Ind. Eng. Chem. Res. 2001, 40, 2971-2981
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Local Composition Structured Fluid Model for the Excess Gibbs Energy of Liquid Mixtures Walter W. Focke† IAM, Department of Chemical Engineering, University of Pretoria, P.O. Box 35285, Menlo Park, Pretoria 0102, South Africa
An n-fluid local composition model is derived for the excess free energy of multicomponent liquid mixtures. Taking into account that differences in fluid structure affect the intermolecular interactions leads to the following expression for a binary mixture:
[
A21x2 A12x1 GE )x1x2 + RT x1+R12x2 R21x1+x2
]
The parameters Aij ) ln γ∞j and Rij can be expressed in terms of molecular properties, and the model should therefore have general validity. For most systems tested, the approximation Rij ) 1/Rji is valid. This leads to a three-parameter version that, with the exception of the alcoholalkane systems, shows greater flexibility than the nonrandom two-liquid equation in correlating vapor-liquid equilibrium (VLE) behavior. As is the case for other local composition models, multicomponent behavior can be predicted from knowledge of the binary VLE alone. Because they can be shown to be special forms, the local composition structured fluid model also provides a proper framework for the extension of the Porter, Van Laar, Margules, and ScatchardHildebrand equations to multicomponent systems. Introduction The nonideal behavior of liquid mixtures is conventionally described in terms of the excess Gibbs energy.1-3 It reflects the differences between the intermolecular interactions in the pure components and in the mixture and also the structural changes that accompany mixing.4 Knowledge of its temperature, pressure, and composition dependence provides a complete description of the thermodynamic properties of the solution. The concept of activity is introduced in order to maintain the functional form of the equations derived for ideal solutions. It allows all of the deviations from ideality to be lumped together into a single factor, i.e., the activity coefficient. Activity coefficients are conveniently derived from the excess Gibbs free-energy equation.3 Unfortunately, thermodynamics does not provide an explicit functional form for the excess Gibbs free energy GE ) GE(T,P,x). Recourse must be taken to empirical and semirational approaches.3 A succession of mostly two-constant equations has been proposed.3,5-7 The most important equations are those of Margules, Van Laar, and Wilson5 as well as nonrandom two liquid (NRTL)6 and UNIQUAC.7 Walas3 has analyzed the comparative merits of the corresponding activity coefficient correlations by considering the goodness of fit to 3563 data vapor-liquid equilibrium (VLE) sets contained in the DECHEMA VLE data collection.8 The resulting ranking of the equations, with the frequency of best fit in brackets, was Wilson (30%), NRTL (23%), Margules (20.6%), UNIQUAC (13.3%), and Van Laar (13.1%). This shows that none of the existing equations are capable of representing the wide range of behavior encountered in practice. The failure of the three-parameter NRTL †
Fax: +27 12 811 1174. E-mail:
[email protected].
equation to provide substantial improvement over the two-parameter models is also puzzling. In addition, when taken together, the Van Laar and Margules equations are best more often than the Wilson equation. Yet, it is not agreed how these models are to be extended to multicomponent mixtures. The implicit temperature dependence of the classic models is generally inadequate to predict experimentally observed variations, and the parameter values often do not vary smoothly with temperature.3 They also fail to cross-correlate VLE with heats of mixing and VLE with liquid-liquid-phase separation.2,3,9,10 Despite the fact that the heat of mixing should be a simple derivative of the Gibbs energy, the available equations are often unable to simultaneously correlate both data sets within experimental accuracy.2,9,10 Activity coefficients obtained from VLE data are generally not suitable for predicting liquid-liquid-phase separation and vice versa.2,3,10 Loblen and Prausnitz10 concluded that these shortcomings are fundamental and not due to any uncertainties in experimental data. Clearly, there is a need for improved expressions for the excess Gibbs energy. Model Development A liquid is a condensed phase characterized by a degree of short-range molecular order and the absence of any long-range order.4 Accordingly, the local liquid structure about a central reference molecule may be defined by the cell formed by the shell of nearest neighbors.11,12 The molecular size and shape as well as the strength of the intermolecular interactions determine the cell size and the relative spatial orientations of the adjacent molecules.2,4,13 Orientation structuring is important when the molecules possess a dipole
10.1021/ie000693s CCC: $20.00 © 2001 American Chemical Society Published on Web 05/23/2001
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moment. For example, liquid water is highly structured. A strong preference is shown for “tetrahedral” orientation of hydrogen atoms about oxygen atoms owing to directional hydrogen bonding.4 Additional complications arise when liquid mixtures are considered.1 Differences in the shape or size of the unlike molecules are expected to affect the molecular arrangement in the mixture.2 However, another important factor is that the attractive forces between like and unlike molecules and those between like molecules are not the same.5 This leads to competition among the various molecular species in their interaction with the central molecule. The result is that local compositions generally differ from the bulk concentration. Wilson5 used a Boltzmann-type distribution function to describe the redistribution of molecules in the shell of nearest neighbors. His postulate leads to the following expression for the local volume fraction of component i:
φi )
xiνi°e-ii/RT
∑j
(1)
xjνj°e-ij/RT
Here νi° is the molar volume of the pure component i and ij is the interaction energy for pair ij. To obtain an expression for the excess Gibbs energy, Wilson substituted the local volume fractions into the Flory-Huggins equation for the excess entropy. While there is no a priori justification for the last step, the Wilson model has proven to be very effective for correlating VLE with binary and multicomponent mixtures.3 However, it suffers from the deficiency that it is incapable of predicting liquid-liquid-phase separation.5 The local composition structured fluid (LCSF) model considers local compositions but, in addition, incorporates differences in fluid structure. It is constructed using the following assumptions and approximations: a. The nonrandom binary mixture is made up of fluid clusters that resemble the parent liquids in their properties13 (Figure 1). b. The unlike interactions in the clusters differ owing to differences in the associated fluid structure.13 c. It is postulated that the magnitude of these interactions can be quantified in terms of the corresponding activity coefficients at infinite dilution. d. The effective volume fractions of the two types of clusters in the mixture are given by Wilson’s local composition model, i.e., eq 1. The global expression for the excess Gibbs energy is obtained by summing the contributions of the two types of liquid clusters:
G
E
RT
)
∑i ∑j φiAijxixj
(2)
and
A12 ) ln γ∞2
if one combines a generalized version of the regular solution model with Scott’s two-liquid theory. The importance of that derivation lies in the fact that the parameter values can be expressed in terms of molecular quantities. This implies that the LCSF should have more general validity than empirical interpolation models.24 Activity Coefficients. For a binary mixture eq 2 can be rewritten as
[
A21x2 A12x1 GE ) x1x2 + RT x1 + R12x2 R21x1 + x2
(3)
Note that the first subscript of a parameter indicates the nature of the parent fluid or “solvent”, whereas the second subscript denotes the nature of the “solute”. In general, and unlike the situation for other local composition models, ij need not necessarily equal ji. In the appendix we show that the same equation is obtained
]
(4)
The structure persistence parameters Rij are defined in terms of the molar volumes and the intermolecular interaction energies:
Rij )
υj exp[-(ij - ii)/RT] υi
(5)
The binary activity coefficients corresponding to eq 4 are
[
ln γ1 ) x22
Here Aij are related to the activity coefficients at infinite dilution according to
A21 ) ln γ∞1
Figure 1. Parent fluids (A and B) each containing infinitesimal admixtures of the second component. The solution (C) is an ideal mixture of these two parent fluids.
