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Langmuir 2007, 23, 2362-2368
Tautomerization Equilibria in Aqueous Micellar Solutions: A Spectrophotometric and Factor-Analytical Study Hamid Abdollahi* and Vahideh Mahdavi Faculty of Chemistry, Institute for AdVanced Studies in Basic Sciences, Zanjan, Iran ReceiVed September 16, 2006. In Final Form: NoVember 28, 2006
The keto-enol equilibria of benzoylacetone (BZA) as a model for 1,3-dicarbonyl compounds are studied in aqueous acid and cationic micellar solution. Evolving factor analysis (EFA), multivariate curve resolution-alternating leastsquares (MCR-ALS), and rank annihilation factor analysis (RAFA) are used for complete resolving of measured spectrophotometric data. The acidity constants of the enolic, KaE, and ketonic, KaK, forms of BZA and also the tautomerization constant, Kt, and its related thermodynamic parameters have been determined by using EFA and MCR-ALS methods and spectral variation of BZA solutions in various pHs and temperatures. The concentration and spectral profiles of all species were calculated without any assumption about chemical models. The spectral variation of BZA solutions as a function of cationic micelle concentration sufficiently beyond its critical micelle concentration is analyzed according to the partition model for distribution between water and micellar pseudo-phase and RAFA. The outputs of using RAFA on measured rank deficient data are the spectrum of enolic form in the micellar pseudophase, free from contribution of the enolic form in the aqueous phase, the partition coefficient of enolic form, KdE, between the micelle and water phases, and the tautomerization constant in the micellar pseudo-phase, Ktm.
Introduction 1,3-dicarbonyl compounds have one ionizable proton and three possible tautomeric forms (keto, enol, and enolate), whose equilibrium ratios may be severely affected by changes in the media in which they are dissolved. The tautomeric forms (enol and keto) are both present in aqueous acid or neutral solutions in measurable proportions. Due to the difference in electrical dipole moments of tautomeric forms, the keto-enol equilibrium of b-dicarbonyl compounds are extremely medium-sensitive, and the enol amount increases in apolar and/or organic (non-hydrogenbond donor or acceptor) solvents, being, in some cases. the predominant species. In aqueous alkaline medium, the enolate is rapidly generated, which, in some situations, is the only existing species.1-3 Tautomerization studies have been frequently reported in general4-6 or in particular for 1,3-dicarbonyl compounds7 using a great variety of techniques, being carried out in water, organic solvent, aqueous micellar,1,8 and cyclodextrin9 solutions. Aqueous micellar solutions are highly anisotropic solvents whose properties change gradually between those of pure water to those of hydrocarbon-like liquids upon going from the bulk water phase to the micellar core.10-13 Because of this, the presence of micelles affects the tautomerization equilibria. The keto-enol equilibrium * Corresponding author. E-mail address:
[email protected]. (1) Iglesias, E. J. Phys. Chem. 1996, 100, 12592-12599. (2) Iglesias, E. Curr. Org. Chem. 2004, 8 (1), 1-24. (3) Iglesias, E. J. Inclusion Phenom. Macrocyclic Chem. 2005, 52, 55-62. (4) Cortijo, M.; Llor J.; Sanchez-Ruiz, J. M. J. Biol. Chem. 1988, 263, 1796017969. (5) Llor, J.; Sanchez-Ruiz, J. M.; Cortijo, M. J. Chem. Soc., Perkin Trans. 2 1988, 951-956. (6) Iglesias, E.; Williams, D. L. H. J. Chem. Soc., Perkin Trans. 2 1988, 1035-1040. (7) Toullec, J. AdV. Phys. Org. Chem. 1982, 18, 1-77. (8) Iglesias, E. J. Chem. Soc., Perkin Trans. 2 1997, 431-439. (9) Iglesias, E. J. Org. Chem. 2000, 65 (20), 6583-6594. (10) Quina, F. H.; Chaimovich, H. J. Phys. Chem. 1979, 83, 1844-1850. (11) Bunton, C. A.; Nome, F. J.; Quina, F. H.; Romsted, L. S. Acc. Chem. Res. 1991, 24, 357-364. (12) Bunton, C. A.; Savelli, G. AdV. Phys. Org. Chem. 1986, 22, 231. (13) Iglesias, E. New J. Chem. 2005, 29, 457-464.
