J. Phys. Chem. B 2009, 113, 3071–3079
3071
Modeling Preferential Solvation in Ternary Solvent Systems Ne´lson Nunes,†,‡ Cristina Ventura,‡,§ Filomena Martins,‡,⊥ and Ruben Elvas Leita˜o*,†,‡ Departamento de Engenharia Quı´mica, Instituto Superior de Engenharia de Lisboa, R. Conselheiro Emı´dio NaVarro 1, 1900-014 Lisboa, Portugal, Centro de Quı´mica e Bioquı´mica, Ed. C8, Campo Grande, 1749-016 Lisboa, Portugal, Instituto Superior de Educac¸a˜o e Cieˆncias, Alameda das Linhas de Torres 179, 1750 Lisboa, Portugal, and Departamento de Quı´mica e Bioquı´mica, Faculdade de Cieˆncias, UniVersidade de Lisboa, Ed. C8, Campo Grande, 1749-016 Lisboa, Portugal ReceiVed: May 11, 2008; ReVised Manuscript ReceiVed: NoVember 27, 2008
The theoretical solvent exchange model of Bosch and Rose´s for binary solvents was extended to ternary solvent mixtures. The model was applied to ENT values for the mixture methanol/1-propanol/acetonitrile, in terms of 48 new values in a total of 79, measured at 25 °C over the whole range of solvent compositions. It was also applied to the mixture methanol/ethanol/acetone at the same temperature using 93 ENT values obtained from literature. Very good fits between experimental and calculated values, substantiated by external validation methods, were achieved for both sets of data. The use of the developed extended model allowed the interpretation of measured solvatochromic shifts in terms of solute-solvent and solvent-solvent interactions in the local environment of the solute’s dye for the two ternary systems and the underlying binary mixtures. It also provided the identification of various complex solvent entities and the quantification of their relative concentrations in the probe’s cybotactic region, thus leading to new and significant physicochemical insights at a molecular level, regardless of the nonconsideration of the formation of solvent complexes in the bulk. Results clearly showed a different solvent composition in the vicinity of the solute. The further extension of the model to four and five components is also presented. 1. Introduction The study of solvation processes is a key topic in physical chemistry since solute-solvent and solvent-solvent interactions can explain many equilibrium and kinetic phenomena. By comparison with a pure solvent, the process of solvation in mixtures of solvents becomes necessarily much more complex. In fact, molecules of different solvents can establish all sorts of solvent-solvent interactions that often differ from those in pure solvents, and solutes can interact differently and to different degrees with each solvent or mixed solvent entities. This brings about changes in the solvent composition in the vicinity of the solute molecules (the so-called cybotactic region), which becomes therefore different from that of the “bulk” solvent, a phenomenon known by the name of preferential (or selective) solvation.1-3Evidence of preferential solvation phenomena has been extensively presented especially for binary solvent mixtures.4-6 These include experimental studies based on thermodynamic,7 IR,8,9 NMR,10,11 or UV-visible measurements5,12-16 and theoretical approaches such as molecular dynamics,17,18 Monte Carlo calculations,19 and molecular solvation theory.20,21 One of the most widely used methods in preferential solvation studies is the measurement of molar transition energies by UV-visible spectroscopy of molecular solute probes, for which the characteristic absorption bands undergo shifts depending on the solvent’s composition (solvatochromism).1 Several models have been proposed to describe preferential solvation in binary mixtures, namely, theoretical models such * To whom correspondence should be addressed. E-mail: rleitao@ deq.isel.ipl.pt. † Instituto Superior de Engenharia de Lisboa. ‡ Centro de Quı´mica e Bioquı´mica. § Instituto Superior de Educac¸a˜o e Cieˆncias. ⊥ Universidade de Lisboa.
