ACTIVITY COEFFICIENTS AND MOLECULAR S T R U C T U R E Activip Coej7cient.s in Changing EnuironmentsColutions of Groups G . M . W I L S O N A N D C. Shell Development Go., Emeryville, Calv.
This article treats a solution os a mixture of groups which make up the components of the mixture and provides a means
to be the sum of two contributions-one associated with differences in molecular size and the other with interactions of structural “groups.” For molecular solute j in any solution:
o f estimating the activity coefficients when they are not known.
estimates.
log
Duta for other molecules in other environments,
where the same groups are involved, form the basis for the For example, measurements o f a single alcohol-
paraffin binary system can b e used t o estimate the activity
H. D E A L
YI
= log
r:
+ log rF
(11
Rule 2. The contribution associated with molecular size differences is assumed to be given by a Flory-Huggins relation expressed in terms of the numbers of constituent atoms other than hydrogen. For solutej in solution of components i:
coefficients, a t any concentration, for any paruffin-alcohol system t o about 10% o f the logarithm o f the activity coefficient. It is a valuable means of relating activity coefficients and molecular structure and o f making practical estimates of phase equilibria-e.g.,
for screening separation processes,
for systems difficult to measure, and for checking individual experimental points.
HE FOREGOING ARTICLES O F THIS SERIES ( 2 , 7) have outTlined schemes for correlating infinite dilution activity coefficients on the basis of solute and solvent structures. These schemes suggest that in any solvent environment the partial molal excess free energy of solution of a solute molecule can be taken as the sum of individual contributions from each of its structural groupings and that these contributions depend in some way upon the numbers and kinds of structural groupings which make up the environment. Redlich, Derr, and Pierotti ( 8 ) and Papadopoulos and Derr (6) have recently taken a similar approach and developed a detailed “group” interaction model for heats of solution which correlates heats for hydrocarbon mixtures quite satisfactorily and which may prove useful in polar systems. S o extension of a “group” approach to excess free energies or activity coefficients has, however, appeared recently (5). The present report suggests an approach to solutions of “groups” from which activity coefficients can be estimated on the basis of group contributions, not only a t infinite dilution but at finite concentrations. These estimates are generally somewhat less precise than Pierotti estimates but require fewer base data.
Solutions of Groups With the idea that a ”group” is any convenient structural unit such as -CH-, -OH, --CHzOH, the present approach rests on the following four assumptions: Rule 1. The partial molal excess free energy or, simply, the logarithm of the activity coefficient of a component is assumed 20
I&EC FUNDAMENTALS
ni = number of atoms (other than hydrogen) in molecular component i xi = molecular mole fraction of component i and the summation is to be made over all the components in solution. Rule 3. The contribution from interactions of molecular “groups” is assumed to be the sum of the individual contributions of each solute “group” in the solution less the sum of the individual contributions in the conventional standard state environment. For molecular solutej containing groups k: log 7; =
ZkVkj
(log rk - log
rt)
(3)
where Ykj
rk I$
= number of groups of type k in solute component j
k in the solution environment referred to an arbitrary standard state
= activity coefficient of group
= activity coefficient of group k in the standard state
environment. This activity coefficient is referred to the same state as rk Although the new kind of standard state to which the individual “group” activity coefficients are referred is arbitrary and in fact need not be defined, it will normally be imagined as a mixture of pure “groups,” just as the normal standard state for the molecular solutej is pure solute. Rule 4. Finally, the individual group contributions (rr) in any environment containing groups of given kinds are assumed to be only a function of group concentrations.
where
&
= group fraction of group k =
zjvjgj ~
zjvjxj
Thus, for example, the solution environment presented by pure n-heptyl alcohol is considered to be the same as that of a n equimolal mixture of methanol and hexane, since the hydroxyl and methylene (methyl = methylene) group concentrations are the same in the two mixtures.
2 . 20 . 0 t t o 2.4 1 I
DIFFERENC
50
0
100
METHANOL, MOLE 70
HYDROXYL GROUP, %
ETHYL ALCOHOL, MOLE yo
Figure 1. Activity colefficients in methanol-n-hexane binary a t 45" C. (3) 0 Methanal 0 n-Hexane
Figure 2. Activity coefficients of groups in binary of hydroxyl and methylene groups a t 45" C.
