Ind. Eng. Chem. Res. 1996,34, 4351-4363
435 1
MATERIALS AND INTERFACES A Modified UNIQUAC Model That Includes Hydrogen Bonding Yuan-Hao Fu, Stanley I. Sander,* and Hasan Orbey Center for Molecular and Engineering Thermodynamics, Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716
Based on the association theory of Wertheim, a UNIQUAC association (UNIQUAC-A) model has been developed in which the hydrogen bond is treated as a strong interaction rather than as a chemical reaction. Compared with the original UNIQUAC model, the UNIQUAC-A model results in better correlations of binary vapor-liquid equilibrium data for self-associating mixtures containing either a n alcohol or a n acid. However, both the UNIQUAC and UNIQUAC-A models produce small errors in pressure and vapor-phase mole fractions for cross-associating mixtures. We report interaction parameters and association parameters for the UNIQUAC-A model that can be used for the extrapolation and prediction of the binary vapor-liquid equilibrium in associating mixtures.
Introduction Hydrogen bonding is usually described by a chemical theory with the assumption that molecules interact to form new species that are in chemical equilibrium. Nagata and Kawamura (1979) used a chemical theory of hydrogen bonding to develop a UNIQUAC association activity coefficient model that was of limited success. Brandani (1983) used the concept of homomorphs, similar molecules but which do not hydrogen bond, t o calculate association constants for alcohols from pure component properties and to improve the UNIQUAC association model (Brandani and Evangelista, 1984). In the model of Brandani the activity coefficient is a sum of combinatorial, residual, and chemical terms, with the first two terms being the same as in the original UNIQUAC model, and a chemical term that arises from hydrogen bonding. Usually, in chemical theories, either monomer-dimer or continuous association models are used to simplify the mathematics. .However, for some compounds, the monomer-dimer model is too simple to represent the association effects, and the continuous model is very complicated for mixtures of components that cross associate. Instead of using chemical theory, here we develop a UNIQUAC association activity coefficient model based on physical interactions. We consider hydrogen bonding to be a strong physical interaction, and use the theory developed from statistical thermodynamicsby Wertheim (1984a,b, 1986a,b) for association. Wertheim derived an expression for the Helmholtz energy of association by using topological reduction and thermodynamic perturbation theory, and Chapman et al. (1990) extended this theory to mixtures. Here we derive an activity coefficient contribution due to the association based on this theory, and by combining this term with the effects of weak physical interactions and molecular size, a UNIQUAC association (UNIQUAC-A) model is obtained that is the sum of combinatorial, residual, and association terms. The combinatorial and residual terms, which result from molecular size differences and
the weak interactions between molecules, are the same as in the original UNIQUAC equation. We test this UNIQUAC association model with both self-associatingand cross-associating binary and ternary mixtures, and compare the results with the original UNIQUAC model. We also develop a correlation for the UNIQUAC association parameters of some associating mixtures that allows the model to be used for extrapolation over a range of temperatures.
Theory Considering the hydrogen bond t o be a strong attractive force, we separate its effect from those of weak physical interactions so that, in the proposed UNIQUAC association model, the activity coefficient for a compound is expressed as the sum of combinatorial, residual, and association contributions. The combinatorial term is due to differences in molecular size and shape, the residual term is due to weak interactions, and the association term is the result of hydrogen bonding. If none of the components in a mixture hydrogen bonds, the association term disappears. The UNIQUAC association equation for compound i is
In this equation, the combinatorial and residual terms are same as in the original UNIQUAC equation and are given by
and
with
* Author to whom correspondence should be addressed. 0888-5885/95/2634-4351$09.00/0 0 1995 American Chemical Society
4352 Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995
where z is the coordination number (taken to be lo), ri and qi are the volume and surface area parameters for compound i, and 6i and 4i are the area and volume fractions, respectively
and
xiri 4. = -
(6)
Therefore we can calculate the excess Gibbs free energy of association using the approximation
GCJrj j
and
TO
is equal to
(7) where u~ is an UNIQUAC adjustable parameter. To derive the contribution of association to the activity coefficient, we begin with the residual Helmholtz free energy of a mixture due to association derived by Chapman et al. (1990) from Wertheim's theory
where i designates a compound, A, is association site A in compound i, XAlis the fraction of association sites A, that are not bonded, and Mi is the total number of association sites in compound i. To obtain the activity coefficient of association from the residual Helmholtz free energy, we consider the following procedure. At a pressure P and temperature T,we have u l p as the volume of pure compound 1 and upp as the volume of pure compound 2, and we then mix the compounds a t ideal solution volume UM,ISE defined as UM,ISE
= x i ~+i 3 ~~ 2 ~ 2 ,
(9)
(17) The contribution to the activity coefficient due to association for compound i then is
where
mixture. The fraction of sites Aj that are not bonded is equal t o
We define the excess Helmholtz free energy of mixing a t constant temperature and ideal solution volume aFsEto be
1 XAj
For mixing at constant pressure and temperature, the excess Gibbs free energy is g; = a; P VE,
+
- xla1(U1,> - x + 2 ( ~ 2 ~-)
+
RTCX,In x1 + x2 In x 2 ) PV; (11)
Combining the above two equations, we have E
gp - &E
= ~ M ( v M , > - aM(vM,IsE> + pupE
(12)
with
VF =
- vM,ISE
(13) The excess volume of mixing for a liquid is small, and a t low pressures the last term in eq 12 can be neglected. Therefore, we can use the approximation VM, e UM,ISE (14) so that vMP
z X%rmpAAJBm
+ m
RRx, In x1 + x 2 In x,) (10)
= aM(vM,)
=
1
= a M ( u M , p E ) - xlal(Vlp)- x2a2(uzp)-
a$E
* indicates a property of pure compound i and
NT is the total number of moles of compounds in the
g; e (15) To find relation between the excess and residual
(19)
Bm
with = 4 r n , p P ~ ~ m f r ~ g ~ , ( rcir )
(20)
where gJm(r)is the radial distribution function, and FbBm = [exp(cAJBm/kT) - 11 where the association energy is E ~ J ~ Since ~ . hydrogen bonding is a short-range effect, the value of rc is similar t o the radius of the molecules. At present the interaction potentials between real molecules is not known, and even if they were it would not be possible to calculate the gJ,(r) from statistical ~(r) mechanics. However, the value of 4 ~ r ~ , p J ? r ~ g ~dr is the number of molecules m that are at a distance smaller than rc from molecule j , which is the partial coordination number. Using the same model as in the UNIQUAC equation (Abrams and Prausnitz, 19751,we assume that the number of external nearest neighbors for a molecule of compound j that are molecules of compound m is given by zqjlmJ,where 6 , is the local area fraction. Therefore, we assume that 4 ~ , p ~ ~ ' r ~ g ~d,r (=r )b6, (21) where b is a constant, and Omj, the local area fraction,
Ind. Eng. Chem. Res., Vol. 34,No. 12, 1995 4353 is
