On the Theory of Solute Solubility in Mixed Solvents

Abstract

A series of equations are developed for the study of the effects of cosolvents on the solubility of a solute in mixed solutions where the solute displays a finite solubility. The equations differ depending on the scale used for the solute (and cosolvent) concentrations. The expressions use Kirkwood-Buff integrals to relate the changes in solubility to changes in the local solution composition around the solute, and can be applied to study any type of ternary system including electrolyte cosolvents. The expressions provided here differ from previous approaches due to the use of a semi open ensemble, and the extension to finite solute solubilities.

Introduction

Cosolvents (anything other than the primary solvent) are well known to affect the solubility of solutes in solution. A rigorous theory of the effects of cosolvents is desirable in order to understand the molecular level interactions responsible for the changes in solubility, and therefore improve our ability to manipulate the properties of solutions. Recently, the Kirkwood-Buff (KB) theory of solutions¹ has been used to rationalize the effects of cosolvents on the solubility of sparingly soluble solutes in terms of intermolecular distributions.^2–13 KB theory is an exact theory of solution mixtures and therefore provides a solid foundation for the understanding of such effects.

Previous studies have investigated many aspects and applications of the theory, from small molecules to large biomolecule solutes. The general idea is to use KB fluctuation theory to extract information about the local microscopic distribution of molecules from macroscopic experimental results. The theory has nothing directly to say about the interactions that bring about the microscopic structure; but, of course, it is the intermolecular interactions which are ultimately responsible for them. Apparently, Hall and O’Connell were the first to suggest the use of KB theory to study changes in solubility in ternary systems with a finite solute solubility.^2,14 In particular, KB theory has been applied to study low solute solubility in water by deriving KB based expressions for Henry’s constant.^2,4 These studies typically used closed ensembles, sparingly soluble solutes, with solubilities and cosolvent concentrations expressed using mole fractions. Lee studied finite solubilities using KB theory and closed ensembles.³ However, Hall and more recently Mazo recognized that the most appropriate ensemble for these studies was one where T, P, and the chemical potential of the solute are held constant.^7,14

The approach presented here builds on the ideas of Hall and Mazo.^7,14 We extend the range of applicability to include solutes with finite solubilities in ternary systems of any type. The observed relationships differ from previous approaches primarily due to the use of an open system ensemble in the current study. Futhermore, we investigate in detail the differences observed when solubilities and cosolvent concentrations are defined using different concentration scales, and argue that the molarity scale provides the simplest and clearest picture of the cosolvent effect.

General Theory of Solubility

We will adopt the convention of a primary solvent (1), solute of interest (2), and cosolvent (3) at a fixed temperature (T) and pressure (P), together with the approach previously outlined by Mazo.⁷ The chemical potential of the solute (μ₂) can be expressed by the statistical mechanical relationship,¹⁵

$$\beta {\mu }_2=\beta {\mu }_2^{*}+\text{ln}({\mathrm{\Lambda }}_2^3{\rho }_2)$$ (1)

where Λ is thermal de Broglie wavelength, μ* is the pseudo chemical potential, ρ₂ is the solute number density (n₂/V), and β = 1/RT. The pseudo chemical potential (pcp) was introduced by Ben-Naim and its uses have been discussed in detail.¹⁵ The current use of the pcp, and thereby the molar concentration scale for the solute solubility, represents one of the major differences from previous studies and the consequences of this choice will be discussed later. At saturation, the chemical potential of the solute is the same as the pure solid (or gaseous) solute at the same T and P for any solvent and cosolvent mixture. Hence, if we follow the saturation equilibrium (ρ₂ = ρ₂^sat = S₂) for a particular system then we have,

$$\beta d{\mu }_2^{*}=-d\text{ln}{S}_2$$ (2)

for all solvent and cosolvent solution compositions. To relate changes in the solubility to changes in the cosolvent concentration (ρ₃) it will prove convenient to define the following derivative,

$$K_{23}^c=-{\left(\frac{\partial \text{ln}{S}_2}{\partial {\rho }_3}\right)}_{T,P,{\mu }_2}=\beta {\left(\frac{\partial {\mu }_2^{*}}{\partial {\rho }_3}\right)}_{T,P,{\mu }_2}$$ (3)

The superscript c for K₂₃ indicates that the solubility and cosolvent concentration are measured in terms of number densities (molarities). The value of K₂₃^c clearly depends on the solution composition and represents the slope of the saturation (solubility versus cosolvent molarity) curve. In general, if the solute solubility is increased on addition of a cosolvent then the value of K₂₃ will be negative for most compositions, and vice versa. Experimentally, this slope is often determined to be independent of cosolvent concentration. However, we do not make that assumption here. The objective is to determine an expression for K₂₃^c in terms of the distribution of both the cosolvent and solvent molecules around the solute.

Kirkwood-Buff Theory

In order to relate the thermodynamics of the salting in/out process to changes in the distribution of cosolvent and solvent molecules in solution we will use the Kirkwood-Buff theory of solutions.^1,15 KB theory relates integrals over radial distribution functions (g_ij) to properties of the solution. The main components are the KB integrals between the different species,

$$G_{ij}=G_{ji}=4\pi {\displaystyle {\int }_0^{\mathrm{\infty }}[g_{ij}^{\mu VT}(r)-1]r^{2}dr}$$ (4)

and the number densities or molar concentrations of each species (ρ_i = n_i/V). An excess coordination number can be defined (N_ij = ρ_jG_ij ≠ N_ji) which characterizes the excess number of j molecules around a central i molecule in an open system above that observed within an equivalent volume of a bulk solution. It is a measure of how the addition of a single i molecule affects the distribution of i and j molecules around it compared to the bulk reference distribution. Generally, a positive value of N_ij indicates an increase in the local density of j around i above that of their bulk ratios. This can be viewed as the result of some favorable net interaction or affinity between the two species. However, fluctuation theory is mute about specific details of the nature of these interactions.

Combinations of KB integrals can be used to provide information on solution properties in either open or closed systems. Before doing so it will be useful to define several expressions which appear repeatedly in the following analysis. In particular, the previous notation introduced by Smith for binary and ternary solutions will be used.^16,17 We define,

$$a_{33}({\rho }_2)=\frac{1}{1+{\rho }_3(G_{33}-G_{13})}$$ (5)

with,

$${\eta }_{13}({\rho }_2)={\rho }_1+{\rho }_3+{\rho }_1{\rho }_3(G_{11}+G_{33}-2G_{13})$$ (6)

and,

$$A_1({\rho }_2)=1+{\rho }_1(G_{11}+G_{23}-G_{12}-G_{13})$$ (7)

Note that we have emphasized that the above expressions depend on the solute concentration even though the expressions themselves do not explicitly contain the solute concentration. There is no requirement that the G_ij values be independent of concentration.

Many interesting cosolvents are salts. KB theory can be applied to salt solutions as long as one does not treat the individual ions as independent thermodynamic variables.¹⁸ There are several approaches to this problem.^19–24 We will use the indistinguishable ion approach in the case that the cosolvent is a salt.²⁵ This involves treating the salt as a collection of ions, without acknowledging the differences between anions and cations. This has particular advantages when analyzing computer simulation data (see the Discussion).²⁵ Therefore, we will distinguish between the traditional molar salt concentration c_s and the total ion number density ρ₃. The following relationships then hold: ρ₃ = n_± c_s, n_±dμ₃ = dμ_s = n₊dμ₊ + n₋dμ₋, and γ₃ = γ_±, etc, where n_± = n₊ + n₋ is the total number of ions produced on dissolving the salt. In our opinion, this is the simplest approach for salts and provides equivalent expressions to other methods.^20,21,26 However, care must be taken when using the mole fraction concentration scale for salts (see Appendix 1).