of 1,3-dicarbonyl compounds is largely influenced in aqueous micellar media due to its extremely sensitive dependence on the solvent nature.1,2,13-15 The origin of this phenomenon resides in the difference of molecular interactions between the solvent and the keto or enol tautomers. The spectrophotometric study of tautomerization equilibria of 1,3-dicarbonyl compounds such as benzoylacetone (BZA) has been reported by univariate classical methods1 and prior assumption that the molar absorptivity of the enol in the micelle pseudo-phase is the same as that in water. The feasible abilities of multivariate chemometrical methods based on factor analysis for studying tautomerization equilibria in water and micellar solutions are considered in this investigation. Thousands of publications bear witness to the power and utility of factor analysis (FA) in chemistry.16 Following Malinowski’s definition, FA is a multivariate technique for reducing matrices of data to their lower dimensionality by the use of orthogonal factor space and transformations that yield predictions and/or recognizable factors.17 In general, FA has been proposed to solve the mixture analysis problem.18,19 Mixture analysis implies the estimation of the number of chemical species simultaneously present in the mixture, the identification of these species, and the determination of their concentration. Among the computational and statistical methods used to solve mixture analysis problems, FA, principal component analysis (PCA),20,21 and singular value decomposition (SVD)17 techniques play a key role, especially in the estimation of the number of species contributing significantly to the experimental data variance. FA, (14) Iglesias, E. New J. Chem. 2002, 26, 1352-1359. (15) Iglesias, E. Langmuir 2000, 16, 8438-8446. (16) Jiang, J. -H.; Liang, Y.; Ozaki, Y. Chemom. Intell. Lab. Syst. 2004, 71, 1-12. (17) Malinowski, E. R. Factor Analysis in Chemistry, 3rd ed.; Wiley: New York, 2002. (18) Liang, Y. Z.; Kvalheim, O. M.; Manne, R. Chemom. Intell. Lab. Syst. 1993, 18, 235-250. (19) Brown, S. D.; Bear, R. S. Crit. ReV. Anal. Chem. 1993, 24, 99-131. (20) Wold, S.; Esbensen, K.; Geladi, P. Chemom. Intell. Lab. Syst. 1987, 2, 37-52. (21) Wold, S.; Geladi, P.; Esbensen, K.; Ohman, J. J. Chemom. 1987, 1, 4156.
10.1021/la0627112 CCC: $37.00 © 2007 American Chemical Society Published on Web 01/24/2007
Tautomerization Equilibria in Micellar Solutions
PCA, and SVD are very similar techniques with slightly different formalisms for the selection of dimensions and for changes in coordinate axes (rotation). Self-modeling curve resolution (SMCR) describes a set of mathematical tools for estimating pure component spectra and concentration profiles from data matrices of mixture spectra recorded from an evolving system. Evolving systems are complex chemical systems that change in systematic, nonrandom ways as a function of time, pH, temperature, and so forth.22-25 SMCR, in principle, does not require a priori any specific information concerning the data to resolve the pure variables. The only assumptions are a certain bilinear model for the data and some generic knowledge about the pure variables, such as non-negativity and unimodality. In common practice, these premises are naturally satisfied for two-way data obtained from multivariate measurements on mixtures with an evolutionary variation of compositions. SMCR provides a useful tool for exploring multicomponent phenomena in complex chemical systems. Several soft-modeling and hard-modeling algorithms have been developed that analyze bilinear data obtained from chemical systems. Examples include kinetics, equilibrium studies, and chromatography.26-29 Soft-modeling methods range from very general approaches with minimal demands on the structure of data, such as EFA,30 HELP,31 SIMPLISMA,32 and ALS,33 to methods that rely on trilinearity, such as PARAFAC,34 GRAM,35 or TLD.36 Hard-modeling approaches to fitting multivariate response data are based on mathematical relationships, which describe the measurements quantitatively.37,38 In chemical equilibria, the analysis is based on the equilibrium model, which quantitatively describes the reaction and all concentrations in the solution under investigation. Recently a mixed approach including hard- and soft-modeling has been