Calorimetric Study of the Micellization of Alkylpyridinium and AlkyltrimethylammoniumBromides in Water

Jurij Lah, Ciril Pohar, and Gorazd Vesnaver*Faculty of Chemistry and Chemical Technology, UniVersity of Ljubljana, AsˇkerceVa 5,1000 Ljubljana, SloVenia

ReceiVed: August 12, 1999; In Final Form: December 29, 1999

Titration calorimetry was employed to study the thermodynamic parameters of micellization ofN-undecyl-,N-tridecyl-, N-tetradecyl-, andN-pentadecylpyridinium bromides and tetradecyl- and hexadecyltrimethyl-ammonium bromides in salt-free aqueous solutions at 25°C. For this purpose a model equation for thecalorimetric titration curve based on the mass action model of the micellization process was derived. Asadjustable parameters it contains the aggregation numbern, the degree of micelle ionizationp/n, and theenthalpy of micellization∆HM. Critical micelle concentrations cmc were determined by applying the Philipscriterion to the experimental titration curves and so obtained cmc values are in excellent agreement with theliterature data. Fitting of the model titration curves to those obtained experimentally shows that this proceduregives reliable values only for∆HM while it fails to provide any useful information on thep/n data. In addition,for surfactants with short alkyl chains and thus with low aggregation numbers (n < ≈50) this fitting methodprovides also aggregation numbers that are close to those reported in the literature.

Introduction

Models developed to describe the thermodynamic propertiesof surfactant solutions have often been used to interpretcalorimetric data resulting from the isothermal titrations ofsurfactant solutions into pure solvent.1-9 Most often, the purposeof calorimetric titration experiments is a direct determinationof the enthalpy of micelle formation∆HM and the critical micelleconcentration cmc. Numerous studies of surfactant systems haveshown that∆HM can be evaluated within a two-state ap-proximation from the calorimetric titration curve as a differenceof its parts above and below cmc extrapolated to cmc.10-16 Onlyfew authors, however, have applied titration calorimetry tosolutions of ionic surfactants and tried to find a relationshipbetween the composition of surfactant solution in the titrationcell and the individual heat effects accompanying each injectionof the titrant solution.17,18

In the study we are presenting here, we employed highsensitivity titration calorimetry to determine cmc and∆HM forseveral cationic surfactants. Usually in a titration calorimetryexperiment all heat effects caused by injecting the titrant solutioninto the titration cell, except the first one, result from mixingof two solutions of different concentrations. Therefore, theycannot be discussed simply in terms of enthalpies of dilution.In this work the simulated calorimetric titration curves werederived from the expression for the surfactant partial molarenthalpyHh 2 in the titration cell expressed in terms of the massaction model of the micellization process and in terms of themeasured heat effect accompanying a single injection of thetitrant solution. In the premicellar region,Hh 2 was expressed interms of the temperature derivative of the surfactant’s meanactivity coefficient γ(, described by the Guggenheim’s ap-proximation.4,5,19,20For all the measured surfactants the criticalmicelle concentrations determined by employing Philips criter-

ion21,22on the experimental titration curves agree well with thecorresponding values obtained by other methods.11,12,23Fittingof the simulated calorimetric titration curves to those determinedexperimentally showed that the applied fitting procedure canprovide only ∆HM values while it is not able to give anyinformation on the micelle ionization parameterp/n. Our resultsalso show that for surfactants with a low aggregation numbernsuch fitting may lead to good estimates of then values.

Experimental Section

Materials. N-Undecyl (C11PyBr), N-tridecyl (C13PyBr), N-tetradecyl (C14PyBr), andN-pentadecylpyridinium (C15PyBr)bromides were synthesized from the corresponding 1-bromo-alkanes and pyridinium24 and purified by repeated recrystalli-zation from acetone, while tetradecyl (C14TABr) and hexadecyl(C16TABr) trimethylammonium bromides were obtained fromMerck and used without any further purification.