Kirkwood-Buff Theory of Semi Open Ternary Systems

One can express several properties of semi open ternary solutions using the above definitions. The following expressions were determined using the approach of Smith.¹⁷ The change in cosolvent activity with cosolvent molarity is provided by,

$$\beta {\left(\frac{\partial {\mu }_3}{\partial \text{ln}{\rho }_3}\right)}_{T,P,{\mu }_2}={\left(\frac{\partial \text{ln}{a}_3}{\partial \text{ln}{\rho }_3}\right)}_{T,P,{\mu }_2}=a_{33}({\rho }_2)$$ (8)

where a₃ is the cosolvent activity. Alternatively, using the molality and mole fraction concentration scales one finds,

$$\beta {\left(\frac{\partial {\mu }_3}{\partial \text{ln}{m}_3}\right)}_{T,P,{\mu }_2}=\frac{{\rho }_1}{{\eta }_{13}({\rho }_2)}$$ (9)

and,

$$\beta {\left(\frac{\partial {\mu }_3}{\partial \text{ln}{x}_3}\right)}_{T,P,{\mu }_2}=\frac{{\rho }_1}{{\eta }_{13}({\rho }_2)}\frac{1}{1-x_3-x_2{\rho }_3{A}_1({\rho }_2)/{\eta }_{13}({\rho }_2)}$$ (10)

respectively. The last equation was obtained from Equation 9 using an expression for the required derivative (∂x₃/∂m₃)_T,P,μ2 which was derived using a similar approach to that in Appendix 1, together with previously existing expressions.¹⁷ All three equations reduce to the expected binary system results when n₂ tends to zero. In the case of the molarity and molality derivatives the expressions themselves do not change, whereas the mole fraction derivative expression simplifies considerably. The additional complexity in Equation 10 arises from the fact that n₂ can vary in a system open to the solute and this directly affects the cosolvent mole fraction, whereas it does not directly affect the cosolvent molarity or molality.

Throughout this paper we will refer to molalities according to m_i = n_i/n₁. Hence, our molalities are dimensionless. They are related to the standard experimental definition of molality by a simple conversion factor of 1000/M₁ or 55.5 mol/kg for water. In real applications this conversion factor has to be included.

Kirkwood-Buff Theory of Closed Binary Solutions

In the limit that the solute concentration approaches zero the above expressions are related to properties of the binary solvent and cosolvent solution. In this case, Equation 8 and Equation 9 adopt the same functional form but are now applicable to a closed binary system of 1 and 3 at constant T and P. Therefore, we can then use several additional established relationships derived for binary systems under constant T and P conditions. For instance, the partial molar volumes (pmv) in the binary solution can be written as,^1,15

$$\overline{V_3}=\frac{1+{\rho }_1(G_{11}-G_{13})}{{\eta }_{13}(0)}\text{and}\overline{V_1}=\frac{1+{\rho }_3(G_{33}-G_{13})}{{\eta }_{13}(0)}=\frac{1}{a_{33}(0){\eta }_{13}(0)}$$ (11)

Alternatively, the cosolvent pmv can be expressed by,^14,17

$$\overline{V_3}=RT{\kappa }_T-{\rho }_1{G}_{31}\overline{V_1}-{\rho }_3{G}_{33}\overline{V_3}$$ (12)

where κ_T is the isothermal compressibility of the solution. The above equation can then be used to eliminate G₁₃ from Equation 5 to generate,²⁷

$$a_{33}(0)=\frac{\varphi }{1+{\rho }_3(G_{33}-RT{\kappa }_T)}$$ (13)

and ${\varphi }_1={\rho }_1\overline{V_1}$ is the volume fraction of solvent. Furthermore, the pmv of an infinitely dilute solute in a cosolvent and solvent mixture is given by,¹⁵

$${\overline{V_2}}^{\mathrm{\infty }}=RT{\kappa }_T-{\rho }_1{G}_{21}\overline{V_1}-{\rho }_3{G}_{23}\overline{V_3}$$ (14)

Finally, we note that,

$$A_1(0)={\rho }_1(G_{23}-G_{21})+{\eta }_{13}(0)\overline{V_3}$$ (15)

All of the above relationships can be used when the solute concentration is low.

General Kirkwood-Buff Theory of Solubility

Having provided a variety of background information we return to Equation 3, which is the starting equation for the present discussion. An expression for the first derivative in Equation 3 in terms of KB integrals has been determined previously for semi open ternary mixtures.¹⁷ Therefore, using this expression (Equation 17 of Reference ¹⁷ with an appropriate change of indices) for the required derivative one finds,

$$K_{23}^c=-\frac{G_{23}-G_{21}}{1+{\rho }_3(G_{33}-G_{31})}=-(G_{23}-G_{21})a_{33}({\rho }_2)$$ (16)

The above expression holds for any concentration of solvent or cosolvent, and also any solubility of the solute. It also holds for electrolyte cosolvents. The G_ij values depend on the particular solute, solvent, and cosolvent composition. All the above KB integrals should be evaluated at saturation of the solute.

Many studies use molality (m) and not molarity for both the solubilities and the cosolvent concentration. In this case one obtains a different expression for K₂₃ given by Equation 20 of Reference ¹⁷ with an appropriate change of indices. The result is,

$$K_{23}^m=-{\rho }_1\frac{1+{\rho }_1(G_{11}+G_{23}-G_{12}-G_{13})}{{\rho }_1+{\rho }_3+{\rho }_1{\rho }_3(G_{11}+G_{33}-2G_{13})}=-{\rho }_1\frac{A_1({\rho }_2)}{{\eta }_{13}({\rho }_2)}$$ (17)

where we have used the definitions provided by Equation 6 and Equation 7. The above equation is essentially that suggested by Hall but derived using a different approach.¹⁴ In addition, concentrations may be measured using mole fractions (x). To derive an expression on this concentration scale we use the following thermodynamic relationship,

$${\left(\frac{\partial x_2}{\partial x_3}\right)}_{T,P,{\mu }_2}=\frac{-x_2+(1-x_2)m_{23}}{1-x_3-x_3{m}_{23}}$$ (18)

where m₂₃ = (∂m₂/∂m₃)_T,P,μ2 and which was derived from the corresponding mole fraction definitions using a similar manner to the approach outlined in Appendix 1. From the above relationship, and the molality derivative used to develop Equation 17, one finds that the correct expression when the solute and cosolvent concentrations are measured using mole fractions is given by,

$$K_{23}^x=-\frac{-1+(1-x_2)\rho A_1({\rho }_2)/{\eta }_{13}({\rho }_2)}{1-x_3-x_2{\rho }_3{A}_1({\rho }_2)/{\eta }_{13}({\rho }_2)}$$ (19)

where ρ = ρ₁ + ρ₂ + ρ₃ is the total number density of the solution. The above expression is applicable to salts. However, care has to be taken with the mole fraction definition using the indistinguishable ion approach. In particular, dx₃ ≠ n_± dx_s and the exact relationship between the two approaches is somewhat complex for ternary systems open to one component (see Appendix 1).

