Ind. Eng. Chem. Res. 1992,31, 2033-2039 Makhatadze, G. I.; Privalov, P. L. Partial specific heat capacity of benzene and of toluene in aqueous solution determined calorimetrically for a broad temperature range. J. Chem. Thermodyn. 1988,20,405-412. McAuliffe, C . Solubility in water of CI-Co hydrocarbons. Nature 1963,200,1092-1093. Molyneux, P.; Frank,H. P. The interaction of polyvinylpyrrolidone with aromatic compounds in aqueous solution. Part I. Thermodynamics of the binding equilibrium and interaction forces. J. Am. Chem. SOC.1961,83,3169-3174. Muller, N. Search for a realistic view of hydrophobic effects. Acc. Chem. Res. 1990,23,23-28. Nemethy, G.;Scheraga, H. A. Structure of water and hydrophobic bonding in proteins. I. A model for the thermodynamic properties of liquid water. J. Chem. Phys. 1962a,36,3382-3400. Nemethy, G.;Scheraga, H.A. Structure of water and hydrophobic bonding in proteins. II. Model for the thermodynamic properties of aqueous solutions of hydrocarbons. J. Chem. Phys. 1962b,36, 3401-3417. Payne, G. F.; Shuler, M. L. Selective adsorption of plant products. Biotechnol. Bioeng. 1988,31,922-928.
2033
Payne, G. F.; Ninomiya, Y. Selective adsorption of solutes based on hydrogen bonding. Sep. Sci. Technol. 1990,25,1117-1129. Payne, G. F.;Payne, N. N.; Nmomiya, Y.; Shuler, M. L. Adsorption of non-polar solutes onto neutral polymeric sorbenta. Sep. Sci. Technol. 1989,24,453-465. Pirkle, W. H.; Pochapsky, T. C. Considerations of chiral recognition relevant to the liquid chromatographic separation of enantiomers. Chem. Rev. 1989,89,347-362. Rebek, J. Molecular recognition and biophysical organic chemistry. Acc. Chem. Res. 1990,23,399-404. Street, I. P.; Armstrong, C. R.; Withers, S. G. Hydrogen bonding and specificity. Fluorodeoxy sugars as probes of hydrogen bonding in the glycogen phosphorylaae-glucose complex. Biochemietry 1986, 25,60216021. Tanford, C. The hydrophobic effect: Formation of micelles and biological membranes, 2nd ed.; Wiley-Interscience: New York, 1980.
Received for review December 9, 1991 Accepted April 29, 1992
Equation of State Mixing Rule for Nonideal Mixtures Using Available Activity Coefficient Model Parameters and That Allows Extrapolation over Large Ranges of Temperature and Pressure David S. H. Wong Department of Chemical Engineering, National Tsing Hua University, Hsin Chu, Taiwan 30043, R.O.C.
Hasan Orbey and Stanley I. Sandler* Center for Molecular and Engineering Thermodynamics, Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716
We demonstrate that a new mixing rule allows cubic equations of state to be used for a broad range of nonideal mixturea which previously could only be described by activity coefficient models. Further, we show that there is no need to recorrelate phase equilibrium data to do this; activity coefficient model parameters currently reported, for example in the DECHEMA Data Series, can be used directly in our model. Perhaps most important is that we also find that we can use the parameters in our model obtained from one low pressure-low temperature isotherm to make accurate predictions a t conditions which are hundreds of degrees and hundreds of bars above the experimental data used to obtain those parameters. Consequently, this new model provides a way of being able, with confidence, to use data for nonideal mixtures obtained at moderate laboratory conditions for design a t harsh processing conditions.