[
ln γ2 ) x12
A12x1(R12 + x1 + R12x2) (x1 + R12x2)2
+
]
(6a)
]
(6b)
A21(x22 - R21x12) (R21x1 + x2)2 A21x2(R21 + R21x1 + x2) (R21x1 + x2)2
+
A12(x12 - R12x22) (x1 + R12x2)2
Ind. Eng. Chem. Res., Vol. 40, No. 13, 2001 2973
Special Forms. A disadvantage of the LCSF model is the fact that it requires four adjustable parameters. That is inconvenient in practice, and we therefore introduce an additional approximation by setting
RijRji ) exp
(
)
ii - ij - ji + jj )1 RT
(7)
This expression is exact for ij ) ji ) (ii + jj)/2. Defining σ ≡ R21 reduces eq 4 to a three-parameter version that was previously proposed on purely empirical grounds:14
σA12x1 + A21x2 GE ) x1x2RT σx1 + x2
(8)
It can be shown14 that the Porter, Margules, and Van Laar equations are special cases of this version and therefore also of the LCSF model. For example, the Porter equation is obtained when both R12 ) R21 ) 1 and A12 ) A21. The Scatchard-Hildebrand regular solution model is obtained if one sets
Rij ) υj/υi
and
ln γ∞i ) υi(δi - δj)2/RT
(9)
Because it only requires information about the purecomponent properties, it also represents the simplest predictive version of the LCSF model family. Prediction of Binary VLE. A rating system is required to put VLE correlation performance into perspective and to allow meaningful comparisons to be made. The mean absolute deviation between predicted and measured vapor compositions (∆y) is used here as a measure for the goodness of fit of vapor composition predictions on the following (arbitrary) scale:
Excellent: ∆y < 0.005 Good: 0.005 < ∆y < 0.015 Poor: ∆y > 0.015 Where possible, infinite-dilution activity coefficients were estimated by fitting the experimental data reported in the literature10,16-20 to equations of the form16,17
ln γ∞i ) a + b/T
(10)
Table 1 lists the binary systems used for testing. Systems with near-ideal behavior were purposefully avoided. The low-pressure VLE data were taken from the DECHEMA data collections8 and also from Hirata et al.15 The DECHEMA data set is very convenient because it also presents the parameter values obtained by fitting the data to the Margules, Van Laar, Wilson, NRTL, and UNIQUAC equations. Mean and maximum deviations from predicted compositions and pressures are also given. When attempts were made to fit the LCSF model, it was found that it was rarely necessary to use the full four-parameter version. In the majority of cases, the data could be adequately represented by the threeparameter version of the model and sometimes even by the Van Laar, Margules, or Porter models. When the individual data sets were considered in isolation, it was found that the three-parameter version was usually able to provide an improved fit. This is hardly surprising in
view of the additional parameter that is available. However, there were notable exceptions. The NRTL equations tended to provide a better fit for alkaneprimary alcohol combinations, and Wilson did somewhat better with nonpolar aromatic-primary alcohol binaries. It was therefore decided to rather evaluate the predictive capability of the three-parameter version of the model. For this purpose the experimental infinitedilution activity coefficients (Table 2) were used as input. The parameter σ was estimated by setting it equal to the ratio of the molar volumes of the binary components. This means that all of the parameters in the model were fixed. The results are presented in Table 1 for 25 binaries. In each case the mean errors in the predicted vapor composition and system pressure for the best-fit DECHEMA model, obtained by fitting the actual VLE data, are also reproduced. Surprisingly, for cyclohexane-heptane, heptane-benzene, acetone-heptane, acetone-chloroform, acetone-water, and methanolbutanone, this crude approach actually provided an improved prediction for the vapor composition when compared to the best DECHEMA system. This illustrates the powerful correlating ability of the new model. Table 1A-C separately compares the predictions for nonpolar-nonpolar, nonpolar-polar, and polar-polar binary combinations. As shown in Table 1A, the model predictions are excellent for nonpolar binaries. With the exception of the chloroform-methanol system, the predictions for polar-polar systems are good (Table 1C). They are excellent for acetone-methanol, chloroformethanol, methanol-butanone, and methanol-water. Table 1C shows that the assumption that σ ≡ v1/v2 can lead to large errors for binaries consisting of combinations of highly polar and nonpolar compounds. The failure is especially dramatic for the methanoltetrachloromethane system. It is also evident for benzene-ethanol, ethanol-toluene, and cyclohexane-1butanol systems. Remarkably, the prediction for the cyclohexane-2-propanol binary is good. The predictions for acetone in combination with benzene or heptane are also good. The common denominator for poor predictive performance appears to be a primary alcohol in combination with a nonpolar compound. It is known that in such systems the alcohol tends to form linear, hydrogenbonded oligomers.23 This increases the effective molar volume of the alcohol and thus requires a corresponding adjustment in the value of the parameter σ. The bestfit σ values provide crude estimates for the degree of polymerization of the alcohol. This gives, at the corresponding temperatures, values of 8.8 for ethanol in benzene, 7.8 for butanol in cyclohexane, 10.5 for ethanol in toluene, and 19 for methanol in tetrachloromethane. Correlating Binary VLE: Effect of Temperature. Better agreement with experimental data is obtained when σ is made an adjustable parameter. Doing this improves the data fit to excellent for the chloroformethanol, chloroform-methanol, THF-water, and waterpyridine systems. The influence of temperature was studied by considering binaries for which data are available over a range of temperatures. In most cases σ is found to be temperature independent. Such systems include acetoneethanol (σ ) 1.25), acetone-methanol (σ ) 1.81), and acetone-water (σ ) 2.00). However, Figure 2 shows that
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Table 1. Low-Pressure VLE for Nonpolar-Nonpolar, Polar-Nonpolar, and Polar-Polar Binary Systems: Comparing the Prediction of the Three-Parameter Version of the Model with the Best-Fit DECHEMA Systems correlation no.