developed.39,40 There is a mutual benefit in the use of such mixed methodology. Hardsoft-modeling equals or overcomes the performance of hardand soft-modeling in any case. The potential usefulness of the novel methodology in kinetics and equilibrium studies has been reported by some researchers.24,40,41 Rank annihilation factor analysis (RAFA) is an efficient chemometric technique based on rank analysis for two-way spectrum data and can be employed to analyze the gray system with an unknown background (22) Dewok, J.; de Juan, A.; Maeder, M.; Tauler, R.; Lendl, B. Anal. Chem. 2003, 75, 641-647. (23) Gemperline, P. J.; Cash, E. Anal. Chem. 2003, 75, 4236-4243. (24) Abdollahi, H.; Sorouraddin, M. H.; Naseri, A. H. Anal. Chim. Acta 2006, 562, 94-102. (25) Mason, C.; Maeder, M.; Whitson, A. Anal. Chem. 2001, 73, 1587-1594. (26) Diaz-Cruz, J. M.; Agullo, J.; Diaz-Cruz, M. S.; Arino, C.; Esteban, M.; Tauler, R. Analyst 2001, 126, 371-377. (27) Esteban, M.; Arino, C.; Diaz-Cruz, J. M.; Diaz-Cruz, M. S.; Tauler, R. Trends Anal. Chem. 2000, 19, 49-61. (28) Bijlsma, S.; Boelens, Hans, F. M.; Hoefsloot Houb, C. J.; Smilde, A. K. Anal. Chim. Acta 2000, 419, 197-207. (29) Gargallo, R.; Tauler, R.; Cuesta-Sanchez, F.; Massart, D. L. Trends Anal. Chem. 2000, 15, 279-286. (30) Maeder, M.; Zuberbuhler, A. D. Anal. Chim. Acta 1986, 181, 287-291. (31) Liang, Y. Z.; Kvalheim, O. M. Chemom. Intell. Lab. Syst. 1993, 20, 115-125. (32) Windig, W.; Guilment, J. Anal. Chem. 1991, 63, 1425-1432. (33) Tauler, R.; Barcelo, D. Trends Anal. Chem. 1993, 12, 319-327. (34) Bro, B. Chemom. Intell. Lab. Syst. 1997, 38, 149-171. (35) Tauler, R.; Smilde, A.; Kowalski, B. R. J. Chemom. 1995, 9, 31-58. (36) Gui, M.; Rutan, S. C. Aghodjan, A. Anal. Chem. 1995, 67, 3293-3299. (37) Gui, M.; Rutan, S. C. Anal. Chem. 1994, 66, 1513-1519. (38) Molloy, K. J.; Maeder, M.; Schumacher, M. M. Chemom. Intell. Lab. Syst. 1999, 46, 221-230. (39) de Juan, A.; Maeder, M.; Martinez, M.; Tauler, R. Chemom. Intell. Lab. Syst. 2000, 54, 123-141. (40) de Juan, A.; Maeder, M.; Martinez, M.; Tauler, R. Anal. Chim. Acta 2001, 442, 337-350. (41) Bezemer, E.; Rutan, S. C. Anal. Chim. Acta 2002, 459, 277-289.
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quantitatively. RAFA is comparable with the hard-soft-modeling approach in solving some chemical problems.42-46 In this work, a systematic spectrophotometric study of the keto-enol tautomerization of BZA as a model for 1,3-dicarbonyl compounds in water in the absence and presence of a cationic micelle, cetyltrimethylammonium bromide (CTAB), is investigated by using FA methods. The thermodynamic characteristics and acid-base dissociation behavior of BZA in aqueous solution are studied by evolving factor analysis (EFA) and multivariate curve resolution-alternating least-squares (MCR-ALS) methods. The measured spectrophotometric data is completely resolved, and the concentration and spectral profiles are calculated without any particular assumption. RAFA is used for resolving the effects produced by surfactant addition on the absorption spectra of BZA solution based on a partition model. Experimental UV-vis absorbance digitized spectra were collected using a CARY 100 spectrophotometer (Varian, Australia) centrally controlled by a PC with Windows NT operating systems, 1 cm quartz cells, a scan rate of 100 nm/min, and a slit width of 2 mm. The recorded spectra were digitized with one data point per nanometer. Measurements of pH were made with a Metrohm 713 pH-meter using a combined glass electrode. BZA (1-phenyl-1,3-butanodione) and CTAB were purchased from Merck and used without farther purification. All solutions were prepared fresh daily. Solutions were allowed to remain in thermostated sample compartments for a minimum of 10 min before the spectra were collected. The temperature was maintained at 25.0 ( 0.1 °C using a Pharmacia Biotech Multi Temp III temperature circulator. Under working conditions, the critical micelle concentration (cmc) of CTAB is considered to be 9.2 × 10-5, and Vf ) 0.361 L mol-1.47,48 Specific details are given in the Results and Discussion section. All calculations were performed in MATLAB 6.0 (Math works, CoChituate Place, MA). MCR-ALS codes are available on the Internet due to Tauler et al.,49 and RAFA codes were written in our laboratory.
