0ne.of the
areas in which thermodynamics finds an important application is in the development and
design of processes for the recovery and purification of the relatively simple organic compounds. Since many such processes exploit small differences in the distribution of a number of compounds between equilibrium phases, the needs in this area are generally for quite accurate descriptions of the distributions of components between phases over some range of temperature and pressure. Indeed, the needs for accuracy are frequently great enough and the systems are complex enough that strictly theoretical means of predicting equilibria have been of limited use. As a consequence, the practice of depending primarily upon direct experimental measurement of equilibria is widespread. Since such measurements are frequently difficult, tedious, and costly, the number of measurements must be held to a minimum. Toward this end, the further practice of exploiting whatever sound theoretical, thermodynamic, and empirical
APPUEY) THERMODYNAMICS SYMPOSIUM
GROUP CONTRIBUTIONS
CARL H. DEAL ELWOOD 1. DERR
I '
p'= I y
U
The use of thermodynamic relationships
is extremely valuable in interpolations and extrapolations that can be made from systems whose components are structurally related
relations can be brought to bear in interpolating and extrapolating from these direct measurements is also quite widespread. For present purposes, this general area of applied thermodynamics can be somewhat arbitrarily separated into two subordinate areas. I n the first, the needs are for data adequate for process design, that is, for activity coefficients in a definite system of specific components. I n this case, demands for an accurate and self-consistent description of the system are severe. This is particularly true since the advent of automatic computational capabilities and consequent greater dependence on computation. Good practice in this area has involved an essentially thermodynamic description of the system in question. Although the primary base remains largely the direct experimental measurement of equilibrium as a function of composition, temperature, and pressure, the measurements necessary are minimized by the use of thermodynamically sound correlations. The validity of the concentration dependency of the experimental activity coefficients at constant temperature (pressure) is assessed on the basis of their consistency with the Gibbs-Duhem relations ; interpolations and extrapolations through the concentration field are made by using one of the consistent relations experience has shown to be adequate. In the second subordinate area, the need is for guides toward arriving at some reasonable processing scheme and for data to serve as a basis for screening estimates which are both fairly simply and rapidly applied. I n this area it is impractical to obtain phase data for every system of components, as well as to use the approach mentioned above in assessing their validity, but whatever interpolations and extrapolations can be made from system to system are extremely valuable. Although these interpolations and extrapolations are not based on rigorous thermodynamic relationships, for the sake of compatibility with the first area, there is a strong incentive that they should deal with the activity coefficients.
The present paper is concerned with the second of these two areas. I t is intended to review some of the ways in which activity coefficients of simple organic compounds can be estimated by interpolating and extrapolating from one system to another system using the intuitively attractive idea of characteristic structural group contributions. Such procedures are useful not only in making estimates when no direct data are available, but also in assessing data for new systems in terms of data for related systems. I n the latter, they are nearly as valuable as the traditional thermodynamic consistency tests. Group Contributions
I n dealing with mixtures of molecules in terms of their constituent groupings of atoms, it should be clear at the outset that in some way account must be taken of (1) the interactions of the various groups which can occur in solutions and standard state, (2) the restrictions imposed upon these interactions by the organization of the groups into molecules, and (3) the organization of the molecules in the solution and in the standard state. The detailed theories of mixtures take these effects into account in terms of some model and attempt to evaluate the result by statistical mechanical procedures. With mixtures of even simple molecules, however, the effects are so complicated that completely satisfactory models have yet to be developed. Although such detailed theories provide a considerable understanding, they have not yet been of very much direct use in fulfilling the applied needs mentioned above. What is needed are perhaps less detailed and simpler models which nonetheless isolate and take account of the important effects. As will be illustrated below, surprisingly far-reaching and precise estimates of activity coefficients in mixtures of simple organic molecules can be made on the basis of appropriate systematic observation and correlation with contributions from structural groupings. VOL. 6 0
