Faraday Symp. Chem. SOC., 1982 17 11-24 Precise Vapour-pressure Measurements of the Solubilization of Benzene by Aqueous Sodium Octylsulphate Solutions BY EDWINE. TUCKER AND SHERRILD. CHRISTIAN Department of Chemistry The University of Oklahoma Norman Oklahoma 73019 U.S.A. Received 11th August 1982 Accurate vapour-pressure-solubility results have been obtained for aqueous solutions of sodium octylsulphate(SOS) and benzene at 15,25,35 and 45 "C. Thermodynamic constants are reported for reactions of benzene with SOS monomers and micelles ; hydrophobic effects are important in stabiliz- ing the aggregates formed between monomeric benzene and SOS species. The observed non-linearity of degree of solubilization with benzene activity indicates that benzene molecules interact cooperatively within the micelles.A mass-action model employing a form of the Poisson distribution equations modified to account for cooperativity provides an excellent fit of all the solubilization data at each temperature. Activity coefficients are reported for benzene in the micellar species as functions of temperature and micellar composition. Among the salient properties of aqueous micellar systems is the ability of these solutions to dissolve large concentrations of hydrocarbons. There have been numerous studies of the extent of solubilization of hydrocarbons by aqueous surfac- tants and many attempts to characterize the solubilized components by physical methods. -3 From a thermodynamic point of view little is known about the dependence of the free energy energy or entropy of solubilized hydrocarbon molecules on either concentration or temperature.Except for recent reports from this laborat~ry,~~~ only a few publications provide any information about the variation of the extent of solubilization with hydrocarbon activity or fugacity. It has generally been assumed either explicitly or implicitly that the solubilization of hydrocarbons occurs as predicted by Henry's law; that is that the concentration of the solubilized species varies linearly with solute activity.6 The large majority of solubilization studies have in fact been restricted to investigations of solutions prepared by allowing the aqueous surfactant solutions to equilibrate directly with pure organic liquids or liquid mixtures.The purpose of the present research has been to provide detailed information about the thermodynamics of solubilization of a hydrocarbon benzene by an aqueous surfactant sodium octylsulphate (SOS) at concentrations below and above the critical micelle concentration (c.m.c.). A high-precision automated vapour-pressure apparatus '** has been used to investigate the formation of discrete complexes between benzene and SOS. Below the c.m.c. the adducts which must be considered include the 1 1 complex and other small molecular species; above the c.m.c. solubiliz- ation occurs primarily through formation of aggregates containing one or more benzene molecules and n surfactant anions. A detailed mass-action model based on an extension of the Poisson distribution equations is employed to correlate the many hundreds of sets of equilibrium vapour-pressure data obtained at benzene fugacities varying from 0 to 70% of saturation at temperatures from 15 to 45 "C.SOLUBILIZATION OF BENZENE BY sos EXPERIMENTAL VAPOUR-PRESSURE MEASUREMENTS An automated vapour-pressure apparatus and techniques described previously '9' were used to measure equilibrium pressures of benzene above aqueous solutions contained in a thermostatted central reservoir. In each series of measurements a known volume of SOS solution was introduced into the central reservoir which had a total volume of 510.9 cm3 and the system was evacuated to the equilibrium vapour pressure of the aqueous solution. The small amount of water volatilized from the sample during the evacuation step was collected and accurately weighed.Under microcomputer control successive samples of benzene stored in an external reservoir at 50 "C,were volatilized into the central reservoir through a 6-port high-pressure chromatography valve; the quantity of benzene added per increment is highly reproducible equalling (2.135 rf 0.002) x mol in the present series of experiments. The total vapour pressure was monitored continuously and after each sample addition comparisons of successive pressure values were made at regular intervals. Approximately 30 to 45 min were required to reach equilibrium which was judged to have been attained when the pressure varied by less than 3 mTorr over 5 min. The primary data consisted of several series of measured equilibrium vapour-pressure values (p) at constant temperature (T),for equal increments of benzene added to a system of known liquid and vapour volume and total composition.Table 