Ind. Eng. Chem. Process Des. Dev. 1984, 2 3 , 251-258
25 1
Improved Calculation of Infinite Dilution Activity Coefficients by the Solution of Groups Model for Nonpolar Compounds in Water Introducing an Additional Group Type Andreas Rlzzl and Josef F. K. Huber' Institute of Analyficai Chemistry, University of Vienna, A- 1090 Vienna, Austria
In order to obtain universal interaction parameters between the groups CH2-H20 and ACH-H,O within the solution of groups model, a careful estimation is required for In yls,the term not associated with groups. A new approach is presented to account for association effects in the region of infinite dilution which is based on the definition of two additional groups which only occur in alkanes and aromatic hydrocarbons, respectively. This procedure aiiows one to keep constant the value for the most important CH,-H,O and ACH-H20 interaction parameters in all s stems and gives results similar to those expected by the introduction of an additional correction term, In yicoR;!
Introduction The calculation of activity coefficients of organic compounds in liquid mixtures has become an important tool in chemical engineering. It was investigated intensively during the past few years in order to find an alternative to the experimental evaluation of phase equilibria. If systems have to be treated for which no or not many experimental data are available, it is advantageous to use models which are based on additive contributions of structural units, which are called groups. Much effort was directed to the calculation of activity coefficients by the solution of groups model developed by Wilson and Deal (1962), since it allows easy generalization to multicomponent liquid mixtures. Especially, calculations of vaporliquid equilibria by means of this model have been carried out during recent years by Fredenslund and co-workers (1975, 1977a,b). This paper deals with calculations of activity coefficients in the region of infinite dilution of one component in liquid mixtures. Although this region is known to be especially tedious, it is of special importance for problems which are common in analytical chemistry, especially in chromatography. Our interest in partition data of organic solutes in liquid-liquid systems under conditions used in chromatographic separation processes makes it necessary to deal with the region of high dilution. Model The basic assumptions and the equations of the solution of groups model have been repeatedly described in the literature (Wilson and Deal, 1962; Derr and Deal, 1969; Fredenslund et al., 1975) and are well known. The activity coefficient is built up by two terms In yi = In yiG + In y?
(1)
One of them, In yiG,is associated with group contributions and counts mainly for the effects which arise from the interaction between groups. The other one, In yp,should account for entropic effects but is mainly restricted to contributions which arise from differences in the size and shape of the molecules. Ln yiGis built up additively by incremental contributions, the group-activity coefficients, rk, which are functions of the group-fraction vector in the mixture, X , the interaction matrix between the groups, A , and temperature, T In rk = ~ G ( X , A , T ) (2) where X = {Xl], A = (A(,), and in the particular case of the
UNIFAC equation also a function of the group surface area, Ql. For In yis equations are used in general, which give this contribution as a function of the molecular volumes and the mole fractions In Y? = f ~ ( x , v ) (3) where x = (xi),vi = (vi),and in particular cases also a function of the molecular surfaces. Within this paper all calculations are performed by the ASOG equation (Derr and Deal, 1969) using parameter sets which were optimized to the region of strong dilution. This optimized parameter matrix is given in Table I. For the evaluation of the size contribution, In y?, a Flory-Huggins equation is used according to the ASOG model. The molecular volumes, ui, were built by summation over group volume increments, Rk,which are listed in Table 11. In a few cases, the ASOG results are compared with results of the UNIFAC equation (Fredenslund et al., 1975) using parameter sets from literature. Problems. The solution of groups model provides a formalism to predict the solution behavior of many compounds by the knowledge of the interaction behavior of a few structural units. In other words, predictions of data of compounds are essentially extrapolations from other compounds which are homologous with respect to certain groups, e.g., CH2, aromatic CH (ACH), COH, CO, NH,, etc. A basic limitation in the formalism of the solution of groups model is the required additivity of the group contributions. Compounds homologous in the groups COH, NH,, and CO (polyols, polyamines, and polyketones) generally do not fit to this condition. The most important feature of the model lies therefore in the interpolation and extrapolation power with respect to the number of CH2 and ACH groups and eventually some other groups where dispersion interaction is dominating. The proper evaluation of the group activity coefficients of CH2 and ACH seems therefore of primary importance. This means that those parameters which describe the interaction with CH2 and ACH groups have to be specially accurate and in a high sense universal. This fact is more important, the greater the influence of the group interaction parameters on the value of FCHPand rACH is. The largest contributions arise from CH2-H20 interaction parameters. to rCH2 Universal parameters can only be obtained if the contribution to the activity coefficient which is not associated with group interaction, In y?, is estimated carefully. It is shown in this paper that different systems containing