NO. 4 A P R I L 1 9 6 8
29
Langrnuir Model
ECHS~JZ
I n the present context, probably the most significant early description of simple mixtures in terms of groups was given by Langmuir (75) some 40 odd years ago. Langmuir began with the basic premise that the force field around a group or radical which is accessible to other molecules is characteristic of that group or radical and is largely independent of the nature of the rest of the molecule-that is, he began with his “principle of independent surface action.’’ As a first approximation, he neglected any local orientation and segregation of a molecule in a liquid which would give rise to an excess entropy and considered instead the various interfacial energies he might expect for a molecule in a liquid or liquid mixture. By summing these interfacial energies (each one characteristic of the pair of dissimilar groups in contact), weighted according to the surface fraction of the respective kinds of surface in the respective molecules and according to the overall surface fraction of solute and solvent in a binary mixture, he derived expressions for the partial pressures of the components. The expression for binary mixtures which resulted was equivalent to a van Laar expressed in terms of surfaces: R T In Y I C Z = ) s1e1zSz’ R T In yz(1) = SZEIZSI~
(1)
where the s’s are molecular surface areas, the S’s are surface fractions and € 1 2 is an interfacial energy/unit area characteristic of the pair of molecules which depends upon the kinds of groups and group fractions in the respective components (1) and (2). Although Langmuir dealt explicitly only with molecules of the sort R-X in which a nonpolar R group is considered a single group and X is a polar group, it is of interest to extend the calculation which led to Equation l to the methylene (methyl) groups in homologous series. For homologs H(CH2)nlyl in homologs H ( C H 2 ) nzyz with additive groups surfaces such that
+ sz = nza +
s1 =
nla
uyl
uu2
and the associated interfacial energies : h J z ECHZYI
30
INDUSTRIAL AND E N G I N E E R I N G CHEMISTRY
ECH,-CH,
one obtains for solute 1 at infinite dilution in Equation 2 below. The terms identified by the brackets are, respectively : (1) a contribution which depends on the identities of both y l andyz, which is roughly inversely proportional to n2 and which is independent of n l ; (2) a contribution which depends only on the identity of yz, which is proportional to n1 and which is inversely proportional to 722; (3) a term which depends only on the identity of y l , which is inversely proportional to n1 and which is independent of n z ; and finally (4) a term which is independent of the polar interaction energies and is a function of nl and n?. Thus, when viewed as an expression for describing binaries formed by the members of two homologous series, the Langmuir expression gives a relatively simple form: Fi,2 R T l n yio = n2
+ Bz nz + CI + n1
-
nl
E C H , - C H , * ~ ( ~ Inz) ,
(3)
where the subscripts 1 and 2 are used to indicate that the coefficient depends upon the solute series, the solvent series, or both. Langmuir indicated that the magnitudes of €12 are reasonable in the light of interfacial energy data, that reasonable behavior within a homologous series was indicated, that Equation 1, perhaps not surprisingly, in retrospect, could fit experimental data for binary systems with moderate deviations, and that reasonable mutual solubility predictions could be made for a few systems with large deviations from Raoult’s law. Although Langmuir indicated an intent to pursue the approach further, apparently he did not do so-perhaps as a result of inadequate experimental data. Smythe and Engel (25), however, considered a group of binary systems formed by volatile paraffins, alkyl halides, and alcohols from the viewpoint of a Langmuir model. They conceded an approximate concordance of their data and the model but were led to the conclusion that the concept lacked general applicability. In retrospect it appears that this discouraging conclusion resulted from perhaps too strict an interpretation of the model. Although the Langmuir model itself was not improved for wider use, there is little question but that Langmuir
1
-.o I
I 2
3
.. .