Introduction Traditionally, there have been two classes of thermodynamic models for phase equilibrium calculations: equations of state and activity coefficient (free energy) models. The ranges of applicability of these models are schematically illustrated as regions in Figure 1,where the axes are density and complexity of the mixture represented by the exceas free energy in a high-density state. The virial equation can be used to describe any mixture (providing virial coefficients are available), but only at low densities. Activity coefficient models can also be used to describe mixtures of any complexity, but only as a liquid well below its critical temperature. Cubic equations of state, with the usual van der Waals one-fluid mixing rules, can be used at all densities, but only for relatively simple mixtures (i.e., hydrocarbons or hydrocarbons and inorganic gases) with small to moderate excess free energies. As is evident from the figure, there has not been a satisfactory thermodynamic model for a large portion of *Towhom all correspondence should be addressed.
the molecular complexity-density range. As a consequence, the proper description of the phase behavior of other than hydrocarbon mixtures over large ranges of temperature and pressure was either not possible, or possible only with difficulty and numerous parameters which may be temperature dependent. Equations of state applicable to both vapor and liquid phases have obvious advantages over activity coefficient models, especially when one is interested in large ranges of temperature and pressure. In particular, one does not have to worry about hypothetical and supercritical components or standard states, the model has built-in temperature and pressure dependencies, and since the same model is used for all phases, an equation of state will predict a critical point. Further, other thermodynamic properties, such as enthalpy, can also be computed simultaneously and in a consistent way. While simple cubic equations of state can describe pure fluids reasonably well, they give satisfactory descriptions only for relatively simple mixtures. It has long been recognized that the source of this difficulty must be the mixing and combining rules which are used to obtain the
0888-588519212631-2033$03.O0/0 0 1992 American Chemical Society
2034 Ind. Eng. Chem. Res., Vol. 31, No. 8, 1992 6-6
-
rehe
compressed liquid
CI~SIIC~I aclivity coellicirnt (excess free energy) models
T
1 density
h
I
.q".tIO..
____------
With "4% rnl"l.1
___-------
c1.111.11
ol mt* ."le.
specific to each fluid. We refer to this as the PRSV equation of state. The most common method of obtainingequation of state parameters for a mixture from those of the pure components is to use the van der Waals one-fluid mixing rules
c1111caI density
a = CCx,xlall
(6)
b = CCx,xlb,
(7)
and
1 dilule gas
y-0 r e g m e
virial equation of 1.1~
with the combining rules a, = (a,,a,)05(1 - k,)
(8)
and
b, =
Si(b11
+ 41)
(9)
When this is done, the equation of state can describe mixtures such as hydrocarbons, but not mixtures of polar and other nonhydrocarbon chemicals. A justification for the van der Waals one-fluid mixing rules comes from the known composition of the second virial coefficient. In particular expanding, for example, eq 1 in density we obtain that B = b-a/RT
(10)
and since it is known from statistical mechanics that for a mixture
B(T,x,) = CCx,xlB,(T)
(11)