Equation 16, Equation 17, and Equation 19 represent the primary results from the application of KB theory to quantify the effects of cosolvents on the solubility of solutes where all concentrations are finite. The difference between Equation 16 and Equation 17 or Equation 19 arises from two factors. The numerator is different due to the different solubility scales adopted for the solute. The denominators are different due to the change in cosolvent concentration scale and thereby the corresponding derivative. The latter difference disappears as ρ₃ decreases, while the former does not. The reasons for this will be discussed later. We will focus primarily on the molarity based expression in the subsequent sections.

Sparingly Soluble Solutes

The above expressions are valid for any stable ternary mixture with the solute component 2 at saturation. Many solubility studies involve a solute that is sparingly soluble in both the pure solvent and the cosolvent-solvent mixtures. In this case one can take the limit that ρ₂ tends to zero. This does not change the expressions provided in Equation 16 or Equation 17. However, in this limit the previous ensemble (T, P, μ₂) now approaches that of a constant T and P ensemble. Therefore, using Equation 16 one finds that,

$$K_{23}^c=-(G_{23}-G_{21})a_{33}(0)$$ (20)

where the activity derivative a₃₃ is now a property of the binary solution mixture alone. It should be noted that although the above expression is identical in form to Equation 16, the KB integrals themselves will be different as the presence of a solute at finite concentrations can alter (both directly and indirectly) the intermolecular distributions. For K₂₃^c to be constant over all cosolvent concentrations one requires G₂₃ − G₂₁ to be proportional to 1/a₃₃. Whether or not K₂₃^c is constant, the value of a₃₃ is always positive.¹⁴ Hence, the difference in KB integrals (G₂₃ − G₂₁) determines the sign of K₂₃^c and therefore the direction of the salting in/out effect.

In many cases one can replace some of the (often unknown) KB integrals with corresponding properties of the solution which are known or can be easily approximated. Elimination of G₂₁ using Equation 14 provides,²⁷

$$K_{23}^c=-\frac{G_{23}+{\overline{V_2}}^{\mathrm{\infty }}-RT{\kappa }_T}{{\varphi }_1}{a}_{33}(0)$$ (21)

where the infinity superscript indicates the pmv of the solute corresponds to that at infinite dilution in the solution mixture. Additional elimination of G₁₃ via Equation 12 then gives,

$$K_{23}^c=-\frac{G_{23}+{\overline{V_2}}^{\mathrm{\infty }}-RT{\kappa }_T}{1+{\rho }_3(G_{33}-RT{\kappa }_T)}$$ (22)

The above equations are exact for an infinitely dilute solute. In the final expression the value of G₃₃ is a property of the mixture in the absence of solute and is the same for all solutes. The value of G₂₃ quantifies the distribution of cosolvent molecules around an infinitely dilute solute molecule. The latter can be obtained from the slope of the solubility curve if the corresponding properties (pmv of the solute and G₃₃) of the solution mixture are known.

When one expresses the solubility and cosolvent concentrations using molalities the sparingly soluble solute limit provides an identical expression to that of Equation 17, although one can now use the properties of the cosolvent and solvent mixture to provide,

$$K_{23}^m=-{\rho }_1\left[{\rho }_1(G_{23}-G_{21})/{\eta }_{13}(0)+\overline{V_3}\right]$$ (23)

or equivalently,

$$K_{23}^m=-{\rho }_1\left[(G_{23}-G_{21})a_{33}(0){\varphi }_1+\overline{V_3}\right]$$ (24)

Alternatively, the sparingly soluble solute approximation for the mole fraction concentration scale provides,

$$K_{23}^x=-\frac{1}{x_1}\left[-1+\rho \frac{A_1(0)}{{\eta }_{13}(0)}\right]$$ (25)

or using the properties of the solution mixture,

$$K_{23}^x=-\rho \left[\rho (G_{23}-G_{21})a_{33}(0)\overline{V_1}+\overline{V_3}-\overline{V_1}\right]$$ (26)

The last two equations can be used for electrolytes. Again, care has to be taken with the mole fraction definition using the indistinguishable ion approach. In particular, one finds that dx₃ = (x₃/x_s)(1−x₃)/(1−x_s) dx_s ≠ n_± dx_s for a closed binary system (see Appendix 1). This differs from the much simpler molarity and molality based cases.

Electrolyte Cosolvents

The above equations refer to sparingly soluble solutes in any type of cosolvent solution. In the case of electrolytes the KB integrals are related to the individual ion integrals through a series of electroneutrality conditions. For the cosolvent we have,^21,28

$${\rho }_3{G}_{33}={\rho }_3{G}_{+-}-1$$ (27)

and,

$$G_{13}=G_{1+}=G_{1-}$$ (28)

In addition, for the salt distribution around the solute one can write,

$${\rho }_3{G}_{23}={\rho }_{+}{G}_{2+}+{\rho }_{-}{G}_{2-}$$ (29)

which combined with the electroneutrality constraint equation,

$$z_{+}{\rho }_{+}{G}_{2+}+z_{-}{\rho }_{-}{G}_{2-}=0$$ (30)

provides,

$$G_{23}=\frac{1}{{\rho }_3}\left({\rho }_{+}-\frac{z_{+}}{z_{-}}{\rho }_{+}\right)G_{2+}=\left(\frac{n_{+}}{n_{\pm }}-\frac{z_{+}}{z_{-}}\frac{n_{+}}{n_{\pm }}\right)G_{2+}=G_{2+}=G_{2-}$$ (31)

Ben-Naim has pointed out that the above expressions for salts are not due to the ionic nature of the interactions per se, but the fact that the cation and anion concentrations cannot be varied independently.¹⁸ Hence, combining Equation 22, Equation 27, and Equation 31 provides,

$$K_{23}^c=-\frac{G_{2+}+{\overline{V_2}}^{\mathrm{\infty }}-RT{\kappa }_T}{{\rho }_3(G_{+-}-RT{\kappa }_T)}$$ (32)

which holds for any solvent and cosolvent concentration, and where we have focused on the cation as the single independent species.

To relate our expressions to the salting out constants commonly presented in the literature one can integrate Equation 3 using the assumption that K₂₃^c (= K_S^c) is independent of cosolvent concentration (as observed for many salts) to give,

$$\text{ln}\left(\frac{S_2^{\mathrm{o}}}{S_2}\right)=K_S^c{\rho }_3$$ (33)

where S₂^o is the molar solubility of the solute in pure solvent. Traditionally, experimental solubility curves have used the common logarithm for the solute solubility, and often the ionic strength in place of concentration. The ionic strength on the molarity scale is defined by,

$$I_3^c=\frac{1}{2}(n_{+}{z}_{+}^2+n_{-}{z}_{-}^2)c_3=\lambda c_3$$ (34)

where z₊ and z₋ are the charges on the cation and anion, respectively, and c₃ is the cosolvent molarity. Consequently, the use of the indistinguishable ion approach provides,

$$\text{log}\left(\frac{S_2^{\mathrm{o}}}{S_2}\right)=\frac{n_{\pm }}{2.303\lambda }{K}_S^c{I}_3^c$$ (35)

which can be directly compared to the experimentally observed slope.⁷

The interpretation of G₂₃ provided by Equation 32 and Equation 35 depends on the treatment of the salt. For the present indistinguishable ion approach, the value of G₂₃ quantifies the deviation in the ion distribution around the solute. Alternatively, one could use the traditional molarity (c₃) of the salt solution and set n_± = 1 during the analysis. The corresponding value of G₂₃ then quantifies the deviation in the distribution of salt “molecules” around the solute.