1 lists the entire collection of 588 vapour-pressure values obtained at 4temperatures (15,25,35and 45 "C)for accurately measured volumes of solutions having known initial concentrations of SOS. In the initial processing of data it is necessary to convert the measured total vapour pressures to fugacities of benzene at known total concentrations of benzene and SOS in the aqueous solution. This requires information about the virial coefficients of benzene in the vapour phase,' as well as partial molar volume data for benzene in the aqueous phase. The volume of benzene added is in each case small compared with the volume of the condensed phase; consequently the partial molar volume of benzene need not be known with great accuracy.A complete table of the derived solubilization results available from the authors lists values of the total aqueous concentrations of SOS and of benzene the fugacity of benzene the concentration of benzene monomer in the aqueous phase (inferred by assuming that monomeric benzene obeys Henry's law on the molarity basis and using a Setchenow coefficient to account for the salting-out effect of added Na+ on the solubility of monomeric benzene) and the absolute temperature for all of the samples investigated. CHEMICALS Sodium octylsulphate (Eastman Kodak Reagent Grade) was purified as described pre- vio~sly.~ Benzene (Mallinckrodt A.R.Grade) was fractionally distilled and carefully dried before use. RESULTS INITIAL EXAMINATION OF SOLUBILIZATION Before developing a detailed mathematical model for the solubilization data we shall consider some general features of the vapour-pressure-solubility results. Fig. 1 shows some of the data obtained at 25 "C for pure water for [SOS]z 0.15 mol dm-3 and for [SOS]z0.29 mol dm-3. In the absence of dissolved SOS the partial pressure of benzene in equilibrium with its aqueous solutions increases nearly linearly with benzene concentration because only a small fraction of the dissolved benzene exists as self-associated species.8 However in the SOS solutions the partial pressure of benzene increases much less rapidly as the concentration of benzene [B] increases.Qualitatively this seems to imply that there are cooperative interactions between E. E. TUCKER AND S. D. CHRISTIAN 75 60 45 h c, .I 0 M 2 30 15 0.02 0.04 0.06 benzene molarity FIG.1.-Benzene fugacity plotted against molarity for solutions at 25 "Cin HzO(-) SOS (X) and 0.29 mol dm-3 SOS (0). 0.15 mol dm-3 4.0 3.0 2 .o d 1.0 0 FIG.2.-Ratios of increments in solubilized benzene concentration to increments in benzene monomer concentration at 15 "Cas functions of benzene monomer concentration for varying SOS concentra-tions curve A 0.3 rnol dm-3 SOS; curve B 0.21 mol dm-3 SOS; curve C 0.15 rnol dm-3 SOS; curve D 0.1 mol dm-3 SOS.c- P TABLE 1.-PARTIAL PRESSURES OF BENZENE ABOVE AQUEOUS SOLUTIONS OF SODIUM OCTYLSULPHATE a -~~ ~=25.00OC T=l5.00 "C Tz25.00 OC T=25.00 "C Ts35.00 "C T=45.00 "C T=l5.00 "C T=25.00 "c T=25.00 "c T=35.00 "c Tz45.00 "c n-w=20.925 n-w=l1.691 n-wz12.869 n-w=11.650 n-w=l2.489 n-w=11.957 n-w=7.9318 n-w=8.7455 n-w=7.8929 n-w=8.4696 n-w=8.1083 2.1700€+00 2.1620~90 2.955M40 3.09SOE*00 4.027OE+OO 5.1 51 OE40 2.3140EMO 3.24SDE90 3.1540E90 4,26mE+00 5.lc6oE+oo 4.3360E90 4.3180E90 5.88mE90 4.1 870E90 8.0380E90 1.0261E41 4.59loE.90 6.4620E90 6.287OE+00 6.4260~*00 1.0270€+01 6.4970E90 6.4660E+00 8.81 70E90 9.2730E90 1.2030E91 1.5355E91 6.8240E90 9.6240€+00 9.376oE+Oo 1.2544E91 1-556CE91 8.6580E+00 8.61 60E90 1.1757E91 1.2362E91 1. S998E91 2.0457E91 9.0471 E90 1.2783E91 1.2437E91 1.6626€+01 2.0421 E91 1.0805E91 1.0761E91 1.C684E41 1.541X91 1.W77E91 2.5552E91 1.1236E91 1.5883E+01 1.5461E91 2.0696E 91 2.548X+01 1.2966E91 1.2907E91 1,7603E91 1.0469E91 2.3965E91 3.0649E91 1.%03E91 1.8WOE+O1 1.8474E91 2.473% *01 3.0507E41 1.5121E+01 1.5045E+Ol 2.051 3E91 2.1539€91 2.7947E*01 3.5737E91 1.5547EtOl 2.2034E91 2.1 63X+O1 2.8757E91 3.5506€+01 1.7272E91 1.71 94E91 2.342SE91 2.4581E91 3.1 9lZE91 4.0854E91 1.7652E91 2.5045E91 2.4370E91 3.271 4E91 4.0464EMl 1.941 7E91 1.9333E91 2.6342E91 2.7631E91 3.5867E91 4.5933E91 1.9743E41 2.8042E91 2.7282E91 3.665X91 4.543QE91 2.1 571 E91 2.1 467E 41 2.9263E91 3.0677E91 3.9828E91 S.lWSEt01 2 .1782E91 3.1024E91 3.01 SEE91 4.0584E41 S.0329€*01 2.3723E91 2.361 lE41 3.2148Ei-01 3.3714E91 4.3774E91 5.6076E41 2.381 1EM1 3.5WE41 3.2994E91 C,647OE91 5.522XdJl 2.5882E91 2.5742E41 3.5051E91 3.6755E91 4.773X91 6.1 132E+O1 2.5807E91 3.6855EMl 3.57QIE91 C.832K91 4.0092E*01 2.8026E91 2.7857E91 3.7962E41 3.9785691 5.1685E*01 6.6177E91 2.7756E91 3.9734E91 3.8561E91 5.21 42E41 4.4924E91 3.0179€91 2.W67E91 4.0854E91 4.2816E91 5.5620E91 7.1228E91 2.9701 €91 4.2583E91 4.1298E91 5.5937E91 6.9702E91 3.233OE91 3.M81 EN9 4.374SE91 4.5856E91 5.95?7E91 7.6247E41 3.1604E91 