it follows that eqs 6 and 7 ensure that eq 11 is satisfied. However, they are not the only choices which do this, as we will discuss shortly. Much effort in recent years has been directed toward developing alternate mixing rules, and it would not be productive to review here the enormous amount of work done in the past two decades, or the limited success. We will mention only two classes of alternative mixing rules here, both of which use eqs 7 and 9 for the "b" parameter. The first mixing rule is that of Panagiotopoulos and Reid (1986) and others (Stryjek and Vera, 1986; Adachi and Sugie, 1986; Schwartzentruber and Renon, 1991) which, in ita density independent form, replaced the binary interaction parameter k,] of eq 8 with
k, = X J l J + x,m,
(12)
The successes and problems with this mixing rule have been discussed elsewhere (Michelsen and Kistenmacher, 1990; Mathias et al., 1991). Huron and Vidal (1979) developed a mixing rule by requiring that the excess Gibbs free energy (GE)at infinite pressure (or V = b) computed from an equation of state be equal to that computed from an activity coefficient model resulting in a = b[Cx,(a,/b,) - aGE]
(13)
where u is a numerical constant which depends on the particular equation of state used. While this model, and ita variations, has been successfully used for some complex mixtures, it is no better than the van der Waals one-fluid mixing rule for hydrocarbon systems (Shibata and Sandler, 1989). Further, both the Panagiotopoulos-Reid and Huron-Vidal mixing rules suffer from the fact that they do not satisfy the second virial coefficient boundary condition, eq 11. Recently, Wong and Sandler (1992) have shown that it is possible to satisfy the second virial condition and ala0 demand that the equation of state predict the same excess
Ind. Eng. Chem. Res., Vol. 31, No. 8,1992 2038 Table I. Correlation of Vapor-Liquid Equilibria
EOS + AE model
activity coefficient model
T,K
NRTL APWb Ay
van Laar
AP%
NRTL
Ay
AP%
Type 1. Nonpolar
3.49 3.99 1.45
Van Laar
AP%
Ay
Ay
+ Polar:
kij
NRTL
VanLaar
datasource'
n-Pentane-Acetone 0.0068 3.16 0.0118 0.0056 3.89 0.0107 0.0052 1.74 0.0086
0.190 0.198 0.203
0.192 0.199 0.207
1/3+4/188 1j3+4jlSS 1/3+4/19O
238.15 258.15 298.15
1.49 1.40 0.42
0.0028 0.0029 0.0031
2.16 2.48 1.21
0.0136 0.0109 0.0081
298.15 323.15 348.15 393.15
0.89 1.85 0.91 1.58
0.0047 0.0056 0.0043 0.0050
0.91 1.19 0.83 1.76
Type 2. Hydrogen Bonding: Ethanol-Water 1.88 0.0080 0.0047 2.03 0.0080 0.0055 1.15 0.0070 0.0059 1.48 0.0039 1.04 0.0044 1.03 0.0061 0.0038 0.0054 2.28 2.46 0.0062
0.236 0.249 0.239 0.234
0.232 0.243 0.239 0.236
lllbJ108 lJlbJ106 lJlbJ107 lllbJ93
298.15 311.65 328.15
2.75 0.50 2.69
0.0096 0.0137 0.0128
Type 3. 5.39 0.60 8.38
Hydrogen Bonding + Nonpolar: Methanol-Cyclohexane 0.0292 6.83 0.0159 10.39 0.0493 0.332 0.0259 0.49 0.0274 0.41 0.0289 0.367 0.0394 7.18 0.0370 3.21 0.0341 0.365
0.353 0.373 0.375
1/2cJ208 1J2c1209 1J 2a1242
298.15 308.15 318.15 328.15
1.32 0.48 1.08 0.61
0.0086 0.0031 0.0038 0.0033
Type 4. Lewis Base + Acidic Proton: 0.0083 0.0086 1.31 1.32 0.48 0.0031 0.69 0.0043 1.56 0.0044 1.08 0.0038 0.0040 0.61 0.0033 1.22
-0.085 -0.123 -0.120 -0.123
1/3+4/98 1/3+4/94 1/3+4/95 1/3+4/96
303.15 323.15 343.15
0.75 0.88 0.96
0.0073 0.0075 0.0085
0.56 0.47 0.52
-0.048 -0,039 -0.032
1/I1409 1/7/411 1171413
Acetondhloroform 1.31 0.0084 -0.085 0.69 0.0042 -0.123 1.54 0.0044 -0.121 0.0039 -0.122 1.24
Type 5. Aromatic: Hexafluorobenzene-Toluene 0.75 0.0074 0.54 0.0055 0.0054 0.0070 0.52 0.0047 0.0047 0.71 0.71 0.0068 0.51 0.0045 0.0053
-0.049 -0.041 -0,034
"Data taken from Gmehling and Onken (1980);numbers correspond to the pages in the order of volumeJpartJpage. order to compare results with that in the DECHEMA Data Series without recalculation, but yet to report results in percent (rather than mmHg as reported in the DECHEMA Data Series), we have defined hp% as the average absolute deviation in pressure divided by the arithmetic mean of the pure component vapor pressures.