as those based on the quasi lattice-quasi chemical theory,6,22 the competitive preferential solvation approach,23-25 the Kirkwood-Buff integrals formalism,26,27 the dielectric enrichment,28 or the stepwise solvent exchange concept29 and also some empirical models.30,31 The model developed by Skwierczynski and Connors32 and later extended by Bosch and Rose´s,33 to which we shall refer hereafter to as the Bosch and Rose´s model, is based on the step-by-step solvent exchange concept and is among the most successful models to describe preferential solvation in binary systems. It combines a straightforward mathematical approach and an easy interpretation of results and has been applied with success to describe the behavior of many solvatochromic dyes4,33-39 and also that of rate constants and even of pKa in binary mixtures.40-46 In the scope of our work with binary and ternary solvent systems,47 we came across the need to quantitatively describe solvent-solvent and solute-solvent interactions in these media. With this purpose, we have been using transition energies of Reichardt’s ET(30) betaine dye in various solvents as probes to monitor those interactions at a molecular level, that is, in the cybotactic region of the dye. Very recently, Bagchi et al. have proposed an extension of a two-phase model of solvation to explain the behavior of some ternary solvent mixtures, also based on the monitoring of the CT absorption band of solvatochromic probes.48-53 Also Dı´az et al.54 have tried to characterize a ternary system based on its acidity, basicity, and dipolarity/polarizability properties but strictly from a solvents’ perspective, that is, outside of the context of preferential solvation models. To the best of our knowledge, these are the only attempts tried so far to address ternary systems using solvatochromic probes. In this paper, we intend to describe an extension of the Bosch and Rose´s model to a three-component solvent system and use it to interpret measured solvatochromic
10.1021/jp804157b CCC: $40.75 2009 American Chemical Society Published on Web 02/16/2009
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shifts for the ENT parameter in terms of solute-solvent and solvent-solvent interactions in the local environment around the solute dye for the ternary mixture methanol/1-propanol/ acetonitrile (MeOH/1-PrOH/MeCN).47 To further validate the proposed model extension, we have also applied it to ENT literature values for the mixture methanol/ethanol/acetone (MeOH/EtOH/Me2CO). 49,54-57 The ENT parameter is the normalized value of the ET(30) parameter, also known as the Dimroth-Reichardt polarity parameter, defined by the excitation energy (in kcal mol-1) of the 2,6-diphenyl-4-(2,4,6-triphenyl-1-pyridinio)-1-phenolate dye, calculated from the wavenumber of the maximum of its absorption spectrum (νjmax in cm-1) in a given solvent.1 This parameter reflects a blend of nonspecific solvent dipolarity interactions and specific solvent hydrogen bond acidity interactions. The normalization is made with reference to tetramethylsilane (TMS) (ENT ) 0) and water (ENT ) 1), according to the equation shown below
ENT ) )
ET(30) - ET(30)TMS ET(30)H2O - ET(30)TMS
constants, the authors of the so-called solvent-exchange-based models32,57 postulate that these complexes are formed in the solute’s cybotactic region according to Scheme 2. However, other authors58-62 consider this assumption as a thermodynamic frailty since Sij complexes in the solvation microsphere have to be in equilibrium with the same species in the bulk. Nevertheless, and since the same interaction patterns in the solute’s cybotactic region of selected binary mixtures (see Figures S1 and S2 in Supporting Information) emerge from application of the Bosch and Rose´s model and of that of El Seoud,58-62 the mentioned hypothesis is accepted as a reasonable one throughout this work. SCHEME 2
m m S2 h I(S12)m + S1 (a′) 2 2 m m I(S1)m + S3 h I(S13)m + S1 (b′) 2 2 m m I(S2)m + S3 h I(S23)m + S2 (c′) 2 2
I(S1)m +
ET(30) - 30.7 32.4
2.8591 × 10-3ν¯ max - 30.7 ) 32.4
(1)
In order to apply the proposed model extension to the referred ternary systems, the solvation characteristics of the corresponding binary mixtures are also required. As such, these values were also collected from the literature.47,49,54-57 Solvation Model. For a ternary mixture composed of solvents 1 (S1), 2 (S2), and 3 (S3), and a solvatochromic indicator I) the following simple equilibria can occur: SCHEME 1
I(S1)m + mS2 h I(S2)m + mS1 (a) I(S1)m + mS3 h I(S3)m + mS1 (b) I(S2)m + mS3 h I(S3)m + mS2 (c)
Here, I(S1), I(S2), and I(S3) represent solute (I) fully solvated by solvents S1, S2, and S3, respectively, and m is interpreted as the number of solvent molecules involved in the exchange process in the solvation microsphere of the solvatochromic indicator and affecting its transition energy57,58 and should not be confused with the total number of molecules that solvate the indicator.59 Each of these equilibria (a)-(c) represent the one-step solvent exchange model35,57 applied to solvents S1 and S2, S1 and S3, and S2 and S3, respectively. In addition, solvent molecules can also interact with each other, forming solvating complexes such as I(S12), I(S13), and I(S23). For the sake of simplicity, it is considered that the different solvent molecules interact in the ratio of 1:1.57 Since ENT is very sensitive to the local solvent composition but not to the bulk solvent composition, the estimation of the macroscopic equilibrium constants for the formation of these Sij complexes based on the transition energies of the solute probe would have a very high associated error. Therefore, in order to avoid constraints related to the estimate of these equilibrium
The simultaneous consideration of equilibria (a) and (a′), (b) and (b′), or (c) and (c′) represents what is widely referred to as the two-step solvent exchange general model.35,57 In a ternary mixture, there is also the possibility of formation of a solvating ternary complex with the indicator I(S123), which can result from the following equilibria: SCHEME 3
m 2m m S2 + S3 h I(S123)m + S1 3 3 3 2m m m I(S2)m + S1 + S3 h I(S123)m + S2 3 3 3 m 2m m I(S3)m + S1 + S2 h I(S123)m + S3 3 3 3
I(S1)m +
For each of the above equilibria, we can define an equilibrium constant which relates the solvents’ mole fractions in the S S S , x13 , x23 , solvation microsphere of the indicator, x1S, x2S, x3S, x12 S , and their mole fractions in the mixture’s “bulk”, x10, x20, x30. x123 These equilibrium constants can be related to a preferential solvation parameter, f, as follows. In the case of the simplest exchange of solvent molecules, for instance for solvents S1 and S2, f2/1 is a parameter that measures the tendency of the indicator to be preferentially solvated by solvent S2 rather than by solvent S1. This parameter results from the quotient of f2 with f1, each of these representing the mole fraction distribution of the solvent between the solute’s cybotactic region and the bulk mixed solvent, eq 2
xS2 f2/1 )
xS2 /xS1 f2 (x02)m xS2(x01)m ) S ) S 0m ) 0 0m f1 x1 x1 (x2) (x2/x1)
(x01)m
(2)
Modeling Preferential Solvation in Ternary Solvent
J. Phys. Chem. B, Vol. 113, No. 10, 2009 3073
The same kind of reasoning can be applied to the other two possible simple equilibria involving solvents 1 and 3 and solvents 2 and 3 to obtain f3/1 and f3/2, respectively.