0196-4305/84/1123-0251$01.50/00 1984 American Chemical Society
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Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 2, 1984
Table I. Structural Increments and the Corresponding Group Interaction Parameter Matrix Used in All Calculations with ASOG Equation at T = 298 K‘ CH, CH , CH;* COH
co
H,O ACH* ACH
1.o
1.o 0.00668 0.0518 0.00001 1.1861 1.1861
CH,
COH
co
H,O
ACH*
ACH
1.0287 1.0287 1.o 0.7844 0.15158 1.6218 1.6218
0.08083 0.08083 0.44024 1.0 0.02262 0.91049 0.9 1049
0.3026 0.19540 2.2144 3.5070 1.o 0.6078 0.72626
0.7 1126
0.71126 0.71126 0.02009 0.15597 0.0001 34 1.o 1.o
*
1.o
1.o 0.00668 0.0518 0 .OO 069
-
1S 8 6 1
Table 11. Structural Increments Used in the Calculations and Their Group-Volume Parameters, R k , and Group Surface Area Parameters, Qk, According to Fredenslund e t al. (1975) CH,
Rk Qk
0.75‘ 0.643’
COH
CO
1.20 1.124
0.77
H,O
ACH
CH,*
ACH*
0.92 1.0Ob 1.40
0.53 0.40
0.75 0.643
0.53 0.40
Average values for the CH, increments in n-heptane. Value determined by optimization.
a
the group species CH2 and HzO, and ACH and H20, respectively, may lead to different values for the interaction parameters if one applies the simple formalism of the group of solution model and the usual equations for In 7:. So a new approach is suggested for the region of infinite dilution in order to achieve a high universality for these most important parameter pairs. Theory Adjustable Parameters in the Equation for In y?. In the usual formalism described in the literature the following characteristics have to be considered. (i) Only the term In yiGdeals with a solution of groups whereas In 7: deals with a solution of molecules. This means that effects which are not dependent on the group fraction in the mixture but rather on the mole fraction should be treated by the second term. Effects which are specific to a particular system, e.g., association, are moleor volume fraction-dependent. (ii) The use of the Flory-Huggins equation for athermal mixtures (Flory, 1953; Huggins, 1942) or the use of the combinatorial part of the UNIFAC equation (Fredenslund et al., 1975) fixes this contribution without variability. In the common solution-of-groups concept, all variable parameters are included in the first term and therefore associated with the group fraction scale. This lack in variability generally creates systematic errors in such cases, where the first term has to correct errors of the second term and further on has to account for effects which cannot be taken into consideration by the usual y? equations. In those cases the resulting regression parameters for the group interactions are not really group-specific variables alone but describe in addition effects specific for particular systems. Parameters determined in such systems are consequently not applicable universally. This point is of special importance where specific effects are large, for instance in mixtures in which the components show a strongly different tendency of association or in which complex formation takes place. To overcome this problem eq 3 can be modified by the introduction of additional parameters, B,, which are specific for the particular systems or specific for the particular components. In 7,s= fs(x,v,B) (4) where B = (B,]. Such an additional set of parameters, B,, has already been introduced by Donohue and Prausnitz (1975) and
-
0.02009 0.15597 0.00214 1.o 1.o
Ronc and Ratcliff (1975). In both cases functions fs of the Flory-Huggins type were used. Kikic et al. (1980) introduced a modification analogous to the Donohue-Prausnitz concept in the combinatorial part of the UNIFAC equation. Equation 4 is especially useful in a model working with group contributions if the molecular or system specific parameters, Bi, can easily be obtained either from molecular data or from increments or if the parameter values do not change very much within a series of compounds, e.g., for homologous compounds. This aspect is subjected to a numerical investigation later on using the modification of Donohue and Prausnitz. Although the Donohue and Prausnitz concept is reported to lead to significant improvement in systems where In y I Gis small (Kikic et al., 1980), it is not the approach to solve the problems of proper accounting for association