Helmholtz free energy at infinite pressure AWEas a function of composition as is obtained from an activity coefficient model with the following mixing model:
b, =
(14)
and (15) where u, which is dependent on the equation of state used, is equal to [h(P.5- 1 ) ] / p for the Peng-Robinson equation of state and x is a vector of compositions. Also,
( - &) + - &) (bj
bi
( b - &)ij
=
2
('
- kij)
(16)
where kij is now a binary interaction Parameter for the second virial coefficient. The Appendix contains a disamion Of the Why -4 the Helmholtz free energy is to be preferred over the Gibbs free energy, and the relevance of the second virial coefficient interaction parameter. The fugacity Coefficient eXpreSSiOnS which reSdt from the miring model have been given elsewhere (WOW and Sandler, 1992).
Test of the New Mixing Model Using Existing Activity Coefficient Parameters We have already shown that the equation of state mixing model preaented above can be used to accurately correlate vapor-liquid equilibrium (VLE) data for a range of systems
(Wong and Sandler, 1992). However, a large amount of low-pressure VLE data have been compiled and correlated using activity coefficient models; the DECHEMA Data Series (Gmehling and Onken, 1977)is the best example of such a compilation. Here we show that such currently existing parameters can be used directly to make equation of state predictions. That is, there is no need to regress once again existing data for use with equations of state; the activity coefficient parameters in the DECHEMA Data Series can be used directly in equations of state with our mixing model. Further, we show in the next section that we need activity coefficient model parameters from only one low-pressure isotherm (or isobar) to make good phase equilibrium predictions at conditions as much as 200 "C higher in temperature and at much higher pressures. To test our model for a spectrum of mixtures representing varying degrees of complexity, we first identified the five general classes of mixtures in Table I and then, for each class, chose a representative mixture for which thermodynamicallyconsistent data at several isotherms appear in the DECHEMA Data Series. For simplicity, we then considered only two activity coefficient models, the NRTL (Renon and Prausnitz, 1968) and Van Laar (Sandler, 1989) models. Table I shows the mixtures we considered and the reported errors in the correlation of pressure and mole fraction when these activity coefficient models were fit to the experimental data as reported in the DECHEMA ~~hseries. Next, in two separate calculations, we used the same NRTL and vanLaar models and parameters for AE with the PRSV equation of state and our mixing rule. This leaves only the second virial coefficient binary parameter unspecified. We chose ita value so that the excess Gibbs free energy calculated from the equation of state at the reported temperature and pressure would match as closely as possible to the excess Gibbs free energy calculated from the activity coefficient model. [The need for a value of kij arises from the distinction between the excess Helm-
2036 Ind. Eng. Chem. Res., Vol. 31, No. 8, 1992
holtz free energy at infinite pressure and the excess Gibbs free energy at low pressures. This point is discussed in the Appendix.] Using these parameters, we obtain the equation of state results shown in Table I. There are two important observations to be made from this table. First is that the results from the activity coefficient model directly and from the equation of state which incorporates this same model and parameters are comparable, and in both cases the deviations from the experimental data are quite small. Thus we can conclude that, with our mkhg rule, simple cubic equations of state can be used with about the same degree of accuracy as activity coefficient models for many different types of mixtures. Further, precisely the same excess free energy parameters can be used. Therefore, the large body of correlated data and parameter values already available in the DECHEMA Data Series and other sources can be used directly in our mixing model. No refitting of VLE data is needed! The second observation is that the value of the second virial coefficient binary interaction parameter, kij, is approximately a constant for each binary mixture, reasonably independentof both temperature and the activity coefficient model used. We will exploit this observation in the next section. There is one point about our mixing model that we should clarify. At first glance it appears that our model contains all the parameters in the excess free energy model as well as the additional kij parameter. In fact, the model contains no more information than that in the free energy model. That is, we use the reported GE function and parameters directly in AE at infinite pressure, and then ad'ust kij to ensure that we also get essentially that same G at the experimental conditions. For example, if the two-parameter Van Laar model was used for both the activity coefficient correlation of data and in our mixing rule, our three parameters are gotten from only the two reported Van Laar parameters, not from any additional data or correlation. In the example here the Van Laar parameters are identical, and the kij parameter is calculated as discussed above. Thus, if one is satisfied with the accuracies reported here, the mixing model has only the same number of parameters as the imbedded activity coefficient model. It is only if one is interested in even higher accuracy that kij should be treated as an adjustable parameter, in which case the mixing model does have one more parameter than the free energy model.