.4 . . 5 . .6
7
8
influenced the development of the theories of solution (23). Apparently as a result of Langmuir‘s paper, there has been some use of the parachor as a measure of molecular volume and the two-thirds power of the parachor as a measure of molecular surface. Although these have generally been molecular rather than group approaches to mixtures of molecules and are not of primary interest here, several do bear on the problem of making estimates. McGowan (76),for instance, in connection with the correlation of the toxicity of organic molecules used the parachor of a molecule in rough correlations of their solubilities in water. For the usually large and relatively insoluble materials of interest, these may be viewed as roughly equivalent to a ScatchardHildebrand formulation in which the solubility parameter difference between solute and solvent is relatively large and constant so that the limiting partial molal excess free energy is proportional to the solute molecule volume (measured by parachor). Den0 and Berkheimer (9) pursued such correlations further but, in ScatchardHildebrand terms, found them inadequate for systems involving less extreme differences in solubility parameters. Further in this general connection, Hala cf al. (72) have derived relations which give the binary van Lam eoefficients for binary systems formed from a single solvent and the members of a homologous series in terms of the parachors and van Lam coefficients for two cases in point and two adjustable parameters. Using the two-thirds power of the parachor as a measure of molecular surface, Erdos (70) has developed a relation which gives the van Laar coefficients for binary mixtures formed by two components involving the same polar grouping. Here the relations involve the parachors and a single adjustable parameter characteristicof the polar grouping. As in the previous examples, the parachor, though it is taken as the sum of group characteristic quantities, is treated as a molecular quantity. Limiting Activity Codficienls
The work of Butler et al. (5,6) is a second piece of early work which is basically a group approach. Butler
departed kom the more conventional study of conceneation effects within a single system of components and considered the infinitely dilute solution of a series of solutes in a given solvent as the simplest case for study. He systematically measured Henry‘s Law constants for a wide range of solutes within a given family. Through these he, in effect, observed a simple relation between solute carbon number and ita activity coefficient as illustrated in Figure 1. He noted that the partial molal excess free energies of solution increase by roughly constant increments through the homologous series and attributed this primarily to the dominant water-water interactions or, in Scatchard-Hildebrand terms, the cohesive energy density of the solvent less some opposing contribution from methylene-water interactions. He also indicated that thin roughly constant increment depends upon the nature of the polar grouping. Pierotti Cornla9ions
This approach to group contributions was apparently not pursued further. Some 10 years later, however, Pierotti and coworkers (79, 20) at Shell Development Co., began a more extensive systematic study of homologous series. Here, as had Butler, these workers considered the infinitely dilute solution as that which is conceptually the least complicated and most promising for study of group effects (even though experimentally most difficult). Insofar as application is concerned, however, there are additional reasons for emphasis upon the dilute region. First, the dilute regions are themselves of considerable technological importance since applications are frequently in purifications; second, in view of the usual shape of log y-concentration curves, interpolations from dilute regions to the standard state can be expected to be more precise than extrapolations into the dilute regions from data in the middle range of concentrations; third, effects are usually at their maximum in the dilute regions so that the maximum solution effects are generally defined directly by the limiting log yo’s; fourth, the extensive experience with thermodynamically consistent correlations gives a fairly good basis for interpolation into regions of high concentrations. The approach in this work was the phenomenological one of experimentallydetermining the dependence of the limiting log yo’s of families of systems of fixed structural type upon solute and solvent structure at several fixed temperatures. In practice, this involved measuring y’s at high dilution for homologous series of solutes in Iixed solvent and k e d solute in homologous series of solvents and inspecting the dependence of the limiting log yo’s upon the carbon numbers of solute and solvent. The procedure can be thought of as simply determining the response surface of log 7”’s as a function of the number of methylene (methyl) groups, nl and nr, of the solute and solvent molecule for the case in which the groups are connected as in homologous series. For n-alkyl homologs, the relations were laid out by successive choices of more and more complicated cases. Thus, for paraffin-paraffin mixtures, that is, the series VOL 6 0