T, °C
NN 1
25
benzene
NN 2
30
benzene
NN 3
20
CCl4
NN 4
25
cyclohexane
NN 5
45
heptane
PN 1
25
acetone
PN 2
45
acetone
PN 3
50
acetone
PN 4
25
benzene
PN 5
50
cyclohexane
PN 6
60
cyclohexane
PN 7
50
ethanol
PN 8
55
methanol
PP 1
20
acetone
PP 2
40
acetone
PP 3
45
acetone
PP 4
35
acetone
PP 5
60
butanol
PP 6
55
chloroform
PP 7
35
chloroform
PP 8
40
ethanol
PP 9
50
methanol
PP 10
40
methanol
PP 11
70
THF
PP 12
80
water
a
component 1
component 2
model
σ ) v1/v2
A. Nonpolar-Nonpolar Binary Systems cyclohexane Wilson Sigma 0.827 Margules C6F6 Sigma 0.774 benzene UNIQUAC Sigma 1.179 heptane UNIQUAC Sigma 0.737 benzene Wilson Sigma 0.610 Wilson B. Polar-Nonpolar Binary Systems benzene UNIQUAC Sigma 0.822 CCl4 Wilson 3.50 Sigmab 0.761 σ ) v1/v2 heptane Wilson Sigma 0.501 ethanol Wilson Sigmab 0.172 1.523 σ ) v1/v2 1-butanol Wilson Sigmab 0.151 σ ) v1/v2 1.181 Wilson 2-propanol NRTL Sigma 0.171 toluene Wilson Sigmab 5.77 0.551 σ ) v1/v2 Wilson CCl4 Wilsonc Sigma 7.56 σ ) v1/v2 0.384 Wilson C. Polar-Polar Binary Systems chloroform Van Laar Sigma 0.761 ethanol Wilson Sigma 1.251 Wilson methanol Wilson Sigma 1.863 Wilson water NRTL Sigma 4.07 water NRTL Sigma 5.08 ethanol Margules Sigmab 0.935 1.363 σ ) v1/v2 methanol NRTL Sigmab 0.549 σ ) v1/v2 1.975 water UNIQUAC Sigma 1.68 σ ) v1/v2 3.26 butanone NRTL Sigma 0.452 water NRTL Sigma 2.25 water NRTL Sigmab 6.4 σ ) v1/v2 4.5 pyridine NRTL 0.162 Sigmab σ ) v1/v2 0.223
mean errors γ∞1
γ∞2
103y1
1.55 1.58 0.74 0.776 1.25 1.25 1.10 1.10 1.44 1.46 1.46
1.66 1.69 1.12 1.204 1.21 1.21 1.15 1.15 2.21 1.90 1.90
0.5 1.9 2.4 3.3 0.7 0.8 1.3 1.3 3.1 2.4 2.8
0.22 0.19 0.24 0.56 0.32 0.31 0.45 0.44 1.27 0.71 0.81
1.72 1.72 2.85 2.88
1.52 1.60 2.05 2.09
4.85 4.85 5.07 5.17 5.17 3.66 4.23 4.23 4.23 6.04 4.83 9.18 9.88 9.88 9.88 34.6 28.0 28.0 28.0
6.74 6.74 16.5 18.4 18.4 23.46 21.35 21.35 21.35 12.97 15.3 6.13 5.51 5.51 5.51 7.95 7.01 7.01 7.01
2.7 4.1 6.6 6.5 12.3 9.8 8.6 2.6 4.8 76.2 5.4 7.2 36.2 8.0 6.4 9.2 3.8 4.5 40 4.4 6.8 5.3 97 4.5
0.48 1.09 1.8 4.4 5.5 4.9 4.8 0.2 1.3 23 2.9 2.2 30 2.3 5.95 8.3 0.51 0.54 11 2.62 4.9 3.4 187 12.1
3.5 3.5 4.6 9.7 12.1 2.6 2.8 3.0 7.3 5.1 10.9 11.3 4.9 4.6 10.4 6.3 6.6 37.4 5.5 5.4 12.6 4.4 3.1 1.3 2.3 5.6 3.0 16.7 3.4 5.4 10.9
4.6 2.0 3.3 2.8 4.0 2.5 3.2 4.1 4.0 7.3 2.8 4.1 1.3 2.9 7.2 2.1 1.7 26 1.8 1.7 3.0 6.4 9.3 1.3 1.6 22 24 41 3.4 3.4 8.6
0.29 0.336 2.27 2.28 2.28 1.96 1.99 7.00 7.35 72.41 52.98 1.50 1.75 1.75 2.55 2.55 2.55 5.06 5.20 5.20 2.09 2.09 2.05 1.90 30.83 48.18 48.18 2.95 2.815 2.815
0.40 0.482 1.98 2.138 2.138 1.79 1.84 5.55 6.03 5.36 4.32 5.31 4.37 4.37 7.30 7.30 7.30 2.69 2.67 2.67 2.35 2.35 1.65 1.65 7.80 7.35 7.35 20.41 20.84 20.84
Ptota
In units of mmHg. Limiting activity coefficients determined from Table 2. b Best fit σ-value. c Data obtained from Hirata et al.15
Ind. Eng. Chem. Res., Vol. 40, No. 13, 2001 2975 Table 2. Infinite-Dilution Activity Coefficient Correlations for the Binary Systems Tested ln γ∞1 ) a + b/T
binary system no.
