Langmuir 1985, I, 347-351 corrects for the low-angle displacement of peak maxima from which one otherwise might erroneously deduce a lattice expansion. However, in the case of aggregates in fully random or azimuthally random (“textured”) orientation, the reflections are not resolved into discrete Debye-scherrer rings if the aggregates consist of fewer than about 200 atoms. Therefore, the lattice parameter cannot be derived from a conventional analysis of the scattering pattern (where the first ring is assumed to be the 111 diffraction ring but is, in fact, a superposition of the 111 and 200 rings), not even after dividing by fA2. The only means to derive an accurate unit cell parameter is then to Fourier transform the scattered intensities to obtain a
347
radial distribution of the atoms, indicating the exact interatomic distances. The calculation of scattering patterns of model particles can also be useful for analyzing the morphology of strongly anisotropic aggregates. For example, due to their different scattering patterns, very flat aggregates (rafts) consisting of only a few atom layers can in principle be distinguished from spherical aggregates.
Acknowledgment. This work was supported by NASA under Grant NCC 2-283. We also gratefully acknowledge financial support by Alcoa Foundation to Eloret Institute for this work.
Group Contribution Model of Mixed Micellization. 1. Prediction of Critical Micelle Concentration in Mixtures Tsuyoshi Asakawa,* Kazuhiro Johten, Shigeyoshi Miyagishi, and Morie Nishida Department of Industrial Chemistry, Faculty of Technology, Kanazawa IJniuersity, Kanazawa 920, Japan Received December 14, 1984 A group contribution method is proposed to predict the cmc of aqueous binary mixtures of surfactants. The interaction parameters of the molecule’s functional groups, including hydrophilic groups, were determined from the cmc values of mixed systems. The calculated cmc’s were in good agreement with the experimental ones. The cmc curves in a series of mixed systems, e.g., lithium perfluorooctanesulfonate (LiFOWlithium dodecyl sulfate (LiDS),-lithium tetradecyl sulfate (LiTS),and -lithium hexadecyl sulfate (LiHS), were fitted by the same interaction parameters. The mutual solubilities of fluorocarbon and hydrocarbon surfactants in the micellar phase were also studied. The hydrocarbon-rich micelles solubilized the fluorocarbon surfactants to a certain extent, whereas the fluorocarbon-rich ones solubilized the hydrocarbon species sparingly. We could predict the cmc of nonideal binary mixtures by the use of cmc, Kg,molecular structure data of pure components, and the group interaction parameters.
Introduction The properties of the mixed surfactant systems are interesting from the viewpoint of both technology and solution theory.1-8 Many workers investigated the mixed systems by surface tension methods, fluorescence probe techniques, and NMR methods, etc.189s*7,p15In the mixed micelles of surfactants with different hydrophobic and hydrophilic groups, significant deviations from ideal mixing were ~ b s e r v e d . ~ JSuch ~ nonideal behaviors were treated by applying the regular solution theory to pseudo-phaseseparation m i c e l l e ~ . ’ ~This ~ ~ ~approach ~ ~ J ~ has been successfully applied to fitting the cmc curves.1 The formation of mixed micelles is known to be governed by the hydro(1) Shinoda, K.; Nomura, T. J. Phys. Chem. 1980,84, 365. (2) Funasaki, N.; Hada, S. J.Phys. Chem. 1980,84, 736. (3) Holland, P. M.; Rubingh, D. N. J. Phys. Chem. 1983, 87, 1984. (4) Rubingh, D. N. In ”Solution Chemistryof Surfactants”; Mittal, K. L., Ed.; Plenum Press: New York, 1979; Vol. 1, p 337. (5) Kamrath, R. F.; Franses, E. I. Ind. Eng. Chem. Fundam. 1983,22, 230. (6) Kamrath, R. F.; Franses, E. I. J. Phys. Chem. 1984, 88, 1642. (7) Hua, X. Y.; Rosen, M. J. J. Colloid Interface Sci. 1982, 90,212. (8) Mysels, K. J. J. Colloid Interface Sci. 1978, 66, 331. (9) Mukerjee, P.; Handa, T. J. Phys. Chem. 1981, 85,2298. (10) Almgren, M.; Lafroth, J. E. J. Colloid Interface Sci. 1983,91,256. (11) Fontell, K.; Lindman, B. J.Phys. Chem. 1983,87, 3289. (12) Carlfors, J.; Stilbs, P. J. Phys. Chem. 1984, 88, 4410. (13) Funasaki, N.; Hada, S. J. Phys. Chem. 1983, 87,342. (14) Harada, S.; Sahara, H. Chem. Lett. 1984, 1199. (15) Asakawa, T.; Miyagishi,S.; Nishida, M. J. Colloid Interface Sci.,
in press. (16) Lange, H.; Beck, K. H. Kolloid 2.2.Polym. 1973, 251, 424.