f3/1 )
f3/2 )
Y ) Y1xS1 + Y2xS2 + Y3xS3 + Y12xS12 + Y13xS13 + Y23xS23 +
xS3 /xS1 x03 /x01 m
(
)
xS3 /xS2
(x03/x02)m
(3)
Y123xS123 (10)
(4)
Mole fractions in the cybotactic region must be converted to known variables, on the basis of eq 10 and considering that the sum of all mole fractions in the cybotactic region and in the solvent’s bulk must be equal to unity.
When considering the formation of binary complexes, we must also quantify their preferential solvation relative to a reference solvent. For example, the preferential solvation of complex S12 relative to solvent S1, which is defined as f12/1 S and in which x12 stands for the mole fraction of the binary complex S12 in the solvation sphere of the indicator, is given by
f12/1 )
xS12 /xS1
√(
(5)
f23/2 )
xS13 /xS1
√(x03/x01)m xS23 /xS2
√(x03/x02)m
xS123 /xS1 3
√(x02x03/(x01)2)m
(6)
(7)
(8)
This representation of the various equilibria by the preferential solvation parameters allows the determination of preferential solvation parameters not expressed by any of the previously shown equilibrium relations; an example is the determination of the preferential solvation of complex S23 in relation to S1:
f23/2 × f2/1 )
xS123 ) 1 (11) This is accomplished through the use of the preferential solvation parameters, f, defined above. The x1S, for example, is given by
xS1 )
)
Finally, we must also consider the formation of a ternary complex which can also interact with the indicator. On the basis of this assumption and considering, as an example, the preferential solvation of S123 relative to solvent S1, an S equilibrium constant f123/1 can also be defined, in which x123 represents the mole fraction of the ternary complex solvating indicator I.
f123/1 )
x03 + x02 + x01 ) xS1 + xS2 + xS3 + xS12 + xS13 + xS23 +
x02 /x01 m
Extending this concept to the other two binary mixtures, we obtain two further relations, f13/1 and f23/2, also in terms of the S and corresponding mole fractions of the binary complexes, x13 S x23
f13/1 )
cybotactic region, which is represented by the product of its mole fraction by the property’s value for that entity, Yi
f23 f2 × ) f23/1 f2 f1
(9)
A given solvatochromic property, Y, in a mixture results from the sum of the contributions of each solvent entity in the
(x01)mf1 (x01)mf1 + (x02)mf2 + (x01)mf3 + √(x01x02)mf12 (x01)mf1
√( )
x01x03 mf13
+
√( )
x02x03 mf23
+
3
√(
+
(12)
x01x02x03 mf123
)
Converting all mole fractions in the cybotactic region in a similar way and after making the necessary simplifications, the following expression is obtained
Y)
[
m
Y1 f1(x01)m + Y2 f2(x02)m + Y3 f3(x03)m + Y12 f12(x01x02) 2 + m
m
m
Y13 f13(x01x03) 2 + Y23 f23(x02x03) 2 + Y123 f123(x01x02x03) 3
[
m
f1(x01)m + f2(x02)m + f3(x03)m + f12(x01x02) 2 + m
m
m
f13(x01x03) 2 + f23(x02x03) 2 + f123(x01x02x03) 3
]
] (13)
Dividing all terms of the above expression by f1, we obtain the preferential solvation equation in its final form, eq 14
Y)
[
m
Y1(x01)m + Y2 f2/1(x02)m + Y3 f3/1(x03)m + Y12 f12/1(x01x02) 2 + m
m
m
Y13 f13/1(x01x03) 2 + Y23 f23/1(x02x03) 2 + Y123 f123/1(x01x02x03) 3
[
m
(x01)m + f2/1(x02)m + f3/1(x03)m + f12/1(x01x02) 2 + m
m
m
f13/1(x01x03) 2 + f23/1(x02x03) 2 + f123/1(x01x02x03) 3
]
]
(14)
The interpretation of the several preferential solvation parameters can be made in the same way as in the original Bosch and Rose´s model; values of fa/b close to 1 represent an ideal mixture; values lower than 1 imply a preferential solvation of the indicator by component b by comparison with component a, and the opposite is true if the value of fa/b is higher than 1. Finally, since all parameters refer to solvent 1, it is possible to establish a preferential solvation order for the mixtures’ constituents in terms of the measured property, Y.