solute
solvent
T range, °C
a
PN 1
acetone benzene acetone CCl4 acetone chloroform acetone ethanol acetone heptane acetone methanol acetone water benzene cyclohexane benzene ethanol benzene C6F6 butanol water chloroform ethanol chloroform methanol cyclohexane butanol cyclohexane 2-propanol ethanol toluene ethanol water heptane benzene methanol butanone methanol cyclohexane methanol CCl4 methanol water THF water water pyridine
benzene acetone CCl4 acetone chloroform acetone ethanol acetone heptane acetone methanol acetone water acetone cyclohexane benzene ethanol benzene C 6 F6 benzene water butanol ethanol chloroform methanol chloroform butanol cyclohexane 2-propanol cyclohexane toluene ethanol water ethanol benzene heptane butanone methanol cyclohexane methanol CCl4 methanol water methanol water THF pyridine water
25-77 25-56 23-74 28-55 32-50 34-55 49-75 25-54 40-60 70-100 20-70 20-70 25-100 35-100 10-79 10-77 25-73 25-75 30-70 30-70 25-100 60-110 55 35-60 35 35 76-108 40-80 40-82 40-80 20-108 35-76 25-70 40-70 25-80 20-93 35-60 35-64 10-60 20-60 20-80 20-80 20-60 40-60 20-50 25-70 50-100 50-100
0.0938 0.0423 -0.1361 0.739 2.7844 0.0967 -0.8789 -1.7606 0.6535 2.9623 -0.7535 -0.9233 3.1631 0.4435 -0.4535 -0.4558 0.3556 -2.8668 0.9344 0.187 5.9816 -1.1676 0.5596 -0.9393 0.9361 1.9879 0.1595 -4.9489 0.5768 -3.8003 -3.2595 0.7132 5.9816 0.981 -1.4336 -0.7765 -0.8451 -0.3203 -4.9086 0.8123 -3.1961 1.3566 2.1698 1.7578 10.1385 -3.5851 1.5073 1.034
PN 2 PP 1 PP 2 PN 3 PP 3 PP 4 NN 1 PN 4 NN 2 PP 5 PP 6 PP 7 PN 5 PN 6 PN 7 PP 8 NN 5 PP 9 PN 9 PN 8 PP 10 PP 11 PP 12
Figure 2. Temperature dependence of the σ parameter for the systems methanol-tetrachloromethane, benzene-ethanol, ethanol-water, and pyridine-water.
σ varies for the methanol-tetrachloromethane, benzeneethanol, pyridine-water, and water-ethanol binaries. The observed temperature dependence is in accordance with that implied by eq 5.
b 133.17 128.26 379.52 -1136.4 -242.1 533.16 790.17 374.39 -580.4 459.28 489.6 -360.0 417.0 270.91 292.64 383.83 1723.3 -359.48 -1364.94 876.63
r (s) 0.9809 0.9990 0.9949 s ) 0.022 -0.9276 0.9999 0.9982 0.9996 -0.9965 0.9666 0.753 -0.9502 0.9195 0.9639 0.9311 0.7629 0.9501 0.9996 s ) 0.001 -0.9951 0.5450
ref 16 16, 17 16, 17 16 20 16, 19, 20, 25 8, 19 8, 16 8, 16, 19 8 10, 19, 20 16, 19
797.0
0.8412 8
415.0 2588.0 332.0 2174.0 1793.27 320.94 -1364.94 659.9 368.16 526.38 352.59 2731.84 667.54 2142.4 194.0 -478.66 -393.50 -214.95 1914.65 540.28 -
0.9959 0.9993 0.9659 0.9981 0.9947 0.9908 -0.9951 s ) 0.015 0.9858 0.9732 0.9970 0.9920 0.9970 0.8505 0.9987 0.9921 -0.8617 0.9965 -0.9999 0.9792 0.6988 s ) 0.049
17 17 16, 19, 22 18, 19 16, 25 16, 17 10, 17, 18 8 8, 18, 19 8, 18 8, 10
Methanol-Tetrachloromethane. Experimental γ∞i data for this system are rare. It was therefore decided to generate it from available VLE data. The pressure-liquid-phase composition (P-x) data of Wolff and Hoeppel8 was used, and the three-parameter version was fitted at each temperature using Barker’s indirect method.24 The results are presented in Figures 2 and 3. Figure 3 shows a good agreement with the limiting activity coefficient for CCl4 in methanol reported by Landau et al.19 The σ parameter was found to have a temperature dependence as shown in Figure 2. These data were then used to predict the P-x-y data at 55 °C, and the results were compared with the experimental data reported in Hirata et al.15 They report a mean deviation of 0.0165 mole fraction for a direct fit of the Wilson equation to the experimental data. The corresponding mean deviations for the threeparameter LCSF model are only 0.0145 mole fractions and 1.8% in pressure. However, as is evident from Figure 4, the data include a clear outlier. If it is
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Figure 3. Limiting activity coefficients for the methanol (1)tetrachloromethane (2) system. The open data points indicate values obtained by fitting the three-parameter model to the P-x VLE data of Wolf and Hoeppel.8 The solid symbol indicates the experimental value reported by Landau et al.19 for tetrachloromethane in methanol at 25 °C.
Figure 4. VLE for the methanol-tetrachloromethane system at 55 °C. The experimental data of Scatchard et al. as reported by Hirata15 are compared to the predictions of the three-parameter version of the LCSF model.
Figure 5. Excess Gibbs energy for the system benzene (1)hexafluorobenzene (2). Temperatures: 0, 70 °C; O, 60 °C; 4, 50 °C; ], 40 °C; ×, 30 °C.
neglected, the mean deviations improve to 0.0074 mole fraction and 0.70% in system pressure. Benzene-Hexafluorobenzene. Figure 5 shows the experimental data for the benzene (1)-hexafluorobenzene (2) system in the temperature range 30-70 °C. The plot of GE/x1x2RT against mole fraction shows that the three-parameter version of the model provides an exceptionally good fit. The data are well correlated by σ ) 0.569, γ∞2 ≈ 1.20, and γ∞1 ) 2.546 exp(-359.48/T). The VLE data fit is virtually perfect for both composition
Figure 6. Limiting activity coefficients for the system acetone (1)-chloroform (2). Experimental data (filled symbols and solid lines) reported by Thomas et al.16 and Trampe and Eckert17 are compared to the values obtained by extrapolation of VLE data using the Margules equation as reported in the DECHEMA Data Series8 (open symbols).