1
0 '2
02
0 4 06 ti8 MOL FRACTION ACETONE
10
Figure 2. Acetone-water binary system. Points are data of Griewold and Wong (1952) and lines are predidiona with Van Laar parameters from DECHEMA (l/la/194) at 373 K, kI2 = 0.270. 6 -
5 4 -
d
Extrapolation of Vapor-Liquid Equilibrium Using the New Mixing Model As equations of state have a built-in temperature dependence, and since we found that the second virial coefficient binary interaction parameter kij was largely temperature (and model) independent, our next test was to determine whether we could use our mixing rule and an equation of state to extrapolate vapor-liquid phase behavior over large ranges of temperature and pressure. The procedure we used is as follows. We considered four mixtures for which VLE data were available over a large range of temperature and pressure, and also for which thermodynamically consistent data and activity coefficient model parameters were reported at low pressure in the DECHEMA Data Series. We then used the reported activity coefficient parameters for only one isotherm (or in one case along an isobar) to obtain a value of k- using the procedure described in the previous section, ancfthen used this fixed parameter value to predict the phase behavior at other temperatures and pressures. The results are shown in Figures 2-7. For these calculations we have used only the Van Laar model.
v
1
1 00
02
0 4 06 08 MOL FRACTION METHANOL
10
Figure 3. Methanol-benzene binary system. Points are data of Butcher and Medaui (1968) and lines are predictions with Van Laar parametere from DECHEMA (1/2a/225) at 373 K; k12 = 0.308.
Figure 2 shows the P z - y for the acetonewater system at elevated temperatures. The experimental data are from Griewold and Wong (1959). The parameters of the Van Laar model we used are those reported in DECHEMA Series for correlating the 100 OC isotherm of the Griswold and Wong data The lines are the equations of state + mixing rule predictions at higher pressures and temperatures. As can be seen, the agreement between prediction and experiment is very good, indeed, much better than the resulb of previous equation of state studies of this mixture (Michelsen, 1990). Of come, had we correlated the data at each temperature (resulting in
Ind. Eng. Chem. Res., Vol. 31, No. 8, 1992 2037 [
"
"
"
"
'
I
7 1
02
00
04
06
08
10
00
MOL FRACTION METHANOL
Figure 4. Methanol-water binary system. Points are data of Griewold and Wong (1952)and lines are predictions with Van Laar parameters from DECHEMA (1/1/49)at 373 K;k12 = 0.084.
0 2
0 4 06 08 MOL FRACTION ACETONE
10
Figure 6. Acetone-methanol binary system. Points are data of Griswold and Wong (1962)and lines are predictions with Van Laar parameters from DECHEMA (1/2a/77)at 373 K;klz = 0.102.
I " " " " ' 1 2
100 8 7 6
100
5 4
6
cc 4
m
5
10
6 5 5
1
Y
4 1
4
00
02
0 4 06 08 MOL FRACTION ETHANOL
10
00
"
"
"
"
02 0 4 06 08 MOL FRACTION 2-PROPANOL
'
10
Figure 5. Ethanol-water binary system. Points are data of BarrDavid and Dodge (1959)and linea are predictions with Van Laar parameters from DECHEMA (1/1/176)at 4.13 bar: klz = 0.211.