Salting Out Constants

Salting out constants are usually defined by the slope of the solubility curve as the solute and cosolvent concentrations tend to zero. Applying the zero cosolvent concentration limit for the case of sparingly soluble solutes we have,

$$K_S^c=-(G_{23}-G_{21})$$ (36)

where it is implied that the KB integrals correspond to the distributions observed at infinite dilution of both solute and cosolvent. Using our relationships for the solute pmv and the properties of salt solutions (Equation 14 or Equation 32) this can be expressed in an equivalent form,

$$K_S^c=-(G_{2+}+{\overline{V_2}}^{\mathrm{\infty }}-RT{\kappa }_T^{\mathrm{o}})$$ (37)

where we have used the fact that ρ₃G₊₋ → 1 as ρ₃ → 0,²⁸ and the zero superscript indicates a property of the pure solvent. Alternatively, using the molality concentration

$$K_S^m=-\left[1+{\rho }_1^{\mathrm{o}}(G_{23}-G_{21}+G_{11}-G_{13})\right]$$ (38)

which can be written,

$$K_S^m=-{\rho }_1^{\mathrm{o}}(G_{2+}+{\overline{V_3}}^{\mathrm{\infty }}+{\overline{V_2}}^{\mathrm{\infty }}-RT{\kappa }_T^{\mathrm{o}})$$ (39)

Equation 38 and Equation 39 were previously derived by Mazo.⁷ Finally, the solubility can also be measured in mole fraction units in which case,

$$K_S^x=-{\rho }_1^{\mathrm{o}}(G_{23}-G_{21}+G_{11}-G_{13})$$ (40)

or,

$$K_S^x=-{\rho }_1^{\mathrm{o}}(G_{2+}+{\overline{V_3}}^{\mathrm{\infty }}-{\overline{V_1}}^{\mathrm{o}}+{\overline{V_2}}^{\mathrm{\infty }}-RT{\kappa }_T^{\mathrm{o}})$$ (41)

Hence, a series of limiting expressions are obtained which depend on the concentration units adopted for measuring the solubility of the solute. We note that Equation 38–Equation 41 involve properties of the cosolvent even at infinite dilution of both the solute and cosolvent.

The Relationship Between Expressions for K₂₃ Using Different Concentration Scales

The expressions for the salting out constants derived using the different solubility scales are clearly similar. A general relationship can be developed for sparingly soluble solutes. Equating the solute chemical potentials using the different concentration scales one can write,

$$\beta {\mu }_2^{\mathrm{o},\mathrm{m}}+\text{ln}{\gamma }_2{m}_2=\beta {\mu }_2^{*}+\text{ln}{\mathrm{\Lambda }}_2^3{\rho }_2=\beta {\mu }_2^{\mathrm{o},\mathrm{x}}+\text{ln}{f}_2{x}_2$$ (42)

for the molality scale, the pcp (or molarity) scale, and the mole fraction scale, respectively. Here, γ₂ and f₂ are the molal and mole fraction activity coefficients and the standard states correspond to the infinitely dilute solute and the pure solute for the molality and mole fraction scales, respectively. Taking derivatives with respect to the cosolvent molality at infinite dilution of the solute one obtains,

$${\left(\frac{\partial [\beta {\mu }_2^{\mathrm{o},\mathrm{m}}+\text{ln}{\gamma }_2]}{\partial m_3}\right)}_{T,P}^{\mathrm{\infty }}=\beta {\left(\frac{\partial {\mu }_2^{*}}{\partial m_3}\right)}_{T,P}^{\mathrm{\infty }}-{\rho }_1\overline{V_3}={\left(\frac{\partial \text{ln}{f}_2}{\partial m_3}\right)}_{T,P}^{\mathrm{\infty }}-x_1$$ (43)

Using the standard thermodynamic relationships of (∂ρ₃/∂m₃)_T,P = ρ₁φ₁ and (∂x₃/∂m₃)_T,P = x₁² for binary mixtures of 1 and 3 provides,

$${\left(\frac{\partial [\beta {\mu }_2^{\mathrm{o},\mathrm{m}}+\text{ln}{\gamma }_2]}{\partial m_3}\right)}_{T,P}^{\mathrm{\infty }}={\rho }_1{\varphi }_1\beta {\left(\frac{\partial {\mu }_2^{*}}{\partial {\rho }_3}\right)}_{T,P}^{\mathrm{\infty }}-{\rho }_1\overline{V_3}=x_1^2{\left(\frac{\partial \text{ln}{f}_2}{\partial x_3}\right)}_{T,P}^{\mathrm{\infty }}-x_1$$ (44)

Therefore, replacing the corresponding derivatives with the appropriate K₂₃, defined in analogous fashion to Equation 3, one finds the final relationship,

$$K_{23}^m={\rho }_1({\varphi }_1{K}_{23}^c-\overline{V_3})=x_1(x_1{K}_{23}^x-1)$$ (45)

which is general for any cosolvent concentration. The above relationship (Equation 45) relates the mole fraction based approach of Shulgin and Ruckenstein,⁶ to the molality based approach of Mazo,⁷ and the pseudo chemical potential approach presented here – all for an infinitely dilute solute. An alternative (more general) derivation is also provided in Appendix 2. The corresponding salting out constants are therefore related by,

$$K_S^{\mathrm{m}}={\rho }_1^{\mathrm{o}}(K_S^{\mathrm{c}}-{\overline{V_3}}^{\mathrm{\infty }})=K_S^x-1$$ (46)

which only involves properties of the pure solvent and an infinitely dilute cosolvent.

Symmetric Ideal and Ideal Dilute Solutions

Symmetric ideal (SI) and ideal dilute solutions provide a useful reference point and so we will briefly discuss the results obtained from Equation 16, Equation 17, and Equation 19 when the cosolvent mixture is ideal. Ideal solutions display unit activity coefficients for all compositions. However, the conditions for ideal behavior depend on the concentration scale for finite cosolvent concentrations. Ideality arises when the lhs of Equation 8, Equation 9, and Equation 10 equals unity. On the molar concentration scale an ideal dilute solution is characterized by a₃₃ = 1 or G₃₃ − G₁₃ = 0 for all compositions. Alternatively, ideal dilute behavior on the molality scale is provided when η₁₃ = ρ₁ or ρ₁(G₁₁ + G₃₃ − 2G₁₃) = −1 for all compositions. These are the same conditions for ideal behavior in closed binary systems of 1 and 3. Consequently, Equation 16 and Equation 17 reduce to,

$$K_{23}^c=-(G_{23}-G_{21})$$ (47)

and,

$$K_{23}^m=-A_1({\rho }_2)$$ (48)

respectively. Both are valid for finite solute concentrations.

The condition for ideality on the mole fraction scale is more involved. Ben-Naim has shown that symmetric ideal solutions are characterized by ΔG_ij = G_ii + G_jj − 2G_ij = 0 for all i-j pairs.¹⁸ In this case, one has η₁₃ = ρ₁ + ρ₃ with A₁ = 1 and therefore K₂₃^x = 0, as expected. If we restrict ourselves to just symmetric ideal solutions of 1 and 3 then Equation 19 reduces to,

$$K_{23}^x=-\frac{{\rho }_1+{\rho }_3}{{\rho }_1}(-1+A_1({\rho }_2))$$ (49)

for any solute concentration. Hence, for symmetric ideal and ideal dilute solvent and cosolvent solutions the expressions involve only the terms in the numerators of Equation 16, Equation 17, and Equation 19.