yi@i(TQyi)P= y~i@~p(T,Py"p)Py"p (29)
(22)
'mj
We can then simplify the equation for the association strength to
(23) and we consider the quantity bFAJBm = a A J B m to be a n adjustable, temperature-dependent parameter. Consequently, the final equation for the association part of the model is
In this equation P is the total pressure, P y p is the component vapor pressure, @i is a fugacity coefficient, and yi is a vapor-phase mole fraction all for species i. We use the UNIQUAC association model derived above for the liquid-phase activity coefficient of the associating compound. For the nonassociating compound, the activity coefficient contribution in this model that includes the effect of the association of component 1 is
Xmp~A~Bm = emjaAJBm
with
where the summation is over all the association sites of compound 1. The combinatorial and residual terms are the usual ones for the UNIQUAC model. In the vapor phase, we assume that the nonideality of the apparent mixture (as, for example, seen in the virial coefficient behavior) is also caused by association, and because of the low density and the availability of vapor-phase association parameters, we used a monomer-dimer model for associating compounds (except for hydrogen fluoride, which we consider separately). Further, since we restrict our interest here t o low-pressure systems, we assume that the actual vapor-phase mixture of monomer, dimer, and nonassociating compound to be an ideal gas. To obtain the fugacity coefficients @pi and myp,we use the result derived by Prigogine and Defay (1954)that
where &'I is the partial molar Gibbs free energy of compound 1 in the apparent mixture and is the partial molar Gibbs free energy of the monomer of compound 1 in the real mixture. From eq 31 the fugacities of the apparent compound and its monomer should be equal, so that
8
Ti = rpv
F e n exp( -
(32)
where is the fugacity of apparent compound 1in the apparent mixture, and is the fugacity of monomer in the real mixture. From the defining equation for the fugacity coefficient we have
T)
(33) (27)
where zf:and @f: are the mole fraction and the fugacity coefficient of the monomer in the real mixture. Since we have assumed the real mixture to be an ideal gas, the fugacity coefficient of the monomer is equal to unity. We then have that the fugacity coefficients are equal to ,M
@
n
Self-Associating Mixtures. A self-associating binary mixture contains an associating compound and a nonassociating or inert compound. "he index 1will be used to designate the associating compound and 2 the nonassociating compound. "he vapor-liquid equilibrium relation for compound i is
1
= 41 Y1
(34)
Following same procedure as above, for the fugacity coefficient of the pure apparent compound, y1 = 1, and we obtain
-
avap M* 1 -21
(35)
where z y is the mole fraction of monomer in pure component 1. For the nonassociating compound, we
4354 Ind. Eng. Chem. Res., Vol. 34,No. 12,1995 have that
Table 1. Coefficients of Eq 53 for the Calculation of the Vapor Phase Association Constants compound AK BK methanoic acid" -18.103 6995 acetic acid" -17.362 7290 propanoic acid" -18.334 7635 butanoic acid" 7000 -16.706 pentanoic acidn -16.404 6892 ethanol -11.584 3028 2140 propanol -8.963 1540 butanol -7.162 8660 -42.38 hydrogen fluoride" 1422 -7.874 water ~~~
22
=-
@2
(36)
Y2
and @"ap 2
= 2%= 1
(37)
Defining the association constant (using an ideal gas 1 atm pressure standard state) to be
~
" Gmehling et al. (19821,and Dymond and Smith (1960)used the equation logl&(Torr) = AK' BK'/T and the coefficients listed here are for the equation In K = AK &/T.
+
and using the mass balance given by 2:
Y1=
+ 22:
1 +z:
and using eqs 44 and 47, we obtain
zf: =
+ ,/I + 4y1(2- yl)K(P/l atm) 2(2 - y,)K(P/l atm)
2B NM=l+-
(39)
u
(48)
and
the monomer mole fractions are
-1
+
N D = - B(40)
u
(49)
and
Since the mixture is ideal gas, using ideal gas equation of state, we get
For the nonassociating compound, the more fractions are
and substituting NM and IF'into the expression for K yields
M D z2 = 1 - z1 - z1 = 1 - z y - (z?I2K(P/1 atm) (42)
in the real mixture and, 2;
=1
(43)
in pure compound 2. The association constants of organic acids in the vapor phase have been reported by Gmehling et al. (1982) based on regressing second virial coefficient data. For other weaker associating compounds, such as alcohols, we have also estimated the association constants in the vapor phase from reported second virial coefficients by following procedure. For a pure associating compound at a low pressure, assume that we start with 1mol of a compound that associates to form an ideal vapor-phase mixture containing two actual components: monomer and dimer. Therefore
1=N'+2ND
(44)
PU = NM+ ND -
(45)
and
RT
where NMand NDare the number of moles of monomer and dimer, and u is the molar volume of the system. If the apparent compound follows the virial equation of state, then
(46) where B is the second virial coefficient. Therefore, from eqs 45 and 46 we have (47)
-B (51) (1 ~ B / ~ ) ~ ( R atm) T/I From the virial equation of state, the molar volume of alcohol is given by
K=
u=
+
-1
+ .J1+ (4PB/RT)
(52) (PIRT) Second virial coefficients were obtained from the compilation of Dymond and Smith (1980) [we only used lowand moderate-temperature data because of the temperature range of interest here1 together with eqs 51 and 52 to compute the association constant. Further, assuming that the relation between the association constant and temperature follows the equation
In K = A,
+ BK/T
(53) and using the least squares method, we obtained the coefficients AK and BK given in Table 1. For hydrogen fluoride we instead considered a monomer-hexamer model for association in the vapor phase
-
6HF (HF), with an association constant defined by
(54)
(55) where is the partial pressure of hexamer, since such a model is k n o w n t o better describe this fluid (Pimentel and McClellan, 1960). The vapor-liquid equilibrium relation is same as eq 33 with fugacity coefficients as in eqs 34 and 35. Using the association constant and the mass balance, the monomer fraction
Ind. Eng. Chem. Res., Vol. 34,No. 12, 1995 4355
is obtained by solving the equation
zy(6 - 5 ~ f : ) ~66K(P/1 atm)5b, - 2:)' = 0 (56) The values used for the association constant of hydrogen fluoride in the vapor phase were those reported by Pimentel and McClellan (1960). Cross-AssociatingMixtures. A cross-associating mixture contains two hydrogen-bonding compounds. The vapor-liquid equilibrium relation is the same as eq 33, and in the liquid phase the UNIQUAC association model is used t o obtain the activity coefficients of both compounds. For the vapor phase, we again use the monomer-dimer model. However, since cross-association occurs, the equations are more complicated. With both compounds 1 and 2 being associating compounds, following the derivation for the self-associating mixture, we find that the expressions for the fugacity coefficients are M
@.=I
zi Yi
(57)
and Here the two self-association constants are
Table 2. Type of Association in Self-Associating Mixtures compound
association tvpe
site A
0-H acid site B C-b'
a-
\H site A alcohol site C H-b' site B \H
* 0 and aAA= aBB= 0
aAC = aBC # 0 and aAA = a B B = ,CC
= aAB = 0
site A water
Table 3. Coefficients for the Temperature Dependence of the Self-AssociationParameter A,
compound ethanol propanol butanol acetic acid propanoic acid hydrogen fluoride water acetone chloroforma
+
- 1.924 -6.376 -4.914 -1.284 -2.214 -4.163 0.937 -5.084
BCl 2066.5 3334.2 2596.9 2658.7 2552.7 2869.5 784.62 1673.4
+
a The mixture of acetone chloroform can only cross-associate and the coefficients are for the cross-association parameter.
