J. Phys. Chem. B 2007, 111, 3069-3071
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COMMENTS Comment on “A Critique of Some Recent Suggestions to Correct the Kirkwood-Buff Integrals” Enrico Matteoli* and Luciano Lepori Institute for Physical-Chemical Processes of National Research Council, Research Campus of Pisa, Via Moruzzi, 1, I-56124 Pisa, Italy ReceiVed: December 12, 2006 Introduction The main subjects of the critique1 are the following: (1) the elimination of a background of interactions in the calculation of the corrected Gij; (2) the lack of meaning of eq 2.9; (3) and the lack of meaning of the function Gij - Gid ij , as a result of the example in section 3.1 These three points are here discussed and confuted with the aid of Gij-related quantities calculated for actual mixtures purposely chosen. Table 1 shows the mixtures considered, their relevant thermodynamic properties, as well as the quantities calculated. Mixtures 1-6 have GE ≈ 0 and VE ≈ 0 but have different values of the component volume difference, ∆V ) VB - VA. In particular, the systems in couples 1, 2 and 3, 4 have the same ∆V value but are different as to functional groups and their polarity. In systems 5 and 6, ∆V is still larger, whereas mixture 7 represents a typical nonideal system. Other points of critique are clearly the result of a forced extrapolation by the author of the meaning of “molecular interactions” to pair potential curves. In their papers,2,3 the present authors have never attributed that meaning to molecular interactions and have never claimed to obtain information on them from Gij. Also, with “indistinguishable interactions”, never did they mean that in ideal systems the potential curves are equal for all pairs. Thus, all of the argumentations in the Introduction1 are irrelevant and gratuitous, and the calculation of Gij of model systems having various forms or parameters of the pair potential curves (section 31) is a brilliant exercise but does not provide valid support to the critique. Discussion 1. On the Subtraction of a Background of Interactions. The main scope of the papers where the contested procedure was first proposed2,3 and then followed with or without slight modifications by others4-6 was to get information on the prevailing type(s) of interactions in complex liquid mixtures of binary or ternary polyfunctional organic compounds by using data of local composition. It is important to mention that the approach consists of two steps: first, the calculation of local composition and/or preferential solvation from Gij or a function of them, and second, identifying what the particular interactions are among all those due to the different polarities of the functional groups, such as hydrogen bonding, dipole-dipole, * To whom correspondence should be addressed. E-mail: matteoli@ ipcf.cnr.it.
dipole-induced dipole, and so on, that explain the observed preferential solvation. 1.1. Thermodynamic Quantities Determining the Gij. The equations to calculate Gij (eqs 2-4 of ref 3) depend on RTκT, V h i (that is, Vi and VE), and D, which is related to GE. For the liquid mixtures considered in refs 2 and 3 and here, RTκT can always be neglected, and VE is also so small that V h i ≈ Vi; in practice, Gij are functions of Vi and D only. For mixtures that show GE ≈ 0 (i.e., D ≈ 1), there are many and very different as to chemical nature, the Gij will depend only on Vi. The explicit dependence is given by eqs 8-10 of ref 3 and is depicted in Figure 1a of ref 3. It is worth pointing out that this dependence propagates also to other quantities, in particular to the number of excess of moles of i in the surroundings of the central j molecule, ∆nij, and to the preferential solvation. We think it is important to discuss the consequences thereof. 1.2. ∆nij in Mixtures with Very Low |GE|. ∆nij is given by ∆nij ) ciGij. By substituting eqs 8-10 of ref 3 in the expression, we realize that ∆nij depend only on the volumes, whatever the chemical nature of the compounds; this implies that for this type of system this quantity has no utility for investigating interactions. Moreover, it is found that ∆nij values are always negative for all couples ii, jj, and ij (except for ii, i being the component with the smaller volume, and in a limited range of molar fraction, when Vi < Vj/2) and that being ∆nij (i * j, xi ) 1) = -Vj/Vi, the excess number of molecules of i in the surrounding of j can be extremely and abnormally large and negative when Vj . Vi. Examples are aqueous solutions of polyetilenglycols3 