Titration Microcalorimetry. The heat effects resulting frommixing of aliquots of titrant solution from the syringe with thesolution (solvent) in the titration cell were measured using TAM2277 calorimeter from Thermometric AB, Sweden (Figure 1).Triply distilled water (2 mL) was titrated at 25°C by a surfactant

* Corresponding author. Phone:+38661 1760 504. Fax: 38661 1254458. E-mail: gorazd. vesnaver@uni-lj.si.

Figure 1. Typical calorimetric titration of surfactant into water; 2 mLof water was titrated with aqueous solution of C13PyBr (0.088 mmol/kg) at 25°C. Each peak corresponds to 6µL injection of the titrantsolution.

2522 J. Phys. Chem. B2000,104,2522-2526

10.1021/jp9928614 CCC: $19.00 © 2000 American Chemical SocietyPublished on Web 02/26/2000

solution placed in a 250µL syringe. Surfactant concentrationsin the syringe were about 20-times higher than their cmc’s. Thereference cell was filled with water. The instrument waschemically calibrated by known heats of dilution of aqueoussucrose solutions25 and before each experiment also by anelectrical calibration. All experiments were performed at thecalorimeter amplifier set between 10 and 100µW and werecompleted within 24 h. The area under a peak, which followseach injection of the titrant solution is proportional to theresulting heat effect∆H (expressed per mole of added surfac-tant), which is then plotted against the total surfactant molalitym in the titration vessel to construct the corresponding∆H vsm curve (Figures 2A and 3A). The integration of peaks fromthe directly recorded calorimetric plots was performed usingthe Digitam program software.

Mass-Action Model.According to the mass-action approach,the process of micellization for cationic surfactants may bedescribed as21,22

where S+ represents the surfactant ions, C- the correspondingcounterions, and Mp+ is the micellar aggregate ofn surfactantmonomers with an effective charge ofp. A surfactant solutionhaving the concentration greater than cmc can be consideredas a mixed electrolyte solution with a total surfactant molalitym, expressed as

and a degree of micellization,R,

wheremS is the molality of surfactant ions in monomeric formandmM is the molality of surfactant in the micellar form. The

apparent equilibrium constant for this micellization processKM

can be expressed as

wheremC is the molality of the free counterions. The equilibriumbetween the monomeric surfactant ions, their micelles and thecorresponding simple counterions can be described also in termsof composition variablesn1 (number of moles of solvent),n2

(total number of moles of surfactant in solution),nS (numberof moles of surfactant ions in monomeric form),nM (numberof moles of surfactant ions in micellar form),nC (number ofmoles of free counterions), and the corresponding partial molarenthalpiesHh 1, Hh S, Hh M, Hh C. The enthalpy of the solution in thetitration cell obtained afterith injectionHi can thus be expressedas

and sincen2 ) nS + nM andnC ) nS + (p/n) nM, eq 5 becomes

where the enthalpy of micelle formation∆HM, defined as

is the enthalpy change accompanying the process in which 1mol of surfactant ions and (1- p/n) mole of counterions passin the solution from the unaggregated form into the micellarform. Partial differentiation of eq 6 with respect ton2 at constantP, T, andn1 combined with the Gibbs-Duhem equation leadsto the following expression for the partial molar enthalpy of

Figure 2. Calorimetric titration curve for C11PyBr at 25 °C, (A)determination of∆HM by extrapolation method, (B) fitting of eq 14 toexperimental titration curve usingp/n ) 0.2 (∆HM ) -2.6 kJ/mol,n) 39), C-minimization ofø2(n, p/n) function overn at differentp/nvalues.

nS+ + (n - p)C- S Mp+ (1)

m ) mS + nmM (2)

R )nmM

m(3)

Figure 3. Analysis of calorimetric titration curve obtained forC13PyBr at 25°C, (A) fitting of eq 14 to experimental titration curveusingp/n ) 0.2 (∆HM ) -5.0 kJ/mol,n ) 50), (B) composition ofthe surfactant solution below and above cmc calculated from eqs 2and 4, (C) the corresponding degree of micellizationR (eq 3) and thederivative (∂nM/∂n2)n1,P,T (eq 8) as a function of the surfactant concentra-tion.