Comparison with the Hall Approach

Hall derived the equations of KB theory using a totally different approach from Kirkwood and Buff. In doing so Hall produced two primary equations from which many of the expressions presented here can be generated. These have the advantage of being directly applicable to semi open systems and are relatively simple to use for binary and ternary systems. His approach, however, was still somewhat involved. Here we present a simpler derivation of the Hall equations. The first focuses on changes in the molar concentrations of any component. If we consider the species molarities in the grand canonical ensemble to be functions of T and all chemical potentials then we can write,

$$d{\rho }_i={\displaystyle \sum _{j=1}^{n_c}{\left(\frac{\partial {\rho }_i}{\partial {\mu }_j}\right)}_{T,{\mu }_{k\ne j}}}d{\mu }_j$$ (50)

for any component i at constant T. The summation is over all n_c components of the solution. The above derivatives in terms of KB integrals are provided directly from the fact that,¹⁵

$${\left(\frac{\partial {\rho }_i}{\partial {\mu }_j}\right)}_{T,{\mu }_{k\ne j}}=\beta {\rho }_i({\delta }_{ij}+{\rho }_j{G}_{ij})$$ (51)

which is the starting equation for KB theory. Here, δ_ij is the Kroenecker delta function. From the these two equations one finds,

$$d\text{ln}{\rho }_i=\beta {\displaystyle \sum _{j=1}^{n_c}({\delta }_{ij}+N_{ij})d{\mu }_j}$$ (52)

which is valid for composition changes of any component in any multicomponent system and any thermodynamically reasonable ensemble with T constant. This is the equation derived by Hall but using a much longer route. From this equation one easily finds,

$${\left(\frac{\partial \text{ln}{\rho }_2}{\partial {\rho }_3}\right)}_{T,P,{\mu }_2}=N_{21}\beta {\left(\frac{\partial {\mu }_1}{\partial {\rho }_3}\right)}_{T,P,{\mu }_2}+N_{23}\beta {\left(\frac{\partial {\mu }_3}{\partial {\rho }_3}\right)}_{T,P,{\mu }_2}$$ (53)

and,

$${\left(\frac{\partial \text{ln}{\rho }_3}{\partial {\mu }_3}\right)}_{T,P,{\mu }_2}=N_{31}\beta {\left(\frac{\partial {\mu }_1}{\partial {\mu }_3}\right)}_{T,P,{\mu }_2}+\beta (1+N_{33})$$ (54)

for our semi open ternary system. These expressions, when combined with the Gibbs-Duhem derived relationships,

$${\left(\frac{\partial {\mu }_1}{\partial {\mu }_3}\right)}_{T,P,{\mu }_2}=-\frac{{\rho }_3}{{\rho }_1}\text{and}{\left(\frac{\partial {\mu }_1}{\partial {\rho }_3}\right)}_{T,P,{\mu }_2}=-\frac{{\rho }_3}{{\rho }_1}{\left(\frac{\partial {\mu }_3}{\partial {\rho }_3}\right)}_{T,P,{\mu }_2}$$ (55)

provide Equation 16. This procedure replaces many of the thermodynamic transformations required by other approaches,^1,15,17 with relatively simple algebraic manipulations.

In addition, Hall also provided the following general expression for changes in molal concentrations at constant T and P,

$$d\text{ln}{m}_i=\beta {\displaystyle \sum _{j=2}^{n_c}({\delta }_{ij}+N_{ij}^{+})d{\mu }_j}$$ (56)

where N_ij⁺ = N_ij + m_j (1 + N₁₁ − N_i1 − N_j1). Equation 56 can be generated from Equation 52 by noting that d ln m_i = d ln ρ_i − d ln ρ₁, and then eliminating dμ₁ via the corresponding Gibbs-Duhem relationship at constant T and P,

$${\displaystyle \sum _{j=1}^{n_c}{\rho }_{j}d{\mu }_j}=0$$ (57)

Hence, the additional constraint of constant P. Using Equation 56 one immediately finds,

$${\left(\frac{\partial \text{ln}{m}_2}{\partial m_3}\right)}_{T,P,{\mu }_2}=N_{23}^{+}\beta {\left(\frac{\partial {\mu }_3}{\partial m_3}\right)}_{T,P,{\mu }_2}$$ (58)

and,

$${\left(\frac{\partial \text{ln}{m}_2}{\partial {\mu }_3}\right)}_{T,P,{\mu }_2}=\beta (1+N_{33}^{+})$$ (59)

for our semi open ternary system which, after combining and rearranging, provide Equation 17. We note that η_1i = ρ₁(N_ii⁺ + 1) and A₁ = ρ₁ N₂₃⁺/ρ₃.

The primary advantage of the Hall approach is that it can be applied to any number of solution components in any constant T ensemble (molarity version) or constant T and P ensemble (molality version). However, Hall did not provide a starting expression for mole fractions. This can be achieved by noting that,

$$d\text{ln}{x}_i=d\text{ln}{\rho }_i-{\displaystyle \sum _{j=1}^{n_c}{x}_{j}d\text{ln}{\rho }_j}$$ (60)

Therefore using Equation 52 we have,

$$d\text{ln}{x}_i=\beta {\displaystyle \sum _{j=1}^{n_c}({\delta }_{ij}+N_{ij})d{\mu }_j-\beta {\displaystyle \sum _{j=1}^{n_c}{x}_j}{\displaystyle \sum _{k=1}^{n_c}({\delta }_{jk}+N_{jk})d{\mu }_k}}$$ (61)

which is somewhat involved but very useful as it applies to any number of components in any constant T ensemble. The above equation leads to the mole fraction results for our ternary system, although the algebra involved makes it no more efficient than our previous approach. It does confirm the relationships derived in the present manuscript, as well as many of those derived earlier.^16,17

The Direct Correlation Function Approach

In several previous applications of KB theory the direct correlation approach has been used.^29,30 The expressions provided here in terms of KB integrals, based on the total correlation function, can be converted to equivalent expressions using the direct correlation function (c_ij). The direct correlation functions are defined by the relationship,

$$(\mathbf{1}+\mathbf{\rho }\mathbf{G})={(\mathbf{1}-\mathbf{\rho }\mathbf{C})}^{-1}$$ (62)

This is merely the Ornstein-Zernike (OZ) equation in matrix form in wave number space evaluated at zero wave number. Here, G is the matrix (G_ij), C is the matrix (C_ij), and ρ is a diagonal matrix with diagonal elements (ρ₁, ρ₂, …). The C_ij integrals are given by,

$$C_{ij}=4\pi {\displaystyle {\int }_0^{\mathrm{\infty }}{c}_{ij}(r)r^{2}dr}$$ (63)