and system pressure with mean errors in y1 and in the system pressure of less than 0.001 and 0.1%, respectively. Corresponding average error values for the NRTL and UNIQUAC equations exceed 0.01 and 1%, while for the Wilson equation, they amounted to 0.0075 and 1%, respectively. Thus, in this particular case, the new model provides an order of magnitude improvement in accuracy. Acetone-Chloroform. According to DECHEMA, the Margules equation provided the best fit for the acetone (1)-chloroform (2) system. However, Figure 6 shows that this requires the use of infinite-dilution activity coefficients that are much lower than the independently determined experimental values. The behavior is highly nonlinear, and it is necessary to use the full LCSF model. Figure 7 shows the GE/x1x2RT plot and Figure 8 the variation of the model parameters with temperature. Dioxane-Water. The behavior of this system is such that it is necessary to use the full LCSF model. Another approximation to the LCSF model applies in this case, namely, that R12 ) R21. In the temperature range 3570 °C, the data are well represented by the following parameter values: R12 ) R21 ) 1.163, γ∞1 ≈ 7.46, and γ∞2 ) 1.208 exp(549.55/T). The mean deviations over the three data sets at 35, 50, and 70 °C are 0.0085 mole fractions and 1.97 mmHg. The corresponding values for the best-fit DECHEMA model are 0.0080 mole fraction and 1.95 mmHg. Note that the LCSF model allowed us to fit the data over the whole temperature range using only four constants. In contrast, the NRTL parameter values reported in the DECHEMA data set do not vary smoothly with temperature. Extension to Multicomponent Mixtures. Local composition models consider pairwise interactions only. That makes it possible to predict multicomponent solution behavior from binary data alone. Extending the LCSF model in this sense is straightforward. Equation 2 also applies to multicomponent mixtures with the Rij defined as before and with
φi )
xi
∑k Rikxk
(11)
Ind. Eng. Chem. Res., Vol. 40, No. 13, 2001 2977 Table 3. Prediction of Ternary and Quartenary VLE from Binary Data 1. Cyclohexane (1)-Heptane (2)-Toluene (3) at 25 °C mean errors component 1
component 2
cyclohexane cyclohexane heptane ternary
heptane toluene toluene
max errors
model
A12
A21
103yi
Ptota
103yi
Ptota
Margules Margules Margules from Margules binaries DECHEMA best fit (NRTL)
0.1376 0.5030 0.4168
0.0936 0.4528 0.4933
1.3 3.3 2.9 11.0 2.8
0.44 0.63 0.73 1.70 0.78
2.9 18.4 7.6 19.7 10.8
0.81 1.73 2.17 3.36 2.43
2. Methyl Acetate (1)-Chloroform (2)-Benzene (3) T ) 50 °C component 1 methyl acetate methyl acetate chloroform ternary
chloroform benzene benzene
mean errors
max errors
model
A12
A21
103yi
Ptota
103yi
Ptota
Margules Margules Margules from Margules binaries DECHEMA best-fit (NRTL)
-0.5955 0.3269 -0.2268
-0.8695 0.3246 -0.0419
3.5 3.9 5.3 7.9 2.9
3.45 2.72 4.29 9.6 2.37
15.0 6.9 14.0 33 20
7.93 5.36 9.1 19.9 9.06
component 2
3. Acetone (1)-Chloroform (2)-Methanol (3) at 25 °C mean errors prediction using LCSF binaries DECHEMA best-fit system (UNIQUAC)
max errors
i
Ptota
103yi
Ptota
38.6 39.2
3.69 7.48
145 155
11.8 21.6
103y
4. Acetone (1)-Chloroform (2)-Methanol (3)-Ethanol (4) at 760 mmHg mean errors prediction using LCSF binaries (parameters evaluated at 60.8 °C) a
max errors
103yi
T, °C
103yi
T, °C
10.2
0.36
88
1.08
In units of mmHg.
Figure 7. Excess Gibbs energy for the system acetone (1)chloroform (2). The solid lines show the fit obtained with the LCSF model to the VLE data of Roeck and Schroeder.8
The corresponding activity coefficients are given by
ln γk ) 2xk
() [ pk
+
qk
∑i xi2
]
Aikqi - Rikpi - piqi qi2
(12)
where
pi )
∑j Aijxj
and
qi )
∑j Rijxj
(13)
Note that the LCSF model also provides a framework for the extension of the classic “random” mixture models to multicomponent solutions. This premise was tested using the cyclohexane-heptane-toluene and methyl acetate-chloroform-benzene systems. The Margules parameters reported in the DECHEMA data set were used as input. The results are reported in Table 3 and
Figure 8. Temperature dependence of the structure persistence parameters for the system acetone (1)-chloroform (2).
shown in Figure 9. The predictions for the ternary VLE from binary Margules parameters are good. For the acetone (1)-chloroform (2)-methanol (3) ternary and its quartenary with ethanol (4), a different approach was used. The infinite-dilution activity coefficients in Table 2 were used to fix Aij. Rij were determined by fitting the available VLE data for the binaries over a range of temperatures. Ternary and quartenary VLE were predicted using the binary constants. The experimental data for the quartenary system were determined at a constant pressure of 760 mmHg. Scrutiny of the data revealed that the temperature did not vary much. Consequently, the parameter values were evaluated at a mean temperature of ca. 60 °C. Table 3 and Figure 9 again show good agreement between the experimental yi values and those predicted using binary data alone. In fact, the predictions made with the binaries for the ternary are better than those reported in the DECHEMA data set.8 Note that the
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Table 4. Binary Parameter Values Used for Predicting Multicomponent VLE for the Acetone (1)-Chloroform (2)-Methanol (3)-Ethanol (4) System T ) 60 °C Rij
Aij component j
1
acetone (1) chloroform (2) methanol (3) ethanol (4)
-0.6268 0.6251 0.6110
2
3
4
-0.6299
0.5463 1.9879
0.7215 1.4422 -0.0124
0.9361 0.6981
-0.0204
1 1.6986 1.8100 1.2500
2
3
4
0.2548
0.5525 1.0000
0.8000 1.0000 1.0000
1.0000 1.0000
1.0000
T ) 25 °C Rij
Aij component j
1
acetone (1) chloroform (2) methanol (3)
-1.0272 0.7870
2
3
-0.7152
0.7188 1.9590
1 0.9107 1.8100
0.8130
2
3
0.4303
0.5525 1.0000
1.0000
Figure 10. Heat of mixing data for benzene (1)-hexafluorobenzene (2). The solid lines show the fit of the three-parameter version of the model with σ ) 0.569.