and, as shown very recently by Ruckenstein and Shulgin,5 dilute solutions of proteins in water. There they have also shown that the implausible values of ∆nij change noticeably and become acceptable if either the criticized procedure (use of eqs 15 and 16, ref 3) or the new one proposed by them5 is followed to calculate ∆nij. 1.3. Preferential SolVation in Mixtures with Very Low |GE|. The coefficients of preferential solvation, δij, are defined by eq 13 of ref 3. Again, for mixtures with GE ≈ 0, from eqs 8-10 and 13 of ref 3, we can easily obtain δij ) xixj(Vj - Vi) and δii ) xixj(Vj - Vi); that is, these coefficients, like ∆nij, depend on ∆V and not on the chemical nature of the compounds. Thus, two (or more) mixtures that have the same ∆V will have the same values of the corresponding preferential solvation coefficients, whatever their chemical nature. The experimental results in Table 1 illustrate this behavior. For the couple of systems 1 and 2, δAB and δAA are spread very near the value 0.7 cm3/mol (that is, average ∆V/4), and for the couple 3 and 4, they are around 4.5, about 7 times larger. How can these findings be explained on the basis of the possible interactions taking place between the components? In particular, how can it be explained that the two chemically different mixtures of each couple may show the same coefficients? They are very different as to the interactions taking place: strong hydrogen bonding in mixture 3, weak van der Waals forces in system 4, strong dipole-dipole interactions and dispersion forces in 2, and weak van der Waals forces in 1. It might be that there is compensation among interactions, but then, how can the
10.1021/jp068532a CCC: $37.00 © 2007 American Chemical Society Published on Web 02/24/2007
3070 J. Phys. Chem. B, Vol. 111, No. 11, 2007
Comments
TABLE 1: Relevant Thermodynamic Properties and Gij-Related Quantities for a Number of Selected Mixturesa #
A+B
T (°C)
GE b
VA
VB
V B - VA
1 2 3 4 5 6 7
octane + 2,2,4-trimethylpentane chlorobenzene + bromobenzene methanol + ethanol hexane + 2,4-dimethylpentane hexane + octane octane + hexadecane water + 1-propanol
98 132 30 30 25 25 25
-11.7 7.3 -7.0 27.7 0.9 -39.6 837.8
163.5 102.2 40.7 131.6 131.6 163.5 18.1
166.1 105.5 58.7 149.9 163.5 294.1 75.1
2.5 3.3 18.0 18.3 31.9 130.6 57.1
VE c
δAAd
δABe
∆δAAf
∆δABg
-0.04 0.003 1.2 -0.620 0.615 0.04 1 0.58 0.24 -0.28 0.009 4.3 4.6 -0.15 0.10 -0.10 6.2 3.1 1.67 -1.49 -0.025 8 7.9 0.057 -0.04 -0.16 28 35 -4.5 2.6 -0.64 208 -31 197 -45
βh
β′ i
ωj
-160 0.04 -0.0622 -99 -0.03 0.26 -38 -0.01 -0.12 -129 0.09 1.58 -129 -0.01 0.049 -161 0.04 -3.5 -15 0.84 121
a The units are cm3 mol-1, except J mol-1 for GE. κT was taken as 0.5 × 10-5 bar-1 for water and 1.0 × 10-5 bar-1 for all other substances. All composition dependent quantities refer to a mole fraction of x ) 0.5. b Sources of data and parameters of the equations for GE: systems 1, 2, and 4, Gmehling, J.; Onken, U. Vapor-Liquid Equilibrium Data Collection; Dechema: Germany; Vol. I; system 3, ref 3; systems 5 and 6, Weiguo, S., Qin, A. X., McElroy, P. J., Williamson, A. G. J. Chem. Thermodyn. 1990, 22, 905; system 7, ref 7. c Sources of data and parameters of the equations for VE: systems 2 and 6, Battino, R. Chem. ReV. 1971, 71, 5; system 5, Itsuki, H., Terasawa, S., Shinoara, K., Ikezawa, H. J. Chem. Thermodyn. 1987, 19, 555; system 3, ref 3; system 7, ref 7; systems 1 and 4, estimated from similar systems. d δAA ) xAxB(GAA - GAB). e δAB ) id h g xAxB(GAB - GBB). f ∆δAA ) δAA - δid h A + ∆nBAV h B. i β′ ) ∆n′AAV h A + ∆n′BAV h B. j ω ) 1/2xAxB∆AB. AA. ∆δAB ) δAB - δAB. β ) ∆nAAV
largely different values shown by the two hydrocarbon mixtures in couples 1, 4 and 5, 6 be explained? Why no compensation here? The mixtures in each couple are very similar, one linear and one branched component in the former, both linear substances in the latter couple, and all being able to display only dispersive interactions. Another possible explanation could be that when ∆V is large, packing phenomena may give rise to high values of δij; it is well-known though that packing effects are larger the larger the ratio Vj/Vi, while the results show that δAB and δAA depend on the V difference. In any case, packing effects when present should be revealed by large VE values. Looking at systems 5 and 6, we see that while there is a large increase of ∆V, VB/VA, δAB, and δAA in going from 5 to 6, their VE values stay very low in magnitude, about ∆V/1000, suggesting that packing is to be ruled out as a possible cause of the large