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J. Phys. Chem. B 2008, 112, 5876-5877
Reply to “Comment on ‘The Kirkwood-Buff Theory of Solutions and the Local Composition of Liquid Mixtures’” Ivan L. Shulgin† and Eli Ruckenstein* Department of Chemical & Biological Engineering, State UniVersity of New York at Buffalo, Amherst, New York 14260 ReceiVed: October 8, 2007; In Final Form: December 29, 2007 The Kirkwood-Buff (KB) theory of solutions1 relates the local properties of solutions, expressed through the KB integrals, to macroscopic thermodynamic quantities. An important application of this theory is to the excess (or deficit) number of molecules of any type around a central molecule. The calculation of this excess (or deficit) is the matter of our disagreement with the preceding Ben-Naim comment.2 First, let us note that the Ben-Naim assertion that we “sought a ‘correction’ to the interpretation of the Kirkwood-Buff integrals” is not accurate. The latter theory does not require any correction. We do not disagree with Kirkwood and Buff, but we disagree with Ben-Naim, namely with the expression he has employed for calculation of the excess (or deficit). According to the numerous publications of Ben-Naim (as cited in our paper3), in a binary mixture of components R and β, the average excess (or deficit) (∆NRβ) of the number of R molecules [for the sake of simplicity only a binary mixture is considered] around a central β molecule is provided by the expression
∆NRβ ) cRGRβ
(1)
where cR is the bulk molar concentration of component R and GRβ is the KB integral. It should be emphasized that ∆NRβ was considered by BenNaim to represent the difference between the number of molecules of a given species around a central molecule and the number of molecules of the same species at the bulk concentration in the same volume surrounding the central molecule. Indeed, in his 1977 paper,4 he writes “Clearly, cRGRβ reflects the total average excess (or deficiency) of R molecules in the entire surrounding of a β molecule”. We recently3 demonstrated that expression (1) for the excess (or deficit) as defined above is not correct, and we provided a correct expression that involves the KB integral. Ben-Naim in his comment2 disagrees. As a reply to his comment, we present again, in a very simple manner, our arguments. Let us consider a binary mixture R-β and a central molecule β. The excess (or deficit) of R molecules in a sphere of radius R around a central molecule β (see Figure 1) is obviously provided by the difference between nRβ, the average number of R molecules around a central molecule β in the volume between bulk Rβ and R, and nRβ , the number of R molecules at the bulk concentration in the same volume for which nRβ was calculated. In Figure 1, Rβ is the radius of a volume that is not accessible to R molecules because of the presence of the central molecule * To whom correspondence should be addressed. E-mail: feaeliru@ acsu.buffalo.edu. Fax: (716) 645-3822. Phone: (716) 645-2911/ext. 2214. † E-mail:
[email protected].
Figure 1. Illustration to the calculation of excess (or deficit) molecules R around a central molecule β.
β (for the sake of simplicity the shapes of the molecules are considered spherical but our considerations are valid for molecules of any shape). Let us now express the above considerations in more details. The number of molecules nRβ is given by5-6
nRβ ) cR
∫0R gRβ4πr2 dr
(2)
where gRβ is the radial distribution function between species R and β. bulk nRβ is obviously provided by bulk nRβ ) cR
∫RR 4πr2 dr β
(3)
bulk Because5-6 gRβ ) 0 for r eRβ, nRβ and nRβ are calculated for the same volume between the radii Rβ and R. As it will be shown later, eq 1 of Ben-Naim can be obtained if one takes Rβ ) 0 in eq 3. Thus, Ben-Naim’s expression (1) provides the difference between the number of R molecules around a central β molecule and the number of R molecules at the concentration of the bulk in a sphere of radius R that includes the volume of radius Rβ. In other words, in his approach, the volumes involved in the two terms are not the same. It is clear that the physically meaningful and relevant excess is that which involves the same volume surrounding the central molecule. As already noted, BenNaim intended to calculate the excess as defined after eq 1, but obtained instead an “excess” corresponding to Rβ ) 0 in eq 3. The basic equations of the two approaches can be derived starting from eqs 2 and 3. The Ben-Naim’s “excess” of R molecules in a sphere of radius R around a central molecule β is obtained by taking Rβ ) 0 in eq 3
(R) ) cR ∆N Rβ
∫0R gRβ4πr2 dr - cR ∫0R 4πr2 dr
(4)
which can be rewritten as (R) ) cR ∆N Rβ
∫0R (gRβ - 1)4πr2 dr
(5)
Because for sufficiently large R, gRβ ) 1, eq 5 leads to his eq 1
∆NRβ ) cRGRβ where GRβ ) ∫∞0 (gRβ - 1)4πr2 dr is the KB integral.