L539lE+Ol 4.4012E91 5.9707E+01 ?.cc43€91 3.4469E41 3.41 85E+01 4.6655E+01 4.8835E91 6.3506E91 8.1 24SE41 3.3469E91 4.8172E91 4.67W91 6.3448E41 7.9l71E91 3.6618E91 3.4293E91 4. %S2E91 5.1822E91 6.7460E91 8.6296E91 3.531 5E91 5 .OW 4EM1 4.9355E91 6.71 40OE91 8.3807E91 3.8765E91 3.8397E91 5.2436641 5.48Qk91 7.1381 EM1 9.1328E91 3.7121 E91 5.3613E91 5.1 961 €91 7.0809€91 8.8565E91 4.0897E91 4.0486E91 5.5317E41 5.7768E91 7.5M2E 91 9.6339E91 3.8898E91 5.4284E91 5.4546€+01 7.4452E91 9.321 5E91 4.3041 €91 4.2566E91 5.8202E91 6.071 9E91 7.9231E91 1-01 36E92 4.064% 91 5.8937E91 5.7096E91 7.8048E91 9.783SE*01 4.51 89E91 4.4641E91 6.1 046E41 6.3655E91 8.31 38E91 1.0635EN2 4.2347E91 6.1551 E41 5.9602E#1 8.1 627E91 1.024&*02 4.7334E91 6.3930E41 6.65 7% 91 8.7040E41 1.t 131 €92 4,4035E91 6.4112E41 6.2076EtOl 8.51 SOEM1 1.0701E92 4.947X91 6.6794E91 4.9485E41 9.0924E91 1.1630E42 4.5668€+01 6.6645E91 4.451 OEM1 8.8646E91 1.1154E42 5.161 R91 4.9636E91 9.4804E91 1.21 28E92 6.9l62E91 6.6931E91 9.2098691 1.1 60R42 5.3756EMl 9.8680E91 1.2622E92 7.161 1EN1 4.9292Eal 9.5524€91 1.MSlE+02 5.588x91 1.0253E42 1.31 17E92 9.8906E91 1.2494EM2 5.8D27€*01 1.0636E92 1.5609E92 1.0225E92 1.293CEtO2 6.01 74~91 1.41 OOE42 1.0554E92 1-3371 €92 6.231 M91 1.3803E92 6.443X91 6.6574E91 TABLE1.-cont.T= 15.00 "C T= 25.00 "C T= 25.00 "C T= 35.00 "C T= 45.00 "C T= 15.00 "C T= 25.00 "C T= 25.00 "C T= 35.00 "C T=45.00 "C n-w= 5.8398 n-wc6.0523 n-w= 5.81 81 n-w=5.9480 n-w= 5.8475 n-w=4.0130 n-w=4.3478 n-w= 3.9947 n-w=4.2473 n-w=4.0347 2.1 2W)EaO 2.9520Et00 2.WWE+00 3.8530E90 4.7980E+00 1.9490E90 2.7510E90 2.6270E90 3.6060E90 4.491oE+00 4.1 9WE90 5.8700E90 5.7990E90 7.6580E90 9.5920E90 3.8760E90 5.4770E90 5.3020E90 7.179OE90 8.9720E90 6.2680E90 8.76&+00 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of the given moles of water (n-w) and 0.024 15 moles of sodium octylsulphate in a volume of 510.9 cm3.c-r Each successive pressure value represents the addition of 2.135E-04 moles of benzene to the enclosed system.vl SOLUBILIZATION OF BENZENE BY sos benzene molecules dissolved in the micelles although an additional factor to consider is that the concentration of free or monomeric SOS must decrease as benzene-SOS solubilizates are formed. Fig. 2 illustrates the tendency toward greater solubilization at the higher benzene concentrations; the figure shows the dependence of A[B],,,/Ac on C at nearly constant [SOS] (where A[B],, is the change in the concentration of benzene solubilized by SOS corresponding to a change in benzene monomer con- centration AcB). At concentrations of SOS less than the c.m.c. (approximately 0.136 rnol dm-3) the AIB],,l/AcB values are quite small and these values increase relatively little as C increases.Above the c.m.c. a very large percentage increase in A[B],,,/ AcB occurs as C varies from 0 to 0.016 mol dmA3. Note that derivative plots like that shown in fig. 2 indicate the high quality of the data; no solubilization data of comparable precision are available in the literature. Each of the data sets in table I consists of 20-35 measurements of total pressure for solutions prepared by adding equal increments of benzene to an aqueous solution having a nearly constant SOS concentration. For each point in the separate data sets values of the total concentration of benzene [B] and the benzene monoLver concentration c, were inferred in the initial processing of data (vide supra). It is possible to fit the individual data sets {[B] c,> to quartic equations [B] = acg + bck + CC~,+ dci (1) nearly to the precision of the data (ca.5 x lov6mol dm-3 in [B]). The collection of empirical parameters (a byc and d) obtained by fitting the separate sets may then be used to interpolate the data or to arrange them in useful forms for plotting. 0.0 7 I 1 0 0.08 0.16 0.24 0.32 SOS molarity FIG.3.-Curves of aqueous benzene solubility at fixed monomer concentrations of benzene (0.002 0.004 . . . 0.016 mol dm-3) against SOS concentration at 25 "C. It is convenient to use the derived quartic equations to obtain values of [B] corresponding to chosen values of c, at nearly constant values of [SOS]. Fig. 3 displays the results at 25 "C; the smoothed curves passing through the interpolated points indicate the dependence of [B] on [SOS] at values of C varying from 0.002 to E.E. TUCKER AND S. D. CHRISTIAN 0.016 mol dm-3 in increments of 0.002 mol d~n-~. These curves show that small amounts of benzene are solubilized below the c.m.c. and that a substantial increase in solubilization occurs beyond the c.m.c. It is also evident that the value of the c.m.c. (estimated to be equal to the concentration of SOS at the intersection of the two nearly linear portions of each curve) decreases as cB increases. Such an effect has been observed