NO. 4 A P R I L 1 9 6 8
31
Figure 2. E.wnpIes of F'iaofti cardalions
32
INDUSTRIAL AND ENGINEERING CHEMISTRY
.H(CH8)nlH in the series H(CH*)nrH, the deviations are relatively small. Although symmetrical in components 1 and 2, the expression for the congruent solutions of Brfinsted and Koefed (3) was adopted as sufficiently precise-i.e., log y10ao = a
n 1
- nn)'
For the somewhat more complicated cases of n-paraffins in each of several polar solvents, that is for H(CH8)nlH in H(CHp)ngz, it was found that log YI($
- D(n1 -
np)*
=K
+ Bnl
Correspondingly, for a series of solutes of the type H(CHJn1yl in a fixed solvent of the type H(CHt)naH: log y1(go
- D(nl - ns)'
= K'
+ C/n1
and so on. Illustrations of a succession of such cases are shown in Figure 2. From a fairly large number of such cases, it was possible to come to a quite general form for the log y1(q0 = f(n1, n t ) surface which describes a homologous series of solutes in a homologous series of solvents in g o d approximation :
(4)
where the coefficients are temperature dependent. M a t importantly, the coefficients in the expression depend upon the polar grouping of the solute series, H(CH,)nlXl, on the solvent series, H(CHn)nqx, or upon neither as indicated by their subscripts 1, 2, or 0, respectively. Thus, for a given solute series in a given solvent series, only one of the coefficients is completely free for adjustment. As will be apparent h m the development of the relation, zero members of a series-e.g., watex for the alcohol s e r i d o not have infinite terms; rather the
6 '
6
'I
convention is adopted that any terms containing an "n" for the zero member simply becomes incorporated into the constant term+.g., for a series H(CHa)nlyl C h . In addition, in water, log 71' = K B=,on, some variation from n-alkyl series can be incorporated. Moderate branching in an alkyl group located so as not to interfere directly with a polar group gives rise to only second order effect and needs no special means for incorporation. Branching in the vicinity of a polar group is, however, important. As solutes, such materiah-e.g., secondary alcohoLhave been satisfactorilyincorporated by means of modifications of the solute (standard state) characteristic C-term of the abovee.g., Ci(l/nl' l/n1") where the n;s represent the carbon numbers of the branches counted from the polar group. As solvents, such materials have been satisfactorily incorporated by modification of the solvent characteristic "F'term in a similar manner. Solutes with large groups which are considered as strongly interacting (polar) groups as for instance alkyl aromatics may be incorporated by m d i y ing the "C" term to Cl/(nl constant). Contributions from two polar solute groupings can in general not simply be added; however, such a solute can be considered a member of a new solute homologous series. Although the best idea of the validity and utility of this expression is obtained from a detailed inspection of the results, an overall idea of its consistency with experimental data can be obtained from the plots shown in Figures 3 through 5. Various organic solutes with generally small polar groups are shown in Figure 3 for a water solvent. Similar solutes in various polar solvents are shown in Figure 4. Finally, hydrocarbons ranging from normal paraffins to alkyl polyaromatics in a series of solvents ranging from heptane to ethylene glycol are shown in Figure 5. Here for the sake of illustration, log 7's (or in effect the partial molal excess free energies) are plotted and the overall average deviations for some 44 sets of systems (350 individual cases) amount to about
+
+
+
+
,--. 1
2
3
.
4
5
I .o 0.8
0.4
e3B
02
0
0.2
04
06
0.8
10
1 worn& in
3%. The overall average deviations in the 7’s themselves are of the order of 10%. This overall consistency with data is particularly striking when it is recognized that these quantities are not the overall levels of exceas free energy curves, but their limiting slopes in the first case and the antilog of the limiting slopes in the second. It is the latter which are of most technological interest. Similar correlations are obtained by characterizing the response of log yo with respect to other simple groupings-e.g., a “B” t u n with respect to aromatic carbon. Likewise, as pointed out by Deal, Dm, and Papadopoulos (a), from the somewhat simpler pattern of response obtained for the distribution of solutes between two fixed environments (in this case, the reference environment does not change as the solute carbon number changes), the distribution of olefinic stmctum can be estimated with reasonable accuracy from measurements of benzene. More recently, Black (2) has correlated the complex equilibria involved in extractions with a group approach somewhat similar in nature. It is clear that the foregoing correlations of partial molal exfree energies take some implicit a a u n t not only of the energies involved in the interactions of the molecular groupings but also of the frequently quite important associated entropies. As a consequence, there is considerable hazard in attempting to identify the meaning of any specific term of the expression in detail. A fairly straightforward analysis based on