phobic interaction and the electrostatic interaction. The hydrophilic groups play a significant role in the formation of mixed micelles, but there are few treatments that take into account such effects.’J7J8 Langmuir suggested that physical (van der Waals) interactions between polyfunctional molecules may be estimated by summing interactions between the functional groups that constitute the m o l e c u i e ~ . ~I~t should be possible to estimate intermolecular interactions and thermodynamic properties with a suitable model. A group contribution concept was such a model.20 The group contribution method has been used for the prediction of activity coefficients in nonelectrolyte liquid mixtures.%* This method is based on the concept that “a solution is composed of many functional groups” and suited for a prediction of the vapor pressure and the compositions of vapor-liquid mixtures. In this paper, the group interaction method has been extended to the mixed micellar phase of aqueous surfac(17) Miyagishi, S.; Ishibai, Y.; Asakawa, T.; Nishida, M. J. Colloid Interface Sci., 1985, 103, 164. (18) Moroi, Y.; Motomura, K.; Matsuura, R. J. Colloid Interface Sci. 1974, 46, 111. (19) Langmuir, I. Colloid Symp. Monogr. 1925, 3. (20) Fredenslund, Aa.; Johes, R.; Prausnitx, J. M. AIChE J. 1975,21, 1086. (21) Gmehling, J.; Rawmussen, P.; Fredenslund, Aa. Ind. Eng. Chem. Process Des. Dev. 1982, 21, 118. (22) Magnussen, T.; Rasumussen, P.; Fredenslund, Aa. Ind. Eng. Chem. Process Des. Deu. 1981,20,331. (23) Abrams, D. S.; Prausnitz, J. M. AIChE J. 1975,21,116.
0743-7463/85/2401-0347$01,50/00 1985 American Chemical Society
Asakawa et al.
348 Langmuir, Vol. 1, No. 3, 1985 tant solutions on the idea that the micelle formation can be considered by the analogy of a vapor-liquid equilibrium.’ The extension permitted the estimation of the interaction between surfactant functional groups on the formation of mixed micelles. Recently, many workers reported the nonideal behavior of the mixed micelle of fluorocarbon and hydrocarbon In these mixed systems, one would have to take into account their molar volumes, the local mole fractions, and temperature. The group contribution concept, taking into consideration these effects, is expected to be useful for predicting the activities of surfactants in the micellar phase and the mutual solubilities of two kinds of mixed micelles.
based on energy interactions. The activity coefficient of surfactant i in the mixed micelles is given by In Fi = In FP + In F? (7) where In FY is the combinatorial and In FiRthe residual component. The combinatorial part is given by the following:
ai + Z-qi
In Fic= In Xi
In
2
4 ai + li - -Zjxjlj ai Xi
(8)
Z 2
li = -(ri - qi) - (ri - 1)
Experimental Section Hexaethylene glycol n-dodecyl ether, Cl,(EO)B, was obtained from Nikko Chemicals and was used without further purification. The other surfactants were prepared by the same procedure as reported previ~usly.’~~~’ Lithium perfluorooctanesulfonate and perfluorooctanoic acid were obtained from Dai-Nippon Ink and 3M Co., respectively. The cmc values of fluorocarbon and hydrocarbon mixed systems were determined by electrical conductivity measurements. The other cmc values were determined from surface tension method. The surface tension was measured by Wilhelmy technique (Kyowa-Kagaku surface tension meter, Model A-3).