previously,1° but the present results are unique in showing how the entire solubilization curve is displaced as the benzene activity gradually increases. In the detailed analysis of the solubilization data (vide infra) it is inferred that the concentration of monomeric octylsulphate anion c,- must decrease as [SOS] increases beyond the c.m.c.Estimates are made of values of cA-for SOS solutions in the limit of zero benzene activity. Using these cA-values one may calculate the concentration of micellar SOS [SOS],ic at c = 0 for each SOS solution for which the concentration exceeds the c.m.c. The derived values of the parameters for the quartic equations for each data set [see eqn (l)] may then be used to calculate the limiting equilibrium constant for each total concentration of SOS where [BImicis the total concentration of benzene in micellar species i.e. [B],, minus the concentration of benzene in non-micellar complexes with SOS.Kl may be interpreted as the equilibrium constant (divided by the micelle aggregation number n) for the reaction A, + B+A,B (3) where A, represents the SOS micelle containing n octylsulphate anions B represents monomeric benzene in dilute aqueous solution and A,B denotes the micellar species containing only one solubilized benzene molecule. Although the average value of the micelle aggregation number n cannot be obtained with accuracy from the present results Kl can be obtained (once c,-is known) directly from the solubilization data at each total concentration of SOS. Table 2 lists Kl values inferred for the solutions TABLE 2.-EQUILIBRIUM CONSTANTS FOR TRANSFERRING BENZENE INTO SODIUM OCTYLSULPHATE MICELLES temperature/"C Kl/dm3moI-l a 15.00 10.3 25.00 10.8 35.00 10.5 45.00 10.1 Kl represents the equilibrium constant (divided by n) for the reaction A +B+A,B in the limit as [benzene]+O.Standard states for the solutes (B = benzene and A = micelles of n octylsulphate anions) are unit molarity ideal dilute solution states. at the largest SOS concentrations at 15,25,35 and 45 "C; Kl appears to have a maxi- mum at ca. 27 "C. At 25 "C,AH = 0.29 & 0.14 kcal mo1-l and ACp = -95 & 17 cal rno1-l K-l for the reaction [eqn (3)] representing the transfer of a benzene mole- cule from dilute aqueous solution into the interior of the SOS micelle. Note that 27 "C is approximately the temperature at which the c.m.c. for sodium octylsulphate reaches its minimum value.'l At this temperature AH for transferring an octylsulphate anion from aqueous solution into the micelle is approximately zero SOLUBILIZATION OF BENZENE BY sos and ACp is ca.-70 cal mol-I K-I for this transfer.11J2 Thus the micellization process is characterized by enthalpy and heat capacity changes very similar to those pertaining to the transfer of one benzene molecule into the micelle. A MASS-ACTION MODEL FOR SOLUBILIZATION Previously we used mass-action equations similar to those derived from B.E.T. adsorption theory to fit solubilization data for cyclohexane with sodium octylsulphate micelles and for benzene and cyclohexane with sodium deoxycholate micelle~.~ One (or two) discrete equilibrium constants were used to represent the association of the first (or first and second) hydrocarbon molecules with the micelle binding site.A step-wise equilibrium constant was employed to account for the binding of each additional hydrocarbon molecule by micelles already containing solubilized hydro- carbon. In the case of the cyclohexane SOS system an equilibrium constant for the reaction A- + cyclohexane+A-• cyclohexane was adequate to account for the soh- bilization results in the premicellar region. An assumed micelle aggregation number of I5 or 16 provided the best fit of the data for a wide range of cyclohexane activities and SOS concentrations. When attempts were made to use similar simple association models with the benzene SOS solubilization data it became apparent that the size of the micelles could not be assumed to remain constant as the benzene activity varied.Successful modelling required the assumption that the average micelle size gradually increases as the fugacity of benzene is increased. There is also some evidence in the literature that the solubilization of aromatic compounds by alkylsulphate micelles does lead to increases in micelle size.I3 Incorporating a degree of polydispersity in the mass- action model is therefore not unreasonable but it does complicate the mathematical analysis of the data. In trying to develop a mass-action scheme that is both reasonable and economical in its use of adjustable parameters we decided to approach the solubilization problem from a different statistical point of view. The Poisson distribution seems a good zeroth-order model because it quite simply predicts relative concentrations of soh-bilizates containing 1 2 3 .. . i molecules of benzene in the limiting case where no interaction occurs between the bound benzenes.14 Thus for micelles A, the expres- sions for the total micellar SOS and bound benzene concentrations are [SOS]mic= ncAn[l + a + a2/2!