Kirkwood-Buff equation

The partial molar volume, which is a very important quantity to probe the response of the free energy (or stability) of protein to pressure, including the so-called pressure denaturation, is not a canonical thermodynamic quantity for the (V, T) ensemble, since volume is an independent thermodynamic variable of the ensemble. The partial molar volume of protein at infinite dilution can be calculated from the Kirkwood-Buff equation [20] generalized to the site-site representation of liquid and solutions [21,22],... 

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. 

Let us apply eqns (13), (14) and (15) to a real system and compare the results. Fig. 2 provides such a comparison for the binary system isopropanol (l)-water (2) (The Kirkwood-Buff integrals were taken from literature and the van der Waals volumes were calculated as suggested in ref. 39-41). Fig. 2 shows that the excesses (deficits) calculated using all three equations (eqns (13), (14) and (15)) provide quite comparable results for both central isopropanol and water molecules. The differences between the excesses (deficits) calculated with eqns (13) and (14) are small. 

The objective of this paper is to propose a predictive method for the estimation of the change in the solubility of a solid in a supercritical solvent when another solute (entrainer) or a cosolvent is added to the system. To achieve this goal, the solubility equations were coupled with the Kirkwood-Buff (KB) theory of dilute ternary solutions. In this manner, the solubility of a solid in a supercritical fluid (SCF) in the presence of an entrainer or a cosolvent could be expressed in terms of only binary data. The obtained predictive method was applied to six ternary SCF-solute-cosolute and two SCF-solute-cosolvent systems. In the former case, the agreement with experiment was very good, whereas in the latter, the agreement was only satisfactory, because the data were not for the very dilute systems for which the present approach is valid. 2001 Elsevier Science B.V. All rights reserved. 

Recently, a method [5] for the prediction of the solubility of a solute in a SC fluid in the presence of an entrainer has been proposed. The method, based on the Kirkwood-Buff (KB) formalism, was however developed for cases in which the entrainer was in dilute amounts. The present paper is focused on the solubility of a solid in a non-dilute mixture of a SC fluid and an entrainer. The theoretical treatment, which is more complex than for the dilute case, is also based on the KB formalism. In this paper the following aspects will be addressed (1) general equations for the solubility in binary and ternary mixtures will be written for the cases involving a small amount of solute (2) the KB formalism will be used to obtain expressions for the derivatives of the fugacity coefficients in a ternary mixture with respect to mole fractions (3) these expressions will be employed to derive an equation for the solubility of a solute in a SC fluid containing an entrainer at any concentration (4) a predictive method for this solubility will be proposed in terms of the solubilities of the solute in the SC fluid and in the entrainer (5) the derived equation will be compared with experimental results from literature regarding the solubility of a solute in a mixture of two SC fluids. 

A similar effect was observed for the solubility of a solid in a gas mixture composed of two SC fluids. In the latter case, experiments " have revealed that the solubility has a value intermediate between those recorded in the individual SC fluids. Thus, the addition of SC ethane to SC carbon dioxide enhanced the solute solubility compared to that found in pure CO2 but decreased it relative to that in pure C2H6. King and co-workers ii also investigated the effect of the addition of helium on the solubilities of cholesterol and soybean oil in SC CO2 and found that the addition reduces them dramatically. To date, a theory that can predict the effect of a gaseous entrainer on the solute solubility has not yet been developed. The aim of the present research was to derive an equation able to predict the solubility of a solid in a SC fluid + entrainer mixture, when the entrainer is another SC fluid or an inert gas. For this purpose, the Kirkwood—Buff formalism for ternary mixtures was used. In previous papers,the Kirkwood-Buff formalism for ternary mixtures was utilized to describe the entrainer effect however, those methods... 

The Kirkwood—Buff formalism was used to derive an expression for the composition dependence of the Henry s constant in a binary solvent. A binary mixed solvent can be considered as composed of two solvents, or one solvent and a solute, such as a salt, polymer, or protein. The following simple expression for the Henry s constant in a binary solvent (H2t) was obtained when the binary solvent was assumed ideal In = [In f2,i(ln V — In V ) + In i 2,3(ln Vj — In V)]/ (In — In V ). In this expression, i 2,i and i 2,3 are the Henry s constants for the pure single solvents 1 and 3, respectively V is the molar volume of the ideal binary solvent 1—3 and and Vs are the molar volumes of the pure individual solvents 1 and 3. The comparison with experimental data for aqueous binary solvents demonstrated that the derived expression provides the best predictions among the known equations. Even though the aqueous solvents are nonideal, their degree of nonideality is much smaller than those of the solute gas in each of the constituents. For this reason, the ideality assumption for the binary solvent constitutes a most reasonable approximation even for nonideal mixtures. 