Results and Discussion In a binary aqueous surfactant solution, the formation of micelles was treated quantitatively by the analogy of a vapor-liquid equilibrium. The group contribution concept was applied to “a pseudo-phase-separation mixed micelle”. The concentration of each comnonent is given
where C, is the monomer concentration of surfactant in the mixed system, i.e., the mixed cmc, C1 and C2 are the cmcs in the single component systems, Cionis the concentration of added counterion, and x and a are the mole fraction of surfactant in the micelle and in the overall binary mixture, respectively. F is the activity coefficient of surfactant in the micelle, which, in what follows, is described in some detail. K , is the micelle counterion binding parameter, which was obtained by plotting the logarithm of cmc vs. the logarithm of the concentration of the counterion. The Kgvalues were additive as reported previous1y.’J5J7 The subscripts 1and 2 refer to surfactants 1 and 2. The activity coefficient of a surfactant was divided into two parts on the basis of the group contribution concept.20 One part refers to the contribution due to differences in the molecular size and the other refers to the contribution due to molecular interactions. A combinatorial part is essentially based on differences in size and shape of the constituent functional groups of a surfactant. The functional group sizes and interaction surface area are introduced from independently obtained pure component and molecular structure data. A residual part essentially is (24) Mukerjee, P.; Yang, A. Y. S. J. Phys. Chem. 1976, 80, 1388.
where xi is the mole fraction of surfactant i in the micelle, Oi is the area fraction, ai is the segment fraction which is similar to the volume fraction, and Z is the coordination number, which was insensitive to numerical results for Fi. In this work, we consistently used Z = 10 (6 5 Z 5 12). Pure component parameter q and qi are the molecular van der Waals volume and the molecular surface area, respectively, which are calculated as a sum of the group volume and area parameters Rk and Qk Rk = Vwk/15.17
Qk
= A,k/(2.5 x
iog)
(10)
where v i i ) is the number of groups of type k in surfactant i. Group parameters Rk and Qk are obtained from the van der Wads group volume and surface area v w k and A w k given by B ~ n d i .The ~ ~ normalization factors 15.17 and 2.5 X lo9 were given by Abrams and P r a u s n i t ~ .Their ~~ fundamental assumption was additivity of the group molar volume. The contribution derived from one group is assumed to be independent of that derived from another group. The molar volume of a surfactant slightly changes by the formation of micelle, with concentration of the surfactant, with temperature, and by the mixing of the surfactants. But the contribution of volume changes was neglected because of the small quantities of such data and uncertainties of cmc data. We assumed that Rk and Qk were independent of the chain length of a surfactant, the micellar size, and the nature of the micellar phase. Thus we calculated V,k and Awkaccording to Bondi. On the basis of “a mixed micelle composed of functional groups”, the residual part of activity is given by In FiR = CkvLi)[lnr k - In rkCi)]
(11)
where rk is the group residual activity coefficient, and r k C 0 is the group residual activity coefficient of group k in a reference solution containing only surfactants of type i. The term In r k ( i ) is necessary to attain the normalization that activity coefficient Fi becomes unity as xi 1. The group activity coefficient r k is expressed by In r k = Q k [ l - In ( c m e m q m k ) - x m ( e m * k m / c n e n q n m ) l (12)
-
e,
=
Qmxm -
CnQnXn
qmn = exp[ - u m nRT -u n m
Equation 12 also holds for I?$),
]
= exp(
-%)
(14)
8, is the area fraction of
(25) Bondi, A. J.Phys. Chem. 1964,68, 441. (26) Shinoda, K. J. Phys. Chem. 1954, 58, 541.