KM )mM

mSn mC

n-p(4)

Hi ) n1Hh 1 + nSHh S + nCHh C + nMHh M (5)

Hi ) n1Hh 1 + n2(Hh S + Hh C) + nM∆HM (6)

∆HM ) Hh M - Hh S - (1 - pn)Hh C (7)

Micellization of CnPy and CnTA Bromides J. Phys. Chem. B, Vol. 104, No. 11, 20002523

surfactantHh 2:

Analysis of the Heat Effects Accompanying TitrationExperiments.The heat effectqi accompanying theith injectionof the titrant solution with concentration significantly higherthan cmc into the titration cell can be expressed as

where Hi and Hi-1 represent enthalpies of solution in themeasuring cell afterith and (i - 1)th injection, respectively,while Ha represents the enthalpy of the added aliquot of thetitrant solution. If before the first injection the titration cellcontains only solvent, with corresponding enthalpyH1

0, thecumulative heat effect∑1

i qi, after i successive injections of thetitrant solution can be presented as

wheren2 is the total number of moles of added solute afterithinjection and const is a constant. If the amount of solute addedper injection∆n2 is very small, this cumulative heat effect canbe simplified into

where∆H is the enthalpy change accompanying each injectionexpressed per mol of added surfactant. Thus it follows fromeqs 10 and 11 that the enthalpy of solution in the titration cellafter ith injectionHi is:

Finally, by taking the first derivative (∂Hi/∂n2)P,T,n1, it followsfrom eq 12 that the partial molar enthalpy of the soluteHh 2 ateach titration point can be expressed as

The combination of eqs 8 and 13 leads to an expression for themeasured quantity∆H which includes also the enthalpy ofmicellization∆HM:

∆HM can be obtained by a graphical extrapolation method whichcan be applied only in case of sharp transitions of calorimetrictitration curves at cmc. This method is based on the assumptionthat micellization is an one-step process occurring at cmc inwhich the (Hh S + Hh C) contributions just above and just belowcmc are equal. Thus, according to eq 14,∆HM is simply adifference between the∆H value just above cmc where (∂nM/∂n2)n1,P,T ) 1 and the∆H value just below cmc where (∂nM/∂n2)n1,P,T ) 0. It can be obtained from the∆H vs m plots as adifference between the parts of the∆H vsmcurves extrapolated

to cmc from the regions of surfactant concentration above andbelow cmc (Figure 2A).

The other possibility of determing∆HM, which involvesfitting of the model calorimetric titration curve to the experi-mental one, is also based on eq 14. It should be applied whenthe experimental∆H vs m curve is not completed and becauseof that the extrapolation of its part from the region beyond cmcto cmc becomes questionable. Since experimental titrationprocess includes titration of surfactant solution below and abovecmc the model titration curve must be able to cover differentsurfactant behavior in these two regions. It follows from eq 8that below cmc where (∂nM/∂n2)n1,P,T ) 0, the (Hh S + Hh C)contribution can be derived from the expression for thesurfactant chemical potential. Using the Gibbs-Helmholtzrelation and Guggenheim approximation for the surfactant meanactivity coefficient4

in which m is the total surfactant molality, one obtains

where A′(T) and B′(T) are the temperature derivatives of coef-ficients A(T) andB(T), respectively, andHh 2

0 is the partial molarenthalpy of surfactant in its standard state. Above cmc it isassumed that due to small changes in concentrations of S+ andC- ions the (Hh S + Hh C) contribution remains constant and equalto its value at cmc.