Results
and
and the cross-association constant is
where Plz and 212 are partial pressure and mole fraction of cross-associating dimer. The mass balances for the two apparent compounds are given by
and
+ 22; + 212 Y2 = 1+ + + 212 2;
2:
2;
(63)
From the association constant equations and the mass balances, we obtain
y1
Before correlating vapor-liquid equilibrium data for a mixture with the model proposed here, one has to determine which compounds associate and how they associate. Instead of specifylng the association complexes that will appear in the mixture as in chemical theory, here we need to identify the number of association sites and the association parameters. Table 2 lists the types of self-association for the compounds we consider. For example, acids usually form only dimers on association; therefore, we assume that acids have only a single association site. There are three adjustable parameters for self-associating binary mixtures in the UNIQUAC association model we have proposed here; the two UNIQUAC parameters and the self-association parameter. For a binary cross-associating mixture, the number of adjustable parameters increases to five (two UNIQUAC parameters, two self-association parameters, and one cross-association parameter). To reduce the number of parameters, we assume that the self-association parameter, which we take to be related to temperature by the equation
+ b1- 2 ) ( ~ 3 ~ K , , ( P atm) /l + Y ~ ( z ; ) ~ K ~atm) ~ ( P /-~ygYz;K12(P/1 atm) zf;'= 0 (64)
and
By the simultaneous solution of these two equations, we obtain the mole fractions of both monomers, from which the fugacity coefficients can be calculated.
(66)
is a pure component parameter that does not vary with the nature of the other compounds in mixture. Since we are using a physical interaction model for the hydrogen bond (rather than a chemical model), the activity coefficient of a pure compound is equal to unity. Consequently, in the UNIQUAC-A model, we must use mixture data to find the self-association parameter. For example, to obtain the self-association parameter for ethanol we correlate with three parameters (the two UNIQUAC parameters and the one self-association parameter) vapor-liquid equilibrium data for several ethanol alkane mixtures in which ethanol only self-
+
4356 Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995 Table 4. Errors in Pressure and Vapor-Phase Mole Fraction for Self-AssociatingMixtures UNIQUAC-A mixture ethanol ethanol
+ butane + hexane
ethanol
+ heptane
ethanol
+ cyclohexane
+ + + + propanol + hexane propanol + 2,2,44rimethylpentane ethanol nonane ethanol undecane propanol dichloromethane propanol propyl bromide
propanol
+ nonane
propanol propanol
+ decane + undecane
propanol
+ cyclohexane
+ hexane butanol + heptane butanol
error in P error in P T ("C) (mmHg) error in y (mmHg) error in y 25.3 25 40 45 60 30 40 60 70 25 30 35 50 70 80 25 55 65 25 25 55 75 60 90 90 60 80 25 55 65 25 60
50
+
acetic acidb benzene
+
propanoic acidb heptane
+
hydrogen fluorideb chlorodifluoromethane
UNIQUAC
60 75 90 20 25 45 50 25 45
50
- 15
13.76 0.86 2.35 2.83 9.52 0.25 1.30 5.27 3.67 1.70 1.19 1.33 0.57 1.55 6.69 5.52 2.56 5.64 1.02 0.43 0.26 2.18 0.40 1.36 3.09 1.21 4.19 1.09 6.27 4.91 12.49 2.95 2.33 0.69 8.35 2.23 0.96 2.23 0.90 5.56 0.83 0.94 0.64 26.55
0.0013 0.0025 0.0025 0.0123 0.0033 0.0134 0.0101 0.0058 0.0088 0.0102 0.0035 0.0027 0.0027 0.0084 0.0032 0.0089 0.0076 0.0017 0.0007 0.0022 0.0017 0.0027 0.0047 0.0025 0.0031 0.0044 0.0046 0.0137 0.0100 0.0059 0.0014 0.0131 0.0017 0.0058 0.0022 0.0037 0.0022 0.0029 0.0052 0.0053 0.0043 0.0091 0.0075
41.23 4.66 9.81 17.79 17.45 2.72 6.03 10.04 11.15 1.67 2.84 2.81 4.56 13.17 12.93 9.39 6.68 5.80 4.49 4.31 5.53 3.60 9.12 7.77 2.68 6.12 16.54 9.73 6.38 9.53 17.73 12.41 3.52 3.34 9.85 5.74 2.03 1.66 0.92 5.03 1.79 2.92 2.98 121.68
0.0114 0.0259 0.0315 0.0159 0.0249 0.0347 0.0228 0.0137 0.0167 0.0191 0.0137 0.0125 0.0048 0.0092 0.0041 0.0121 0.0071 0.0055 0.0068 0.0111 0.0151 0.0091 0.006 0.0035 0.0031 0.0066 0.0131 0.0174 0.0097 0.0067 0.0058 0.0136 0.0073 0.0107 0.0101 0.0093 0.0232 0.0039 0.0136 0.0123 0.0126 0.0146 0.035
Anderson and Prausnitz error in P (mmHg)
error in y
data source"
16.20 '0.88 2.53 1.62 11.53 0.43 1.25 3.01 3.33 1.92 0.59 1.33 0.90 2.28 6.05 8.96 4.25 2.63 1.03 0.95 0.64 2.70 0.95 3.42 4.02 1.30 4.58 1.14 5.44 6.29 17.13 3.42 2.78 0.98 9.08 3.38 2.03 1.66 0.92 5.03 1.79 2.92 2.98 121.68
0.0020 0.0043 0.0048 0.0049 0.0022 0.0126 0.0057 0.0034 0.0067 0.0056 0.0017 0.0010 0.0017 0.0080 0.0038 0.0090 0.0032 0.0019 0.0016 0.0024 0.0036 0.0029 0.0032 0.0026 0.0030 0.0047 0.0048 0.0141 0.0102 0.0065 0.0019 0.0132 0.0026 0.0065 0.0054 0.0093 0.0232 0.0039 0.0136 0.0123 0.0126 0.0146 0.035
1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 2 4 4 2 5