δAB and δAA values. It is worth also discussing the signs of δAB and δAA. If δAB is positive, it means that component B prefers to be solvated by A rather than by B, or that the local composition in the surroundings of B is richer in A than in the bulk composition. From the point of view of interactions, this means that attractive interactions between A and B are stronger than those between B and B. On the other hand, if δAA is also positive, the local composition in the surroundings of A is richer in A than in the bulk and attraction forces between A and A are stronger than those between A and B. These results are quite unexpected; it would be more plausible to find that when the surrounding of A is richer in A, then the surrounding of B is poorer in A and richer in B. It may be argued that this behavior (as well as that of ∆nij described in the previous section) is the consequence of the Kirkwood-Buff (KB) theory being derived in an open system; right, but this gives no suggestion on how to correlate these strange preferential solvation values to interactions. All of the above considerations apply also to excess local composition, δxij, as expressed by eq 12 of ref 3, as long as Vcor is taken to be sufficiently large. 1.4. Mixtures with Large GE Values. For mixtures with large and positive GE (system 7 in the table is an example of a fairly nonideal mixture), Gij values are large enough in magnitude to make the dependence of ∆nij and δij on the volume difference negligible in a range of molar fractions that is larger the larger GE is. In this range then, we observe that Gii and Gjj, and consequently ∆nii and ∆njj, have equal sign that is opposite to the sign of Gij and ∆nij. For 7, this range is about 0.4 < xA < 1, and at xA ) 0.7, we obtain ∆nAA ) 26, ∆nBB ) 2.7, and ∆nAB ) -15 mol. The calculations in Table 1 show that, for 7, different from all other mixtures, δAB and δAA do have opposite
signs, but this is not so at all mole fractions. Plotting the quantity δAB for 7 in the full x-range, we observe that there is a change of sign at about xA ) 0.4, being δAB > 0 at xA < 0.4. What are the interactions responsible for this change of preferential solvation? This is a difficult question to answer if one considers that plots of VE, V h EA, V h EB, GE, γA, and γB show smooth monotonic trends and no change of sign around xA ) 0.4, unless one recognizes that this change is brought about by a balance between the contributions of ∆V and of the interactions. For other examples of nonideal mixtures that show a similar behavior, see Figure 1 in ref 7 for water + acetonitrile and water + dioxane. 1.5. Conclusion of this Section. It is evident from the above examples based on a variety of actual systems that the dependence of Gij and consequently of the preferential solvation coefficients on component volumes brings about difficulties in the interpretation of ∆nij and δij (as well as δxij) in terms of a prevailing type of interactions among components, especially in the cases where the effects of the interactions if any are overwhelmed by the ∆V contribution. By proposing to consider the function Gij - Gid ij instead of Gij to calculate ∆nij and δij (∆n′ij and ∆δij, respectively, eq 3 of ref 2 and eq 18 of ref 3), we have suggested a way to get rid of the volume difference dependence and make them more and better informative. The proposed operative procedure does not involve any assumption on interactions meant as potential curves and less than ever on interactions in ideal systems being distinguishable or not. Equation 18 of ref 3 for ∆δij can also be written as ∆δij id E E E ) xixj(Gii - Gij - Gid ii + Gij ) ) δij(G , V , ∆V) - δij (G ) 0, id E V ) 0, ∆V) ) δij - δij ; that is, the procedure subtracts from δij a well-defined quantity that is present both in real and in ideal systems, and that is not a background of interactions. It is instead a contribution to preferential solvation that is useless and also misleading particularly for mixtures with small deviations from ideality, because it is due to a physical property, the volume of the components, and not to the interactions among molecules brought about by the polarity of the functional groups. By examining the values of ∆δij in Table 1, we can realize how they are much friendlier than δij from the interpretation point of view. First, the dependence on ∆V is almost absent. Second, ∆δAB and ∆δAA have opposite signs; this means that if A is preferentially solvated by A (∆δAA > 0), then B is not preferentially solvated by A, suggesting that like-like attraction interactions prevail, whereas δAB and δAA values > 0 indicate that both A and B are preferentially solvated by A. For system 7, the deviations from ideality are large enough to overwhelm the volume contribution, so that δAB and δAA have opposite signs in a large x-range. However, by examining the behavior of ∆δAB