Prediction of Mixed cmc
Langmuir, Vol. 1, No. 3, 1985 349
Table I. Mixed Systems 1. CH3(CHJnCOOK (n = 3-7)CH3(CHz),COOK (m = 4-8)’ 2. C ~ ( E O ) & ~ ( E O ) B ~ 3. C1zS04Na-C8(EO)4b 4. C1zS04Na-Clo(CH3)SOe 5. Cl,,S0,Na-Cl2SO4Na 6. C,Fl6COONa-Cl0SO4Na 7. C7F,,COONa-ClrCOONad 8. C;F&OONa-C&O4Na 9. C8F17COONH4-ClzSO4NH4‘ 10. CnFl,SOLi-C, ,SOALi 11. CiF;;SO,Li-C;;SO;Lt 12. C8F17S03Li-ClBS04Li 13. C1zS04Na-C11CONHC8H8COONa 14. CllCONHCHzCOONa-CllCONHC8H8COONa 15. CllCONHC4H8COONa-CllCONHC8H8COONa 16. C11CONHC4H8COONa-C12(EO)~ ‘Cmc data from ref 26. bReference 16. dReference 25. OReference 1.
cReference 3.
group m, and the sums are over all different groups. 0, is calculated in a manner similar to that for Bi. X, is the mole fraction of group m in the mixture. ,@ ,,, is the group interaction parameter. U,, is a measure of the energy of interaction between groups m and n. The group interaction parameters, A,, (two parameters per binary mixture of groups), are the parameters that must be evaluated from experimental cmc data. A,, has units of degrees kelvin and was obtained from a data base using a wide range of experimental results. Nonidealities in a bulk phase were not taken into account because of a rather dilute solution. As described above, we can calculate the mixed cmc values when cmc, Kg,molecular structure data of pure components, and the group interaction parameter are given. We have developed a program on a FACOM M170F computer. On the basis of vapor-liquid equilibrium data,2@22 numerous mixed systems tabulated in Table I were investigated to obtain the group interaction parameters. The cmc and Kgdata are listed in Table 11. The molecular structure data are listed in Table 111, and the obtained parameters were summarized in Table IV (the group interaction parameters which were not tabulated in Table IV were equal to zero). The group interaction parameters derived from the vapor-liquid equilibrium measurements were made to four significant figures, but we expressed the extended parameters with two significant figures because of the uncertainties of experimental cmc values. We assumed that the group interaction parameters of “COO” and “CH,SO” were equal to ones of “COOH” and “(CH&SO” in vapor-liquid equilibrium, respectively.21 The parameters of “(HOCH2CH2)20”and “CONH” were data from a liquid-liquid equilibrium.22 The same group number was used for CF3,CF2,and CF, which were treated as subgroups belonging to the same main group, “CF2”. For the subgroups belonging to the same main group, the interaction parameters were equal to zero (e.g., AcF,,CF = ACF~,CF~ = 0). Although the physical property of tke
1
klmmiscibleRegion 1 1.00
h
9.00
E Y
u
5.00
11111111(11
aoo
0.20
OLO
0.60
0.80
1.00
M o l e Fraction of LiFOS
Figure 1. Cmc’s of mixtures of LiFOS and LiDS at 25 O C . The plotted points are experimental values. The solid line and the dashed line are the cmc curves predicted from the group contribution method and the regular solution method (0 = 2.2), respectively. The dotted-dashed line indicates the micellar composition curve predicted from the group method. 10.00 Immiscitlc Region
-
7
!-.-.-..---.-______. . ;
6.00
i
0.0 0 0.00
0.20
0.40
0.60
M o l e Fraction o f
0.80
1.00
LiFOS
Figure 2. Cmc’s of mixtures of LiFOS and LiTS a t 25 “C. The dotted-dashed line indicates t h e micellar composition curve predicted from the group method.