of the excess enthalpy has the same functional form as the excess Gibbs energy if σ is temperature independent. Comparison with eq 3 shows that the infinite-dilution activity coefficients can be obtained from the intercepts of a plot of HE/x1x2RT versus x1:
-T
Figure 9. Prediction of ternary and quartenary VLE from binary data.
latter reports the performance of the NRTL, UNIQUAC, and Wilson models with the parameters obtained from a fit to the ternary data set. Cross-correlating VLE with Heats of Mixing. The Gibbs-Helmholtz equation links the heat of mixing data with the excess Gibbs energy:3
hE )
(
E
E
)
∂G /RT H ) -T RT ∂T
(14)
P,x
Substituting eq 8 into eq 14 yields (with σ assumed constant)
[
]
σx1A′12(T) + x2A′21(T) HE hE ) ) -T x1x2 x1x2RT σx1 + x2
(15)
Equation 15 shows that the compositional dependence
(
d ln γ∞i HE )h h E,∞ ≡ lim i xif0 x1x2RT dT
)
(16)
Application to the Benzene-Hexafluorobenzene System. No γ∞i experimental data were available for this binary either, and it was therefore decided to use the available heat of mixing data as a crosscheck. Figure 10 shows the fit of eq 15 with σ ) 0.569 to the heat of mixing data reported by Andrews et al.25 that spans the temperature range 25-70 °C. The partial molal excess enthalpies at infinite dilution (h h E,∞ i ) were obtained from the intercepts of this graph. Plotting these against temperature suggested a temperature dependence of the form (Table 4)
) A + B/T h h E,∞ i
(17)
Substituting eq 17 into eq 16 and integrating lead to
ln γ∞i ) C - A ln T + B/T
(18)
Equation 18 provides the temperature trend for the limiting activity coefficient to within a constant of integration. Figure 11 confirms this. Thus, in principle, heat of mixing data can be used to extrapolate the infinite-dilution activity coefficients to other temperatures provided an γ∞i value is known at some tempera-
Ind. Eng. Chem. Res., Vol. 40, No. 13, 2001 2979
Acknowledgment Financial support for this research from the THRIP program of the Department of Trade and Industry and the National Research Foundation of South Africa as well as Xyris Technology CC is gratefully acknowledged. Appendix: Alternative Derivation of the LCSF Model
Figure 11. Infinite-dilution activity coefficients for the system benzene (1)-hexafluorobenzene (2). The experimental data points were obtained from the VLE data. The solid curves are the predictions obtained from heats of mixing data.25 Table 5. Temperature Dependence of the Limiting Activity Coefficients from Heat of Mixing Data25 for the Benzene-Hexafluorobenzene System errors, % hE,∞ i
implies ln
solute
solvent
γ∞i
) C - A ln T + B/T A
B
C
hE,∞ i
ln γ∞i
avg max avg max
benzene C6F6 1.9652 -945.0 14.103 1.5 C6F6 benzene -0.7950 278.2 -5.275 3.5
2.9 5.3
3.5 2.7
7.9 5.6
ture. Equation 18 also shows that the temperature dependence of γ∞i is more complicated than suggested by relation (10). This is not surprising because a linear h ∞i is indepenvariation of ln γ∞i with 1/T implies that H dent of temperature in disagreement with the experimental data. Table 5 shows that the absolute values for the integration constants are very large in comparison to the measured ln γ∞i values. This implies that ln γ∞i results from the subtraction of two very large terms. The temperature dependence of ln γ∞i is also relatively weak. Combined with the typical uncertainty in the experimental data, these factors probably preclude accurate prediction of heat of mixing from knowledge of ln γ∞i alone. A similar observation was previously made in connection with the benzene-cyclohexane system.14 Conclusions The binary LCSF equation is a two-fluid local composition model that, implicitly, also takes differences in fluid structures into account. It is suitable for use with highly nonideal liquid mixtures, e.g., acetone-chloroform. However, for most systems the simplified threeparameter version is more than adequate. With the exception of associating molecules, it provides greater flexibility for correlating binary VLE behavior than the NRTL equation. Similar to other local composition models, the LCSF equation can be extended to multicomponent systems by considering binary interactions only. The Margules, Van Laar, and Scatchard-Hildebrand activity coefficient correlations are shown to be special cases. Thus, the LCSF equation also provides a framework for extending these classic “random mixture” models to multicomponent systems.
According to Sandler,26-29 statistical mechanics, rather than lattice models, provides the proper basis for developing thermodynamic expressions for the excess energy of mixing. It yields theoretical models with adjustable parameters that can be expressed in terms of molecular properties. Such activity coefficient correlations should therefore have more general validity than purely empirical ones.26,27 Because of uncertainties about the intermolecular pair potentials and the intractable nature of the partition function,27 the introduction of simplifying assumptions and even empirical modifications is necessary.26,27 One approach pioneered by Sandler and co-workers26-29 is to use a simplified form of the partition function that is independent of any particular potential model, e.g., the generalized van der Waals partition function (GVDW). We will use as a basis the one-parameter regular solution model1 for the excess Helmholtz energy, initially developed by Van Laar30 and by Scatchard.31 Generalized forms of this equation can be derived from statistical mechanics when certain simplifying assumptions29,32,33 are made:
x1x2C12 AE ) RT v1x1 + v2x2
(A-1)
with
C12 ) c11 + c22 - 2c12 and
cij )
Vij [exp(wij/RT) - 1] 2
The parameter Vij is a volumetric measure of the range of attractive interaction of the pair potential, and wij is the mean value of the free energy of interaction.33 For example, if the square-well potential with well depth ij ) wij and width Rij(σij - 1) describes the intermolecular interaction between spherical molecules, one finds that
Vij )
4π 3 σ (R 3 - 1) 3 ij ij
Let us examine some factors related to the assumptions made in the derivation of this mean-field model. First, Lee33 had to treat the interaction parameter wij as a free-energy quantity. This was necessary to accomplish the transformation of the internal energy into free energy by integration of the Gibbs-Helmholtz equation. This assumption is probably a good one for liquids because local compositions are expected to arise from a combination of molecular size and shape34 as well as differences in molecular forces.35-37 Second, for convenience, spherical potentials were assumed;27,29,33 i.e., a radial pair correlation function was used. This probably constitutes the weakest point