effects. The modification of Ronc and Ratcliff (1975) multiplying In y? by a variable factor needs the knowledge of an optimum factor B, for each binary system. It is therefore somehow difficult to work in practice with this concept within an increment system. The equations of Renon and Prausnitz (1967a,b) have been developed to describe the thermodynamic quantities in binary mixtures where the polar component forms linear chains by self-association in mixtures with nonpolar partners. An incorporation of this concept to the solution of groups model needs much effort in determining all association parameters necessary for a widely spread data reduction. Another way to bring additional variability into the solution of groups model is the introduction of a correction term in eq 1which explicitly counts for effects specific to particular systems. In yi = In
yp + In 7: + In y?ORR
(5)
Generally, it is useful to introduce this term, In ytCoRR, where no proper correlation of data can be obtained by the original equations. In the case of association, In ylCoRR must be a function of temperature, as can be seen from Figure 1. where C = {Cil...,e.g., “associationparameters” or analogous constants. Such a third term has already been introduced by Oishi and Prausnitz (1978) for a free-volume correction treating solutions of macromolecules and by Neau and Peneloux (1979) calculating vapor-liquid equilibria in associating mixtures of aliphatic alcohols and alkanes. Among the various mixture specific effects, the changes in self-associationof one component occur in simple binary systems and should be accounted for because of the numerical importance. Because of the great effort necessary for a widely spread use of the mentioned association corrections within an increment model, this paper presents a new approach, which corrects for effects of association
Ind. Eng. Chem. Process Des. Dev.. Vol. 23, No. 2, 1984
'"3;;
In y*:-
= In y:
+ h(rsm/rrm) - C v k , In ( r k ' * / l ? k l ) k
253
-
In (r;*/r;*) (9) k = 1, ..., r, ..., n excluding s Furthermore, the differences in the concentration variables of yyoRRand rk do not appear in binary systems if only infinite dilution is concerned so that In y,CO" = In (rs-/r,-)-CVk, In ( r k ' * / r k ' ) - In (r;*/r;*) (10) k
2
4
6
8
1
0
9m2
Figure 1. Experimental activity coefficients In yc- and calculated contributions In y p and In ycwT- of homologous n-alkanes in water at infinite dilution in dependence on the carbon number of the molecules, vCH Measurements of 7,- were carried out at temperatures of 289 K ?+, Pierotti et al., 1959) and 298 K (-+-, McAuliffe, 1966). Calculations of In y p (--Am-)are performed by use of the Flory-Huggins equation with values of RCH = 0.75 and R H , = 0.92. The differences between In y and In y p 2at 289 K (-- 0- -) and 298 K (--+ --) are the In ychT' curves which are extrapolated to
.
YCH2 e
0.
in the region of strong dilution, where this problem is very important. Additional Compound-Type Specific Groups. This new approach suggests the definition of additional groups which only occur in certain types of compounds, namely alkanes and aromatic hydrocarbons (PAHs). These compound-type specific groups have their own set of interaction parameters which counts for effects specific to certain systems. This approach is simple to handle since it uses the known analytical function for In,:y fG, instead of the unknown function for a correction term In y y O R R , fCORR, and does not leave the incremental contribution philosophy of the solution of groups model. It is easily applicable to multicomponent mixtures and does not narrow the generality of this method at all. There are only a few new parameters which have to be determined, namely the "interaction parameters" between water and the new groups; this is done by the usual procedures common to each data reduction. It is useful to compare the effect of defining a new compound type specific group, s, of the solute with the theoretically more reasonable way of introducing an additional term. It can be shown that both methods lead to equivalent results in the case of infinite dilution. Introduction of a correction term leads to In y l = In y: + In y,G + In y,CORR (7) 1n 7 : = Cvk, In rk - Evkr In r k ' (74 k
k
k = 1 , ...,r ,..., n Defining a new group, s, of the solute instead of a group of type r and marking this new group mixture by an asterisk gives In y, = In y: + In y,G* (8) In y,G* = C V k l In rk*+ In (rs*/rp*) In rk[*k
k
In (I';*/I';*) k = 1, ..., r, ..., n excluding s At infinite dilution, rk* rkso that
-
(8a)