+ a3/3! + . . .] = nca,,exp(a) and [BImic= cA,[a + a’ + d/2!+ a4/3! + . . .] = acA,,exp(a) (4) where a = nK1cB. (The constant Kl is the solubilization constant introduced in the previous section; note that Kl equals l/n times the equilibrium constant for the reaction A + B +A,B.) Eqn (4) predicts that the ratio of the concentration of benzene bound in the micelle to that of all micellar forms of SOS will increase linearly with cB or with benzene activity.However it is clear that any reasonable analysis of the solubilization data will require accounting for cooperativity in the binding of benzene. That is as cB increases the ratio [B],,c/[SOS],ic/cB increases (see for example fig. 2). Although the unmodified Poisson equations do not provide a quantitative fit of the data they do serve as a logical point of departure in developing a model that accounts for intramicellar benzene-benzene interactions. To represent micelles containing one benzene molecule only the a term is required in the expressions for bound B and micellar SOS [eqn (4)]. But in considering micelles containing two E. E. TUCKER AND S. D. CHRISTIAN benzene molecules we propose adding a multiplicative term eb to account for the free energy of interaction of the B .-. Bpair at the average distance of contact. If b > 0 the A,B2 species will form with a larger equilibrium constant than that predicted by the original Poisson distribution; but if b < 0 the A,B2 species will have a smaller formation constant than that given by the Poisson equations. In considering the micellar species A,B3 it seems reasonable to include the factor eb to the third power to account for an average of 3 times as many B -B interactions (presumably at the same average distance of contact) as in the A,B2 species. Similarly for the A,B species the term (eb)6 is used as a factor to enhance (or inhibit) formation of that solubilizate. Proceeding in this way we may derive the following expressions for the concentrations of micellar SOS and bound benzene [SOS],ic = mA,[l + a + a2eb/2!+ a3(eb)3/3!+ .. . + ai(eb)'(l-l)12/i!+ . . . ] and [B'Jmic (5) = cA,[~ + a2eb+ a3(eb)3/2!+ . . . + ai(eb)'('-l)l2/(i-I)! + . . .I. Eqn (5) have proved to be quite efficient in fitting solubilization data; for a given micelle A, they involve only the two unknowns Kl (needed in calculating a) and b the interaction parameter. Unfortunately we have been unable to derive closed-form expressions that will represent the series. Therefore in practice the partial sums are ob- tained numerically the series being terminated at values of i for which the ai(eb)'('- l)12/i! terms become insignificant. APPLICATIONS OF THE MASS-ACTION MODEL IN FITTING SOLUBILIZATION RESULTS In attempting to correlate solubilization data throughout wide ranges of SOS concentration and benzene activity we have encountered two major problems (1) It appears to be necessary to consider the existence of at least two different micelle sizes and to include separate solubilization constants for each type of micelle.(2) The thermodynamic equilibrium constant for forming each micelle pertains to reactions of the type pNa+ + qA-+Na,AqPd4. Therefore the activities of the monomeric ions and the micelle should be used (rather than concentrations) in any rigorous mass-action model for fitting the solubilization data. Our previous modelling of the cyclohexane SOS data led to a good correlation of results without considering the polydispersity of the micelles or the dependence of activity coefficients on total SOS c~ncentration.~ However considerably smaller amounts of cyclohexane are solubilized (compared with benzene) and the range of SOS concentrations covered is smaller than for the present system.We are able to account for the major effects of micelle polydispersity on solubiliz- ation by assuming that only two micelles exist the species containing 16 A-anions and the aggregate of 22 A-anions. Obviously other combinations of micelle sizes could be used in modelling but the chosen sizes seem reasonable and they lead to an adequate fit of all the data. Problems arising from the variation of ionic activity coefficients with SOS con-centration are not easily solved.However it seems reasonable to ignore activity- coefficient effects in fitting the separate series of solubilization data at nearly constant [SOS] because the addition of benzene alone should not cause significant variations in the ionic strength. We have been able to estimate that at [SOS] = 