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in the model development: It implies that anisotropy effects only contribute in an angle-averaged form to the interaction parameters. Such angle averaging must surely be dependent on the local composition around the molecule of interest. Third, the derivations consider pairwise additive interactions only even though it is well-known that that this assumption is actually incorrect.4 However, Hamad38 argues that the use of an “effective” binary interaction parameter can compensate for the neglected multibody interactions. If this view is correct, the “effective” correction should be composition dependent. This is clearly seen if we consider the two composition limits x1 f 0 and x2 f 0 of a binary mixture. The thirdbody interactions with respect to a molecule of type 1 will in the former case only involve molecules of type 2 and in the latter case mainly involve molecules of type 1. Ruckenstein and Shulgin39 argue that the nonpairwise additivity in real solutions is the primary reason ∞ ∞ * w21 (where w∞ij represents the value of the for w12 interaction energy at infinite dilution of component j in fluid i). It is concluded that intermolecular interactions must have a concentration dependence to correct for the latter two simplifying assumptions. Ruckenstein and Shulgin39 proposed a linear composition dependence for the interaction energy:
w12 ) x1w∞12 + x2w∞21
(A-2)
w∞ij correspond to the values of the intermolecular interaction parameter at infinite dilution of component j in pure fluid i. Several previous studies40-43 also considered composition-dependent corrections similar to the one advocated by Ruckenstein and Shulgin.39 The disadvantage of this approach is that the activity coefficient equations become very complex. In addition, the extension of the models to multicomponent systems is not obvious41,42,44 or unknown.39 Consider instead the use of Scott’s two-liquid theory13 as an empirical correction to eq A-2. In essence, his theory postulates that the solution is made up of two hypothetical liquids that form ideal solutions on mixing.27 This implies that the excess Helmholtz energy of the mixture is simply given by the mole fraction weighted sum of the excess Helmholtz energies of the hypothetical parent fluids:
AE ) x1AE12 + x2AE21
(A-3)
The pure-component liquids, with the other component present in infinitesimal amounts, are the most convenient parent-fluid choices. With respect to the mixture properties, this also puts the emphasis on the infinite-dilution region as suggested by Prausnitz45 and Wang and Vera.46 Thus, AEij correspond to the excess Helmholtz energy at infinite dilution of component j in pure fluid i. This concept was previously suggested by Andiappan and McLean40 and implemented by Nagata and Nagashima44 as a correction to the quasi-chemical expression for the excess Helmholtz energy. Substitution of eq A-1 into eq A-3 yields an expression similar to eq 4 in the main text:
[
]
A21x2 A12x1 AE ) x1x2 + RT x1 + a12x2 R21x1x2
(A-4)
with Aij ) Cij/νi° and Rij ) νj∞ij /νi° with νi° the molar volume of pure component i and νj∞ij the partial molal volume at infinite dilution of component j in fluid i. If either νi°νj° ) νj∞ij νj∞ij or if the volume of mixing behaves ideally, this derivation predicts that R ≡ R12 ) 1/R21. Because the excess Helmholtz energy at ideal solution volume is approximately equal to the excess Gibbs energy at constant pressure,47,48 eq A-4 reduces to the three-parameter model form14 given in the main text. Extension of eq A-4 to multicomponent systems is trivial if we view the n-component solution as an n-fluid mixture in the Scott sense. Note that, for clarity, we have neglected the hardcore contribution to the excess entropy.33 This implies that we were considering molecules that are of similar size and shape. For general application, eq A-4 needs to be modified by the inclusion of appropriate expressions for the combinatorial entropy of mixing.49,50 Literature Cited (1) Prausnitz, J. M. Molecular Thermodynamics of Fluid-Phase Equilibria; Prentice-Hall: Englewood Cliffs, NJ, 1969. (2) Palmer, D. A. Handbook of Applied Thermodynamics; CRC Press: Boca Raton, FL, 1987. (3) Walas, S. M. Phase Equilibrium in Chemical Engineering; Butterworth: Boston, 1985. (4) Maitland, G. C.; Rigby, M.; Smith, E. B.; Wakeham, W. A. Intermolecular Forces; Clarendon Press: Oxford, U.K., 1987. (5) Wilson, G. M. Vapor-Liquid Equilibrium. XI. A New Expression for the Excess Free Energy of Mixing. J. Am. Chem. Soc. 1964, 86, 127. (6) Renon, H.; Prausnitz, J. M. Local Compositions in Thermodynamic Excess Functions for Liquid Mixtures. AIChE J. 1968, 14, 135. (7) Abrams, D. S.; Prausnitz, J. M. Statistical Thermodynamics of Liquid Mixtures: A New Expression for the Excess Gibbs Energy of Partly or Completely Miscible Systems. AIChE J. 1975, 21, 116. (8) Behrens, D., Eckermann, R. Eds. Vapor-Liquid Equilibrium Data Collection; DECHEMA Chemistry Data Series; DECHEMA: Frankfurt, Germany, 1977-1996. (9) Nicolaides, G. L.; Eckert, C. E. Optimal Representation of Binary Liquid Mixture Nonidealities. Ind. Eng. Chem. Fundam. 1978, 17, 331. (10) Loblen, G. M.; Prausnitz, J. M. Infinite Dilution Activity Coefficients from Differential Ebulliometry. Ind. Eng. Chem. Fundam. 1982, 2, 109. (11) Eyring, H.; Hirschfelder, J. O. The Theory of the Liquid State. J. Phys. Chem. 1937, 41, 249. (12) Hirschfelder, J. O.; Stevenson, D.; Eyring, H. A Theory of Liquid Structure. J. Chem. Phys. 1937, 5, 896. (13) Scott, R. L. Corresponding States Treatment of Nonelectrolyte Solutions. J. Chem. Phys. 1956, 25, 193. (14) Focke, W. W. A New Three-Parameter Activity Coefficient Correlation for Modelling Liquid-Phase Behaviour. Part 1: Theory. SA J. Chem. Eng. 2000, 12, 35-46. (15) Hirata, M.; Ohe, S.; Nagahama, K. Computer Aided Data Book of Vapor-Liquid Equilibria; Kodansha Ltd.: Tokyo, 1975. (16) Thomas, E. R.; Newman, B. A.; Nicolaides, G. L.; Eckert, C. E. Limiting Activity Coefficients from Differential Ebulliometry. J. Chem. Eng. Data 1982, 27, 233. (17) Trampe, D. M.; Eckert, C. A. Limiting Activity Coefficients from an Improved Differential Boiling Point Technique. J. Chem. Eng. Data 1990, 35, 156. (18) Pividal, K. A.; Birtigh, A.; Sandler, S. I. Infinite Dilution Activity Coefficients for Oxygenate Systems Determined Using a Differential Static Cell. J. Chem. Eng. Data 1992, 37, 484. (19) Landau, I.; Belfer, A. J.; Locke, D. C. Measurement of Limiting Activity Coefficients Using Non-Steady-State Gas Chromatography. Ind. Eng. Chem. Fundam. 1991, 30, 1900. (20) Tochigi, K.; Kojima, K. The Determination of Group Wilson Parameters to Activity Coefficients by Ebulliometer. J. Chem. Eng. Jpn. 1976, 9, 267.