k = 1, ..., r, ..., n excluding s In the general case, In yy0" will change with the species i because of the dependence of rkt*and r;* on the group composition of the solutes and therefore C, in eq 6 will be a molecular parameter. In the particular case where the group s has the same interaction behavior as the group r with all groups of the considered solute i, eq 10 reduces to In y,CORR- = In (rs-/rT-) (11) In this particular case C, is constant within a homologous series and yyoRRm is independent of the solute i, so that it can be concluded that in the limiting region of infinite dilution the definition of a compound-type specific group either in the solute or in the solvent give results in regression analysis of binary mixtures equivalent to the introduction of a correction term In y,CoRR. Calculation Algorithm. The nonlinear regression procedure within these calculations has been performed by use of Monte Carlo, SIMPLEX, and Gradient algorithms in this consecutive order (James and Poos, 1971). The functions which have been minimized are the following F(7) = C((r,-,eW- yl-D'calcd) / r , - m ~ t l ) 2 (12) or
F(K)= F,,)
+ E((K,m,exptl - K,m~cdcd) /Kzm~exptl)z (13) I
respectively. Results and Discussion Evidence for Systematic Errors with the Usual Equations. Regression analysis of experimental data (McAuliffe, 1966; Pierotti et al., 1959) of homologous nalkanes in water using the ASOG equation described before shows that a satisfactory correlation of the limiting activity coefficients cannot be obtained by the common formalism of the model if n-alkanes are defined as consisting of CH2 groups only. The experimental 7,-values and calculated contributions in such systems are shown in Figure 1. The differences between the measured In 7,-curves and the calculated In -y: curve give the In yFESTm curves. According to the solution of groups model, these In ypESTm curves should be equal to the function In yIG = V C H , In IICH2, representing a straight line through the origin. %he slope of this line is determined by the two interaction parameters between the CH2 and H 2 0 groups. As can be seen, In yFESTis, as predicted, a linear function of vCH2,, but does not pass through the origin. The magnitude of the intersection at V C H ~ ,= 0 is a function of temperature. Comparing this result with calculations by means of the UNIFAC equation with the group assignment and the parameter sets reported by Fredenslund et al. (1977b),one has a good potential to distinguish between CH, and CH2 groups and to account for effects of terminating groups. Although differences between calculated and experimental
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Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 2, 1984
21
:
1 2
-
6
CfiG'BC:
6
-
ALw8Ee
Figure 2. Experimental activity coefficients In 7; (-e, McAuliffe, 1966) and UNIFAC calculations (-) using group assignment and CH2-H20parameter sets according to: (1)Fredenslund et al. (1975); (2) Fredenslund et al. (197713) (recommended for alkanes in water); (3) Skjold-Jolrgensen et al. (1979) (recommended for alcohols, ketones, and ethers in water); (4) regression results of Rizzi and Huber (1981) for infinite dilution activity coefficients in systems of alcohols and ketones in water, making no difference between CH3, CH2,and CH groups.
results seem not to be serious considering a few consecutive n-alkanes, it is easy to see from Figure 2 that in principle the same systematic deviation discussed before does still exist, which is seen by the crossing of lines. This systematic deviation leads to an overestimation of the r C H 2 value in water. Skjold-Jsrgensen et al. (1979)split the parameter set using different values for the CH2-H20 interaction whether working on liquid-liquid systems with alkanes or on vapor-liquid systems with alcohols, ketones, or ethers in water. But no parameter set proposed, not even that for alkanes alone (Fredenslund et al., 1977b), proves to be optimal for the alkanes due to the systematic error discussed. Additional Groups in Alkanes and PAHs. The definition of a new compound-type specific group, CH2*, in alkanes allows the use of a universal set of parameters for the interaction of the normal, old CH2group with H20 in all systems. These compound-type specific new groups, CH2*,are defined to replace a certain but arbitrary number ofCH2groups in alkanes only,e.g.,hexane (5 CH2*,1CHJ, heptane (5 CH2*, 2 CH2) etc. The interaction matrix of the CH2* group with all other groups is defined in this approach to differ from that of the CH2 group only with respect to the interaction with H20. An excellent fit can be obtained for the -yim data of alkanes in water by this method. An analogous approach is applied with other nonpolar homologous compounds dissolved in water, e.g., with polycyclic aromatic hydrocarbons. This is illustrated by partition data of homologous PAHs in the well investigated 1-octanol-water system (Rekker, 1980) which are plotted in Figure 3. Using one type of group for aromatic C the calculation of the partition data of benzene (6 aromatic C), naphthalene (10 aromatic C), and anthracene (14 aromatic C) leads to unsatisfactory results for the polycyclic compounds using the parameter set determined on limiting activity coefficients of binary mixtures of benzene. Introducing a second type of group, so that we have now ACH* and ACH, a very good correlation can be achieved. Again the parameter set of the new group, ACH*, was
1
-be
'
c
