Kirkwood-Buff theory

The Kirkwood-Buff theory of solutions (Kirkwook and Buff, 1951) doesn t depend on special assumptions about the nature of the intermolecular interactions 

We obtain the KB relations by considering the change in the interaction contribution to the chemical potential of species a upon increasing the density of species y with temperature fixed  

We consider a fluid, and take the density change to be uniform in the system volume 8py(r) = Apy. Noting also Eq. (6.54), this leads to 

10 Consider the case of a spherical solute (A) dissolved at infinite dilution in a molecular solvent (S), and show that the partial molar volume - see Eq. (4.100), p. 97 - can be expressed as 

The Kirkwood-Buff theory of solutions was originally formulated to obtain thermodynamic quantities from molecular distribution functions. This formulation is useful whenever distribution functions are available either by analytical calculations or from computer simulations. The inversion procedure of the same theory reverses the role of the thermodynamic and molecular quantities—i.e., it allows the evaluation of integrals over the pair correlation functions from thermodynamic quantities. These integrals Gy, referred to as the Kirkwood-Buff integrals, were found useful in characterizing mixtures 

Having information on G,yS one can compute the thermodynamic quantities. However, since the quantities Gy are not available from experiment, the theory could have been used only in rare cases where Gy were obtained from theoretical work. In principle, having an approximate theory for computing the various pair correlation functions it is possible to evaluate the integrals Gy and then compute the thermodynamic quantities through the KB theory. Comparison between the thermodynamic quantities thus obtained and the corresponding experimental data could serve as a test of the theory that provides the pair correlation functions. 

In this form the thermodynamic quantities are used as input to compute the molecular quantities Gy. Since it is relatively easier to measure the required thermodynamic quantities, the inversion procedure provides a new tool to investigate the characteristics of the local environments of each species in a multicomponent system. 

A brief outline of the inversion procedure is given below. We consider a two component system, say of water (W) and ethanol (E). The isothermal compressibility Kt, the partial molar volumes Ve, and the derivatives of the chemical potentials are given in Eqs. (6.7.38)-(6.7.41). These are 

Thus in essence we have three independent equations relating the thermodynamic quantities to the three quantities Gfvw, Gee, and Gew= Gwe (for any given temperature and composition of the system). An inversion procedure is possible in which the three GijS may be computed from the above-mentioned thermodynamic quantities. 

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From the point of view of molecular theory, the coefficients dpu jdp2 are fundamentally related to structural properties of these fluids - OZ direct correlation functions (Eq. (6.71), p. 141) - as is discussed in detail subsequently in Section 6.3 on the Kirkwood-Buff theory. Alternatively, these coefficients could be explicitly evaluated if an explicit statistical thermodynamic model, as in the discussion here, were available for the unmixed fluids. Finally, these comments indicate that much of the iirformation supplied by these coefficients is susceptible to measurement... 

The Kirkwood-Buff theory of solutions and the local composition of liquid mixtures. 

Chapter 1 is devoted to the application of the Kirkwood-Buff theory of solutions to the investigation of the structures of binary and multicomponent mixtures. The analysis involves the quantity, which represents the excess (or deficit) number of molecules of species i around a central molecule of species j compared with a hypothetical mixture in which molecules of species i are distributed randomly around a central molecule of species j. 

Many models are available for describing the thermodynamic behavior of solutions. " However, so far no one could satisfactorily simulate the solution behavior over the whole concentration range and provide the correct pressure and temperature dependencies. This generated interest in the thermodynamically rigorous theories of Kirkwood—Buff and McMillan—Mayer. In the present paper, the emphasis is on the application of the Kirkwood—Buff theory to the aqueous solutions of alcohols, because it is the only one which can describe the thermodynamic properties of a solution over the entire concentration range. The key quantities in the Kirkwood-Buff theory of solution are the so-called Kirkwood-Buff integrals (KBIs) defined as... 

The aqueous systems of methanol, ethanol, propanols, and butanols were examined in the framework of the Kirkwood-Buff theory of solution. The Kirkwood—Buff integrals were calculated using thermodynamic equations, in which the derivatives (9 nyi/dxi)pj were expressed in terms of (9 nP/dXi)r, which... 

The Kirkwood—Buff theory of solution was used to investigate the formation of clusters in aqueous alcohol solutions. The correlation volume (volume in which the composition differs from the bulk one) was calculated for the systems 1-propanol—water and fert-butyl alcohol—water and compared with the sizes of clusters determined by various physical techniques. The calculations indicated that two types of clusters, alcohol- and water-rich clusters, are present in the solutions. Their sizes, which depend on composition in a similar way, exhibit maxima in the water-rich region. The calculated values are in a satisfactory agreement with experiment. The composition inside the clusters (the local composition) was calculated as a function of the correlation volume for dilute aqueous methanol, ethanol, propanols, and terf-butyl alcohol solutions. The results were compared with the local compositions provided by the Wilson and NRTL equations. 

The clustering in aqueous solutions of alcohols was examined by combining the Kirkwood—Buff theory of solution with the Wilson and the NRTL equations. The correlation volumes were calculated for the aqueous systems of 1-PrOH and f-BuOH. Two type of clusters, alcohol- and water-rich, were found with similar dependencies of size on composition. Satisfactory agreement was found between the calculated cluster sizes and those provided by the SAXS, SANS, and LS experiments. 