The derivative (∂nM/∂n2)n1,P,T, expressed as a function ofm,can be obtained for givenn and p/n from the apparentequilibrium constantKM (eq 4). Introducing the degree ofmicellizationR (eq 3), one obtains

where

It has been shown that in surfactant solutions with no addedelectrolyte the mass action model of the micellization processcombined with the cmc definition by Philips leads to themicellization constantKM expressed as21,22

According to Philips21 the cmc can be obtained using a criterion(∂3Φ/∂m3)cmc ) 0 expressed in terms of propertyΦ, defined bythe relationshipΦ ) AmS + BmM in which A and B areconstants andmS andmM are the molalities of surfactant ionsand micelles, respectively.21,22Around cmcΦ differs from thecorresponding cumulative heat effect∑1

i qi only for a constantterm, and thus the measured∆H vs m curve is simply the (∂Φ/∂m) vs m curve. Therefore, cmc is obtained as the molalitym

- log γ( ) [ A(T)xm

1 + xm+ B(T)m] (15)

Hh S + Hh C ) Hh 20 + 2RT2( A′(T)xm

1 + xm+ B′(T)m) (16)

(∂nM

∂n2)

n1,P,T)

n[u + g]

1 + n[u + (1 - p/n)g](17)

u ) R1 - R

, g )(1 - p/n)R

1 - (1 - p/n)R

KM-1 )

n(2n - p)(4n - 2p - 1)

(2n - p - 2)×

[ (2n - p)(4n - 2p - 1)

(2n - p - 1)(4n - 2p + 2)cmc]2n-p-1

(18)

Hh 2 ) Hh S + Hh C + ∆HM(∂nM

∂n2)

n1,P,T(8)

qi ) Hi - (Hi-1 + Ha) (9)

∑1

i

qi ) Hi - iHa - H10 ) Hi - n2 const- H1

0 (10)

lim∆n2f0

∑1

i

qi ) lim∆n2f0

∑1

i ( qi

∆n2)∆n2 ) ∫0

n2∆H dn2 (11)

Hi ) ∫0

n2∆H dn2 + H10 + n2 const (12)

Hh 2 ) ∆H + const (13)

∆H ) (Hh S + Hh C - const)+ ∆HM(∂nM

∂n2)

n1,P,T(14)

2524 J. Phys. Chem. B, Vol. 104, No. 11, 2000 Lah et al.

at which the second derivative of the experimental∆H vs mcurve with respect tom is equal to zero.

Once cmc is known, a pair ofn andp/n values is chosen andthe correspondingKM is calculated.∆HM can then be obtainedby fitting of eq 14 to the experimental∆H vs m curve usingthe general linear least (chi)-square fitting procedure based onthe chi-square functionø2 defined as26

where (∆Hi)calc is obtained from eq 14 and∆(∆Hi) representsthe experimental error in the measured quantity∆Hi. For theminimization of the chi-square functionø2(n, p/n) overn at givenvalue ofp/n the standard Simplex algorithm was used.26

Results and Discussion

A result of a stepwise injections of small aliquots of the titrantsurfactant solution (mtit ≈ 20 cmc) into the measuring cellcontaining initially water is an enthalpogram presented in Figure1. If the individual heat bursts∆Hi (∆Hi ) qi/∆n2) are plottedagainst the surfactant concentrationm in the titration cell, atypical sigmoidal curve with an inflection point at the cmcemerges (Figures 2A and 3A). Below cmc this curve describesthe concentration dependence of the heat effects causedpredominantly by the demicellization of micelles injected intosolution while above cmc it reflects mainly the dilution ofmicelles added at each injection.

In the present work we tried to determine the thermodynamiccharacteristics of the micellization process for several surfactantsby fitting the model functions derived from the mass actionmodel of the micellization process (eq 1) to the correspondingexperimental calorimetric titration curves (Figures 2B and 3A).Following the Philips criterion, the most characteristic propertyof micellization, cmc, was obtained as the concentration at which(∂2∆H/∂m2)cmc ) 0. For the measured surfactants such deter-mination of cmc was not problematic because the transitionsof their ∆H vs m curves are rather sharp (Figures 2B and 3A).Comparison with the corresponding cmc data obtained in thislaboratory from the emf measurements of galvanic cells consist-ing of an appropriate surfactant ion selective electrode andreference electrode12 and from the measurements of specificconductivity of surfactant solutions as a function of theirconcentration23 shows an excellent agreement.