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Computation Methods. All calculations were performed with Microsoft’s Excel. The nonlinear fit to eq 14 was carried out with Excel’s add-in SolVer. This iterative routine allowed the estimation of the various parameters by minimizing the sum of squares of the residuals,63 χ2 ) ∑(Yn, exp - Yn,calc)2. In order to assess the quality of the fitted parameters, their standard deviations (si) were also determined through the SolVerstat addin,64 which uses the following function
si )
pii-1χ2 N-M
(15)
where N is the number of experimental points, M is the number of fitted parameters, and pii-1 is the ith diagonal term of the inverse of an M × M matrix containing the partial differentials of the fitting function in order to parameters ai and aj N
pij )
∂Y ∂Y
∑ ∂ani ∂anj
(16)
n)1
2. Methodology The application of the preferential solvation model’s extension to ternary systems implies the consideration of two essential characteristics of nonlinear regression since the analysis of eq 14 shows that there are 13 variables to be adjusted, excluding m, which must assume a positive integer value. First, if all parameters vary freely during the fit procedure, the high intercorrelation among some of the model parameters leads to a very high VIF (variance inflation factor) and to an overparametrization of the system. This process results in incorrect estimated parameters and uncertainties, corresponding to physically meaningless solutions. However, as the mathematical relations between some parameters are known, constraints can be easily introduced in the fitting program, thus significantly reducing the number of variables to be adjusted. Correct uncertainties for the fitted parameters can be obtained directly from eqs 15 and 16. The uncertainties of the remaining parameters are subsequently obtained through error propagation analysis. Second, it is also crucial to be able to estimate correct starting values for the largest possible number of variables, in order to ensure a successful fit convergence since this type of regression can produce multiple solutions (i.e., false minima) in systems where a large number of variables have to be adjusted. Careful scrutiny of eq 14 shows that three of the variables represent solvent property Y in the pure components (Y1, Y2, Y3). Since these values are usually available from literature or can be experimentally obtained, they can be used as adequate initial estimates. Other variables for which it is also possible to make correct guesses for starting values are those corresponding to the three underlying binary mixtures. Generally speaking, the determination of these variables is performed through the application of the Bosch and Rose´s35,57 model to Y values obtained for the binary mixtures of Si and Sj, where i ) 1,2, j ) 2,3, and i * j, according to eq 17 m
Y ) Yi +
fj/i(Yj - Yi)(xj0)m + fij/i(Yij - Yi)(xi0xj0) 2 m
(xi0)m + fj/i(xj0)m + fij/i(xi0xj0) 2
(17)
In the particular case of m ) 2 for Scheme 2, if eqs 18 are valid, then eq 17 simplifies to the mathematical model described by eq 19, which reflects the preferential solvation of the probe in these particular conditions.
Yij )
Yi + Yj fj/i 1 + fj/i
Y ) Yi +
and
fij/i ) 1 + fj/i
fj/i(Yj - Yi)xj0 xi0 + fj/ixj0
(18)
(19)
For each binary mixture, only fj/i, fij/i, and Yij are allowed to vary. The appropriate m value for each mixture is chosen according to the best fit statistics. This procedure enables therefore the determination of appropriate starting values for the fitting of eq 14 for f2/1, f3/1, f12/1, f13/1, and f23/2 and also for Y12, Y23, and Y13. Additionally, the knowledge of f2/1 and f23/2 allows the calculation of f23/1 from eq 9, whose initial estimate is also necessary to fit the data to eq 14. It is thus feasible to obtain good initial estimates for 11 out of the 13 parameters involved in the fitting equation. The use of eq 14 implies the use of a given m value, but it is possible that the application of eq 17 to all binary solvent combinations does not produce a unique value for m. In fact, if any of the underlying binary mixtures shows synergism, that is, if some intermediate compositions of the mixtures show values for the probing property higher than those of the pure solvents, then for that mixture m must be (at least) equal to 2 in eq 17, and its simplified version (eq 19) cannot be used. Thus, in that case, we must assume m g 2 also for the ternary system to be able to fully account for the behavior of the underlying binary mixtures. The first fit to eq 14 should then be made using the lowest value of m that adequately describes the full set of binary combinations. In this fit, Y123 and f123/1 are the only adjustable parameters. Their optimized values constitute proper starting values to be used in a final fit, always performed, where all parameters are allowed to vary simultaneously, subject to the appropriate constraints already referenced, in order to ensure that the best set of parameters and correct uncertainties are always attained. 