0.30moI dm-3 the concentration of the A-anion is 0.120mol dm-3. This value derives from our mass-action simulation of SOS activity data obtained from Kale and Evans in SOLUBILIZATION OF BENZENE BY sos advance of p~b1ication.l~ The simulation includes the effects of ionic strength on activity coefficients and it leads to the result that the degree of counterion binding (of Na+) is ca. 40%. In fitting the solubilization data at each temperaturq we have fixed the value of cA-at 0.120 mol dm-3 (in the absence of benzene) at the highest con- centrations of SOS.At other SOS concentrations we have not forced cA-to have any predetermined value; instead the cA-values are inferred from the complete mass-action correlation of the data. In the analysis the formation constants for the two micelles A16 and AZ2 are assumed to be independent parameters to be determined by least-squares optimization at each SOS concentration. In other words instead of attempting to account quantitatively for activity coefficient effects throughout the range of SOS concentrations we have allowed the individual micelle formation con- stants to vary with concentration. These equilibrium constants thus pertain to the reactions 16 A-+A16 and 22 A-+ A22 where concentrations of species may now be used in place of activities in the mass-action fitting of data.In fitting the data at SOS concentrations less than the c.m.c. it is necessary to introduce the equilibrium constant K,, for the reaction A- + B -+ A-.B plus one other formation constant to account for a species with several A- anions and at least two benzene molecules. The complex A6B2 is assumed to form with an equilibrium constant & and both K, and Kbt are inferred from the solubility data for [SOS] < 0.12 rnol dm-3. The derived values of these constants are incorporated into the com- posite model used to correlate the entire collection of data at each temperature. Given the assumptions and rationale in the preceding several paragraphs we may summarize the features of the actual model used in fitting data (I) The equilibrium constants K16 = cA16/ci!and K22= cA2Jc2! are parameters to be determined separ- ately for each SOS concentration greater than the c.m.c.[except for the largest SOS TABLE 3.-MASS-ACTIONPARAMETERS FOR SODIUM OCTYLSULPHATE-BENZENE COMPLEXES IN AQUEOUS SOLUTION 15.00 "C 25.00 "C 35.00 "C 45.00 "C Kll/dm3 mol-' a K62/105dm21 mol-' K for A16 micelles/ 0.45 i0.02 2.63 & 0.05 9.97 If 0.05 0.62 i0.01 5.72 * 0.10 10.10 f0.03 0.83 i0.02 5.16 i0.06 10.01 & 0.05 0.93 i0.02 5.12 i0.06 8.88 i0.06 dm3 rnol - Kl for AZ2micelles/ 11.19 i0.05 11.74 & 0.05 11.44 i0.05 11.29 f0.06 dm3 mol-l b for A16 micelles 0.036 & 0.002 0.027 i0.002 0.031 f0.002 0.028 i0.004 b for A22micelles 0.055 & 0.001 0.049 i0.001 0.050 5 0.001 0.042 f0.001 r.m.s.d.(mol dm-3) 9.82 x 10-6 2.71 x 10-5 7.97 x 10-6 1.04 x 10-5 a Equilibrium constant for A + B -+ AB (where A denotes octylsulphate and B denotes benzene); equilibrium constant for 6A $-2B -+ A&; equilibrium constant for A16 + B +A16.B divided by 16; * equilibrium constant for A22-f B +A22.B,divided by 22; cooperativity parameter in modified Poisson equations for B * B interactions in AI6 micelle (see text); cooperativity parameter in modified Poisson equations for B * * * B interactions in A22micelle; root-mean-square deviation in total benzene molarity. concentration see (3) below]. (2) At the highest SOS concentrations at each tem- perature cA-is taken to be 0.120 mol dm-3 in the absence of benzene.(3) Values of K16 and K22at the largest values of [SOS] are fixed to force the total concentrations of A- in these two forms to be equal consistent with cA-= 0.120 mol dm-3. (4) The micelles are individually assumed to solubilize benzene according to the modified E. E. TUCKER AND S. D. CHRISTIAN Poisson distribution including a cooperativity parameter b for each size of micelle. The constants Kl and b used in relating [B],, and [SOS],i to the benzene activity are parameters to be inferred in the least-squares analysis of all the data at a given temperature. (5) Formation constants for the 1 1 SOS-benzene complex and the solubilizate assumed to contain 6 A-anions and 2 benzene molecules are the only equilibrium constants introduced to account for the formation of non-micellar aggregates.The least-squares fitting of data requires as a first step developing expressions to relate [SOS] and the total benzene concentration to the concentrations of the mono- mers cB and cA-.4 The expression for [SOS] can be solved numerically for cA-using provisional values for all of the equilibrium constants including Kl and b in the modified Poisson equations for both types of micelle. The calculated value of cA-may then be used together with the same