Since cmc is determined according to Philips criterion21,22

the first micellar aggregates start forming before so defined cmcis reached. The calculation of the (Hh S + Hh C) contribution atcmc from eq 16 may thus become problematic due to thepresence of the already formed micelles which are not takeninto account by the Guggenheim’s approximation on which theeq 16 is based. The model calculations ofmS and mC show,however, that beyond the appearance of the first aggregates theyboth increase linearly with the increasing total surfactantconcentration up to cmc while above cmc they remain almostconstant and equal to cmc. Furthermore, these calculations showthat in comparison withmS andmC the corresponding micelleconcentrationsmM are very small and can be neglected (Figures3B and C). Apparently, as far as the (Hh S + Hh C) contribution isconcerned, the assumption that beyond the appearance of thefirst micelles the micellar concentration may be ignored whilethe mS and mC increase linearly up to cmc and then remainconstant, seems reasonable. Therefore, it was used in evaluatingthe (Hh S + Hh C) contributions from eq 16 in all of the simulated∆H vs m curves.

Enthalpy of micellization has been obtained by many authorsas the enthalpy difference between the parts of∆H vs m curveabove and below cmc extrapolated to cmc.10-16 This widelyused method gives correct results only if the transition in thetitration curve is completed or in other words, if in theconcentration region beyond cmc the ending plateau of thetitration curve is reached. Sometimes single experiment datado not show whether the observed transition ended or not.Usually this happens with surfactants with shorter alkyl chainsfor which the transitions are too wide or when the concentrationsof the titrant solutions used in the experiments are too low. Inany case, the extrapolation of the “nonended” parts of suchtitration curves to the corresponding cmc values may lead tounacceptably large errors in∆HM. Our results suggest that insuch cases the curve fitting method of determining∆HM maybe more appropriate since it makes possible to obtain reliable∆HM also from the titration curves that do not reach the endingplateau. Since the part of the fitting procedure that directly leadsto the determination of∆HM involves interplay of threeparametersn, p/n, and ∆HM, their interactivity was closelyinvestigated. The simulation of calorimetric titration curvesshows that at fixedp/n and∆HM the curves change withn onlyup ton ≈ 50 (Figure 4A). while at fixedn and∆HM they showonly slight dependence onp/n (Figure 4B). In other words, thedetermination ofn by the minimization of the least (chi)-squarefunction at givenp/n is possible only as long as the simulatedcurves show observable dependence onn, that is forn < ≈50(Figure 4A). Such minimization is presented in Figure 2C whereit is shown that for C11PyBr at any reasonable fixed value ofp/n a sharp minimum in the least (chi)-square function appearsat n ≈ 40. Similar result is obtained also with C13PyBr wheresuch minimum in the least (chi) square is observed atn ≈ 50.The relative stability and small mean square deviations ofnand∆HM determined atp/n values between 0.1 and 0.4 (Table1) indicate that for surfactants such as C11PyBr and C13PyBrhaving short alkyl chains (smalln) and well defined cmc’s thedescribed curve fitting procedure can provide quite accurate∆HM andn values. Good agreement of the observed∆HM andn parameters with those recently determined in this laboratoryfrom heat of dilution12 and apparent molar volume23 measure-ments clearly supports this conclusion.

With surfactants having longer alkyl chains and thus largern values the fitting method is less successful. Although thecorrelation between the model (eq 14) and experimentalcalorimetric titration curve is still very good (Tables 1 and 2),it fails to provide reliablen data since the mean square deviations

Figure 4. The dependence of the simulated calorimetric titration curveat given∆HM ) -2.6 kJ/mol, B′(T) ) 0.03 kg/mol K, and cmc) 18.6mmol/kg at 25°C: on (A) the parametern (varying between 5 and100) at givenp/n ) 0.2, (B) the parameterp/n (varying between 0.1and 0.5) at givenn ) 40.

