expressions for total and partial molar properties of multicomponent systems using the present theory.

Acknowledgement

This research is supported by U.S. National Science Foundation Grant CTR-9108595.

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PrePrint - University of Illinois at Chicagotrl.lab.uic.edu/1.OnlineMaterials/Mixtures/00...In the above equations, the partial molar volume, V; and 8µ;/8x1 are related to the direct