Equating the elements of the two resulting matrices in Equation 62 provides expressions for the G_ij integrals in terms of the C_ij integrals,

$$\begin{array}{l}1+{\rho }_1{G}_{11}=[(1-{\rho }_2{C}_{22})(1-{\rho }_3{C}_{33})-{\rho }_2{\rho }_3{C}_{23}^2]/\left|1-\rho C\right|\\ 1+{\rho }_2{G}_{22}=[(1-{\rho }_1{C}_{11})(1-{\rho }_3{C}_{33})-{\rho }_1{\rho }_3{C}_{13}^2]/\left|1-\rho C\right|\\ 1+{\rho }_3{G}_{33}=[(1-{\rho }_1{C}_{11})(1-{\rho }_2{C}_{22})-{\rho }_1{\rho }_2{C}_{12}^2]/\left|1-\rho C\right|\\ G_{21}=[C_{12}(1-{\rho }_3{C}_{33})+{\rho }_3{C}_{13}{C}_{32}]/\left|1-\rho C\right|\\ G_{31}=[C_{31}(1-{\rho }_2{C}_{22})+{\rho }_2{C}_{21}{C}_{32}]/\left|1-\rho C\right|\\ G_{32}=[C_{32}(1-{\rho }_1{C}_{11})+{\rho }_1{C}_{12}{C}_{31}]/\left|1-\rho C\right|\end{array}$$ (64)

for a ternary system. If necessary, the |1−ρC| determinant appearing on the rhs of the above expressions can also be evaluated. However, this factor will cancel in the majority of expressions we have provided here. The above set of equalities can then be used to transform any of the present expressions into equivalent expressions involving just the C’s.

For example, the major result of this study is Equation 16. In terms of the direct correlation function integrals this can be written,

$$K_{23}^c=-\frac{C_{23}(1-{\rho }_1{C}_{11}-{\rho }_3{C}_{31})-C_{21}(1-{\rho }_3{C}_{33}-{\rho }_1{C}_{31})}{(1-{\rho }_2{C}_{22})(1-{\rho }_1{C}_{11}-{\rho }_3{C}_{31})-{\rho }_2{C}_{12}({\rho }_1{C}_{12}+{\rho }_3{C}_{32})}$$ (65)

For sparingly soluble solutes one then has,

$$K_{23}^c=-\frac{C_{23}(1-{\rho }_1{C}_{11}-{\rho }_3{C}_{31})-C_{21}(1-{\rho }_3{C}_{33}-{\rho }_1{C}_{31})}{1-{\rho }_1{C}_{11}-{\rho }_3{C}_{31}}$$ (66)

and the corresponding salting out constant is then provided by,

$$K_S^c=-\frac{C_{23}(1-{\rho }_1^{\mathrm{o}}{C}_{11})-C_{21}(1-{\rho }_1^{\mathrm{o}}{C}_{31})}{1-{\rho }_1^{\mathrm{o}}{C}_{11}}=-(C_{23}-C_{21}{\rho }_1^{\mathrm{o}}{\overline{V_3}}^{\mathrm{\infty }})$$ (67)

where we have used previously derived expressions for the solution properties in terms of the direct correlation function integrals.³⁰ We have not provided a complete set of expressions in terms of the direct correlation function for reasons outlined in the Discussion.

Discussion

In the previous sections we have outlined a general approach for understanding solubility curves in terms of KB integrals. The primary result is Equation 16 where all concentrations are measured on the molarity scale. This equation can be applied to small solutes (He, Ne, CH₄, etc) or large molecules such as peptides and proteins at any level of solubility. In addition, the KB approach allows one to isolate G₂₃ (or G₂₊) at infinite dilution, and other concentrations if a₃₃ is known. A more physical picture of the difference in KB integrals can be obtained by reference to a similar thermodynamic property which is used to quantify cosolvent effects in biological systems.³¹ In particular, Timasheff and coworkers have used preferential binding to understand equilibrium dialysis experiments and the thermodynamics of transfer of proteins from pure solvent to cosolvent mixtures.³² The preferential binding at infinite dilution of a protein solute is given by,¹⁶

$${\mathrm{\Gamma }}_{23}={\left(\frac{\partial m_3}{\partial m_2}\right)}_{T,{\mu }_1,{\mu }_3}={\rho }_3(G_{23}-G_{21})=N_{23}-\frac{{\rho }_3}{{\rho }_1}{N}_{21}$$ (68)

The last equality indicates that when the ratio of cosolvent to water molecules in the vicinity of the solute is greater than the bulk cosolvent to solvent ratio (ρ₃/ρ₁) the value of Γ₂₃ is positive. This is equivalent to a negative value of K₂₃^c and a salting in effect. When the ratio of the local cosolvent to solvent concentrations is less than the bulk ratio one observes a salting out effect. In the case of hydrophobic solutes and ionic cosolvents, it is expected that the ions will be excluded from the solute surface due to their strong solvation shells. Here, the solutes are preferentially hydrated and Γ₂₃ will be negative. KB theory provides a way to quantify these changes in the ion and solvent distributions. The use of Equation 14 to relate G₂₃ and G₂₁ physically states that the solvent and cosolvent distributions are linked through a constraint on the volume surrounding the solute. Therefore, if a cosolvent is excluded from the vicinity of a solute then an equal volume of water must replace the cosolvent to maintain a constant overall volume.

Our presentation has been given in terms of the KB integrals and not direct correlation function integrals. Indeed, when KB theory was first published the direct correlation function c_ij(r) was rarely used in statistical mechanics. Since that time, however, several authors have preferred to express the results of KB theory in terms of these integrals.^3,29,30 Presumably, the rationale for using the C’s is that their integrands (c_ij) have a shorter spatial range than do the integrands (g_ij −1) of the G’s. Although it is not difficult to use Equation 64 to express Equation 16, Equation 17, and Equation 19 in terms of the C’s rather than the G’s, we have not pursued this approach for two main reasons. First, the G’s have a clear physical meaning in terms of the excess particle numbers, or concentration fluctuations.¹⁸ The C’s have no such direct physical meaning. Second, the c’s are defined by Equation 62. They are not measurable experimentally nor calculable theoretically by any method of which we are aware that does not use Equation 62 or something equivalent to it. The G’s, on the other hand, are measurable (at least in principle) by radiation scattering and have been successfully calculated by simulation.^19,33,34 In addition, the resulting expressions in terms of the G’s are much simpler. When any of the g_ij functions become really long range, such as near a critical point,³⁰ then it becomes advantageous to use the C’s instead of the G’s. But, barring such exceptional circumstances, we advocate the present formulation in terms of the G’s.

The value of G₂₃ − G₂₁ is central to the discussion of the salting in or out effect. Other KB integrals correspond to properties of the solution mixture and are not measures of the effect of a specific cosolvent on a specific solute (they are the same for all solutes in that particular cosolvent solution). This is precisely why the pseudo chemical potential approach is to be preferred. If a cosolvent displays a positive value of G₂₃ − G₂₁ then K₂₃^c is negative and the solute solubility is increased (salting in effect). Conversely, if G₂₃ − G₂₁ is negative then K₂₃^c is positive and the solute solubility is decreased (salting out effect). If one measures the solute solubility using molalities or mole fractions, then additional terms appear in the numerator of the expressions for K₂₃ which refer to properties of the solution. These terms have nothing to do with the solute under investigation. Hence, the molarity or pcp based approach is much simpler and easier to interpret, especially when the cosolvent and water pmvs are unknown. For example, if a cosolvent displays no preference for the solute (G₂₃ − G₂₁ = 0) then an increase in cosolvent molarity leads to no change in the solute solubility (Equation 16) as K₂₃^c is zero. However, if one uses the molality or mole fraction scale results (Equation 17 and Equation 19) then the values of K₂₃^m and K₂₃^x are non zero and there will be a change in solubility even though there is no preference of the cosolvent for the solute. Let us be clear about what this sentence means. We are certainly not asserting that the physical amount of solute in a solution changes when one changes the units of measurement. The point is that the definition of solubility is dependent on these units. The molal and mole fraction based equations are perfectly correct, but they do not, in our opinion, provide the simplest and clearest interpretations of the experimental data.