Results and Discussion β-Dicarbonyl compounds, such as BZA, can exist in solution in three chemical forms: two tautomeric species (enol and keto forms) and a deprotonated form (enolate species). So, the composition of BZA solution is strongly dependent on pH. Measuring the absorbance of an aqueous BZA solution as a function of pH can be used for observing the overall process that has occurred in the solution. Figure 1 shows the variation in absorption spectra of BZA solution in the pH range of 6.0011.00. The absorption band with a maximum near 312 nm increases with the pH since the enolate ion should absorb light much more strongly at this wavelength than does the enol. As it can be seen, the maximum at 312 nm, as a consequence of the enolic form, shifts to 320 nm due to the enolate ion; in addition, the peak centered at 250 nm also shifts to 245 nm. So, there is a clear difference between the spectral shapes of the enolate ion and the resultant spectrum of the enol and keto forms that is observed at low pH values. (42) Beltran, J. L.; Ferrer, R.; Guiteras, J. Anal. Chim. Acta 1998, 373, 311319. (43) Roch, T. Anal. Chim. Acta 1997, 356, 61-74. (44) Zhu, Z. L.; Xia, J.; Zhang, J.; Li, T. H. Anal. Chim. Acta 2002, 454, 21-30. (45) Maeder, M.; Neuhold, Y. M.; Olsen, A.; Puxty, G.; Dyson, R.; Zilian, A. Anal. Chim. Acta 2002, 464, 249-259. (46) Abdollahi, H.; Nazari, F. Anal. Chim. Acta 2003, 486, 109-123. (47) Beltran, J. L.; Prat, M. D.; Codony, R. Talanta 1995, 42, 1989-1997. (48) Safavi, A.; Abdollahi, H. Microchem. J. 2001, 69, 167-175. (49) http://www.ub.es/gesq/mcr/mcr.htm.
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Figure 1. Variation in the absorption spectra of 6.2 × 10-5 M BZA solution recorded in the pH range of 6.00-11.00. Scheme 1
It is possible to propose Scheme 1 for describing the system. In this scheme, KH, EH, and E- refer to keto, enol, and enolate species, respectively; Kt is the tautomerization constant, and KaE and KaK are the acidic dissociation constants of the enol and keto forms, respectively. The equilibrium concentration variation of each species as a function of pH (H+ concentration) can be considered as follows:
[EH] )
[KH] )
[E-] )
CtKt[H+] Kt[H+] + [H+] + KtKaE Ct[H+] Kt[H+] + [H+] + KtKaE CtKtKaE Kt[H+] + [H+] + KtKaE
(1)
(2)
(3)
As can be seen, the concentration profiles of the tautomeric forms are linearly dependent ([EH] ) Kt[KH]), and so the measured data matrix that is shown in Figure 1 is “rank deficient.”50 A data matrix is rank deficient when the number of significant contributions to the data variance, estimated by using SVD or other related FA techniques, is lower than the real number of chemical components present in the system. Complete resolution of the measured data matrix and calculation of the concentration and spectral profiles of the chemical species by using the soft-modeling techniques is not possible. Also, as expected, augmentation of several measured data under different initial concentrations of BZA (Ct) cannot solve the rank deficiency problem in this case. Overall information about the apparent acidic dissociation equilibrium of BZA can be extracted from the measured data. An estimate of the evolution of chemical components at different pH values was obtained by EFA. This approach provides an estimation of the regions or windows where the concentration of different components is changing or evolving, and it also provides an initial estimation of how these concentration profiles change along the experiment. The EFA method is based on the (50) Xu, C. J.; Gourvenec, S.; Liang, Y. Z.; Massart, D. L. Anal. Chim. Acta 2006, 575, 1-8.