methylene group attached to a polar head group was different from that of the main methylene groups, this effect was neglected at the present stage. We assumed that the interaction parameters between the resembled groups were also equal to zero, because the cmc curve can be fitted by ~ ~The ~ interaction pathe assumption (e.g., A c o ~=, 0). rameters of hydrophilic groups were taken into account because the electrostatic interaction was a serious factor in the micelle formation (e.g., A s o , , so ~ ~# O).” Figures 1-3 show the calculated and observed cmc values of the mixtures of fluorocarbon and hydrocarbon surfactants. The activity coefficient of both surfactants in a micellar phase was calculated by the use of eq 7-14 (see Appendix). Then the calculations of mixed cmcs were performed by a similar procedure as reported previously.15J7 The LiFOS-LiDS system was previously described by the regular solution theory.15 The calculated cmc by
Table 11. Experimental Data of Cmc and K, at 25 OC surfactant cmc, m M K. group assignment C7F&OONa (SPFO) 30.1 0.65 lCF3, 6CF2, lCOO C$17S03Li (LiFOS) 7.1 0.60 lCF3, 7CFz, l S 0 3 32.0 CloHzlS04Na(SDeS) 0.62 lCH3, 9CHz, l S 0 4 8.2 0.63 lCH3, 11CH2,l S 0 4 ClzHZ6SO4Na(SDS) 0.58 Cl2HZ6SO4Li (LiDS) 8.6 lCH3, 11CH2,lS04 2.3 0.60 1CH3, 13CHz, l S 0 4 C14HmS04Li(LiTS) 1CH3, 15CHz, lS04 CIBH3,SO4Li(LiHS)’ 0.63 0.61 13.6 0.67 lCH3, 11CH2,ICONH, lCOO C11Hz3CONHCHzCOONab (SLG) 7.3 3CH3,10CHz, 2CH, lCONH, lCOO 0.62 (SLV) CllH23CONHC4H8COONab 2.8 lCH3, llCHz, lCH, lCONH, lCO0,GACH 0.65 C11H23CONHC8H8COONab (SLP) ‘Data at 40 OC. bData in 0.4 mM NaOH.
350 Langmuir, Vol. 1, No. 3, 1985
Asakawa et al. 10.00
Table 111. Grouu Volumes and Surface Area Parameters
-
8.00
1A 1B 1c 2A 2B 2c 3 4 5 6 7 8 9
0.9011 0.6744 0.4469 1.406 1.0105 0.615 0.5313 1.301 1.3039 1.582 son so; 1.826 (HOCH2CH2)2OC 4.0013 1.9255 CHSSO CH3" CHz" CH" CF3b CFZb CFb ACH" COOb CONH
0.848 0.540 0.228 1.380 0.920 0.460 0.400 1.224 1.036 1.280 1.520 3.568 1.624
lb
hydrocarbon chain
fluorocarbon chain E aromatic carbon group carboxylate group acid amide group sulfonate group sulfate group diethylene glycol group methyl sulfoxide group
Reference 20. *Reference 21. Reference 22.
0.00
so3 so4
(HOCH2CH2)20b CH@O"
coo CONH so4
coo so3 so4
coo" CONHb (HOCHzCH2)ZO (HOCH2CHJ20 (HOCH2CH2)ZO CH3S0
32.08 663.5 928.3 600 600 269.2 526.5 100 30 90 537.4 364.2 -500 100 -500 -500
I
,
1
I
*
I
I
.
from ref 16. 10.00
8 .oo
ACH" COO" CONHb
I
Fraction o f S D S Figure 4. Cmc's of mixtures of SDS and C8(E0)4at 25 OC. Data
main group
ACH
I
Mole
Table IV. Group Interaction Parameters
CF2
-
0 200
15.26 315.3 -11.91 300 300 81.49 50.49 500 500
500 62.32 -80.48 230 100 230 10
-
r
600
I
E
u E
400
O D0 0.20
0.00
a80
0.60
040
I!
M o l e Fraction of S L P
Figure 5. Cmc's of mixtures of SLP and SDS in 0.4 mM NaOH at 25 "C.