Recently, Mazo presented an analysis of the effects of salts on the solubility of benzene based on the experimental data of McDevit and Long.^7,35 Here, we have reanalyzed the same experimental data. The new analysis is presented in Table 1 and is different to that of Mazo for several reasons. First, Mazo analyzed the molarity based data of McDevit and Long using the molality based Equation 39. In principle, this is incorrect but is numerically not serious at these low concentrations. Second, Mazo used the pmv of the salt rather than the indistinguishable ion $(n_{\pm }\overline{V_3}=\overline{V_s})$ – but this is also not serious. Third, a misplaced decimal point in the original calculations led to the most serious error in the construction of the previous table. Therefore, R.M.M. would like to withdraw Table 2 of Reference ⁷ and substitute for it Table 1 of the present paper.

Table 1.

KB analysis of the effects of different salts on benzene solubility at 298 K.

Salt	ks M−1	G23 − G21 cm3/mol	G2+ (= G2−) cm3/mol
NaF	0.254	−292	−382
NaCl	0.195	−225	−314
NaBr	0.155	−179	−268
NaI	0.095	−109	−199
LiCl	0.141	−162	−252
NaCl	0.195	−225	−314
KCl	0.166	−191	−281
RbCl	0.140	−161	−251
CsCl	0.088	−101	−191

Salting out constants (k_s = K_S^c = K₂₃^c) were taken from Reference ³⁵. All values correspond to infinite dilution of both solute and cosolvent. Data in column three were obtained using Equation 35 and Equation 36. Data in column four were obtained from Equation 37 using a value of ${\overline{V_2}}^{\mathrm{\infty }}-RT{\kappa }_T=-G_{21}^{\mathrm{\infty }}=89.4{\text{cm}}^3/\text{mol}$ taken from Reference ³⁵.

The results presented in Table 1 demonstrate a decrease in salt exclusion for sodium salts as the anion becomes larger. This is exactly the trend predicted on simple solvation principles. The same trend is observed for alkali metal chlorides as the cation size increases, with the noticeable exception of LiCl. If LiCl were incompletely dissociated in dilute solution this would decrease the salting out effect and explain the anomaly at least qualitatively. On the other hand, we are not aware of any evidence for incomplete dissociation. In conductance studies it appears to be a typical strong electrolyte. An alternative conjecture involves the possibility of significant cation interactions with the benzene pi system. This would favor an increase in G₂₊ for Li⁺ over other cations. However, this remains quite speculative at present.

In the above analysis the value of G₂₁ = −89.4 cm³/mol is the same for all cosolvents at infinite dilution. The corresponding excess coordination number N₂₁ is then −4.9. This can be interpreted as benzene, at infinite dilution, occupies the same space as 4.9 water molecules would occupy in pure water.

The results obtained from Equation 39 and presented in Table 1 should not be interpreted as favoring a picture of cation dominated interactions. The anions play an equally important role. The deviations in the distribution of both cations and anions around a benzene molecule, from the bulk solution distributions, must be the same due to the electroneutrality constraint (Equation 31). It is possible that only one ion interacts directly with the solute. However, exactly which of the ions provides the driving force for the association with, or exclusion from, the solute cannot be deduced from the KB approach. In our opinion, this kind of information can only be obtained from accurate computer simulation data.

There have been several computer simulation studies aimed at understanding the changes in small molecule solubility by different cosolvents.^33,36,37 These approaches are often based on Widom particle insertion calculations.³⁸ In this case the excess chemical potential (μ₂^ex) of the solute is the property to be determined and is defined by the following expression,

$$\beta {\mu }_2=\beta {\mu }_2^{ex}+\text{ln}({\mathrm{\Lambda }}_2^3{\rho }_2{q}_2^{-1})$$ (69)

where q₂ is the internal partition function of the solute. It should be noted that the above excess chemical potential is different to the traditional experimental definition for mixtures on the mole fraction scale. Clearly, the pcp and the excess chemical potential differ by a factor of ln q₂, which in most cases can be safely assumed to be reasonably independent of solution composition. Hence, the particle insertion calculations essentially probe changes in the pcp of the solute. In addition, one of the main advantages of using the indistinguishable ion approach is observed when analyzing computer simulation data concerning the distribution of salt ions around a solute. While it is relatively easy to count the number of ions (either anions or cations) observed around a central solute, it is quite difficult to count salt “molecules” when the salt is fully dissociated.

Many of the relationships provided for finite solute concentrations in our open ensemble take on exactly the same form as those for sparingly soluble solutes in closed ensembles, especially for the molarity and molality cases. They are different because the KB integrals will adopt different numerical values under different compositions. The similarities are immediately understood if one examines the Gibbs-Duhem relationship for these ternary systems. Namely,

$$n_{1}d{\mu }_1+n_{2}d{\mu }_2+n_{3}d{\mu }_3=0$$ (70)

at constant T and P. Clearly, our open ensemble (dμ₂ = 0) and the low solute concentration closed ensemble (n₂ → 0) provide the same relationship (n₁dμ₁ + n₃dμ₃ = 0) between the chemical potentials of the solvent and cosolvent. This relationship generally appears in the expressions occurring in the denominator of Equation 16 and Equation 17 (compare with Equation 8 and Equation 9). From the point of view of the solute, as the solute concentration tends towards that of an infinitely dilute solute (n₂ → 1), the properties of the solution tend towards that of a simple binary solution of 1 and 3 at the same composition, and therefore n₂dμ₂ → 0. This provides the expressions that occur in the numerator of Equation 16 and Equation 17.

The current approach is different from that of other studies which have primarily focused on sparingly soluble solutes.^2,4,10 In particular, the majority of previous approaches have adopted the KB expressions for ternary systems in closed ensembles, even for finite solute solubilities.³ Hence, their expressions differ from the semi open ensemble results developed here. In particular, the finite concentration results quoted here are generally simpler and do not involve G₂₂. Both the open and closed ensemble approaches provide the same results for sparingly soluble solutes. Equation 16 and Equation 19 are new, while Equation 17 was originally derived by Hall.¹⁴ They are the only ones that should be applied to study changes in solubility where the solute appears at a finite concentration.

The equations presented here provide a solid foundation for the analysis and interpretation of solubility in mixed solvents. The required KB integrals can be obtained from theory, simple models, or experimental data where available. The latter is relatively simple to obtain for sparingly soluble salts as one only requires properties of the binary solvent and cosolvent solution (G₃₃ and G₁₃), from which knowledge of the solubility curve provides values for G₂₃ − G₂₁. The required experimental data is more extensive for semi open ternary systems where all species appear at finite concentrations. Fortunately, the KB integrals can be obtained from studies of the corresponding closed ternary system at the same composition, i.e the solute solubility limit, using existing equations for closed ternary systems.^17,39,40 This is the same process as used previously.³ Performing such an analysis will generate all the G_ij values. Hence, one does not require the slope of the solubility curve to provide G₂₃ − G₂₁ in this case, and therefore one could use knowledge of the solubility curve as a consistency check. In either case, we argue that the correct expressions for the interpretation of the solubility data involve Equation 16, Equation 17, or Equation 19.