Figure 3. Comparison of calculated and measured activity coefficients for ethyl alcohol-heptane binary
Data points correspond to those shown in Figure 1 0 -OH 0 -CHz-
-Calculated, 45OC. --Experimental 0 Ethylalcohol 0 Heptane
T h e distinction in Rules 1, 2, and 3 between two kinds of contributions to log y j is necessary, since the excess entropy of mixing caused by size effects cannot be associated with "group" interactions. Thus, Rule 2 represents that part associated with the difference in molecular sizes of the solute and its solvent environment and is a n entropy term; whereas Rule 3 is associated with group interactions and consists of the heat and the remainder of the entropy. Some improvement might be made over Rule 2 in evaluating the size effect; however, empirical evaluation of the contribution of Rule 3 compensates any errors to some extent. Consequently, the Flory-Huggins relation has been adopted for Rule 2, and the volume is assumed to be proportional to the number of nonhydrogen atoms in a molecule. T h e "group" interaction contributions of Rule 3 are derived from the assumption that the total free energy caused by group interactions is dependent only on the group environment and, accordingly, is independent of how or whether the groups are connected together. 'This assumption seems intuitively sound in view of the fact that interatomic forces are short range and significant interaction 'occurs only between nearest neighbors, which are for the most part not bonded groups. I t is noteworthy that only throu'gh Rule 2 is a distinction made between two environments which have the same group constitution but different molecular constitutions. T h e assumption that individual group contributions are functions only of the group concentrations (Rule 4) permits experimental data for one system to be applied to a second system involving the same groups. T h e form of this composition function is left open to avoid unnecessary distortions which might be introduced if a restricted expression, such as the van Laar equation, were to be used. I n practice, a graphical composition relation derived directly from a n appropriate set of experimental d ,ata is satisfactory. Interpolations and extrapolations of log :rn curves are made with the aid of the Gibbs-Duhem restriction, which can be shown to apply if the activity coefficients calculated from the rk curves are to be thermodynamically consistent. Use of the model is fairly straightforward. I t involves applying Rules 1, 2, and .3 to a suitable set of 71 data to establish r points for each group involved; constructa set of log rr-log X ing rncurves for each group, which, for convenience, are extrapolated to include the entire range of group concentrations
(4). 30' C.
o L
lo-'
IO
15
20
25
ALCOHOL CARBON NUMBER
Figure 4. Comparison of calculated and measured limiting activity coefficients for alcohols in heptane and heptane in alcohols at 45" C.
0 Alcohols in heptane
1
0 Heptane in alcohols
10
20
PARAFFIN CARBON NUMBER
Figure 5. Comparison of calculated and measured limiting activity coefficients for paraffins in ethylene glycol and in ethyl alcohol a t 45' C.
0 Paraffins in ethylene glycol 0 Paraffins in ethyl alcohol VOL.
1
NO. 1
FEBRUARY 1 9 6 2
21
gk
Figure 6. Activity coefficients in pentane-acetonitrile binary a t 50" C. 0 Pentane 0 Acetonitrile
and normalized so that r k = 1 when = 1; and, finally, applying these rkrelations with Rules 3, 2, and 1 to estimate y, values in some new system which contains the same groups. Generally. the base system is most suitably chosen as one which covers the broadest possible range of group concentrations so as to minimize subjective extrapolations. O n the other hand, choice of a base system which is as close as possible in characteristics to the system for which estimates are to be made can, of course, be expected to give the best results. Tests of the Model
PER-TANE, MOLE %
'
Figure 7. Activity coefficients of groups in binary of nitrile and methylene groups a t 50' C.
1
0-CrN
0 -CH>--
AITRILE GROLP, %
Table 1. Comparison of Calculated and Measured limiting Activity Coefficients of Alcohols and Paraffins in Water log
Compound Methanol Ethyl alcohol n-Pt-opyl alcohol n-Butyl alcohol n-Amyl alcohol n-Octyl alcohol n-Decyl alcohol n-Pentane n-Hexane n-Heptane n-Octane a
Y O
Measd. ( 7 ) 0.28 0.65 1.19 1.68 2.28 3.97 5.18 3.70" 4.23' 4.740 5.62O
Calcd.
0.29 0.75 1 26 I .77 2.28 3,85 4.88 3.71 4.30 4.88 5.46
76' C.
Table 11.