a Data sources: (1) Holderbaum, T.; et al. Fluid Phase Equilib. 1991,63, 219. (2) Gmehling, J.; et al. DECHEMA Chemistry Data Series, Vapor-liquid Equilibrium Data Collection; DECHEMA: Frankfurt, 1977. (3) Lark, B. S.; et al. J . Chem. Eng. Data 1984,29,277. (4) Lark, B. S.; et al. J . Chem. Eng. Data 1985, 30, 286. (5) Wilson, L. C . ; et al. AIChE Symp. Ser. 1989, 85, 51. In the m I Q U A C model, the acid is assumed to associate in the vapor phase.
associates. Then we use the self-association parameters obtained from the various mixtures a t different temperatures to obtain the coefficients A, and B a of ethanol using the least squares method. Figure 1 shows the self-association parameters for three alcohols obtained by fitting several self-associating mixtures using the procedure described above. The lines in this figure are obtaining by fitting eq 66 to these results using the least squares method. Table 3 contains the self-association parameters A, and B a for the alcohols and acids we considered in this work. Using these correlated self-association parameters, we then recorrelated the data for various mixtures by fitting only the two UNIQUAC adjustable parameters. Table 4 lists the errors in pressure and vapor-phase mole fraction obtained for self-associating mixtures containing an alcohol with both the UNIQUAC model and the UNIQUAC-A model proposed here. In the calculation with the original UNIQUAC model, the vapor phase is assumed an ideal gas mixture without association since the vapor-phase association of alcohols is very small. Figures 2-5 are examples of correlated
results for self-associating mixtures containing an alcohol. From these figures and Table 4, we see that the UNIQUAC-A model presented here generally results in smaller errors than the original UNIQUAC model for these weakly associating mixtures. Perhaps most importantly the UNIQUAC-A model did not produce false liquid-liquid splitting that is found when correlating these same data using the original UNIQUAC model. The original UNIQUAC model with the assumption of an ideal vapor phase does not result in good correlations for mixtures containing strongly associating acids. To improve the correlations, Gmehling et al. (1977)used a monomer-dimer model for acids in the vapor phase, but did not consider any association in the liquid phase. The assumption of association in the vapor phase improves the correlation results for the vapor phase obtained with the UNIQUAC equation; however, the inconsistency of assuming only vapor-phase association introduces a peculiar shape to the phase boundary in the low acid concentration region. Here we have used the Gmehling model (1982)for the vapor phase with the association constants reported by them and the UNI-
Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995 4357
A
01
I
E
E
Y
2 a
cn v)
2 n
1
I
_,I.
1
I
I
1
0.0026
0.0028
0.0030
0.0032
0
100 0.00
I
4
I
0.25
0.50
0.75
1 0
mole fraction of hexane
1/T(K) Figure 1. Association parameter a for self-associating mixtures containing a n alcohol plotted as In a versus l/T. The lines and symbols are as follows: -, 0 , ethanol; - - -,0, propanol; - - -, 0, butanol.
Figure 3. Correlation of the vapor-liquid equilibrium data for the hexane ethanol mixture a t 40 "C. Legend as in Figure 2.
+
A
0)
I
E E v
23 cn cn
e
n
-0.b 0.00
I
I
I
0.25
0.50
0.75
1.00
mole fraction of ethanol Figure 2. Correlation of the vapor-liquid equilibrium data for the ethanol heptane mixture at 60 "C. The solid line results
+
from the correlation using UNIQUAC and the dashed line from the UNIQUAC-A model proposed here. The points are the experimental data.
QUAC-A model for the liquid phase. Table 4 also lists the correlation results for both the Gmehling et al. UNIQUAC model and the UNIQUAC-A model presented here. Figures 6 and 7 are examples of correlated results using both these models. The superiority of the model proposed here is evident from these results. We have also tested the UNIQUAC association model for mixtures containing hydrogen fluoride, which strongly associates in both the liquid and vapor phases. The original UNIQUAC equation does not represent these mixtures very well. The correlated results from the original UNIQUAC model assuming an ideal vapor phase are shown in Figure 8. We also considered a monomer-hexamer model for hydrogen fluoride association in the vapor phase with the original UNIQUAC model for the liquid phase. We see in Figure 8 that this results in an improvement for the vapor-phase compositions; however, the error in the liquid composi-
0.25
0.50
0.b
1.00
mole fraction of hexane Figure 4. Correlation of the vapor-liquid equilibrium data for the hexane propanol mixture at 25 "C. Legend as in Figure 2.