Conclusions

Solubility may be quantitatively defined in several ways using different measures of the relative amounts of solute and solvent(s) present in solution. We have discussed the three most common measures in this paper - molarity, molality, and mole fraction. From the point of view of logic, these are all equivalent, merely involving a (nonlinear) change of scale. From the point of view of interpretation in terms of molecular distributions, we have argued that the molarity scale of solubility is the most convenient one. We have given formulae for solubilities and salting out coefficients for all three concentration scales. All of these rely on the Kirkwood-Buff integrals to express the local structure of the solution, particularly near a solute molecule, as well as auxiliary thermodynamic properties of the solution. It appears that the molarity scale formulae emphasize the structural information, and deemphasize the ancillary part to a greater extent than do the other two scales. We therefore recommend that, as far as possible, the molarity scale be used when measuring and reporting solubilities. Certainly, molalities and mole fractions have their advantages as concentration measures. Nevertheless, for interpretive purposes we recommend molarities.

Appendix 1. The Indistinguishable Ion Approach for Mole Fractions

The use of the indistinguishable ion approach is relatively straight forward for the molarity and molality concentration scales. In this case we find dρ₃ = n_± dc_s and d ln ρ₃ = d ln c_s for the cosolvent molarity, and similarly for molality. Both are independent of the ensemble. However, the same is not true for the mole fraction scale in both closed and, especially, open ensembles. There are two possible definitions for the cosolvent mole fraction in ternary systems,

$$x_s=\frac{n_s}{n_1+n_2+n_s}\text{and}{x}_3=\frac{n_{\pm }{n}_s}{n_1+n_2+n_{\pm }{n}_s}$$ (A1.1)

where n_s is the number of salt “molecules” and the latter refers to the indistinguishable ion definition. Note that the particular definition also affects the solvent and solute mole fractions. Let us consider our ensemble with T, P, and μ₂ constant and therefore n₂ can vary. Using the first expression one can write,

$${\left(\frac{\partial x_s}{\partial n_s}\right)}_{T,P,{\mu }_2}=\frac{1}{n_1+n_2+n_s}-\frac{n_s(1+d{n}_2/d{n}_s)}{{(n_1+n_2+n_s)}^2}=\frac{1}{n_1+n_2+n_{\mathrm{s}}}[1-x_s(1+n_{\pm }{m}_{23})]$$ (A1.2)

where m₂₃ was defined earlier in the text. If we apply the same procedure to x₃ then we have,

$${\left(\frac{\partial x_3}{\partial n_s}\right)}_{T,P,{\mu }_2}=\frac{n_{\pm }}{n_1+n_2+n_{\pm }{n}_s}[1-x_3(1+m_{23})]$$ (A1.3)

Hence, we find that,

$${\left(\frac{\partial x_3}{\partial x_s}\right)}_{T,P,{\mu }_2}=\frac{x_3[1-x_3(1+m_{23})]}{x_s[1-x_s(1+n_{\pm }{m}_{23})]}$$ (A1.4)

and therefore it is clear that d ln x₃ ≠ d ln x_s and also dx₃ ≠ n_± dx_s for this ensemble. The corresponding relationship in a closed ternary system is recovered when m₂₃ = 0, for which one also has dx₃ ≠ n_± dx_s for the cosolvent. The relationship in a closed binary system of 1 and 3 is provided by setting n₂ = 0 and m₂₃ = 0, but one still finds dx₃ ≠ n_± dx_s for the cosolvent. Only when x₃ tends to zero do we find that dx₃= n_± dx_s is valid. Hence, the determination of K_S^x is relatively straight forward, but the determination of K₂₃^x is more involved. One can still use the indistinguishable ion approach in all cases, but care should be taken that the required transformations are performed correctly.

Appendix 2. The Sechenov Constant for Different Concentration Measures – A Unified Approach

The object is still to express the slope of the solubility curves in terms of Kirkwood-Buff integrals. In this appendix we show how all three cases can be treated on a common basis. This is neither more nor less correct than the previous approach, it is merely a way of emphasizing the common idea behind all three representations. The Sechenov coefficient (K_S^z) is defined by,

$$K_S^z=\underset{c_s\to 0}{\text{lim}}{\left(\frac{d}{d{c}_s}\text{log}\left(\frac{S_0^z}{S^z}\right)\right)}_{T,P,{\mu }_2}$$ (A2.1)

where S is the solubility of a solute in a salt solution of salt concentration c_s, measured in concentration units z. The units of z can be either molality (m₂), molarity (c₂), or mole fraction (x₂). From now on, we drop the T and P subscripts on partial derivatives, since these will be held constant throughout. We first write,

$${\left(\frac{\partial \text{log}{z}_2}{\partial c_s}\right)}_{{\mu }_2}=-\frac{n_{\pm }}{2.303z_2}\left(\frac{{(\partial {\mu }_2/\partial c_3)}_{z_2}}{{(\partial {\mu }_2/\partial z_2)}_{c_3}}\right)$$ (A2.2)

where n_± is the total number of ions resulting from the dissociation of one formula unit of the salt, and c₃ = n_±c_s (see text). All we have to do is calculate the derivatives on the rhs. It is the derivatives (∂μ_i/∂N_j)_N' that are easily computed in terms of KB integrals, so we must express the derivatives appearing in Equation A2.2 in terms of the latter. That can be done as follows,

$${\left(\frac{\partial {\mu }_2}{\partial N_j}\right)}_{N^{\prime }}={\left(\frac{\partial {\mu }_2}{\partial z_2}\right)}_{c_3}{\left(\frac{\partial z_2}{\partial N_j}\right)}_{N^{\prime }}+{\left(\frac{\partial {\mu }_2}{\partial c_3}\right)}_{z_2}{\left(\frac{\partial c_3}{\partial N_j}\right)}_{N^{\prime }}j=1,2$$ (A2.3)

These are two linear equations in the two unknowns. These can be solved, say by Cramer’s rule, for the two unknowns, ∂μ₂/∂z₂ and ∂μ₂/∂c₃. It only remains to place the solutions in Equation A2.2 and then pass to the limit c₃ → 0.

The ∂z₂/∂N and ∂c₃/∂N derivatives are easy to calculate. Since we are computing a ratio and then taking a limit, a lot of cancellation occurs. In particular, we may safely set c₃ = 0 everywhere except where it multiplies G₃₃ or where it occurs in a denominator. When the cosolvent is an electrolyte it is not completely obvious that c₃G₃₃ = 0 at infinite dilution of electrolyte since c₃G₊₋ = 1 in this limit.²¹ However, Equation 27 of this paper shows that, in fact, there is no problem.

When c₃ occurs in a denominator, the dangerous looking poles get tamed when taking the ratio but this must be done carefully. It is not necessary to compute the KB determinant since it will cancel on taking the ratio. Similarly, in solving Equations A2.3 using Cramer’s rule it is not necessary to compute the determinant of the coefficients as this will also cancel. These rules greatly simplify the algebra of the computation.

Finally, the results obtained from Equation A2.2 yield Equation 38, Equation 39, and Equation 41 of the main text.