10.1021/jp7098224 CCC: $40.75 © 2008 American Chemical Society Published on Web 04/12/2008
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Comments
J. Phys. Chem. B, Vol. 112, No. 18, 2008 5877
In our approach, the excess of R molecules in a sphere of radius R around a central molecule β is (R) bulk ∆nRβ ) nRβ - nRβ )
cR
∫0R gRβ 4πr2 dr - cR ∫RR 4πr2 dr ) R R cR ∫0 (gRβ - 1)4πr2 dr + cR ∫0 4πr2 dr β
β
(6)
For sufficiently large values of R, gRβ ) 1, and eq 6 becomes
∆nRβ ) cRGRβ + cR 4πRβ3/3 ) cRGRβ + cRV β
(7)
where Vβ is the volume inaccessible to R molecules because of the presence of the central molecule β. Comparing eqs 1 and 7, it is clear that in the Ben-Naim treatment the volume Vβ is ignored. The difference between the two quantities ∆nRβ and ∆NRβ is particularly important for large central molecules such as the biomolecules. Let us consider a very instructive example:3 a protein molecule, ribonuclease A (denoted 2), at infinite dilution in a binary mixture of water (denoted 1) and glycerol (denoted 3). Is the ribonuclease A molecule hydrated by water or solvated by glycerol? We calculated3 the excesses using both our approach and Ben-Naim’s eq 1. The excess calculated with our expression (for a glycerol volume fraction of 30%) is positive for water and negative for glycerol (∆n12 = 33.3 mol of water per mole of protein and ∆n32 = -8.2 mol of glycerol/mol of protein). With the Ben-Naim expression both are negative (∆N12 = -341.8 mol of water/mol of protein and ∆N32 = -48.1 mol of glycerol/mol of protein). Therefore, the use of Ben-Naim’s eq 1 provides large deficits for both water and glycerol in the vicinity of ribonuclease A. In contrast, many experiments7-9 confirmed that in this mixture water is in excess and glycerol is in deficit in the vicinity of the protein molecule. It is clear why the Ben-Naim excesses are negative: eq 1 ignores the term cRVβ which is comparable in magnitude in the considered example with cRGRβ (β ) 2 and R ) 1,3); when this term is ignored, the number of molecules subtracted (cR ∫R0 4πr2 dr)
from the number of molecules nRβ around a central molecule β becomes too large (see eq 4). There are additional erroneous comments in the Ben-Naim paper that are less relevant. Let us mention one of them. He is not accurate in attributing to us “the claim that a negative KBI is not plausible”. We have not made such an assertion. On the contrary, we provided3 several examples of real mixtures for which all the KBIs are negative. We use such examples because they show that Ben-Naim’s eq 1 can provide negative values for all of the excesses around any central molecule, and we said that this is not plausible. Unfortunately, Ben-Naim’s eq 1 has been used in many papers published over the last three decades. All of them considered erroneously that it provides an excess with respect to the bulk concentration in the surrounding of a central molecule. It should be emphasized that Matteoli and Lepori10 and Matteoli11 were the first to criticize the Ben-Naim excess by introducing a reference state. For a detailed comparison between their approach and ours, one can see ref 3. To close our reply, we emphasize the contradictions between the first and the last parts of Ben-Naim’s comments, as well as the contradictions between most of his published papers concerning this topic (some cited in our paper3) and the present comment. In his previously published papers, he defined the excess as noted after eq 1, but derived an incorrect result: eq 1; in the present paper he interpreted the quantity cRGRβ in a correct manner, but failed to mention that cRGRβ is not the excess defined after eq 1. References and Notes (1) Kirkwood, J. G.; Buff, F. P. J. Chem. Phys. 1951, 19, 774. (2) Ben-Naim, A. J. Phys. Chem. B 2008, 112, 5874. (3) Shulgin, I. L.; Ruckenstein, E. J. Phys. Chem. B 2006, 110, 12707. (4) Ben-Naim, A. J. Chem. Phys. 1977, 67, 4884. (5) Hill, T. L. Statistical Mechanics: Principles and Selected Applications; McGraw-Hill: New York, 1956. (6) Ben-Naim, A. Statistical Thermodynamics for Chemists and Biochemists; Plenum Press: New York, 1992. (7) Gekko, K.; Timasheff, S. N. Biochemistry 1981, 20, 4667. (8) Timasheff, S. N. AdV. Protein Chem. 1998, 51, 355. (9) Courtenay, E. S.; Capp, M. W.; Anderson, C. F.; Record, J. M. T. Biochemistry 2000, 39, 4455. (10) Matteoli, E.; Lepori, L. J. Chem. Soc. Faraday Trans. 1995, 91, 431. (11) Matteoli, E. J. Phys. Chem. B 1997, 101, 9800.