Ind. Eng. Chem. Res., Vol. 40, No. 13, 2001 2981 (21) Bao, J.-B.; Han, S.-J. Infinite Dilution Activity Coefficients for Various Types of Systems. Fluid Phase Equilib. 1995, 112, 307. (22) Knoop, C.; Tiegs, D.; Gmehling, J. Measurement of γ∞ Using Gas-Liquid Chromatography. 3. Results for the Stationary Phases 10-Nonadecanone, N-Formylmorpholine, 1-Pentanol, mXylene, and Toluene. J. Chem. Eng. Data 1989, 34, 240. (23) Pimentel, G. C.; McClellan, A. L. The Hydrogen Bond; W. H. Freeman and Co.: San Francisco, 1960. (24) Raal, J. D.; Muhlbauer, A. L. Phase EquilibriasMeasurement and Computation; Taylor & Francis: Philadelphia, 1984. (25) Andrews, A.; Morcom, K. W.; Duncan, W. A.; Swinton, F. L. J. Chem. Thermodyn. 1970, 2, 95. (26) Sandler, S. I.; Lee, K.-H. A Proper Theoretical Basis for Local Composition Mixing Rules and a New Class of Activity Coefficient Models. Fluid Phase Equilib. 1986, 300, 135. (27) Sandler, S. I. The Generalized van der Waals Partition Function. I. Basic Theory. Fluid Phase Equilib. 1985, 19, 233. (28) Lee, K.-H.; Sandler, S. I. The generalised van der Waals Partition Function. IV. Local Composition Models for Mixtures of Unequal Size Molecules. Fluid Phase Equilib. 1987, 34, 113. (29) Sandler, S. I.; Lee, K.-H.; Kim, H. The Generalized van der Waals Partition Function as a Basis for Equations of State, Their Mixing Rules and Activity Coefficient Models. In Equations of State: Theories and Applications; Chao, K. C., Robinson, R. L., Jr., Eds.; ACS Symposium Series 300; American Chemical Society: Washington, DC, 1986; p 180. (30) Van Laar, J. J. On the Vapor Pressures of Binary Mixtures. Z. Phys. Chem. 1910, 72, 723. (31) Scatchard, G. Equilibria in Nonelectrolyte Solutions in Relation to the Vapor Pressures and Densities of the Components. Chem. Rev. 1931, 8, 321. (32) Mollerup, J. A Note on Excess Gibbs Energy Models, Equations of State and the Local Composition Concept. Fluid Phase Equilib. 1981, 7, 121. (33) Lee, L. L. A Molecular Theory of Solubility parameters: Generalization to Polar Fluid Mixtures. Fluid Phase Equilib. 1987, 35, 77. (34) Bernal, J. D. The Structure of Liquids. Proc. R. Soc. London 1964, A280, 299. (35) Lee, K.-H.; Patel, N. C.; Sandler, S. I. The Generalized van der Waals Partition Function. 3. Local Composition Models for a Mixture of equal Size Square-Well Molecules. Fluid Phase Equilib. 1986, 25, 31-49. (36) Hoheisel, C.; Kohler, F. Local Compositions in Liquid Mixtures. Fluid Phase Equilib. 1984, 16, 13. (37) Gierycz, P.; Tanaka, H.; Nakanishi, K. Molecular-Dynamics Studies of Binary Mixtures of Lennard-Jones Fluids with Differing Component Sizes. Fluid Phase Equilib. 1984, 16, 241.
(38) Hamad, E. Z. Exact Limits of Mixture Properties and Excess Thermodynamic Functions. Fluid Phase Equilib. 1998, 142, 163. (39) Ruckenstein, E.; Shulgin, I. Modified Local Composition and Flory-Huggins Equations for Nonelectrolyte Solutions. Ind. Eng. Chem. Res. 1999, 38, 4092. (40) Andiappan, A. N.; McLean, A. Y. Prediction of VaporLiquid Equilibria: Derivation of a new Expression for VaporLiquid Equilibrium Correlations. Can. J. Chem. Eng. 1972, 50, 384. (41) Sabarathinam, P. L.; Andiappan, A. N. A New Expression for Predicting Multicomponent VLE Data. J. Chem. Eng. Jpn. 1982, 15, 229. (42) Noda, K.; Ishida, K. A Modified quasi-chemical Theory based on Local Composition. J. Chem. Eng. Jpn. 1980, 13, 334. (43) Nishimura, Y.; Iwai, Y.; Arai, Y.; Nagatani, M. A Correlation of Ternary Liquid-Liquid Equilibria by a Modified Wilson Equation. J. Chem. Eng. Jpn. 1985, 18, 377. (44) Nagata, I.; Nagashima, M. Correlation and Prediction of Vapor-Liquid and Liquid-Liquid Equilibria. J. Chem. Eng. Jpn. 1976, 9, 6-11. (45) Prausnitz, J. M. Phase Equilibria for Complex Fluid Mixtures. Fluid Phase Equilib. 1995, 104, 207. (46) Wang, W.; Vera, J. H. Liquid-Liquid Equilibrium Calculations with Excess Gibbs Energy Models Based on Renormalization of Guggenheim’s Canonical Partition Function. Fluid Phase Equilib. 1995, 104, 207. (47) Sandler, S. I.; Fisher, J.; Reschke, F. Free Energies of Mixing. Fluid Phase Equilib. 1989, 45, 251. (48) Scatchard, G. Change of Volume on Mixing and the Equations for Non-Electrolyte Mixtures. Trans. Faraday Soc. 1937, 33, 160. (49) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (50) Staverman, A. J. The Entropy of High Polymer Solutions. Generalisation of Formulae. Recl. Trav. Chim. Pays-Bas 1950, 69, 163.
Received for review July 26, 2000 Revised manuscript received March 15, 2001 Accepted March 15, 2001 IE000693S