provisional values of all the equilibrium constants to predict a value of the total benzene concentration for each of the data sets. The non-linear least-squares program NLLSQ l6 is used to minimize the sum of squares of deviations between the calculated and experimental total benzene con- centrations and to infer values of the equilibrium constants and their standard devia- tions.Table 3 summarizes some of the extensive results obtained from the least- squares analysis of data at all four temperatures. DISCUSSION AND CONCLUSIONS The values of the root-mean-square deviation (r.m.s.d.) in total benzene concen- tration given in table 3 indicate the excellent goodness of fit achieved by utilizing the mass-action model to correlate all of the solubilization data. Except for the results at 25 "C,for which all 245 data sets are fitted to an r.m.s.d. of 2.71 x lo-' mol dm-3 the model in general yields r.m.s.d. values that are within a factor of two of the value corresponding to the extremely high precision of the measurements (ca.5 x mol dmM3). Considering the fact that the total concentration of benzene in each series of measurements attains values of the order of several hundredths mol dm-3 we conclude that the modelling quite successfully reproduces the major features of the solubilization phenomena. The values of the equilibrium constants for forming the 1 1 complex increase with temperature but at a decreasing rate as may be expected for a typical hydrophobic association complex. Using the K, values we may calculate that AH = 5.6 & 0.6 kcal mo1-I at 25 "C for the reaction A-+ B -+ A-B. We also obtain the value ACp = -170 & 80 cal K-l mol-' a more negative quantity than that obtained for similar complexes of hydrocarbons with non-ionic molecules.* The K62 values do not show any clear trend but they appear to pass through a maximum in the temperature range 15-35 "C.Of major interest are the values of the constants Kl and b needed to fit the solu- bilization data to the modified Poisson model [eqn (5)]. K for the A, micelle is consistently larger at each temperature than K for the AI6 species and the co- operativity parameter b is also considerably larger for A, than for A16. At each temperature the average of the Kl values for the AI6and micelles are equal to within 3% to the Kl values in table 2 which were inferred from the limiting solubiliz- ation data using eqn (2) without assuming anything about the size of the micelles. The b values (ca. 0.03 for the A16 micelles and 0.05 for the A2 micelles) correspond to free energies of pairwise interaction of ca.-20 cal mol-' for the smaller and -30 cal mol-1 for the larger micelles. The complete least-squares analysis provides many other results that may be used SOLUBILIZATION OF BENZENE BY sos to characterize the formation of solubilizates in the SOS benzene system; however extensive details cannot be given here because of lack of space. At each temperature the concentration of the octylsulphate anion reaches a maximum of ca. 0.14 mol dmm3 at [SOS] z 0.16 mol dm-3 in the absence of added benzene and cA-decreases uni- formly as [SOS] increases further reaching the assumed value of 0.120 mol dm-3 at ca. 0.30 mol dm-3. The simulation of activity data for aqueous SOS solutions described above leads to quite comparable values of c,- and recent analyses of activity data for aqueous solutions of sodium dodecylsulphate and sodium decanoate yield anion concentrations that vary similarly with the total surfactant concentra- tion."-I9 A decrease in c,-of ca.10% occurs for each series of results at constant [SOS] as the benzene fugacity varies from 0 to 70% saturation. There is a sub-stantial increase in the relative proportion of the A22solubilizates compared with the A, species as the benzene activity increases at constant [SOS]. There is also a pronounced increase in the relative proportion of A22micelles as [SOS] increases at constant benzene activity. The general view emerging from the rather complex analysis presented here is that as the benzene activity increases the benzene molecules are solubilized preferentially within the A22species reflecting the tendency of benzene to interact more strongly with the larger micelles and also with other benzene mole- cules within the micelle.A comparison may be made of the thermodynamic constants in table 2 with constants for the transfer reaction benzene (dilute aqueous solution) -+benzene (pure liquid).' For the latter reaction AH = -0.56 kcal mol-' at 25 "Cand ACp = -61 cal mo1-I K-l compared with the values AH = 0.29 kcal mo1-' and ACp = -95 cal mol-I IS-'obtained here for the reaction A, + B -+A,,B in dilute aqueous solution. Both reactions have large negative values of ACp,presumably owing to the abnormally