ø2 ) ∑i [∆Hi - (∆Hi)calc

∆(∆Hi) ]2

(19)

Micellization of CnPy and CnTA Bromides J. Phys. Chem. B, Vol. 104, No. 11, 20002525

of n at n > ≈50 become too large. On the other hand, thecomparison with the corresponding literature data11,12,23showsthat it still gives accurate∆HM values characterized by smallmean standard deviations and only weak dependence on theparametersn and p/n. As shown in Table 2 these values arealso close to those obtained by extrapolation method. Since suchextrapolations are safe for surfactants with longer alkyl chainsdue to the sharp transitions of their calorimetric titration curvesone can expect that, for C14PyBr, C14TABr, C15PyBr, or C16-TABr, the extrapolation method will give reliable∆HM values.In other words, for surfactants with longer alkyl chains nocomplicated fitting procedure is needed to determine their∆HM

values.Taken together our results show that fitting of eq 14 to the

experimental calorimetric titration curves cannot provide allparameters needed for a full thermodynamic description of themicellization process. It can only lead to relatively safe∆HM

values and for surfactants with smaller aggregation numbersalso to approximaten values but fails to give any usefulinformation on thep/n values. Since according to eq 19 themicellization constantKM can be obtained only from cmc,n,andp/n data it follows thatKM and the corresponding standardGibbs free energy of micellization∆GM

0 , together with therelated entropy of micellization∆SM

0 , cannot be determinedonly by isothermal titration calorimetry. To obtain∆GM

0 valuesthe correspondingn andp/n parameters have to be determinedby some other method.12,23 They can then be combined withthe measured∆HM values (assuming∆HM ) ∆HM

0 ) to give thecorresponding∆SM

0 values.

Conclusions

Isothermal titration microcalorimetry has become a powerfultechnique for determining cmc’s and enthalpies of micellizationin surfactant solutions. In our contribution an attempt was madeto simulate the calorimetric titration curves on the basis of asimple mass action model of the micellization process and thePhilips definition of cmc. In the premicellar region these titrationcurves were simulated only on the basis of electrostaticinteractions between the surfactant ions and their counterionsdescribed in the form of the Guggenheim’s approximation ofthe surfactant mean activity coefficient. As a result a modeltitration curve was derived which contains adjustable parametersn, p/n and ∆HM. It was shown that for all the measuredsurfactants the Philips criterion provides cmc values that agreewell with the literature data. Fitting of the model titration curvesto those obtained experimentally showed that this procedure canprovide reliable values only for∆HM and for surfactants withlow aggregation number (n < ≈50) also forn. On the otherhand, it fails to give any information on the degree of the micelleionization p/n. To conclude, our results show that a fullthermodynamic description of the micellization process dis-cussed in terms of a simple mass action model cannot be realizedonly on the basis of the titration calorimetry data.

Acknowledgment. This work was supported by the Ministryof Science and Technology of the Republic of Slovenia.

References and Notes

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1964, 60, 996.(4) Woolley, E. M. and Burchfield, T. E.J. Phys. Chem.1984, 88,

2155.(5) Desnoyers, J. E.; Caron, G.; DeLisi, R.; Roberts, D.; Roux, A.;

Perron, G.J. Phys. Chem.1983, 87, 1397.(6) Gruen, D. W. R.J. Phys. Chem.1985, 89, 146.(7) Gruen, D. W. R.J. Phys. Chem.1985, 89, 153.(8) Archer, D. G.J. Solution Chem.1987, 16, 347.(9) Blandamer, M. J.; Cullis, P. M.; Soldi, L. G.; Engberts, J. B. F.

N.; Kacperska, A.; van Os, N. M.; Subha, M. C. S.AdV. Colloid InterfaceSci.1995, 58, 171.

(10) van Os, N. M.; Daane, G. J.; Haandrikman, G.J. Colloid InterfaceSci.1991, 141, 199.

(11) van Os, N. M.; Haak, J. R.; Rupert, L. A. M.Physico-ChemicalProperties of Selected Anionic, Cationic and Nonionic Surfactants;Elsevier: Amstedam, 1993.

(12) Bezan, M.; Malavasˇic, M.; Vesnaver, G.J. Chem. Soc., FaradayTrans.1993, 89, 2445.