3. Experimental Section All chemicals employed were supplied by Sigma-Aldrich. Solvents were HPLC grade, and their purity was confirmed by ascertaining that values obtained for measured physical properties for pure solvents (namely, densities, refractive indices, and ET values) were in agreement with literature values.1,65 Precautions were taken to avoid evaporation and contamination by humidity. Mixtures were carefully prepared by mass with an uncertainty of (2 × 10-4 g. No correction was incorporated for the water content of the alcohol in the calculation of the mole fraction. The pyridinium-N-phenolate betaine used was the commercially available Reichardt’s dye (95%). Spectral measurements were taken on a Shimadzu UV-vis 1603 spectrometer, at 25.0 ( 0.1 °C, using quartz analytical cells with a 10 mm path length and the apparatus software. Four to six measurements were performed to obtain a mean wavelength value for the maximum absorption band of Reichardt’s dye at each mole fraction. The precision of replicate measurements was better than (1 nm. The dye’s concentration was about 10-4 mol dm-3. All
Modeling Preferential Solvation in Ternary Solvent “blank” solutions were identical to the sample solutions except for the absence of the indicator. 4. Results and Discussion Experimental ENT values for the ternary mixture methanol(S1)/1-propanol(S2)/acetonitrile(S3) are shown in Table S1 in Supporting Information. Forty-eight new ENT values, both for the ternary mixture and the corresponding binary mixtures, were determined in the context of this work, whereas the remaining values refer to our previous work on these solvents.47 Table S1 of the Supporting Information shows that the polarity of the MeOH/1-PrOH/MeCN ternary mixture, as given by ENT, is, in general, higher than that of the binary mixture 1-PrOH/MeCN and lower than that of MeOH/MeCN. It seems therefore that the addition of the third component to the mixture 1-PrOH/MeCN leads to a larger stabilization of the ground state of the betaine dye (and also to a slight destabilization of the excited state) and to the opposite effect in the case of MeOH/ MeCN. In any case, the addition of a third component to alcohol/ MeCN mixtures gives rise to a ternary mixture with polarity properties clearly distinct from pre-existing ones. The value of a certain measured property in an ideal mixture of solvents is given by the weighted average of the contributions of each pure solvent for that property. In the case of the transition energy of Reichardt’s dye in a ternary system, one should then have ENT(ideal) ) x1ENT(1) + x2ENT(2) + x3ENT(3). An evident deviation between ENT(mixt) and ENT(ideal) is usually interpreted as an indication of preferential solvation phenomena resulting from solute-solvent and/or solvent-solvent interactions. A close inspection of Table S1 (Supporting Information) reveals that ENT values for the ternary mixture show a clear positive deviation from ideality for the whole domain of mole fractions, thus confirming the existence of a differentiation between the solute’s cybotactic region and the mixture’s bulk. Maximum deviation is generally attained when the mixture is rich in acetonitrile and equimolar in the two alcohols. Data in Table S1 (Supporting Information) also show that the binary mixtures alcohol/MeCN exhibit a somewhat more polar behavior than either of their cosolvents in a large range of mole fractions and are said, for this reason, to show a synergetic behavior. This could be rationalized by accepting the existence of a third, more polar, solvent entity (Sij), preferentially solvating the indicator. Surprisingly, the addition of the third component to the alcohol/MeCN mixtures eliminates the synergetic behavior visible in the binary systems. With the purpose of interpreting (and quantifying) these aspects, namely, the nature of the solute-solvent and solventsolvent interactions present in the ternary mixture MeOH/ 1-PrOH/MeCN and the synergism observed in some of the corresponding binary solvent systems, we have applied the preferential solvation model equations, previously presented in the Introduction section (eqs 14 and 17 for the ternary and binary mixtures, respectively), to this system. To further validate the proposed extended model, eq 14, we have also applied it to 93 ENT values collected entirely from literature,49,54-57 pertaining to the ternary system methanol(S1)/ethanol(S2)/acetone(S3) (Table S2 in Supporting Information). This system is particularly interesting because there is no consensus about the presence or absence of synergism in alcohol/Me2CO mixtures. As mentioned earlier, the first step of the referenced methodology is always the determination of the parameters’ values for the various binary equilibria. For this purpose, ENT values for the binary data were fitted either to eqs 17 or 19, and the resulting parameters are presented in Table 1. For the
J. Phys. Chem. B, Vol. 113, No. 10, 2009 3075 TABLE 1: Fitted and Calculated Parameters for the Binary Mixturesa solvent (S1)
cosolvent (S2)
f2/1
f12/1
ENT12
R2
eq
methanol methanol 1-propanol
1-propanol acetonitrile acetonitrile
1.190 0.532 0.236
2.190 9.544 5.656
0.683 0.776 0.679
0.998 0.999 0.998
19 17 17
methanol methanol ethanol
ethanol acetone acetone
1.103 0.125 0.129
2.103 1.125 1.129
0.705 0.728 0.624
0.998 0.996 0.998
19 19 19
a
Fitted parameters are shown in bold type.