large heat capacity of monomeric benzene dissolved in water.20 The transfer enthalpy for benzene (liquid) -+ benzene (in the micellar interior) is ca.0.85 kcal mol-l at 25 "C implying that the environments of the benzene molecules are quite similar in the two states. The present results do not seem to support the conclusion that arom- atic hydrocarbon molecules solubilize preferentially in the vicinity of the polar-ionic region of the micelle rather than in the hydrocarbon core.21 It may be mentioned that the enthalpy of transfer of benzene into the micelle does become more exothermic as the benzene activity increases reflecting the enhancement in benzene-benzene inter- actions within the micelle as the total intramicellar concentration of benzene increases.The availability of values of [SOS],, and [BImicfrom the analysis of the solubiliz- ~~~, ation data makes it possible to examine the dependence of Y~ the activity co- efficient of benzene in SOS micelles (on the pure-component standard-state basis) on XB,mic, the average mole fraction of benzene in all the micellar forms. Thus XB,mic = [Blmic/([Blmic + [sos]~ic) and YB,mic =fB/(XB.micfi> wherefB is the fugacity of benzene in equilibrium with the micellar solution andfi is the fugacity of pure liquid benzene at the same temperature. Fig. 4 is a plot of yB,,, against XB,,, for the solutions at [SOS]z 0.3 mol dm-3 at 15,25 35 and 45 "C. The activity coefficient values indicate a considerable degree of positive deviation from Raoult's law the deviation decreasing with increasing temperature.There is also a decrease in yB,micwith increasing XB,mic at each temperature reflecting cooperative interactions between benzene molecules within the micelles. Recent research from this laboratory has shown that high-precision vapour-pressure measurements can provide detailed information about the thermodynamic quantities for forming complexes between hydrocarbons and other aqueous solutes. The results 4.4 < 4.2 m ?-4.0 3.8 0 0.05 0.10 0.15 0.20 XB,rniC FIG.4.-Activity coefficient of micellar benzene plotted against benzene mole fraction in sodium octylsulphate micelles. reported here are the most extensive ever obtained for an aqueous surfactant system. Hydrophobic association effects are examined quantitatively in a way not possible with other physical methods for studying solubilization.We are using the techniques and models described in the present article to investigate other surfactant systems; a report on the effect of added NaCl on properties of benzene-SOS solubilizates is in preparation. The research described here has been supported by the National Science Found- ation (grant CHE-8103084) the United States Department of Energy Bartlesville Energy Technology Center (contract DE-AT 1981 BC10476) the University of Oklahoma Energy Resources Center and the University of Oklahoma Mining and Mineral Resources Research Institute. M. E. L. McBain and E. Hutchinson Solubilization and Related Phenomena (Academic Press New York 1955).’P. H. Elworthy A. T. Florence and C. B. McFarlane Solubilization by Surface Actiue Agents and Its Application in Chemistry and the Biological Sciences (Chapman and Hall London 1968). P. Mukerjee in Solution Chemistry of Surfactants ed. K. L. Mittal (Plenum New York 1978) vol. 1. S. D. Christian E. E. Tucker and E. H. Lane J. Colloid Interface Sci. 1981 84 423. S. D. Christian L. S. Smith D. S. Bushong and E. E. Tucker J. Colloid Interface Sci. in press. A. Wishnia J. Phys. Chem. 1963 67 2079; I. B. C. Matheson and A. D. King J. Colloid Interface Sci. 1978,66,464; S. A. Simon R. V. McDaniel and T. J. McIntosh J. Phys. Chem. 1982,86 1449. ’E. E. Tucker and S. D. Christian J. Chem. Thermodyn. 1979 11 1137. E. E. Tucker E.H. Lane and S. D. Christian J. Solution Chem. 1981 10 1. M. B. King Phase Equilibrium in Mixtures (Pergamon Oxford 1969) p. 260. lo S. J. Rehfield J.Phys. Chem. 1967 71 738. l1 E. D. Goddard and G. C. Benson Can. J. Chem. 1957 35,986. l2 G. L. Musbally G. Perron and J. E. Desnoyers J. Colloid Interface Sci.,1974 48 494. SOLUBILIZATION OF BENZENE BY sos l3 H. W Offen D. R. Dawson and D. F. Nicoli J. Colloid Interface Sci. 1981 80 118; H. Coll J. Phys. Chem. 1970,74 520. l4 S. S. Atik M. Nam and L. A. Singer Chem. Phys. Lett. 1979 67 75. K. Kale and D. F. Evans unpublished work. l6 S. D. Christian and E. E. Tucker Am. Lab. 1982 14(9) 31. l7 T. Sasaki M. Hattori J. Sasaki and I(.Nukina Bull. Chem. Soc. Jpn 1975,48 1397. S. G. Cutler P.Meares and D. G. Hall J. Chern. Soc. Faraday Trans. 1 1978 74 1758. l9 E. Vikingstad J. Colloid Interface Sci. 1979 72 68. 'O F. Franks in Water-A Comprehensive Treatise ed. F. Franks (Plenum New York 1973) chap. 1. '' P. Mukerjee and J. R. Cardinal J. Phys. Chem. 1978 82 1978.