Comparison of Calculated and Measured limiting Activity Coefficients of Water in Alcohols HzO = 1.5 -OH
group
log Alcohol Methanol Ethyl alcohol n-Propyl alcohol n-Butyl alcohol n-Octyl alcohol n-Decyl alcohol
Y O
Measd. ( 7 ) 0.25 0.42 0.49 0.55 0.63 0.63
Calcd. 0.23 0.42 0.52 0.60 0.75 0.79
Tests of the model with a limited number of cases which, nonetheless, cover a quite broad range of solution types are shown in Tables I to V and Figures 1 to 7. As is apparent, estimates generally fall within about 1Oy0of log y. Mixtures of Methylene and Hydroxyl Groups. O n the basis of the binary hexane-methanol data of Ferguson ( 3 ) in Figure 1, the log rrt0s. gkcurves shown in Figure 2 were determined by means of Rules l to 4. From these curves good estimates of log y values are obtained in a variety of systems. Estimates for binary mixtures of ethyl alcohol-heptane compared in Figure 3 with the data of Ferguson, Freed, and Morris ( 4 ) are within about 15% for log y throughout the concentration range, the largest deviations occurring a t the limits of infinite dilution. Limiting values (infinite dilution) for a series of alcohols in heptane and of heptane in a series of alcohols are in agreement with the data of Pierotti, Deal, and Derr (7) as shown in Figure 4, although there are some systematic deviations. Limiting values for paraffins in ethyl alcohol and ethylene glycol are compared in Figure 5 ; fair agreement is obtained for paraffins in ethyl alcohol, but the agreement in ethylene glycol is somewhat poorer, being only within about 25y0 of log y. This latter discrepancy might be due to the assumption that the methylene groups in ethylene glycol are equivalent t o the methyl groups in methanol. Since the r curves in Figure 2 include 100% hydroxyl groups, it proves interesting to compare this environment with pure water. Therefore, with the assumption that water is equivalent to 100% hydroxyl groups, a comparison of calculated and measured limiting activity coefficients of alcohols a n d hydrocarbons in water is given in Table I. I n view of the approximations which have been made, the agreement is surprisingly good, thus confirming both this assumption and the validity of the r curves for this case. Water solvent corresponds to pure hydroxyl groups; it is further assumed that one water molecule is equivalent to 1.5 hydroxyl groups, and a comparison of calculated and measured activity coefficients of water in alcohols is given in Table 11, where again the agreement is good.
Table IV. Comparison of Calculated and Measured Limiting Activity Coefficients in Nitrile Paraffin Systems 50' C.
log Table 111. Comparison of Calculated and Measured Activity Coefficients in a Homogeneous Ternary Mixture n-Propyl alcohol-n-octane-H20,
Compound n-Propyl alcohol n-Octane Water
22
Liquid, Mole % 52.3 20.7 27.2
l&EC FUNDAMENTALS
45' C.
log Y
Measd. 0.08 0.63 0.52
Calcd. 0.00 0.69 0.59
Solute Pentane Heptane Cetane Docosane Dotriacontane Heptane Valeronitrile Heptane Cetane
SoIuent Acetonitrile Acetonitrile .4cetonitrile Acetonitrile Acetonitrile Capronitrile Heptane Adiponitrile Adiponitrile
Measd. 1.19 1.29 2.57 3.23 4.72 0.54 0.91 1.58 3.12
Yo
Calcd. (1.19) 1.55 2.88 3.66 4.91 0.46 1 .oo
1.05 2.36
Table V.
Temp., O
c.
17
Comparison of Calculated and Measured Limiting Activity Coefficients of Polyfunctional Solutes in Various Solvents Solute Groups, log Yo log T’, A - B Poiyfunctional Solvent C A or B A or B (‘7) Calcd. Measd. Solute, A - B 0.86 1.79 1 .75a Ethyl acetoacetate Water Acetone
60
60 25 80
25 25
CHjCOCHzCOOCzHj Butyl Cellosolve C4Hg-O-C2HqOH Paraldehyde (CHzCHO)? Ethyl adipate (CHZ-CHLCOOCZH~)~ Diethyl formal CHz(O C Z H S ) ~ Glycol dimethyl ether (CHz--OCHs)2 Diethvl formal
CHdO(ZeHd2 5
Calculated from solubility data ( 9 ) .