Figure 7. 2-Propanol-water binary system. Points are data of Barr-David and Dodge (1969)and lines are predictions with Van Laar paramekere from DECHEMA (1/1/336)at 4.12 bar;klz = 0.326.
temperature-dependent parameters), rather than made predictions as we have done here, the agreement would have been even better. Figure 3 shows similar results for the methanol + benzene system measured by Butcher and Medani (1968). The Van Laar parameters, taken from DECHEMA, are based on data of Schmidt (1926)at 100 "C. This system was ale0 examined recently by Michelsen (1990)using an equation of state with less success than found here. The results for the methanol + water system (Griswold and Wong, 1959)are shown in Figure 4, with predictions based on the parameters reported in the DECHEMA Data Series for the 100 OC isotherm. Figure
+ water system (Barr-David and Dodge, 1959)using Van Laar Parameters obtained by correlating 4.13-bar isobar data of O t h e r et al. (1951),as reported in the DECHEMA series. Our results for this system are much better than predictions obtained earlier using an equation of state by others (Heidemann and Kokal, 1990;Michelsen, 1990). The results for the acetone-methanol system are shown in Figure 6. For this case Van Laar parameters are taken from the correlation of 100 OC isotherm reported in DECHEMA (Gmehling and Onken, 1977). Finally, in Figure 7,we show results for the 2-propanol-water system measured by 5 shows the results for the ethanol
2038 Ind. Eng. Chem. Res., Vol. 31, No. 8, 1992
The next obvious question is how to proceed in the absence of any experimental data. For many mixtures, group contribution methods such as UNIFAC (Fredenslund et al., 1977) and others can be used to make reasonably good activity coefficient model predictions of vapor-liquid and liquid-liquid equilibria, but only over limited ranges of temperature and pressure. Much effort has been devoted to extend those group contribution methods so that they can be used over wide ranges of temperature and pressure. The mixing model/equation of state approach used here provides a method of doing this, and we are currently working on this.
0 450 GEX AT 1 BAR, AEX AT 1 & l 000 BARS
Acknowledgment 0 0 0 0 ~' 00
' 02
' ' ' ' ' 0 4 06 08 MOL FRACTION METHANOL '
\\ '
10
Figure 8. Calculated excesa Gibbs (---I and Helmholtz (-1 free energies of mixing for the methanol + benzene mixture at T = 373 K. On the scale of this figure the exceee Gibba free energy at P = 1 bar and the excess Helmholtz free energies at 1 and lo00 bar are
identical.
Barr-David and Dodge (1959), and predictions are based on Van Laar parameters obtained by correlating 4.1-bar isobar data of Wilson and Simons (19521, reported in the DECHEMA Data Series. We find it remarkable that so simple an equation of state such as the PRSV equation used here, and our mixing model with temperature-independent AE and kij parameters, can lead to such accurate predictions at temperatures as much as (and perhaps more than) 200 OC higher and pressures as much as 200 bar higher than the data used to obtain these parameters.
Conclusion In a previous paper we have developed a mixing rule which allows one to combine equations of state and activity coefficient (Gibbs free energy) models in a simple, theoretically correct way which greatly extended the range of application of equations of state. Here we have made two further observations which establish that this new mixing rule is even better than we had expected. First, we have shown that there is no need to recorrelate all existing vapor-liquid equilibrium data to get the parameters in our mixing rule. One can merely take already published parameters, for example from the DECHEMA Data Series, and with a simple calculation to obtain the kij parameter, have all the information necessary to describe the mixture with an equation of state. The results obtained this way are of comparable accuracy to correlating the phase behavior directly with an activity coefficient model. Second, and more important, we have shown here with six examples for highly nonideal mixtures, once the mixing rule parameters are obtained in this way at a single temperature (or pressure), the results can be extrapolated with very good accuracy over 200 "C in temperature and hundreds of bars in pressure. Thus, the equation of statemixing rule combination presented here also provides the first reliable method of extrapolating phase behavior for mixtures which traditionally had to be described by activity coefficient models over wide ranges of temperature and pressure. Consequently, by using this thermodynamic model, it appears that one can measure phaee behavior for a mixture at conditions which are convenient and safe in the laboratory, and then extrapolate with accuracy and confidence to extreme processing conditions.