+
tions is still large, as is shown in the figure. Using the UNIQUAC-A model in the liquid phase and the monomer-hexamer model in the vapor phase, we can accurately correlate the observed vapor-liquid equilibrium data as shown in Figure 8. For cross-associating mixtures, both the type of crossassociation and the cross-association constant need t o be specified. Table 5 lists the association type and assumed cross-association constant for different classes of cross-associating mixtures. For the liquid phase of such mixtures, the self-association parameter for each of the pure components was obtained by the method described earlier, and the cross-association parameter was taken t o be the geometric mean of the selfassociation parameters. In the vapor phase, as before, we allowed only for dimerization, but now including cross-association with the cross-association constant again assumed to be the geometric mean of two selfassociation constants. Since the self-association constants are known, there are only the two UNIQUAC adjustable parameters to be determined. The errors in correlating the pressure and vaporphase mole fractions when determining the two param-
4358 Ind. Eng. Chem. Res., Vol. 34,No. 12, 1995 125,
100 A
0)
0)
I E
-
I E
E
75
L3
La
u)
u)
50
u)
u)
2 n
n
E
Y
L
25
",
0.00
,
0.25
I
I
0.50
0.75
0 1 D
mole fraction of hexane Figure 5. Correlation of the vapor-liquid equilibrium data for the hexane butanol mixture at 60 "C. Legend as in Figure 2.
+
100,
1
0.b
0.25
0.h
0.75
1.60
mole fraction of propanoic acid Figure 7. Correlation of the vapor-liquid equilibrium data for the propanoic acid heptane mixture at 45 "C. Legend as in Figure 6.
+
r*c h
ZOOOJ
0)
I
A
m
E E
I E E
-
f!a
1500-
!!
u) u)
a
2
u)
1OOO-
u)
0.
?!
P
500-
-,
0.00
0.2s
0.50
0,;s
1.b
mole fraction of acetic aicd Figure 6. Correlation of the vapor-liquid equilibrium data for the acetic acid benzene mixture a t 25 "C. The solid line results from the correlation using UNIQUAC and the dashed line from the UNIQUAC-A model proposed here, both of which include vapor-phase association as described in the text. The points are the experimental data.
+
eters in both the original UNIQUAC and UNIQUAC-A models are listed in Table 6. Figure 9 is an example of the comparison between the experimental data and the correlated results of an alcohol-alcohol mixture with both of these activity coefficient models. In this case both the UNIQUAC and UNIQUAC-A models correlate the data quite well, and are virtually indistinguishable for this almost ideal, weakly associating system. Figure 10 is an example for a more highly associating acidalcohol mixture, and we see that the correlated results using the UNIQUAC-A model with the geometric mean for the cross-association constant produces a slightly larger error than the original UNIQUAC model in which only the acid is assumed to associate, and then only in the vapor phase. Since the difference in the association constants in the vapor phase between acid and alcohol is very large (at 25 "C, K = 1198 for acetic acid and K = 0.240 for ethanol), the acid is more likely to selfassociate than to associate with the alcohol. Therefore, it is probably not appropriate to use the geometric mean
" I
0.00
1-
0.25
T
1
0.50
0.75
-
1
mole fraction of HF Figure 8. Correlation of the vapor-liquid equilibrium data for the HF + chlorodifluoromethane mixture at -15 "C. The line
- - - results from the correlation using UNIQUAC, the line from UNIQUAC accounting for association in the vapor phase, and the line - - - from the UNIQUAC-A model proposed here. The points are the experimental data.
of the self-associationconstants for the cross-association constant in the vapor phase. If we assume that there is no vapor-phase cross-association in the acid-alcohol mixture, the correlation results of the UNIQUAC-A model are much improved. Therefore, the geometric mean combining rule suggested above should be used with caution when the difference in pure component association constants is large. There are few data for aqueous self-associating mixtures covering a wide composition range. Consequently, we have had to use data for cross-associating mixtures to obtain the self-association parameter of water. Using a similar procedure as for self-associating mixtures, we initially correlated vapor-liquid equilibrium data for cross-associating mixtures of ethanol and water with three adjustable parameters: the two UNIQUAC parameters and the self-association parameter of water, with the assumption that the cross-association parameter is the geometric mean of the two self-association
Ind. Eng. Chem. Res., Vol. 34, No. 12,1995 4359 Table 5. Type of Association and Association Parameters for Cross-AssociatingMixtures component 1
component 2
acid
association type
acid
site 01
site 82
C-b'
C-b'
\ti
\H site A2
site Al alcohol
c": 0-H
cross-association parameters
}
aA1Bi #
0; a 4 B z # 0;
aA1Bz
= abB> t 0; others = 0
aA:A:
0;
&Az
= aA&t 0; others = 0
alcohol site 03 site A1
acid
C-b'
\H
f
a4Bz #
0;
site A2 '
alcohol
site C1
H-b' site B~
site A2 site A,
:'0
= aBlclf 0; aAZA2
aAlcl
f
0; aA14= a B I A z = a C I A z
f
0; others = 0
d
= a ~ l= ~ a ~ z = l ~ z
a ~ 1 ~ 2
acid
water site C,
I"-"\
site 0, C-b'
H-b' site B~
\ti site A,
site A2
water
alcohol
Table 6. Errors in Pressure and Vapor-Phase Mole Fraction for Cross-AssociatingMixtures UNIQUAC-A mixture ethanol
+ -DroDanol -
+ butanol ethanolb + acetic acid ethanol + water ethanol
acetic acid"
+ water +
propanoic acid" water acetone chloroform
+
T ("C)
error in P (mmHg)
50 70 80 50 70 35 45 25 40 50 55 40 70 60 15 30 35 40 50
2.34 7.09 6.60 1.42 1.06 0.48 0.57 0.22 1.58 3.19 2.81 0.59 3.53 2.40 0.33 0.40 1.07 0.92 1.57
UNIQUAC
error in y
error in P (mmHg)
0.0083 0.0091 0.0053 0.0056 0.0020 0.0061 0.0041 0.0038 0.0052 0.0063 0.0065 0.0070 0.0136 0.0173 0.0028 0.0019 0.0021 0.0022 0.0017
2.45 7.67 7.60 1.50 0.97 0.48 0.58 0.40 1.82 3.24 2.09 0.49 3.07 3.27 0.49 1.09 1.69 1.62 2.13
Anderson and Prausnitz
error in y
error in P (mmHg)
error in y
0.0087 0.0098 0.0059 0.0051 0.0014 0.0059 0.0057 0.0063 0.0055 0.0073 0.0067 0.0095 0.0174 0.0240 0.0029 0.0033 0.0030 0.0034 0.0028
2.45 7.68 7.61 1.51 0.96 0.52 0.46 0.41 1.85 4.44 2.06 0.49 2.98 3.25 0.49 1.09 1.69 1.62 2.13
0.0087 0.0098 0.0059 0.0051 0.0014 0.0057 0.0052 0.0064 0.0056 0.0069 0.0067 0.0106 0.0175 0.0358 0.0029 0.0033 0.0030 0.0034 0.0028
In the UNIQUAC-A model, it is assumed that there is no vapor-phase cross-association in mixtures containing acids. In the UNIQUAC model, it is assumed that only the acid associates in the vapor phase in acid-containing mixtures. Data source: Cmehling, J.; et al. DECHEMA Chemistry Data Series, Vapor-Liquid Equilibrium Data Collection; DECHEMA: Frankfurt, 1977.