Data from liquid-vapor equilibrium.20*21 Data from liquidliquid equilibrium.22 (I
I F l m m i s c i b l e Region
I
0.00 0.00
0.00
0.20
040
0.60
0.80
1.00
Fraction of L i F O S Figure 3. Cmc's of mixtures of LiFOS and LiHS at 40 O C . The dotted-dashed line indicates the micellar composition curve predicted from the group method. Mole
the group contribution method exhibited an excellent agreement with the observed values as compared with ones by the regular solution method. The parameters were checked by the cmc curves in the temperature range of 15-40 "C (the standard deviation; the group contribution method, 0.123 mM, the regular solution method, 0.236 mM). This method predicted that the two kinds of mixed micelles were formed in the solution (see Figure 1). The hydrocarbon-rich micelles solubilized the fluorocarbon surfactant to a certain extent. These results were similar to ones by the regular solution method.I5 On the other hand, the fluorocarbon-rich micelles solubilized the hy-
t
I
0.20
t
I
0.40
I
I
0.60
I
I
OB0
I
I 1.00
M o l e Fraction o f S D S Figure 6. Cmc's of mixtures of SDS and CIo(CH3)SO in 1 mM Na2C03at 24 "C. Data from ref 3.
drocarbon species sparingly. This mutual solubility behavior of two kinds of mixed micelles resembled that for the mixed micelles of NF and STS reported by Funasaki and Hada.2 The cmc values of LiFOS-LiTS and LiFOS-LiHS could be calculated by the use of the same interaction parameters and molecular structure data as in the LiFOS-LiDS system. The results mean that the series of mixed systems containing the same functional groups can be described if only the values of cmc and Kgof pure surfactants are given. As shown in Figures 1-3, the immiscible region increased with the increases in carbon number of hydrocarbon surfactants. This result was consistent to the mutual immiscibilities of fluorocarbon and hydrocarbon liq~id.~~~~~ (27) Bedford, R. G.; Danlap, R. D. J.Am. Chem. SOC.1958,80, 282.
Langmuir, Vol. 1, No. 3, 1985 351
Prediction of Mixed cmc
The important features of this method are its applicabilities to the mixed system composed of surfactants with a complicating structure such as branched hydrocarbon chains and the mixed systems of ionic-nonionic surfactants (see Figures 4-6). The calculated cmc curves of ionicnonionic mixed systems could not be fitted to experimental ones without the interaction parameters between the different hydrophilic groups (e.g., AS0,,CH3S0 # 0). The large deviation from the ideal mixing would be mainly derived from the interaction between these hydrophilic groups. The mixed systems 15 and 16 in Table I deviated from the ideal model as reported previous1y.l' Although the contribution of the substituent groups would be different from ones of the main groups, the cmc c w e s could be fitted by the use of the same parameters as one of the main groups. This group contribution method is worthy of a prediction of mixed cmc significantly deviated from ideal model. It is possible to estimate which groups would be the main factor relating to the nonideal behavior in micellar phase. Applications. The group contribution method will be a very useful concept toward alleviating the difficulty of predicting the properties of surfactants solutions, e.g., the mixed cmc, the mutual solubilities of two kinds of mixed micelles, the adsorption on air-water interface, and so on, even though the group interaction parameters are the adjustable parameters which are not appreciably assigned a significance in terms of physical inter,actions. This method can be used to advantage in describing the mixed systems extremely deviated from ideal mixing by using the extended parameters. It is possible to attribute a nonideality in the micellar phase to the interaction energies ( Umn)between the functional groups. Holland and Rubingh predicted the cmcs in the multicomponent systems by using net interadioh pyameters of pairs which were determined independently in binary system^.^ The group contribution model will be easily expanded to multicomponent mixtures only by using the parameters of the binary systems without another paraneter. Most of surfactants have been used as the multicomponent mixtures in the chemical industry. It is expected to compile the experirdental data of cmcs and to elucidate the properties of the multicomponent mixtures. The group contribution model will provide a powerful tool for the analysis of such data and the chemical design of new surfactants.