Comments as a function of x, we find that its sign is always negative in the whole x-range, contrary to the δAB behavior described in the previous section, suggesting that at xA ) 0.4 there is no change of preferential solvation, a result that is not in contrast with the other thermodynamic properties. 2. Meaning and Validity of Eq 2 (Eq 4 of Ref 2). The excess (or deficit) of moles of j in the neighborhood of i, ∆nji, can be considered as being located in a volume surrounding the central molecule i called the correlation volume, Vcor. In Vcor, the composition may be different from the bulk composition, and ∆nji gives the values of the difference between nj in Vcor and nj in a volume of bulk, Vbk, equal to Vcor. These nj molecules occupy a volume V* given by njV h j, and with respect to Vbk, they replace a number of molecules of i that occupy the same volume V*, that is, niV h i. Mathematically, this can be expressed as ∆niiV h i ) -∆njiV h j, or ∆niiV h i + ∆njiV h j ) 0.8 However, by checking this equation with whatever actual mixture, it is found that it is very far from being satisfied. The examples in Table 1 clearly show this; the calculated values of ∆nAAV h A + ∆nBAV h B, reported in column 13 (β), are undoubtedly different from zero. On the other hand, if the proposed function Gij - Gid ij is used, the above equation becomes ∆n′iiV h i + ∆n′jiV h j ) 0 (that is, eq 4 h A + ∆n′BAV of ref 2); to check it, the values of ∆n′AAV h B have been calculated for the systems in Table 1 and collected in the second-to-last column. The very small deviations from zero, all except one being smaller than 0.1 cm3, are proof of the validity of eq 4 of ref 2. Moreover, its general exact validity can be easily demonstrated by substituting in eq 4 the equation for ∆n′ij (eq 3 of ref 2) with Gid ij given by eqs 8-10 of ref 3 and performing simple algebraic rearrangements with the usual assumptions κET ≈ 0 and VE ≈ 0. See also ref 4 for a thorough discussion of the general validity of this equation. 3. The Gas Mixture with Small Deviation from Ideality. In section 3 of ref 1, the author calculates the preferential solvation coefficients with the contested procedure for a gas mixture slightly deviating from ideal behavior (eq 3.18). In the
J. Phys. Chem. B, Vol. 111, No. 11, 2007 3071 derivation, eq 3.11 is not an exact equality but contains an approximation. This implies that eqs 3.12-3.16 are not exact but carry the same approximation contained in eq 3.11. Indicating with R the quantity neglected in the approximation, the exact eqs 3.14-3.16 are the following: ∆GAA ) -R, ∆GBB ) -R, and ∆GAB ) -1/2∆AB-R. This invalidates the critique of the author regarding the meaning of the corrected Gij integrals. As a further assessment of the validity of the critique, it may be interesting to check if the actual mixtures here considered obey eq 3.18. By comparing the values in column 11 (∆δAA) with the corresponding ones in the last column (1/2xAxB∆AB), we see that in no case eq 3.18 is satisfied. By recalling that ∆δBB ) -∆δAB, it can be seen that the same result is found for ∆δBB. The author’s critique based on the validity of eq 3.18 is therefore not applicable to actual systems. Conclusions The main points of criticism regarding the use of the function Gij - Gid ij to calculate preferential solvation coefficients and local compositions are either irrelevant or unfounded. The local quantities that are obtained using this procedure are more informative on solute-solute-solvent interactions than those obtained from Gij. The contested approach is therefore fully justified and can still be profitably used. References and Notes (1) Ben-Nam, A. J. Phys. Chem. B 2007, 111, 2896. (2) Matteoli, E.; Lepori, L. J. Chem. Soc., Faraday Trans. 1995, 91, 431-436. (3) Matteoli, E. J. Phys. Chem. B 1997, 101, 9800-9810. (4) Shulgin, I.; Ruckenstein, E. J. Phys. Chem. B 1999, 103, 24962503. (5) Shulgin, I.; Ruckenstein, E. J. Phys. Chem. B 2006, 110, 1270712713. (6) Marcus, Y. Monatsh. Chem. 2001, 132, 1387-1411. (7) Matteoli, E.; Lepori, L. J. Chem. Phys. 1984, 80, 2856-2863. (8) This is for a binary mixture. For a k-component system, this k equation reads ∑j)1 ∆n′jiV h j ) 0.