34
INDUSTRIAL A N D ENGINEERING CHEMISTRY
the assumption that the polar-polar group interactions dominate (p-ya andy1-y. in solution andylyl in standard state) leads to a reasonable accounting for the various terms. Thus, the “B” term has to do with generation of an nrsized “hole” in the solvent environment; it is proportional to “1, is inversely proportional to na through a “dilution” effect in the y e s interactions, and depends only upon the nature of the solvent ( 8 s ) . As is apparent, this empirical relation approximates the relation in Equation2 alreadyshownfor the Langmuir model. If the Langmuir model is extended to recognize interactions between individual ‘‘like’’ groupings (apparently neglected since they would give no interfacial energies) (see equations below). The primary differencesbetween Equations2 and 5 are the occurrences of an contribution to that linear in n1 and an tvlrr contribution to that reciprocal in nl. The form of the above suggests that Pierotti correlations might be somewhat improved, particularly when the polar grouping is large, if n l and nt be replaced by the companding n+ constant when they occur in the denominator. This has only been tested in the F’ierotti “C” term where such a technique is necessary in treating such large groupings as the phenyl group as a solute polar group. Aside from its evident direct utility as a guide to stmctural effectsin solutions,the Pierotti work shows that, although detailed group models (such as the Langmuir model or other more sophisticated ones) may become so complex or restricted as to be of limited use, one can nonetheless go a long way toward treating the behavior of alsolute molecule as a simpleseries of contributionscharae teristic of the solute standard state and solvent group environments. Even if it is not feasible to calculate these characteristic contributions fmm pure component properties of solute or solvent, they may be fairly simply determined by direct measurements. Although derived for the special case of solute at infinite dilution in solvent, the succe8s with which very broad ranges of solutesolvent c a m have been dealt suggests that any envuonment, such as the range of finite concentrations in a binary or more complex mixture, might be similarly treated. There has been no systematic attempt to define Pierotti characterizations for molecular mixtures, but the viewing of a molecular environment as an environment of its constituent groups has proved a valuable
Figwc 6. Actimo cw”& hepane Rdxhrrss
of hqtancfrom alcohd andfrom dhaml-
guide in making estimates of activity coefficients in mixtures. The plot shown in Figure 6 represents a case in point. Here the activity coefficients of n-heptane from various alcohols and of n-heptane from mixtures of ethanol and heptane are plotted as a function of the -OH group concentrations and fall on essentially the same curve. Such a result is, roughly, the polar mixture analog of the congruency principle originally proposed by Bransted and Koefed for normal paraffin mixtures. Group Inbradion
With the success of the foregoing, Redlich, Derr, and Pierotti (22) developed a group interaction model which calculates the heat of solution as the sum of contributions from interacting groups proportional to the number of groups present, a group cross section characteristic of each kind of group and an interaction energy characteristic of each group pair. As does the original Langmuir model, it neglects any local ordering or segregation except insofar as it is implicitly taken into account thmugh the determination of the parameters from experimental data. Likewise, it pKovides a van Laarlike relation for the partial molar excess heats with concentrations expressed in surface fractions. As a result, the limiting partial molal heats in a binary are in the ratio of the total molecular cross section of the component molecules. I n a companion paper, Papadopoulos and Derr (78) though still limited by suitable test data provided a preliminary test of this model for hydrocarbon systems. To reduce the number of adjustable parameters in this test, group crogp sections were taken as proportional to the two-thirds power of group volumes which were, in turn, established on the basis of volumetric properties of liquid hydrocarbons. With these in hand, the l i t ing partial molal excess heats mostly derived from the change in partial molal excess free energy with temperature of some 29 hydrocarbon systems were inspected. By use of three adjustable parameters and six simple and reasonable but arbitrary rules regarding the similarities of the interaction energies of similar groups, the results shown in Figure 7 were obtained. The average