Figure 2. (a) Estimated concentration profiles by EFA and (b) calculated apparent concentration profiles by MCR-ALS for the rank deficient measured absorbance data shown in Figure 1.
evaluation of the magnitude of the singular values (or of the eigenvalues) associated with all the submatrices of a matrix built up by adding successively all the rows of the original data matrix. The calculation are performed in two directions: forward (in the same direction of experiment), starting with the two first spectra, and backward (in the opposite direction of the experiment), starting with the last two spectra. Figure 2a shows the EFA plot for measured data, which is identical to the estimated concentration profiles of a monoprotic acid. Really, tautomeric forms were observed as a single component. The concentration profiles from EFA were used as the initial estimate for the concentration matrix input in the constrained ALS optimization. When an initial estimation of the concentration profiles is available, the best least-squares unconstrained solution of the spectral profiles is estimated from
S ) C+D
(4)
where the C+ is the pseudo-inverse of the concentration matrix. The least-squares solutions obtained in this way are pure mathematical solutions that probably will not be optimal from a chemical point of view. The UV absorptivities must be positive. This constraint is applied accordingly during the least-squares optimization. A new estimate of concentration profiles is obtained by least-squares:
C ) S+D
(5)
where now S+ is the pseudo-inverse of the spectral matrix. Therefore, an optimization procedure is started by iteratively resolving the two last equations and constraining, at each stage of iterative optimization, the solution to be non-negative. Other constraints implemented during the ALS optimization were closure (the sum of the concentrations of the two detected components at different pHs is equal to the total concentration of BZA) and the unimodality (the concentration profiles have a unimodal peak or a cumulative shape). Figure 2b shows the concentration profiles obtained from the MCR-ALS procedure. The apparent pKa value (8.80 ( 0.03) was calculated by using five points of these profiles.
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Figure 3. Variation in UV absorption spectra of 6.2 × 10-5 M BZA solutions measured over a temperature range of 5-79 °C with an increment of 4 °C in (a) 0.033 M HCl and (b) pH ) 9.0.
The incorporation of the temperature as an external controlled variable and a multiple-process run strategy were used for overcoming the rank deficiency problem in the considered system. Figure 3 shows the UV absorption spectra of BZA solutions measured over a temperature range of 5-79 °C with an increment of 4 °C in 0.033 M HCl (Figure 3a) and pH ) 9.0 (Figure 3b). According to the calculated apparent acidic dissociation constant of BZA (Figure 2b), in acidic solution (0.033 M HCl), only the two tautomeric forms exist, and, in pH ) 9.0, the solution contains enol, keto, and enolate species. Simultaneous analysis of several experiments giving data matrices obtained for the same chemical system under different experimental conditions improves the resolution power of the MCR-ALS method. So, the data matrices shown in Figure 3a,b were arranged in an augmented columnwise data matrix. This augmented data matrix has a number of rows equal to the total number of temperatures in two different experiments, and it has a number of columns equal to the number of wavelengths in the measured spectra (the same wavelengths were measured in the two data experiments). This columnwise data matrix augmentation assumes that common vectors span the column vector spaces of individual data matrices.
Daug ) [D1;D2] ) [C1;C2]ST + [R1;R2] ) Caug ST + Raug (6) The unconstrained ALS solutions of Caug and ST are
Caug ) [C1;C2] ) [D1;D2] (ST)+ ) Daug (ST)+
(7)
ST ) ([C1;C2])+ [D1;D2] ) (Caug)+Daug
(8)
where Daug and Caug are the corresponding columnwise augmented data and concentration matrices, respectively. The augmented concentration matrix Caug contains the concentration profiles resolved for data matrices D1 and D2 on top of each other, and the single matrix ST contains the pure individual spectra of the detected contributions in the two experiments. The ALS iterative process needs to start with an initial estimation of either matrix C or ST. Usually estimates are obtained
Figure 4. Initial estimates for the spectral profiles of the enol, keto, and enolate forms (a) and calculated concentration profiles for data matrix corresponding to (b) Figure 3a and (c) Figure 3b.