parameters. The cross-associationconstant in the vapor phase was also assumed to be the geometric mean of the two vapor-phase self-association constants. We then assumed that the self-association parameter of water in the liquid phase was the function of temperature given by eq 66, and used the self-association parameter for water obtained from a three-parameter correlation of vapor-liquid equilibrium data for mixtures of ethanol and water at different temperatures to obtain the coefficients A, and B , for water. We then recorrelated the vapor-liquid equilibrium data for the aqueous mixtures to obtain the UNIQUAC interaction parameters. Table 6 lists the correlation errors in the equilibrium pressure and vapor-phase mole fraction for mixtures containing water. For the water-acid mixture, we assume that there is no vapor-phase cross-association in the UNIQUAC-A model, and that only the acid associates in the vapor phase in the UNIQUAC model.
Figures 11-13 are the correlated results for the UNIQUAC and UNIQUAC-A models. For these crossassociating mixtures, which are known to be easier t o describe than mixtures with only a single associating component, we see that the correlation errors with the UNIQUAC-A model are similar those of the original UNIQUAC model. For mixtures such as acetone chloroform that can only cross-associate, we use a similar procedure as for self-associating mixtures to determine the cross-association parameter. First, we correlated the experimental chloroform mixture at each data for the acetone temperature with three adjustable parameters (the two UNIQUAC interaction parameters and the cross-association parameter). We then used these cross-association parameters in eq 66 to obtain their relation to temperature. Finally we calculated smoothed values of the cross-association parameter as a function of temperature and refit the two UNIQUAC interaction
+
+
4360 Ind. Eng. Chem. Res., Vol. 34,No. 12, 1995
200
-
OI
I
A
m I
E E
E
150
E
-
W
t!
3 u) u)
t!
n
--
I
0.00
0.25
0.b
mole fraction 0 0.00
0.25
0.50
0.75
o.is
1.00
of ethanol
Figure 11. Correlation of the vapor-liquid equilibrium data for the water ethanol mixture at 50 "C. Legend as in Figure 2.
+
1.00
mole fraction of ethanol Figure 9. Correlation of the vapor-liquid equilibrium data for the ethanol + butanol mixture at 50 "C. Legend as in Figure 2.
100
n
A
-
.4
m
I E E
D
W
I
E E
?!
a u) u)
Y
E
n
0.00 -"
!
0.00
1 -
0.25
1
I
0.50
0.75
0.25
0.50
0.75
1.00
mole fraction of water Figure 12. Correlation of the vapor-liquid equilibrium data for
,
1.00
+
mole fraction of ethanol Figure 10. Correlation of the vapor-liquid equilibrium data for the ethanol acetic acid mixture a t 35 "C. The line - results
+
the water acetic acid mixture a t 40 "C. The line - results from correlation using UNIQUAC, and the line - - is from the UNIQUAC-A model without cross-association in the vapor phase. The points are the experimental data.
from correlation using UNIQUAC, the line - - - is from the UNIQUAC-A model with cross-association in the vapor phase, and the line - - - is from the UNIQUAC-A model without crossassociation in the vapor phase. The points are the experimental data.
+
parameters. The errors in correlation of the acetone chloroform mixtures at different temperatures using the original UNIQUAC and UNIQUAC-A models are shown in Table 6. From Table 6, we see that the UNIQUAC-A model consistently produces smaller errors than the original UNIQUAC model. It should be pointed out that there have been other modifications of the UNIQUAC model to account for association. For example, Anderson and F'rausnitz (1978) modified the original UNIQUAC model by using different area parameters q in the combinatorial and residual terms for water and the alcohols. In the combinatorial term, the value of q used for a n alcohol or water was the same as in the original UNIQUAC model, while in the residual term the values used for q for each alcohol and for water were determined by fitting a variety of mixtures containing these compounds. After determining the value of q, they refitted
-
200
n
D
I
E
E
v
0.00
I
I
I
0.25
0.50
0.75
1.00
mole fraction of water Figure 13. Correlation of the vapor-liquid equilibrium data for the water 12.
+ propanoic acid mixture at 60 "C. Legend as in Figure
Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995 4361 Table 7. UNIQUAC Parameters Used in the Predictions UNIQUAC-A
UNIQUAC
compound 1
compound 2
uiz (K)
uzi (K)
~ i (K) z
~ z (K) i
ethanol ethanol ethanol propanol propanol propanol propanol propanol propanol propanol butanol acetic acid acetic acid acetic acid propanoic acid propanoic acid propanoic acid propanoic acid ethanol ethanol propanol ethanol propanol acetic acid acetic acid propanoic acid hexane
hexane cyclohexane heptane propyl bromide hexane cyclohexane heptane 2,2,4-trimethylpentane nonane undecane heptane Cc4 propyl bromide benzene Cc4 propyl bromide benzene heptane propanol butanol butanol water water propanoic acid water water heptane
45.648 26.944 80.297 357.503 32.937 74.460 60.627 3.237 42.400 -6.008 -23.724 71.432 193.085 101.829 185.190 443.801 63.423 17.054 -20.316 -29.169 -42.404 239.738 249.467 -117.855 222.007 421.767 66.757
36.543 76.518 -20.845 -290.446 18.350 -30.617 -13.122 57.248 0.594 82.055 107.077 31.795 -97.023 -25.371 -257.128 -305.833 -78.123 92.161 32.055 70.085 53.121 5.836 78.754 168.553 85.471 57.980 -66.740
-94.077 -61.110 -87.204 -91.830 -75.374 - 109.847 -76.255 -107.204 -110.097 - 139.642 -117.550 -72.232 58.076 -85.286 131.530 168.231 -227.580 -98.332 -73.055 -63.626 -58.779 74.262 126.619 -118.561 388.432 519.283 66.757
619.787 504.289 585.668 376.772 390.177 486.188 404.177 436.576 432.178 525.636 396.535 360.487 162.996 351.157 -140.487 -35.655 507.783 367.898 91.525 102.949 71.630 54.579 106.533 169.731 -203.599 -171.301 -66.740
100,
1
s Y
Q
""
1
I
I
I
0.00
0.25
0.50
0.75
'-
1.b
I
0.00
+
UNIQUAC model and the dashed line from the UNIQUAC-A model proposed here. The points are the experimental data.