three binary mixtures underlying the MeOH/EtOH/Me2CO ternary system and for the MeOH/1-PrOH binary mixture, f12/1 and ENT12 values are also presented in Table 1, although they are not fitted parameters. They are shown simply because they are needed for the fitting procedure of the ternary systems. The application of the model showed that it is sufficient to use m ) 2 in eq 17 to describe the binary solvent combinations underneath the MeOH/1-PrOH/MeCN solvent mixture. However, for any binary solvent combination such as MeOH/ 1-PrOH, for which no synergistic behavior is observed47 and for which, therefore, eq 19 is applicable, the use of eq 17 with m ) 2 may seem inappropriate. Yet, since the statistical figures obtained upon fitting the data to both equations are very similar, the choice of the simplified model, eq 19, is strictly due to the need to avoid overparametrization of the mathematical fit. In fact, in case of eq 17 with m ) 2, if one uses as the run’s initial estimates parameter values obtained considering eq 19, one arrives at essentially the same numerical results for the preferential solvation parameters, the use of either equation, from this perspective, being therefore irrelevant. The fitted parameters for the alcohol/MeCN binary mixtures presented in Table 1 clearly show that the equilibria represented by Scheme 2 are in fact all shifted toward a preferential solvation of the indicator by the implicitly accepted Sij complexes (f12/1 . 1), which is particularly evident for the binary mixture MeOH/MeCN (f12/1 ∼ 9.5). This means that the betaine dye shows a higher tendency to interact with the solvent complexes rather than with the pure solvents (also f12/2 ) f12/1/f2/1 . 1). The enhanced polarity and polarizability47 properties of these Sij complexes, whose components are linked together by specific solvent-solvent interactions of the type -O-H · · · NtC-, may facilitate the establishment of nonspecific interactions Sij · · · betaine, thus leading to higher transition energies and therefore higher ENT values. The order of solvation found on the basis of the values presented in Table 1 (S1-S2 complex > alcohol > MeCN) is different from what we have reported before47 for MeOH/MeCN mixtures, for a smaller number of mole fractions. In fact, in this case, the slight difference between ENT12 and ENT for methanol has formerly led us to consider this solvent mixture as nonsynergetic and therefore described by Scheme 1. The present perspective is in line with the tendency reported by Bosch and Rose´s et al. for MeCN-cosolvent mixtures.35,36,57 On the other hand, the binary mixture MeOH/1-PrOH shows only a small deviation from ideality, as can be noted from the magnitude of the f2/1 parameter in Table 1. Given the synergism observed for two of the underlying binary mixtures, we have accordingly used a value of m ) 2 for the resultant ternary system, for the reasons already mentioned. In the case of the binary mixtures underlying the MeOH/ EtOH/Me2CO ternary system, the application of the model shows that it is sufficient to consider the simplified form of eq
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TABLE 2: Fitted Parameters for the Ternary Mixtures MeOH(S1)/1-PrOH(S2)/MeCN(S3) and MeOH(S1)/EtOH(S2)/ Me2CO(S3) MeOH /1-PrOH/MeCN parameter f2/1 f3/1 f12/1 f13/1 f23/1 f123/1 ENT1 ENT2 ENT3 ENT12 ENT13 ENT23 ENT123
std. error
1.256 0.06 0.262 0.03 2.256 0.06 4.70 0.5 6.21 0.8 2.79 0.8 0.7646 0.001 0.6172 0.001 0.4608 0.003 0.6831 0.002 0.7761 0.002 0.6800 0.002 0.765 0.02 R2 ) 0.998; sfit ) 0.003
17, that is, eq 19, to describe these binary combinations. In fact, the application of eq 17 for the alcohol/Me2CO mixtures leads to high errors in the parameters and to ENT12 values lower than ENT values for the pure alcohol component, which is physically meaningless. The observation of a (very) small number of ENT values equal or just somewhat higher than the ENT value in the alcohol, in alcohol very rich regions, seems like not enough to justify the need to consider the existence of synergism. Rose´s and Bosch et al. had already pointed out the low synergism observed for the mixture EtOH/Me2CO due to the much higher ENT value of EtOH by comparison with acetone,57 even if later on they postulated the existence of synergetic behaviors for the mixtures of ethanol with acetone, dimethyl sulfoxide, and acetonitrile.35 Our mathematical treatment gives, however, no evidence for any synergetic behavior in these alcohol/Me2CO mixtures. A preferential solvation order can thus be derived from the f parameters in Table 1, alcohol > S1-S2 complex > Me2CO. On its own, the MeOH/EtOH binary mixture shows an almost ideal behavior, similar in every respect to what was seen for MeOH/1-PrOH. For the ternary system MeOH/EtOH/Me2CO, we applied the extended model, eq 14, imposing as limiting conditions those expressed by eqs 18 since, in this case, the three underlying binary mixtures show no synergism. Table 2 summarizes the results obtained for the two ternary mixtures, with sfit, R2, and the parameters’ uncertainties showing unambiguously the quality of the fits. Figures 1 and 2 illustrate the excellent agreement between the extended model equation predictions and the experimental data. This is highly significant, especially for the second mixture if one considers that the corresponding data set comes from five different sources. If we compare the