D.S.H.W. thanks the National Science Council, Taiwan, ROC, for providing the financial support (Grant No. 29041F) for his visit to the University of Delaware during which this work was completed. This research was supported, in part, by Grant No. DE-FG02-85ER13436from the U.S. Department of Energy and Grant No. CTS89914299 from the National Science Foundation, both to the University of Delaware.
Nomenclature A = molar Helmholtz free energy a = equation of state energy parameter B = second virial coefficient b = equation of state excluded volume parameter G = molar Gibbs free energy k, I, m = binary interaction coefficients P = pressure R = gas constant T = temperature V = molar volume x = mole fraction x = vector of mole fractions Greek Letters a = temperature-dependent equation of state parameter K , K ~ K~ , = constants in a function Q
w
= constant which depends on equation of state used = acentric factor
Superscript E = excess property Subscripts c = critical property i , j = molecular species
m = mixture r = reduced property = infinite-pressure state
Appendix. Distinction between Escess Gibbs and Helmholtz Free Energy Models The molar excess Gibbs and Helmholtz free energies of mixing at constant temperature and pressure are related through the molar excess volume of mixing by GE(T,P,x)= AE(T,P,x) + PP(T,P,x)
(A.l)
The exvolume, especiallyaway from the critical point, is generally very small, so that at low pressure, say P = 1 bar, to an excellent approximation we have that GE(T,P=l bar,x) = AE(T,P=l bar,x)
(A.2)
Next we must examine how GEand AE vary with pressure. This is shown in Figure 8, where we have computed GEand AE for the methanol + benzene system using the PRSV
Ind. Eng. Chem. Res., Vol. 31, No. 8, 1992 2039 equation of state and our mixing rule at T = 373 K for P = 1 and lo00 bar. We see from the results that AE is essentially independent of pressure while GE is much more pressure dependent. Further we see from the figure, we can write GE(TQ=lbar,x) = AE(TQ=lbar,x) = AE(TQ=high pressure) (A.3)
which provides the justification for using exactly the same funtional form for AE at infiite pressure as has been used for GE at low pressures. Further, it also explains why model parameters obtained from low-pressure data can be used, without change, at high pressures in AE. Since we have also shown that GE is a function of pressure, it follows that GE at 1bar is not equal to GE at high pressurea, which exposes an underlying theoretical problem with the Huron-Vidal mixing rule. They tried to avoid this by demanding that VE be zero at infinite pressure, i.e., using eq 9, but this result$ in the violation of a second virial coefficient condition and does not remove the pressure dependence of GE. The fact that GE at low pressure and that at high pressure are different also explains why the parameters in the Huron-Vidal equation of state model are found to be different from thase when the imbedded activity coefficient model is used directly. However, when AE, which is essentially pressure independent, is used as is the case here, the same parameters can be used. Next, to obtain our mixing rule, we required that the excess Helmholtz free energy computed from the equation of state at infinite pressure equal that of an activity coefficient model. Since the excess Gibbs free energy is a function of pressure, as we have already shown,this does not necessarily ensure that GE at low pressure computed from an equation of state will be the same as that from the activity coefficient model. However, by adjusting the value of the kij parameter, we can assure this. Further, the value of kij we find is largely independent of model or conditions used. Its magnitude could be related to that of the excess volume term. Perhaps by equating free energies from an equation of state and an activity model at low pressures, rather than at infinite pressure as we have done here, it might be possible to avoid the use of the kij parameter. However, at finite pressures the molar volumes of the components and mixtures would have to be computed and would then appear in the mixing rule (since then Vi is not equal to bi), and difficulties would arise with supercritical species or components above their boiling point at the temperature of interest. It is simpler to avoid these complexities by using the infinitepressure limit as we have done here, and computing the value of kij as we have described.
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Received for review March 18, 1992 Accepted May 25,1992