mixture data using the two UNIQUAC interaction parameters. However, they did not extend their model to acids which strongly associate. For comparison, we have also shown the results of correlation using the modified UNIQUAC model of Anderson and Prausnitz in Tables 4 and 6. In these tables we see that both the UNIQUAC-A and the modified UNIQUAC of Anderson and Prausnitz are better than the original UNIQUAC model and, in most cases, the UNIQUAC-A proposed here results in smaller errors than the model of Anderson and Prausnitz for self-associating mixtures containing an alcohol and for cross-associatingmixtures of alcohol alcohol or alcohol water. For other mixtures, including those containing acids, the modified UNIQUAC model of Anderson and Prausnitz is exactly the same as the original UNIQUAC model.
+
+
0.50
0.k
1.b
mole fraction of cyclohexane
mole fraction of ethanol Figure 14. Prediction of the vapor-liquid equilibria of the ethanol heptane mixture at 760 mmHg. The solid line results from the
0.k
Figure 15. Prediction of the vapor-liquid equilibria of the cyclohexane propanol mixture at 760 mmHg. The solid line results from the UNIQUAC model and the dashed line from the UNIQUAC-A model proposed here. The points are the experimental data.
+
To correlate vapor-liquid behavior as a function of temperature, we have assumed that the UNIQUAC parameters in the association model are temperature independent, but that the association parameter is temperature dependent as given by eq 66 with the coefficients in Table 3. We then correlated data for mixtures a t different temperatures t o obtain the best value of the UNIQUAC parameters. For comparison, we also assumed that the parameters in the original UNIQUAC model are temperature independent, and followed the same procedure as above. Table 7 lists the values of the parameters of the UNIQUAC and UNIQUAC-A models for different mixtures determined in this way. Figures 14 and 15 are two examples of the results obtained using the UNIQUAC and UNIQUAC-A activity coefficient models for mixtures a t isobaric
4362 Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995 Table 8. Results of Prediction for Ternary VLE Mixtures Using Original UNIQUAC, UNIQUAC-A, and Modified UNIQUAC Model of Anderson and Prausnitz UNIQUAC-A UNIQUAC Anderson and Prausnitz error in error in error in error in error in error in mixture TorP P or T YI andyz P or T YI andyz P or T y1 and yz propanol + hexane + heptane CCld + acetic acid + propanoic acid propyl bromide + acetic acid + propanoic acid benzene + propanoic acid + acetic acid ethanol + propanol + butanol water + acetic acid + propanoic acid ethanol + water + propanol ethanol + water + propanol ethanol + water + propanol
40 "C 30 "C
3.23 mmHg 5.89 mmHg 3.75 mmHg 16.07 mmHg 0.0521, 0.0324 19.65 mmHg 0.0516, 0.0301 19.65 mmHg 0.0516, 0.0301
760 mmHg 2.47 "C
0.0410, 0.0210 2.97 "C
0.0390, 0.0214 2.97 "C
0.0390, 0.0214
760 mmHg 0.71 "C
0.0087, 0.0077 1.08 "C
0.0186, 0.0083 1.08 "C
0.0186, 0.0083
760 mmHg 0.96 "C 760 mmHg 0.78 "C
0.0196,0.0134 1.10 "C 0.0208, 0.0144 1.00 "C
0.0180,0.0129 1.04 "C 0.0245, 0.0162 1.20 "C
0.0184,0.0131 0.0277, 0.0177
50 "C
70 "C 90 "C
data sourceu
5.68 mmHg 0.0103, 0.0103 5.45 mmHg 0.0092, 0.0085 5.03 mmHg 0.0088, 0.0084 7.62 mmHg 0.0123, 0.0112 9.12 mmHg 0.0136, 0.0120 9.25 mmHg 0.0131,0.0125 12.77 mmHg 0.0140, 0.0138 14.30 mmHg 0.0165, 0.0138 13.53 mmHg 0.0126,0.0135
Data sources: (1) Zielkiewicz, J. J . Chem. Thermodyn. 1991,23,605. (2) Tamir, A.; et al. Fluid Phase Equilib. 1983,10, 9. (3) Wisniak, J.; et al. J . Chem. Eng. Data 1982,27,430. (4) Malijevska, I.; et al. Collect Czech. Chem. Commun. 1986,51,2665.(5)Gmehling, J.;et al. DECHEMA Chemistry Data Series, Vapor-liquid Equilibrium Data Collection; DECHEMA: Frankfurt, 1977.
conditions using the parameters in Table 7. In these figures, we see that the UNIQUAC-A model yields better results than the original UNIQUAC model over a range of temperature. Finally, we have tested the accuracy of the UNIQUAC-A model for the prediction of the vapor-liquid equilibria of ternary mixtures using the parameters from binary mixtures listed in Table 7. We compare the predictive results of the UNIQUAC-A model with both the original UNIQUAC model and the modified UNIQUAC model of Anderson and Prausnitz. Table 8 lists the errors of prediction from three models. From Table 8 we see that the UNIQUAC-A model usually results in similar or smaller errors than the original UNIQUAC model and the modified UNIQUAC model of Anderson and Prausnitz. This is not unexpected, since as more components appear in a mixture the components that associate are diluted, and the effect of association diminishes.
Conclusions In this work, we have considered a hydrogen bond to be a strong physical interaction. Using Wertheim's theory, a UNIQUAC association (UNIQUAC-A)activity coefficient model has been developed. The association parameter in this model is obtained from binary vaporliquid equilibrium data for mixtures containing the same self-associating compound. This new UNIQUAC-A model is found to be a significant improvement over the original UNIQUAC model for mixtures containing alcohols or acids in which only one component associates (self-associating systems). For cross-associating mixtures, the original and association UNIQUAC models produce similar errors in pressure and vapor-phase mole fraction, and both models describe such almost ideal mixtures well. Finally, with the assumption of temperature independence of the UNIQUAC parameters and a temperature-dependent association parameter, we found that the UNIQUAC-A model can be used to extrapolate vapor-liquid equilibrium to higher temperatures.