estimates of the solvation parameters obtained on the basis of the binary mixtures calculations (Table 1) with those obtained from the ternary system MeOH/1-PrOH/ MeCN (Table 2), we notice they are extremely similar, except for f3/1 and f13/1 in Table 2 by comparison with fMeCN/MeOH and fMeCN-MeOH complex/MeOH in Table 1, which is a further indication of the ability of the extended model equation to also explain the behavior of the binary components. The fitted parameters for this ternary mixture show that the solvent dipolarity and hydrogen bond acidity interactions both reflected by ENT for the Sij and Sijk entities are always higher than those of 1-PrOH and MeCN. From all different solvent entities present in the ternary system, the MeOH(S1) · · · MeCN(S3) complex is the one showing the highest dipolarity and acidity, the general order being as follows: ENT13 >
MeOH /EtOH/Me2CO parameter
std. error
0.994 0.06 0.1179 0.006 1.994 0.06 1.1179 0.006 1.112 0.09 0.91 0.3 0.7630 0.002 0.6565 0.002 0.3468 0.007 0.7099 0.002 0.7191 0.003 0.624 0.01 0.660 0.02 R2 ) 0.989; sfit ) 0.007
ENT123 ∼ ENT1 > ENT12 ∼ ENT23 > ENT2 > ENT3. The observed differences in fMeCN-MeOH and fMeCN-MeOH complex/MeOH between Table 1 (MeOH/MeCN) and Table 2 (MeOH/1-PrOH/MeCN) reveal that, although, in both cases, the dye interacts preferentially with the solvent complex rather than with MeOH (fcomplex/MeOH >1) and with this solvent rather than with MeCN (fMeCN-MeOH MeOH-MeCN complex > MeOH-1-PrOH-MeCN complex > MeOH-1-PrOH complex > 1-PrOH > MeOH > MeCN. This preferential solvation order agrees with the sorting of the ENT values expressed above for the various solvent complexes (with the exceptions of ENT23 and ENT1), which is not surprising given the fact that the solvated solute (Reichardt’s dye) is particularly sensitive to the dipolarity and hydrogen bond acidity characteristics of the solvent, accounted for by ENT. The order found allows the rationalization of the solvatochromic positive deviations already mentioned. Since the spectroscopic property under analysis depends on the composi-
tion in the cybotactic region of the indicator, the positive deviations can be explained if one accepts that the indicator is actually “surrounded” by solvent entities (Sij and Sijk) with higher polarity/polarizability than those expected from the weighted average of these properties for the pure components. This difference induces an increase in dipolarity in the cybotactic region of Reichardt’s betaine, leading (mainly) to a greater stabilization of its dipolar ground state and therefore to a transition energy higher than that anticipated on the basis of an ideal behavior. As to the ternary mixture MeOH/EtOH/Me2CO, we notice some differences from the previous ternary system. In both cases, the betaine dye has a greater tendency to interact with MeOH than with the nonalcoholic solvent (f3/1 , 1). However, there is a clear preference of the solute to interact with the MeOH · · · MeCN complex rather than with the MeOH · · · Me2CO one (f13/1 ) 4.70 and 1.12, respectively). Also, the solute · · · S123 interaction when compared with the solute · · · MeOH interaction is reversed in the two ternary systems; in the first one, the solute is preferentially solvated by the MeOH-1-PrOH-MeCN complex by comparison with MeOH (f123/1 > 1), while in the second one, it interacts preferentially with MeOH rather than with the corresponding ternary complex (f123/1 < 1). These observations seem reasonable if one looks at the difference in polarity (as measured by µ and ε) between MeCN and Me2CO. The preferential solvation order for this mixture follows the sequence MeOH-EtOH complex > MeOH-Me2CO complex > EtOH-Me2CO complex > MeOH ∼ EtOH > MeOH-EtOH-Me2CO complex . Me2CO. These results agree with those of Bagchi et al.49 for the same mixture. These authors conclude on the basis of a different solvation model that the cybotactic region of the betaine dye is richer than the bulk in methanol and ethanol and poorer in acetone. The preferential solvation order found for this ternary system and derived from the extension of the solvent exchange model of Bosch and Rose´s (eq 14) not only leads to the same conclusions but goes significantly further as it provides additional information at the molecular level by identifying distinct complex solvent entities in the local composition of the solute’s cybotactic region. In order to evaluate the predictive ability of the extended model equation, eq 14, we have applied an external validation procedure. For this purpose, both data sets were divided into training and test sets with similar degrees of variability. The sizes of the training sets were 54 and 63 for MeOH(S1)/ 1-PrOH(S2)/MeCN(S3) and MeOH(S1)/EtOH(S2)/Me2CO(S3),
3078 J. Phys. Chem. B, Vol. 113, No. 10, 2009
Nunes et al.
Figure 3. ENT calculated values versus ENT experimental values using eq 14 for the ternary mixture MeOH/1-PrOH/MeCN at 25.0 °C; ] training set, 2 - test set.
Figure 4. ENT calculated values versus ENT experimental values using eq 14 for the ternary mixture MeOH/EtOH/Me2CO, at 25.0 °C; ] training set, 2 - test set.
TABLE 4: Comparison between Experimental and Calculated ENT Values for the Pure Components ENT solvent
mixture
MeOH
MeOH/1-PrOH/MeCN MeOH/EtOH/ Me2CO MeOH/1-PrOH/MeCN MeOH/1-PrOH/MeCN MeOH/EtOH/Me2CO MeOH/EtOH/Me2CO
1-PrOH MeCN EtOH Me2CO
exp. 0.762 0.617 0.460 0.654 0.355
calc.