In this paper, the Kirkwood—Buff formalism was used to relate the Henry s constant for a binary solvent mixture to the binary data and the composition of the solvent. A general equation describing the above dependence was obtained, which can be solved (analytically or numerically) if the composition dependence of the molar volume and the activity coefficients in the gas-free mixed solvent are known. A simple expression was obtained when the mixture of solvents was considered to be ideal. In this case, the Henr/s constant for a binary solvent mixture could be expressed in terms of the Henry s constants for the individual solvents and the molar volumes of the individual solvents. The agreement with experiment for aqueous solvents is better than that provided by any other expression available, including an empirical one involving three adjustable parameters. Even though the aqueous solvents considered are nonideal, their degrees of nonideality are much lower than those of the solute gas in each of the constituent solvents. For this reason, the assumption that the binary solvent behaves as an ideal mixture constitutes a reasonable approximation. 

The paper is organized as follows first, the Kirkwood—Buff formalism will be used to derive general expressions for the derivatives of the activity coefficients in ternary mixtures with respect to the mole fractions. Then, the obtained expressions will be applied to the gas solubility in dilute and concentrated salt solutions. Numerical calculation will be carried out for several mixtures, particularly for those for which the Sechenov equation failed to provide an accurate correlation. Finally, a criterion will be proposed for the a priori prediction of the kind of salting (salting-in or salting-out). 

The Kirkwood-Buff theory of solutions for ternary mixtures was used to analyze the gas solubility in a mixed binary solvent composed of a high molecular weight and a low molecular weight cosolvent, such as the aqueous solutions of water soluble polymers. A rigorous expression for the composition derivatives of the gas activity coefficient in ternary solution was used to derive the composition dependence of the Henry constant under isobaric and isothermal conditions. The obtained expressions as well as the well-known Kri-chevsky equation were tested for the solubilities of Ar, CH4, C2H6 and C3H8 in the aqueous solutions of PPG-... 

The present paper is concerned with mixtures composed of a highly nonideal solute and a multicomponent ideal solvent. A model-free methodology, based on the Kirkwood—Buff (KB) theory of solutions, was employed. The quaternary mixture was considered as an example, and the full set of expressions for the derivatives of the chemical potentials with respect to the number of particles, the partial molar volumes, and the isothermal compressibility were derived on the basis of the KB theory of solutions. Further, the expressions for the derivatives of the activity coefficients were applied to quaternary mixtures composed of a solute and an ideal ternary solvent. It was shown that the activity coefBcient of a solute at infinite dilution in an ideal ternary solvent can be predicted in terms of the activity coefBcients of the solute at infinite dilution in subsystems (solute + the individual three solvents, or solute + two binaries among the solvent species). The methodology could be extended to a system formed of a solute + a multicomponent ideal mixed solvent. The obtained equations were used to predict the gas solubilities and the solubilities of crystalline nonelectrolytes in multicomponent ideal mixed solvents. Good agreement between the predicted and experimental solubilities was obtained. 

The present paper is devoted to the extension of the theory developed by the authors for the solubility of proteins to the solubility of gases. Because this theory is based on the Kirkwood-Buff fluctuation theory of solutions, the next section summarizes the expressions which are involved. This is followed by a summary of the derivation of an equation for the solubility of proteins and finally its extension to the solubility of gases. 

The Kirkwood-Buff formalism can be also used to derive the composition dependence of the Henry constant for a sparingly soluble gas dissolved in a mixed solvent containing water-r electrolyte [27]. The obtained equation requires information about the molar volume and the mean activity coefficient of the electrolyte in the binary (water-H electrolyte) mixture. Several expressions for the mean activity coefficient of the electrolyte were tested and it was concluded that the accuracy in... 

The main difficulty in predicting the solid solubility in a mixed solvent consists in calculating the activity coefficient of a solute in a ternary mixture In this paper, the Kirkwood-Buff (KB) theory of solutions (or fluctuation theory) (Kirkwood and Buff, 1951) is employed to analyze the solid (particularly drug) solubility in mixed (mainly aqueous) solvents. The analysis is based on results obtained previously regarding the composition derivatives of the activity coefficients in ternary solutions (Ruckenstein and Shulgin, 2001). These equations were successfully applied to gas solubilities in mixed solvents (Ruckenstein and Shulgin, 2002 Shulgin and Ruckenstein, 2002). 