Acknowledgment. We are grateful to Dai-Nippon Ink for providing the fluorocarbon surfactants. Appendix We calculate the activity coefficient of perfluorooctane in an equimolar C8F18 (1)-C12HB(2) mixture. Perfluorooctane has two CF3 groups (2A) and six CF2 groups (2B). Thus Y Z ( ~ = ) 2, ~ 2 ' = ~ )6, R ~ = A 1.406, R ~ = B 1.0105, Q ~ A= 1.380, and Q2B = 0.920. rl = 2(1.406) + 6(1.0105) = 8.8750 q1 = 2(1.380)
+ 6(0.920) = 8.280
Dodecane has two CH, groups (1A) and ten CH2groups (1B). Thus yJ2) = 2, ~1'~)= 10, RIA = 0.9011, Rip = 0.6744, QIA = 0.848, and Q 1 B = 0.540, similarly, r2 = 2(0.9011) + lO(0.6744) = 8.5462 q2 = 2(0.848)
+ lO(0.540) = 7.096
(28) Gilmour, J. B.;Zwicker, J. 0.; Katz, J.; Scott, R.L.J. Phys. Chem. 1967, 71, 3259.
+
= 8.8750(0.5/[(8.8750)0.5 (8.5482)0.5]) = 0.5094 81 = 8.280(0.5/[(8.280)0.5 11
+ (7.096)0.5]) = 0.5385
= Ei(8.875 - 8.280) - 7.875 = -4.900
= 5(8.546 - 7.096) - 7.546 4.296 0.5094 In FIC = In - Ei(8.280) In 0'5380 - 4.900 0.5 0.5094 12
+
~
+
0'5094 [0.5(4.900) + 0.5(0.296)] = 0.06533 0.5 The needed group interaction parameters are obtained from Table IV. \k2,1 = exp( - 147.3 = 0.6102
m)
91,2 = e x p (-2.859 -G)
= 1.0096
For pure perfluorooctane ( X 2 A = 218, X 2 B = 6/8), 2 (1.380) OZA(l) = 0.3333 2(1.380) + 6(0.920) 0pB") = 0.6667 In
r2A'l)
1 - In (0.3333
= 1.380 1
(0.33ATT6667 For
0.6667 0.3333 0.6667
+
xi = 0.5, 0.5(2) = 0.1 = 0.5(2) + 0.5(6) + 0.5(2) + 0.5(10)
X2A
02A
+
+ 0.6667) -
O.l(l.380) = O.l(l.380) + 0.3(0.920) + O.l(O.848) + 0.5(0.540) = 0.1795 0 2 =~ 0.3590 0 1 A = 0.1103 0 1 B = 0.3512 1 - In (0.1795
f
0.1795
I
+ 0.3590 + 0.1103.
+ 0.3590
A
[0.1795 + 0.3590 + 0.4615(1.0096)) '
+
0.1103 0.3512 0.5385(0.6102) + 0.1103 + 0.3512 In
r2~ =
)]
= 0.1421
0.920(0.1030) = 0.09476
In FIR= 2(0.1421 - 0)
+ 6(0.09476 - 0) = 0.8528
+
In F, = In FIC In FIR= 0.06533 + 0.8528 = 0.9181 Thus we obtain F, = 2.505. Similar calculations predict that the immiscible region is between x1 = 0.0415 and x1 = 0.9692. x1 = 0.0415 al = 0.973 a2 = 0.963 x1 = 0.9692 a, = 0.973 a2 = 0.963 where a is the activity of the component. Registry No. SPFO, 335-95-5; LiFOS, 29457-72-5; SDES, 142-87-0;SDS,151-21-3;LiDS, 2044-56-6; LiTS, 52886-143; LMS, 89045-35-2; SLG, 18777-32-7; SLP, 95408-35-8; SLV, 95420-21-6; C12(EO)e, 95408-36-9.