The Kirkwood—Buff theory of solutions was applied to the systems aromatic fluorocarbon/aromatic hydrocarbon. The Kirkwood-Buff integrals and the excess (or deficit) number of molecules around a central one were calculated for five systems (hexafluorobenzene—benzene, hexafluorobenzene—toluene, hexafluorobenzene—cyclohexane, benzene—toluene, benzene-cyclohexane). 

The Kirkwood—Buff Theory of Solutions and the Local Composition of Liquid Mixtures... 

The present paper is devoted to the local composition of liquid mixtures calculated in the framework of the Kirkwood—Buff theory of solutions. A new method is suggested to calculate the excess (or deficit) number of various molecules around a selected (central) molecule in binary and multicomponent liquid mixtures in terms of measurable macroscopic thermodynamic quantities, such as the derivatives of the chemical potentials with respect to concentrations, the isothermal compressibility, and the partial molar volumes. This method accounts for an inaccessible volume due to the presence of a central molecule and is applied to binary and ternary mixtures. For the ideal binary mixture it is shown that because of the difference in the volumes of the pure components there is an excess (or deficit) number of different molecules around a central molecule. The excess (or deficit) becomes zero when the components of the ideal binary mixture have the same volume. The new method is also applied to methanol + water and 2-propanol -I- water mixtures. In the case of the 2-propanol + water mixture, the new method, in contrast to the other ones, indicates that clusters dominated by 2-propanol disappear at high alcohol mole fractions, in agreement with experimental observations. Finally, it is shown that the application of the new procedure to the ternary mixture water/protein/cosolvent at infinite dilution of the protein led to almost the same results as the methods involving a reference state. 

On the basis of the Kirkwood—Buff theory of solution, " one can show that the excess (or deficit) number of molecules i (i = 1, 2) around a central molecule j (j = 1, 2), A y, in a binary liquid mixture can be obtained by using the expression "... 

Shulgin, I. L. Ruckenstein, E. The Kirkwood-Buff theory of solutions and the local composition of liquid mixtures. J. Phys. Chem. B 2006, no, 12707-12713. 

The dilute supercritical mixtures were examined in the framework of the Kirkwood—Buff theory of solutions. Various expressions were employed for the excess number of aggregated molecules of solvent around individual solute molecules to conclude that at infinite dilution the above mentioned excess is zero. This suggested that the density enhancement observed when small amounts of a solute were added to a solvent near the critical point of the latter may not be caused by the aggregation of the solvent molecules around individual solute molecules as usually considered. Further, comparing experimental results, it was shown that the density enhancement caused by the near critical fluctuations in a pure solvent are almost the same, in a wide range of pressures, as those in dilute supercritical mixtures near the critical point of the solvent. 

In this paper, some recent experimental results regarding the density fluctuations in pure SCF are used to show that the local density enhancement in dilute SCR mixtures is mainly due to the near critical fluctuations in the solvent and an explanation is suggested for the negative partial molar volnme of the solute. This conclusion was also strengthened by a discussion, presented in the following section, based on the Kirkwood—Buff (KB) theory of solution. First, the problem will be examined in the framework of the Kirkwood—Buff theory of solution. Second, nsing experimental results about the near critical fluctuations in pure SCF, it will be shown that the density enhancement in dilnte SCR mixtures is mainly caused by the near critical density fluctuations in pure SCF. 

The Kirkwood—Buff Integrals. The Kirkwood—Buff theory of solution relates the so-called Kirkwood—Buff integral (defined below) to macroscopic quantities, such as the compressibility, partial molar volumes, and the composition derivative of the activity coefficient. 

Consequently, in the first and third cases the systems behave in the same way, while in the second case the system behaves differently. It is appropriate to note that in the first case IC22 is small in absolute value, in the third case it is positive and large, while in the second case it is large in absolute value but negative. Obviously, the quantity/ 22 is a measure of the nonideality of the system. In the first case, the system behaves almost as an ideal one in the second case it behaves like a nonideal system with a strong positive deviation from ideality and in the third case like a nonideal system with a strong negative deviation from ideality. In the framework of the Kirkwood-Buff theory of solution the... 

Ben-Naim, A. Inversion of the Kirkwood—Buff theory of solutions apphcation to the water—ethanol system. J. Chem. Phys. 1977, 67, 4884-4890. 

The solubilities of gases in binary, ternary or more complex multicomponent solvents are good examples in which the Kirkwood-Buff theory of solutions provides exceUent results that cannot be obtained using the methods of traditional thermodynamics. Thermodynamics cannot provide explicit pressure, temperature, and composition dependence of the thermodynamic functions, such as the activity coefficients of the components. Therefore, various assumptions regarding the activity coefficients must be made. In contrast, the Kirkwood-Buff theory of solution allows one to establish, in some cases, relations between multicomponent... 

The present authors employed the Kirkwood—Buff theory of solution to obtain expressions for the derivatives of the activity coefficients in a ternary mixture with respect to the mole fractions and applied them to ternary mixtures when the composition(s) of one (or two) component(s) was (were) small. That approach will be used here to derive new expressions that can predict the Henry s constant in a binary solvent mixture in terms of binary data. 

Expressions for the Derivatives of the Activity Coefficients in a Ternary Mixture with Respect to Mole Fractions through the Kirkwood—Buff Theory of Solution. For the present purpose, the following two derivatives, obtained in a previous pa-per,ii are useful... 

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