Acknowledgment This research was supported, in part, by Contract No. DOE-FG02-85ER13436 from the U.S. Department of Energy and Grant No. CTS-9123434 from the U.S. National Science Foundation, both to the University of Delaware.
Nomenclature AK = coefficient to calculate the vapor-phase association constant A, = coefficient to calculate the liquid-phase association parameter ul(ulp)= Helmholtz free energy of pure compound i at a pressure P and temperature T U M ( U M , I S E ) = Helmholtz free energy of mixture at constant temperature and ideal solution volume U M ( U M ~ )= Helmholtz free energy of mixture at constant temperature and pressure aassoc = residual Helmholtz free energy due to association u,asSoc(ulp) = residual Helmholtz free energy of compound i due to association at constant temperature and pressure u ~ ~ ( u M , I S E= ) residual Helmholtz free energy of mixture due to association at constant temperature and ideal solution volume = excess Helmholtz free energy at constant temperature and ideal solution volume a: = excess Helmholtz free energy at constant temperature and pressure B = second virial coefficient BK = coefficient for vapor-phase association constant B , = coefficient for liquid-phase association parameter p,"= fugacity of monomer of compound i in the vapor phase = fugacity of compound i in the vapor phase & = partial molar Gibbs free energy of compound i g, = radial distribution function gF = partial molar Gibbs free energy of monomer of compound i g; = excess Gibbs free energy at constant temperature and pressure K = vapor-phase association constant K11 = vapor-phase self-association constant K ~ =z vapor-phase cross-associationconstant Kzz = vapor-phase self-association constant M,= number of association sites in compound i ND = number of moles of dimer Nu = number of moles of monomer N , = number of moles of compound i NT = total number of moles in mixture P = total pressure = partial pressure of dimer of compound i pk = partial pressure of hexamer of compound i = partial pressure of monomer of compound i Pyp= vapor pressure of compound i qL= area parameter of compound i R = gas constant r, = volume parameter of compound i
e
Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995 4363
T = temperature u, = UNIQUAC binary interaction parameter v = molar volume vlp = molar volume of pure compound i at a pressure P and temperature T UMJSE = molar volume of mixture at constant temperature and ideal solution volume = molar volume of mixture at constant temperature and pressure vE = excess volume at constant temperature and pressure = mole fraction of association sites A, that are not bonded x, = liquid-phase mole fraction of compound i y l = vapor-phase mole fraction of compound i z = coordination number z, = mole fraction of nonassociating compound i in the real mixture zf = mole fraction of dimer of compound i in the real mixture zp = mole fraction of hexamer of compound i in the real mixture zf" = mole fraction of monomer of compound i in the real mixture
yAt
Greek Letters
= association parameter between association sites A, and BJ A k B ~= association strength between association sites A, and B, E ~ & = ~ Jassociation energy between association sites A, and BJ y, = activity coefficient of compound i K ~ =~association J volume between association sites A, and BJ 8,= area fraction of compound i 8, = local area fraction p = total molar density u = diameter of segment t, = exp(-u,lT) @, = fugacity coefficient of compound i @.," = fugacity coefficient of monomer of pure compound i @yp= fugacity coefficient of pure compound i at saturation pressure dl = volume fraction of compound i aAIBJ
Superscripts
* = in pure compound A, = association site A in compound i assoc = association comb = combinatorial D = dimer E = excess H = hexamer IGM = ideal gas mixture IG = ideal gas ISE = ideal solution M = monomer
res = residual V = vapor phase Subscripts
M = mixture P = constant P
Literature Cited Abrams, D. S.; Prausnitz, J. M. Statistical Thermodynamics of Liquid Mixtures: A New Expression for the Excess Gibbs Energy of Partly or ComDletelv - Miscible Systems. AIChE J . 1975y21,116. Anderson, T. F.; Prausnitz, J. M. Application of the UNIQUAC Eauation to Calculation of MulticomDonent Phase Eauilibria. 1.*Vapor-LiquidEquilibria. Ind. End. - Chem. Process Des. Dev. 1978,-17, 552. Brandani. V. A Continuous Linear Association Model for Determining the Enthalpy of Hydrogen-bond Formation and the Equilibrium Association Constant for Pure Hydrogen-bonded Liquids. Fluid Phase Equilib. 1983,12,87. Brandani, V.; Evangelista, F. The UNIQUAC Associated Solution Theory: Vapor-liquid Equilibria of Binary Systems Containing One Associating and One Inert or Active Component. Fluid Phase Equilib. 1984,17,281. Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. New Reference Equation of State for Associating Liquids. Ind. Eng. Chem. Res. 1990,29,1709. Dymond, J. H.; Smith, E. B. The Virial Coeficients of Pure Gases and Mixtures: a Critical Compilation;Clarendon Press: Oxford, 1980. Gmehling, J.; Onken, U.;Arlt, W. DECHEMA Chemistry Data Series, Vapor-liquid Equilibrium Data Collection. Parts l a ; DECHEMA: Frankfurt, 1977. Gmehling, J.; Onken, U.; Grenzheuser, P. DECHEMA Chemistry Data Series, Vapor-liquid Equilibrium Data Collection. Parts 5; DECHEMA: Frankfurt, 1982. Nagata, I.; Kawamura, Y. Thermodynamics of Alcohol-unassociated Active Component Liquid Mixtures. Chem. Eng. Sei. 1979, 34, 601. Pimentel, G. C.; McClellan, A. L. The Hydrogen Bond; W. H. Freeman and Company: San Francisco, 1960. Prigogine, I.; Defay, R. Treatise on Thermodynamics Based on the Methods of Gibbs and De Donder; Longmans, Green: London, 1954. Wertheim, M. S. Fluids with Highly Directional Attractive Forces. I. Statistical Thermodynamics. J . Stat. Phys. 1984a,35, 19. Wertheim, M. S. Fluids with Highly Directional Attractive Forces. 11. Thermodynamics Perturbation Theory and Integral Equations. J. Stat. Phys. 1984b,35, 35. Wertheim, M. S. Fluids with Highly Directional Attractive Forces. 111. Multiple Attraction Sites. J . Stat. Phys. 1986a,42, 459. Wertheim, M. S. Fluids with Highly Directional Attractive Forces. IV.Equilibrium Polymerization. J . Stat. Phys. 1986b,42,477.
Received for review J a n u a r y 20, 1995 Revised manuscript received August 8 , 1995 Accepted September 1, 1995@ IE950066G
@
Abstract published in Advance A C S Abstracts, November
15, 1995.