deviation is in the vicinity of 120 cal/mole which is wJJ thin the experimental errors of the limiting heat dab. Some unpublished work showed promise that the scheme might be successfully applied to heats of m i h g of polar mixtures. Et is of interest in passing that a free energy correlation, analogous to the above heat correlation and different only in that a Flory-Huggins entropy correction is applied to log y’s before the residual free energy is correlated, provides useful estimates of hydrocarbon activity coefficients. I t is still not dear, however, how successful this free energy model can be in Peating polar systems. The implications fiom the Pierotti work are, however, that although an overall polar group contribution can be isolated, multiple polar groups in a single molecule cannot be added in the same way as can relatively inert methylene groups. Nonetheless, at least three adjustable parameters are available for the simplest case, the polar group cross section, and the appropriate interaction energies of polar grouping with at least methylene and methyl groups. The model is, thus, quite flexible in this sense. On the other hand, the model is somewhat restrictive in requiring a van Laartype concentration dependency except for the FloryHuggins correction mentioned in molecular system. Solution of Groups
Wilson (28) has proposed a “solution of groups’’ approach which does not restrict the molecular activity coefficients to a van Laar-type relation but nonetheless estimates the partial molal excess free energy as the sum of group contributions and provides a concentration dependency for these group contributions. In this case, a “group” portion of the partial excess free energy is taken as the difference between contributions in solution and molecular standard state, log 7,” =
Ct n,
(log
r, - log r,*)
(6)
1
i;.I
I
V O L 6 0 NO. 4
APRIL 1968
35
The functions Fk are defined from a set of activity coefficient data and obtained by applying the Equations 6 to 10 to an experimental case. For convenience, they may be thought of as “group” activity c d c i e n t s and normalized and extrapolated to a group standard state such that l’, = 1 in an environment which is pure group
i _
:/:
h
3’
-on
0
:
k.
-cn,-
/ .
0.4
0
0.8
0.6
1.0
6F
5 4
-z??zzt 0
02
04
06
08
IO
Figure 9. Wilson’s solulirm of group estimatts (systmfiom alcohols, hydrmarbons, md wutn based on methanol-hcxat~~)
where the log r’s represent the contribution/group in the solution and standard state environment. I’ is taken as a single function of the group concentrations of environment for both solution and standard state:
rr
fk(x1, x2. .)
(7)
where the X’s are group concentrations:
The molecular activity coefficient is taken as the sum of the group contribution and a contribution having to do with size, log 7*= log 7“ log ys (9)
+
where the size contribution represents the only distinction behveen environments of the same group constitution and different molecular constitutions. It is evaluated with a Flory-Huggins-like expression using only the number of groups in the respective molecules of the mixture:
Wilson applied such an approach to two fairly extreme cases-CH,-OH and C H r C N mixturemaking no distinction between methyl and methylene groups. I n the --OH case, he u d the single hexanemethanol binary to obtain l’-curves shown in Figure 8 and estimated the clmelyrelated ethanol-heptane binary, and the limiting partid excess free energies of (1) alcohols in heptane, (2) heptane in alcohols, (3) paraffins in ethanol, (4) alcohols in water (extrapolated to pure O H environment), (5) paraffins in water (extrapolated to pure OH environment), and finally of (6) water in alcohols (HOH taken as 1.5 -OH groups). In the - - C N case, he used acetonitrile pentane as base and estimated other paraffin-nitrile systems. The results for the -OH case are compared with experimental data in Figure 9. As is apparent, substantial agreement is obtained over an extraordinarily wide range of valua from the single-base system; the logarithms of the activity coefficients are generally estimated to within about 10%. The above, with its assumptions that in some way both the enthalpic and entropic contributions (in excess of the “size” effect) to the log y are simply additive (Equation 6) and that the concentration dependency of these contributions may be charactvized from a base case (Equation 7) is clearly mwt useful. These equations must, however, be applied with some care. Thus, for instance, it is not reasonable to expect that a polar grouping, located in a solute molecule such that there is a large stearic interference with its interactions, would be comparable to one in which there is no such interference. Tertiary butyl alcohol is hardly comparable to n-butyl alcohol; however, it is likely that substituted tertiary alcohols are quite comparable to tertiary butyl alcohol mixtures just as other n-alcohols mixtures are comparable to methanol. Likewise it seems unreasonable to expect that the interactions of two polar groupings attached to the same molecule would be comparable to those in which a single polar group is involved. Ethylene glycol as a solute or as an environment is hardly comparable to methanol; however, substituted glycols are probably quite comparable both as solutes and a8 solvents to ethylene glycol. It is of interest that the foregoing assumptions regarding the additivity of group contributions and the characterizability of solution environments seem to be better when applied to free energies than when applied to