%|error|
0.7645 0.7630 0.6172 0.4608 0.6565 0.3468
0.33 0.13 0.03 0.17 0.38 2.31
respectively. Table 3 shows the resulting fitting parameters for the training and full sets of both mixtures. The remaining experimental results were used as independent test sets to check the model’s true predictive power. The average error (AE) and the absolute average error (AAE) were also calculated for both test sets. The figures of merit for MeOH/ 1-PrOH/MeCN are AE ) 6.58 × 10-4; AAE ) 2.37 × 10-3, and for MeOH/EtOH/Me2CO, they are AE ) 3.16 × 10-5; AAE ) 4.45 × 10-3. Figures 3 and 4 show the excellent agreements between calculated and experimental values obtained in both cases, as illustrated by sfit and R2 in Table 3, and demonstrate the robustness of the model equation. The quality of the predictions is further corroborated by the average percentage error for the calculated ENT values in the pure solvents (0.56%), as shown in Table 4. Moreover, the outstanding results obtained for the external validation procedure further ensure that eq 14 is not overparametrized. 5. Conclusions The extension of the Bosch and Rose´s model to ternary solvent systems was achieved with success. Its application to two distinct sets of data, one of which combines information
from five different sources, resulted in a general equation that adequately describes and rationalizes at the molecular level the behavior of both ternary mixtures and also that of the underlying binary components, in contrast with other model equations and/ or approaches proposed in the literature and referenced in this work. The predictive ability and the robustness of the extended model was tested and assured. The analysis of the preferential solvation parameters allowed, as in the case of the reduced model equation, the establishment of a relative order of selective solvation of the solvatochromic indicator. The outcome reproduced previous results obtained for the binary components investigated with the original Bosch and Rose´s model. Furthermore, we were able to demonstrate that in protic/ protic/aprotic ternary solvent systems, the composition in the solute’s microsphere is clearly distinct from that in the bulk. Therefore, in the solvation process of a probe sensitive to the solvent’s dipolarity and hydrogen bond acidity, in addition to solvent-solvent interactions, one must not neglect the various possible solute-solvent interactions, given their unquestionable relevance. In situations similar to those studied in this paper, the probe is expected to be solvated by different solvent entities, including Sij and Sijk complexes. The local composition of the mixture will, however, be dependent on the nature and relative magnitude of the solute-solvent and solvent-solvent interactions. The same molecular trends would also be found if one considered, in the model’s formulation, the existence of solvent complexes outside of the solute’s cybotactic region. The model proposed in this paper can be applied to solvents of different chemical natures and can be easily extended to encompass mixtures with further components, as shown by Scheme S1 in Supporting Information. Supporting Information Available: Figures containing xeffective versus xanalytical for the three species in solution (Si, Sj, Sij) for the binary mistures underlying the methanol/1-propanol/ acetonitrile ternary system (Figure S1) and ENTexp, ENTcalc (Bosch and Rose´s’s model), ENTcalc (El Seoud’s model) versus xeffective and ENTexp, ENTcalc (Bosch and Rose´s’s model), ENTcalc (El Seoud’s model) versus xanalytical for the same binary mixtures (Figure S2). Tables containing ENTexp values for the mixtures methanol/1-propanol/acetonitrile (Table S1) and methanol/ ethanol/acetone (Table S2). Extension of the proposed model to four- and five-component mixtures (Scheme S1). This material is available free of charge via the Internet at http:// pubs.acs.org. References and Notes (1) Reichardt, C. SolVents and SolVent Effects in Organic Chemistry, 3rd ed.; Wiley-VCH: Weinheim, Germany, 2003. (2) Marcus, Y. SolVent Mixtures: Properties and SelectiVe SolVation; Marcel Dekker: New York, 2002. (3) Reichardt, C. Pure Appl. Chem. 2004, 76, 1903. (4) Bosch, E.; Rived, F.; Roses, M. J. Chem. Soc., Perkin Trans. 2 1996, 2177. (5) Wu, Y. G.; Tabata, M.; Takamuku, T. J. Solution Chem. 2002, 31, 381. (6) Marcus, Y. J. Chem. Soc., Faraday Trans. 1989, 85, 381. (7) Costigan, A.; Feakins, D.; Mcstravick, I.; Oduinn, C.; Ryan, J.; Waghorne, W. E. J. Chem. Soc., Faraday Trans. 1991, 87, 2443. (8) Jamroz, D.; Stangret, J.; Lindgren, J. J. Am. Chem. Soc. 1993, 115, 6165. (9) Meade, M.; Hickey, K.; McCarthy, Y.; Waghorne, W. E.; Symons, M. R.; Rastogi, P. P. J. Chem. Soc., Faraday Trans. 1997, 93, 563. (10) Bagno, A.; Campulla, M.; Pirana, M.; Scorrano, G.; Stiz, S. Chem.sEur. J. 1999, 5, 1291. (11) Bagno, A. J. Phys. Org. Chem. 2002, 15, 790. (12) Mancini, P. M.; Adam, C.; Perez, A. D.; Vottero, L. R. J. Phys. Org. Chem. 2000, 13, 221.
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