As in a previous paper [Int. J. Pharm. 258 (2003) 193-201], the Kirkwood-Buff theory of solutions was employed to calculate the solubility of a solid in mixed solvents. Whereas in the former paper the binary solvent was assumed ideal, in the present one it was considered nonideal. A rigorous expression for the activity coefficient of a solute at infinite dilution in a mixed solvent [Int. J. Pharm. 258 (2003) 193-201] was used to obtain an equation for the solubility of a poorly soluble solid in a nonideal mixed solvent in terms of the solubilities of the solute in the individual solvents, the molar volumes of those solvents, and the activity coefficients of the components of the mixed solvent. 

In a previous paper (Ruckenstein and Shulgin, 2003), the Kirkwood-Buff theory of solutions (Kirkwood and Buff, 1951) was employed to obtain an expression for the solubility of a solid (particularly a drug) in binary mixed (mainly aqueous) solvents. A rigorous expression for the composition derivative of the activity coefficient of a solute in a ternary solution (Ruckenstein and Shulgin, 2001) was used to derive an equation for the activity coefficient of the solute at infinite dilution in an ideal binary mixed solvent and further for the solubility of a poorly soluble solid. By considering that the excess volume of the mixed solvent depends on composition, the above equation was modified empirically by including one adjustable parameter. The modified equation was compared with the other three-parameter equations available in the literature to conclude that it provided a better agreement. 

Equations (6)-(9) allows one to derive expressions for the Kirkwood-Buff integrals Gj2 and G23, and the preferential binding parameter F j for various kinds of ternary mixtures. 

The above equations are valid for any n-component system. For ternary mixtures, the Kirkwood-Buff integrals can... 

Because the preferentitil binding partimeter is a meaningful physical quantity, attempts have been made to relate it to a general theory of solutions, such as the Kirkwood-Buff theory of solutions (9). Severed authors reported results in this direction (10-17). The authors of this Comment derived the following equation for Fj (16) ... 

However, Eqs. 3 and 5 are different equations even though they are based on the same definition of the preferential binding parameter and have the same theoretical basis the Kirkwood-Buff theory of solutions. To make a selection between Eqs. 3 and 5 a simple limiting case, the ideal ternary mixture, will be examined using the traditional thermodynamics, and the results will be compared to those provided by Eqs. 3 and 5. 

In this paper, the Kirkwood-Buff theory of solutions is used to examine the effect of PEG on aqueous protein solutions, the focus being on the local composition of the mixed solvent in the vicinity of the protein molecule and on the protein solubility. The theoretical considerations led to equations that coimect the experimental preferential binding parameter with the excess (or deficit) numbers of water and cosolvent molecules around a protein molecule. Calculations were carried out for various proteins in various PEG solutions. The results showed that in all cases the proteins were preferentially hydrated. Evidence was also brought that the hydration is a result of steric exclusion. 

Equations 3 and 5 allow one to calculate the Kirkwood-Buff integrals G12 and G23 using experimental data regarding the preferential binding parameters r2 and the partial molar volume of a protein at infinite dilution in a mixed solvent 49-50,52 jjjg Kirkwood—Buff integrals Gn and Go can be evaluated on the basis of the properties of protein-free mixed solvent water + cosolvent. It should be mentioned that recently the Kirkwood—Buff theory was used to analyze the effects of various cosolvents on the properties of aqueous protein solutions. " ... 

Gap in eqs A2 and A4 is expressed as volume per molecule. These equations are valid for any -component system. Expressions for the Kirkwood—Buff integrals in ternary mixtures can be obtained from eq A4. ° In particular, one can obtain the following expressions for Gu and G23 for an infinitely dilute solute (component 2)... 

Statistical mechanics gives relationships between the distribution functions and the bulk properties of fluids. The total internal energy of a fluid is given by the energy equation, the pressure is given by the virial equation, and the isothermal compressibility is given by the compressibility equation, see e. g.. Ref. 11. Through the Kirkwood-Buff formulas (0,... 

The spectroscopic data may be interpreted using an equation that was derived form Kirkwood-Buff solution theory. The number of solvent molecules in excess of the bulk value, U2 , may be defined in terms of a local volume about the solute Vi2as... 

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