h
IVater Water
1.72 0.65 2.15 (1.18)~
Ethyl formate Ethyl alcohol Diethyl ether Dimethyl ether
\$later
Ethyl propionate Ethyl propionate Water Dimethyl ether Methyl ethyl ether Methanol Dimethyl ether Dimethyl ether Methanol Dimethyl ether Methyl ethyl ether Calculated from solubility data ( I ) . c Estimated jrom homolog
T h e comparisons made so far demonstrate the validity of the the activity coefficients of components in binary combinations of hydrocarbons, alcohols. and water. T o show that one is not restricted to binary mixtures, the component activity coefficients calculated in the ternary composed of water, n-propyl alcohol, and n-octane agree well with measured values, as is indicatzd in Table 111. Mixtures of Nitrile Groups a n d Methylene Groups. T h e interactions of nitrile groups with methylene and other nitrile groups should differ somewhat from those occurring in the case of hydroxyl groups. Thus, in this case a test of the group solutions model will test certain aspects not encountered in the case of the hydroxyl groups and methylene groups above. T o make this test, the methylene group and nitrile group activity coefficients have been calculated from experimental data in the binary composed of acetonitrile and pentane. T h e acetonitrile-pentane data are shown in Figure 6, and the group activity coefficients appear in Figure 7. Comparisons between calculated amd measured limiting activity coefficients are given in Table IV, where good agreement is again demonstrated. More Complicated Groups. Although only relatively simple groups have bren considered so far, a group may be taken as any convenient structure, as mentioned earlier. Therefore, if a molecule can be divided into two or three groups for which activity coefficients in a given environment are known, one can calculate the activity coefficient of the parent molecule in the environment. Several examples have been calculated, and the comparison with measured activity coefficients (Table V) ‘shows good agreement.
r curves derived from methanol-hexane data in predicting
Conclusions
T h e “solutions of groups” model has been found to represent actual solutions with nearly quantitative results in several cases. Although there are exceptions, such as the behavior of ethylene. glycol, the agreement with measured activity coefficients is generally within about 10% of log y. This “solutions of groups” approach to liquid mixtures permits a wide range of estimates from very few data and should have a number of practical uses. T h e approach should be
2.46 2.46 (1 .20)c (1.68) 0.34 0.34 0.34 (0.43)c considerations.
1.85
1 .745
2.17
2,04O
3.94
3 . 42a
2.22
2,09( 7)
0.37
0.22
0.40
0.68
particularly useful in making estimates of phase equilibria for screening possible separation processes and for evaluating the general validity of individual experimental measurements. Although the precision of estimates is generally not adequate for final design purposes, it may frequently be adequate for estimating the fate of noncritical components in the final design of a separation process. The approach should also be particularly useful in estimating activity coefficients of compounds for lvhich experimental measurements are difficult, as in polymer or biological systems. Further, it appears that the a priori problem of calculating solution nonidealities from statistical mechanical considerations is simplified, since it is possible to compare calculated results with mixture data for systems such as the methylene group-hydroxyl group system wherein the “components” have approximately the same size and simpler interactions. Perhaps with this beginning refinements will be made in this approach permitting one to estimate activity coefficients with even a higher degree of certainty than is possible now. Acknowledgment
T h e authors wish to acknowledge the contributions of E. L. Derr to the development of “solutions of groups” ideas and the general guidance and support of C. L. Dunn and Mott Souders, Jr. literature Cited
(1) Cox, H. L., Cretcher, L. H., J . Am. Chem. SOC. 48, 451 (1926). (2) Deal, C. H., Derr, E. L., Papadopoulos, M. N., IND.ENG. CHEM., FUNDAMENTALS 1, 17 (1962). (3) Ferguson, J. B., J . Phys. Chem. 36, 1123 (1932). (4) Ferguson, J. B., Freed, M., Morris, A. C., Ibid., 37, 87 (1933). (5) Langmuir, I., Colloid Symposium Monograph 3, 48 (1925). (6) Papadopoulos, M. N., Derr, E. L., J . Am. Chem. Soc. 81, 2285 (1959). (7) Pierotti, G. J., Deal, C. H., Derr, E. L., Znd. Eng. Chem. 51, 95 (1959). (8) Redlich, O., Derr, E. L., Pierotti, G. J., J. Am. G e m . SOC.81, 2283 (1959). ( 9 ) Seidell, A., “Solubilities of Organic Compounds,” Vol. 11, Van Nostrand, New York, 1941. RECEIVED for review July 24, 1961 ACCEPTED November 18, 1961 27th Annual Chemical Engineering Symposium, Division of
Industrial and Engineering Chemistry, ACS, St. Louis, Mo., December 1960. VOL. 1
NO. 1
FEBRUARY 1 9 6 2
23