via the previous individual analysis of the different data matrices included in the augmented matrix. The pure spectral profile of enolate ion species was obtained by measuring the spectrum of BZA in a highly basic solution. Pure spectral profile estimates for the enol and keto forms were obtained from resolving the measured data matrix at 0.033 M HCl by the MCR-ALS procedure. Figure 4a shows the estimated pore spectral profiles of all three components as the initial estimate for resolving the augmented data matrix Daug. The application of the ALS procedure to the augmented data matrix, using the non-negativity (concentration and spectral profiles), closure (concentration profiles), selectivity (zero values for concentration of enolate species in correspondence to part of the Caug matrix with measured data at 0.033 M HCl), unimodality (concentration profiles) and equality (known pure spectrum of enolate species) constraints, allowed the estimation of the concentration profiles and spectra associated to each form of BZA. The plot shown in Figure 4b gives the concentration evolution as a function of temperature for three species of BZA under two different conditions. The Kt, KaE, KaK, and enthalpy changes for three considered equilibrium processes were calculated according to the concentration profiles shown in Figure 4b by inserting the values of variables into the equilibrium constant expression for each process at a certain temperature and also the variation of equilibrium constants as a function of temperature according to the van’t Hoff model. All the results are presented in Table 1. All results obtained by using the soft-modeling method
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Table 1 calculated ∆H (kJ mol-1)
Kt
KaE
KaK
0.485 8.4 ( 0.9
2.37 × 10 19.0 ( 3.0
-9
9.85 × 10-10 10.9 ( 2.3
(Kt ) 0.485, pKaE ) 8.62, and pKaK ) 9.01) are in some agreement with those determined by classical methods1 (Kt ) 0.53, pKaE ) 8.30, and pKaK ) 8.52).
Tautomerization Equilibrium of BZA in Micellar Solution Shifts in absorption spectra, acid-base equilibria, and complex formation constants due to interactions of solutes with micelles have been shown frequently.51-53 A micelle can be regarded as a microreactor that influences reaction rates and equilibria by taking up reactants and providing a medium different from that of the bulk solvent. The observations of the micellar effects on chemical reactivity and equilibria lead to the generalization that the aqueous micelles could be regarded as submicroscopic reaction or solubilization media.54,55 There are some reports on the study of the influence of micelles on tautomerization equilibria,1,3,9 but, to our knowledge, no systematic factor analytical study has been performed on the effects of micelles on the enol-keto equilibria. Several models have been proposed to discuss the effects of micelles in chemical and physical properties. Binding equilibria56-58 between solute and micelle aggregates, the partition equilibria57,59,60 of solutes between aqueous and micellar pseudophases, and the ion-exchange equilibria61,62 are the most applicable models. The partition model is one of the most successful models in which it is assumed that micelles act as a separate phase uniformly distributed through the solution, and the distribution of neutral species and ion associates between the aqueous phase and micelle can occur. However, it was shown that the partition model, regardless of specific solute-micelle interaction, can be used in a solution of ionic surfactants, provided that the surfactant concentration is sufficiently above its cmc and that the concentration of the background electrolyte is much higher than those of the solutes.52 Under these experimental conditions, the partition model provides tools for the study of the micellesolute interactions; first of all, the partition model can be applied to ionic and non-ionic surfactants, regardless of the specific solute-micelle interactions. So the comparison of the equilibrium and rate constants for a given system in different surfactant solutions yields a direct comparison in terms of the extent of solute-micelle interaction for the different species or the solubilization ability of the surfactants studied. Figure 5 shows the absorption spectra of 6.2 × 10-5 M BZA in 0.033 M HCl as a function of CTAB concentration sufficiently (51) Beltran, J. L.; Codony, R.; Izquierdo, A.; Prat, M. D. Talanta 1993, 40, 157-165. (52) Marcus, R. A. J. Phys. Chem. B 2005, 109, 21419-21424. (53) Underwood, A. L. Anal. Chim. Acta 1982, 140, 89-97. (54) Bunton, C. A.; Savelli, G. AdV. Phys. Org. Chem. 1986, 22, 231. (55) Weiss, J.; Coupland, J. N.; McClements, D. J. J. Phys. Chem. 1996, 100, 1066-1071. (56) Scilia, D.; Rabio, S.; Perez-Bendito, D. Anal. Chim. Acta 1994, 297, 457. (57) Blatt, E.; Chatelier, R. C.; Sawyer, W. H. Chem. Phys. Lett. 1984, 108, 397-400. (58) Vidya, B.; Porter, M. D.; Utterback, M. D.; Bartsch, R. A. Anal. Chem. 1997, 69, 2688-2693. (59) D’Angela, N. E.; Collette, T. W. Anal. Chem. 1997, 69, 1642-1650. (60) Saiteh, T.; Kinura, Y.; Kamidate, T.; Watanabe, H.; Haraguchi, K. Anal. Sci. 1989, 5, 577. (61) Chaimovich, H.; Bonilha, J. B. S.; Politi, M. J.; Quina, F. H. J. Phys. Chem. 1979, 83, 1851-1854. (62) Bunton, C. A.; Ramsted, L. S.; Sepulveda, L. J. Phys. Chem. 1980, 84, 2611-2618.