H. Deal and .F4wood L. DmareRcsuach Chemists for the Shell Development Co., Emcr.vville, Calq.
AUTHORS Carl
36
I N D U S T R I A L A N D ENGINEERING C H E M I S T R Y
AnolyHcol Solution d Omups
Ipwcntly, one ofthe present authors (E.L. Dem) in an
as yet unpublished work has used an “analytical solutions of groups” approach. He has chosen to represent the log I’ concentration dependency with the Wilaon (27)
0
.I
.3
I
.4 Hydroxyl Group Fraction
2
.5
.6
heats of mixing. As shown in Figure 10, on the basia of Brown’s (4) data for heats of mixing in alcohol-hexane
mixtures, roHc u m based on heats are quite diffarnt for methanol and propanol in hexane. In addition, they lie at different levels from those derived from free energy data. Brown himself satisfactorily incorporates the discrepant lower alcohols in a heat&-mixing correlation by using contributions taken from the heat of mixing of the hydrocarton homomorphs of the various alcohols with hexane. Recently, Scheller (24) has used a “solution of group” approach to produce a correlation of a broad range of mixtures. By the use of mixture data containing watm he has, in Wilson nomenclaturr, defined roHand rCHr curves over the entice range of C H r O H mixtures. In addition, he has calculated the log ys term with molar volumes rather than the group numbers and has incorporated certain polyols as solutes by the use of effective group numbers in very much the same way as water was handled in the original paper. The net result is a quite satisfactory correlation. With six base systems involving hydrocarbons, alcohols, and water (1.6 --OH group), log y’s appeared to be generally within about 10% of experimental values. In comparison with the Pierotti correlations, it appears that the Wilson “solution of groups” provides somewhat lesa accurate limiting activity coefficients. The “solution of group,” however, r e q u h s fewer base system than are required for the development of Pierotti correlations. In addition, it provides not only a built-in concentration dependency for molecular systems encompassing two kinds of groups but, in general, a themnodynamically consistent molecular concentration dependency. If the base data are thermcdynamically consistent or if the same conditions for consistency as apply to molecular mixtures are imposed upon the rcurves, the estimated molecular y-curves are consistent. In comparison with the Langmuir-like Redlich model, the solution of groups does not restrict concentration dependencics to a van Laar-type expression and is thus, in this sense, more flexible. On the other hand, the lack ofan analytical d a t i o n makes it difficult or practically impossible to apply the solution of p u p s as presented to mixtures containing more than two group.
molecular free energy expression, based largely on the o h a t i o n that this relation gives a more suitable shape to the log rr concentration dependencies in binary group systems than do other two twwoefficient expmesions. Preliminary results promise that this procedure gives an adequate concentration dependence in both binary and some multi group mixtures and thus provides a basis for handling complex system in computers. A successful development will be extremely useful in making screening estimates. In addition, however, the technique is particularly attractive for correlating complex systems which contain many more kinds of molecules than kinds of groups, since many fewer adjustable parameters are required. An example comparing the estimated molecular activity coefficients in the three-group binary system formed by acetonitrile and ethanol with the experimental data of Vierk (26) is shown in Figure 11. An example comparing the estimated activity coefficient ratios in the two-group ternary system ethanol-propanol-water with the experimental data of Carlson at al. (7) is shown in Figure 11. In both examples, the base data used to establish the group coefficients were the limiting molecular activity coefficients in binaries formed with a hydrocarbon and the most polar member of the series of materials containing the polar group involved. 0th- Group Comlahns
Recently Irmann (14) has presented a correlation of the solubilities of h y k a r b o n s and halocarbons in water which is essentially a group correlation. He correlates on the basis of the numbers of atoms nl of atom of type i and the number n, of linkages of type j in the solute molecule
- .. . . ...