(13) Paula, S.; S_s, W.; Tuchtenhagen, J.; Blume, A.J. Phys. Chem.1995, 99, 11742.

(14) Sarmiento, F.; del Rio, J. M.; Prieto, G.; Attwood, D.; Jones, M.N.; Mosquera, V.J. Phys. Chem.1995, 99, 17628.

(15) Meagherm, R. J.; Hatton, T. A.; Bose, A.Langmuir1998, 14, 4081.(16) Bijma, K.; Blandamer, M. J.; Engberts, J. B.Langmuir1998, 14,

79.(17) Bijma, K.; Engberts, J. B.; Blandamer, M. J.; Cullis, P. M.; Last,

P. M.; Irlam, K. D.; Soldi, L. G.J. Chem. Soc., Faraday Trans.1997, 93,1579.

(18) Blandamer, M. J.; Cullis, P. M.; Engberts,J. B. F. N.J. Chem. Soc.,Faraday Trans.1998, 94, 2261.

(19) De Lisi, R.; Fiscario, E.; Milioto, S.J. Solution. Chem.1988, 17,1015.

(20) Inglese, A.; De Lisi, R.; Milioto, S.J. Phys. Chem.1996, 100,2260.

(21) Philips, J. N.Trans. Faraday Soc.1955, 51, 561.(22) Hunter, R. J.Foundations of Colloid Science;Clarendon Press:

Oxford, 1987; Vol. 1.(23) Skerjanc, J.; Kogej, K.; Cerar, J.Langmuir1999, 15, 5023.(24) Knight, C. A.; Schaw, B. D.J. Chem. Soc, 1938, 682.(25) Briggner, L. E.; Wadso¨, I. J. Biochem. Biophys. Methods1991,

22, 101.(26) Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T.

Numerical Recipies; Cambridge University Press: Cambridge, 1989.

TABLE 1: Enthalpy of Micellization ∆HM and theAggregation Number n for C11PyBr and C13PyBr at 25 °CObtained from Fitting of eq 14 to the ExperimentalTitration Curves at Different Degrees of Micelle Ionizationp/n

surfactantcmc,

mmol kg-1 naa p/n nb

-∆HM, kJmol-1 rc

0.1 35( 6 2.7( 0.2 0.9940.2 39( 5 2.6( 0.2 0.993

C11PyBr 18.6 42 0.3 41( 4 2.5( 0.2 0.9930.4 44( 3 2.4( 0.2 0.9930.1 40( 20 5.2( 0.3 0.9980.2 50( 10 5.0( 0.3 0.998

C13PyBr 3.6 57 0.3 50( 10 4.8( 0.2 0.9980.4 49( 7 4.7( 0.2 0.998

a na values are obtained from the apparent molar volume measure-ments.23 b The ( values listed are the statistical root-mean-squaredeviations. The relative error of each experimental value of∆H (titrationpoint) was estimated to be(10%. c r values are the statistical correlationcoefficients.

TABLE 2: Enthalpies of Micellization of C 11-, C13-, C14-,and C15PyBr and C14- and C16TABr at 25 oC Obtained fromFits of Eq 14 Usingp/n ) 0.2,23 ∆HM, and by theExtrapolation Method, ∆HM

ex

surfactantcmc,

mmol kg-1-∆HM

ex,kJ mol-1

-∆HM,a

kJ mol-1 rb

C11PyBr 18.6 2.3 2.6( 0.2 0.993C13PyBr 4.6 5.4 5.0( 0.3 0.998C14PyBr 2.7 6.9 6.4( 0.3 0.997C15PyBr 1.3 8.3 7.6( 0.3 0.998C14TABr 3.6 4.3 4.1( 0.2 0.999C16TABr 0.9 8.1 7.7( 0.2 0.995

a The( values listed are the statistical root-mean-square deviations.The relative error of each experimental value of∆H (titration point)was estimated to be(10%. b r values are the statistical correlationcoefficients.

2526 J. Phys. Chem. B, Vol. 104, No. 11, 2000 Lah et al.