Figure 5. Absorption spectra of 6.2 × 10-5 M BZA solutions as a function of CTAB micelle concentration at 0.033 M HCl. Arrows indicate the spectral trends in changing CTAB from 3 to 12 mM. Scheme 2
beyond its cmc. The keto-enol equilibrium of BZA is affected by the CTAB micelle: the maximum absorbance around 312 nm, which is due to the enol form in water, increases with the CTAB concentration above the cmc; meanwhile, the absorption band around 250 nm, due to the ketonic form, decreases as the CTAB concentration increases. This means that, as the enol form is extracted by the micelle, the keto-enol equilibrium in water displaces toward the enol formation. It is then possible to propose Scheme 2. In this model, the subscripts w and m refer to water and micelle pseudo-phases respectively; Kt is again the tautomerization constant in water, KdE is the distribution constant of the enol form between the water and the micelle, and Ktm represents the tautomerization constant in the micellar pseudophase (i.e., Ktm ) KtKdE). The concentration profiles of species in this evolutionary process can be considered as
[Ketow](1 - R) )
[Enolw](1 - R) )
[Enolm](R) )
Ct(1 - R) (1 - R) + Kt(1 - R) + KtmR Ct(1 - R)Kt (1 - R) + Kt(1 - R) + KtmR CtRKtm
(1 - R) + Kt(1 - R) + KtmR
(9)
(10)
(11)
R is the ratio of the micellar volume to the total volume of the solution, which can be written as
R)
Vm ) (CCTAB - cmc)Vφ V
(12)
where CCTAB is the total concentration of CTAB surfactant, cmc is the critical micelle concentration of both in moles per liter, and Vf is the partial molar volume of the micellized surfactant (L mol-1). According to the proposed model, the matrix of concentration profiles corresponding to measured data, which is shown in Figure 5, is rank deficient. As can be seen, the concentration profiles of water tautomeric forms are linearly dependent ([Enolw] ) Kt[Ketow]). The rank of the measured data matrix is two, which
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is calculated based on obtained eigenvalues. If the pure spectral profiles of two enolic species (Enolw and Enolm) are different, the rank deficiency in the absorbance data matrix is due to linear dependency in concentration profiles. For complete resolution of this data matrix, RAFA is proposed. RAFA is an efficient chemometric technique based on rank analysis for two-way spectrum data and can be employed in several forms for solving various problems in chemistry.42-46 RAFA was originally developed by Ho et al. as an iterative procedure.63 It was modified by Lorber to yield a direct solution of a standard eigenvalue problem.64,65 Sanchez and Kowalski extended the method to the general case of several components that are not necessarily present in both the calibration and the unknown samples, obtained the solution by solving a generalized eigenproblem, and called the method the generalized rank annihilation method (GRAM).66 The two-way data matrix D, which collects m spectra in rows (measured spectra in different concentrations of surfactant), each containing n wavelengths, can be described by the following model:
D ) D(Enolm) + D(Enolw) + D(Ketow) + E
Figure 6. The relationship between RSD(1) and the micellar tautomerization constant, Ktm, obtained at Ktm ) 168.