0.4
0.6
.08
1.0
V O L 6 0 NO. 4 A P R I L 1 9 6 8
37
where the (I is a parameter dependent upon the type of compound, 6, is a coefficient dependent upon the type of an atom, and C, is dependent upon the type of linkage. Some 200 solubilities are drawn together into a single quite satisfactory correlation. Most of the materials covered are of low solubility so that the solute does not strongly affect the nature of the solution environment. From the present authors’ viewpoint, however, such a correlation might be better expressed in terms of the activity coefficients of the solute molecule, although the inverse of low solubilities do correspond to limiting activity coefficients. With regard to the foregoing group correlation, it appears that halocarbon group contributions to log 7’scan be treated in much the same way as are methylene groups in the foregoing. A recent article by Meyer and Wagner (77) deala with the cohesive energies of methyl ketones through a
in which direct experimental data are called for in some practical application or, on the other hand, eliminate cases in which the volatility or distribution coefficient in an application is so uncritical that experimental measurements are not worthwhile. Although the approaches to handling mixtures in terms of groups remain essentially empirical in nature, their relatively high precision as well as their intuitive sense suggests that they are basically sound. Their general similarity to some of the more theoretical approaches, such as the quasichemical approximations of Barker (7) and the corresponding states approximations of Prigogine (27), Hijmans (73), and Flory (11), suggests that more detailed theories of mixtures may ultimately become sufficientlyquantitative to meet technological needs. GLOSSARY
A, B, C,D , F = cornlation paramema, homologous safa relations ni nb Ki‘
si
Si
X. xi Y’ et1
ri ri0 Yi(n0
-P 7s
I 0
l
l
0.2
I 0.4
l 0.6
l
1 0.8
rb
1 1.0
at
-
molccule i, 1 aolute, 2 = solvent = number of p u p s of kind k number of p u p k in moleculej surface of molecule i = surface fraction of component i in mixture group fraction of p u p k in mixture = molc fraction of component i = polar grouping of component i = energy or free energy anrociatcd with interaction of p u p k and I = activity cDdfident of component i = limiting activity &dent of component i at infinite dilution = limiting activity weEiacnt of i in mmpomt j = facta of activity coefficient d a t e d with the interactiona of p u p = factor of activity d a e n t samciated with molecular sizes per % = factor of P d a t e d with group t (gmup activity &Cimt) = surface of gmup t [subaeript dropped for methylene (methyl) p u p s ] = number of methylene (methyl) group in
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REFERENCES (1) Barker, J. A,, Smitb, P., J . C h . P h p 21. 375 (1951). (2) BlseL,C.,A.I.Ch.E. SympodumsCriu,in-. I . N., W e d , J., Kg/. D.s& Vi&&&. MtM, M&p.
ct) wid, comparison of the members of the homologous series. Although it deals with pure component properties outside the scope of the present paper, it is of some interest in the present context in that it deals with molecular groupings and leads to a measure of the importance of the contributions to the cohesive energies in the respective liquid phases. The result is that the dipole orientation energy, which is a significant contributor in the low series members, is approximately proportional to the reciprocal of the carbon number. This is analogous to the dependency of the Pierotti “B” term upon na. Summary and Conclusions
From the foregoing review it is apparent that the intuitively reasonable approach of treating mixtures in terms of their constituent groupings to estimate activity codficients can be of considerable practical use. Estimates are generally quite adequate for orientation and at the very least can, on the one hand, point to the areas 38
INDUSTRIAL A N D ENGINEERING CHEMISTRY
1 (1946).
Mdr. 23,
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(4) Bmwn, I., Fxk, W.,Smitb, P.,Aulrdim 3. 17.1106 (1964). (5) Bu rls, J. A. V., R.meh.ndoni C.N., J. Wm. Sor., London, 1995.p. 952.
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&.