(13)
) [Enolm]sTEnolm + [Enolw]sTEnolw + [Ketow]sTKetow + R ) CST + R where D(Enolm), D(Enolw), and D(Ketow) are the bilinear absorbance matrices of pure tautomeric species, and each one can be decomposed into the corresponding concentration profiles [Enolm], [Enolw], and [Ketow] (column vectors) and the molar absorptivity spectra sTEnolm, sTEnolw, and sTKetow (row vectors; superscript T denotes the transpose of a matrix or vector). C and ST represent matrices formed by the concentration profile and the molar absorptivity spectrum of each species, respectively. E is the residual matrix and should contain only noise. The size of matrix D is c × w, where c is the number of R values for which absorbances were recorded, and w denotes the number of wavelengths (c is smaller than w). Obviously, the sizes of matrices C and ST are c × 3 and 3 × w, respectively. According to the results of MCR-ALS analysis, the pure spectra of keto and enolic species in water (sEnolw and sKetow) and the tautomerization constant in water (Kt) are known, so the rank annihilation process can be considered as follows:
F ) D - D(Enolw) - D(Ketow) ) D - [Enolw]sTEnolw + [Ketow]sTKetow (14) The aim of the RAFA approach is to find a suitable Ktm value so that the rank of matrix F can be reduced by 1 from that of matrix D through the introduction of the concentration profiles of tautomeric species in water obtained from eqs 9 and 10. Upon specific operations, the grid searching method can optimize the Ktm value. The optimized solutions can be reached by decomposing matrix F to the extent that the residual standard deviation (RSD)17 of the residual matrix obtained after the extraction of one principal component reaches the minimum. On the basis of PCA, the RSD method is widely used to determine the number of principal components.17,67 The RSD is a measure (63) Ho, C. N.; Christian, G. D.; Davidson, E. R. Anal. Chem. 1978, 50, 1108-1113. (64) Lorber, A. Anal. Chim. Acta 1984, 164, 293-297. (65) Lorber, A. Anal. Chem. 1985, 57, 2395-2397. (66) Sanchez, E.; Kowalski, B. R. Anal. Chem. 1986, 58, 496-499. (67) Elbengali, A.; Nygren, J.; Kubista, M. Anal. Chim. Acta 1999, 379, 143158.
Figure 7. Resolved (a) concentration and (b) spectral profiles for BZA species in micellar solution.
of the lack of fit of a principal component model to a data set. The RSD is defined as c
RSD(n) ) (
∑ gi/[n(c - 1)])1/2 i)n+1
(15)
where gi is the eigenvalue, and n is the number of considered principal components. The number of latent variables was determined by applying PCA to the data matrix shown in Figure 5. The ratio of consecutive eigenvalues reaches a maximum at i ) 2, therefore indicating that there exist two independent systematic variables in the considered tautomerization system. The measured data matrix D was processed by the RAFA method, and the relationship between RSD(1) of the F matrix and the values of Ktm is shown as an RSD curve in Figure 6. For each value of Ktm, concentration profiles of tautomeric species in water were calculated, and the corresponding F matrices were obtained. The clear minimum is observed in the RSD curve shown in Figure 6, which indicates the optimum value of Ktm. Due to rank deficiency in the original data matrix D, rank annihilation occurs after the correct removal of the contribution of two tautomeric species (eq 11). The optimal solution is 168 ( 5. The standard deviation of the calculated
2368 Langmuir, Vol. 23, No. 5, 2007
equilibrium constant value was estimated by the jackknife procedure.68 The concentration profiles of all components were calculated according to resolved equilibrium constants and are shown in Figure 7a. After the rank annihilation, the matrix F only has information about the enolic species in the micellar pseudophase. So, the pure spectrum of the micellar enolic species is simply calculated as a least-squares solution of the F matrix to its obtained concentration profile (Figure 7b).
Conclusions This study showed that FA methods like EFA, MCR-ALS, and RAFA can provide very powerful tools to systematically study the thermodynamic behavior of tautomerization equilibria in water and micellar solutions. In principle, the proposed procedure can be applied to any b-dicarbonyl compound whose percentage of the enolic form varies under any experimental conditions. Soft-modeling methods were used for complete resolution of the measured spectral data at different temperatures for the BZA (68) Caceci, M. S. Anal. Chem. 1989, 61, 2324-2327.
Abdollahi and MahdaVi
tautomerization system. Soft-modeling procedures start with no model and no assumption at all, not even the use of the mass action law. The considered system was thermodynamically characterized by using MCR-ALS as a powerful soft-modeling method. RAFA was used for extracting the concentration profiles, spectral profiles, and the parameters of a partition model for tautomerization behavior of BZA in the micellar system. A proper combination of FA methods such as the proposed method can introduce new abilities for solving chemical problems and improves the resolution results. Our further works will be directed to investigating the possibilities of applying the proposed method to solving more complex problems such as tautomerization kinetics in micellar media and the thermodynamic characterization of tautomerization in the presence of cyclodextrins. Supporting Information Available: Additional details regarding the EFA, MCR-ALS, and RAFA methods. This material is available free of charge via the Internet at http://pubs.acs.org. LA0627112
