Experimental Insight into the Thermodynamics of the Dissolution of Electrolytes in Room-Temperature Ionic Liquids: From the Mass Action Law to the Absolute Standard Chemical Potential of a Proton.

Room-temperature ionic liquids (ILs) are a class of nonaqueous solvents that have expanded the realm of modern chemistry, drawing increasing interest over the last few decades, not only in terms of their own unique physical chemistry but also in many applications including organic synthesis, electrochemistry, and biological systems, wherein charged solutes (i.e., electrolytes) often play vital roles. However, our fundamental understanding of the dissolution of an electrolyte in an IL is still rather limited. For example, the activity of a charged species has frequently been assumed to be unity without a clear experimental basis. In this study, we have discussed a standard component-based scheme for the dissolution of an electrolyte in an IL, supported by our observation of ideal Nernstian responses for the reduction of silver and ferrocenium salts in a representative IL, 1-ethyl-3-methylimidazolium bis(trifluoromethanesulfonyl)imide ([emim+][NTf2 -] or [emim+][TFSI-]). Using this scheme, which was also supported by temperature-dependent measurements with ILs having longer alkyl chains in the imidazolium ring, and the solubility of the IL in water, we established the concept of Gibbs transfer energies of "pseudo-single ions" from the IL to conventional neutral molecular solvents (water, acetonitrile, and methanol). This concept, which bridges component- and constituent-based energetics, utilizes an extrathermodynamic assumption, which itself was justified by experimental observations. These energies enable us to eliminate inner potential differences between the IL and molecular solvents (solvent-solvent interactions), that is, on a practical level, conditional liquid junction potential differences, so that we can discuss ion-solvent interactions independently. Specifically, we have examined the standard electrode potential of the ferrocenium/ferrocene redox couple, Fc+/Fc, and the absolute intrinsic standard chemical potential of a proton in [emim+][NTf2 -], finding that the proton is more acidic in the IL than in water by 6.5 ± 0.6 units on the unified pH scale. These results strengthen the progress on the physical chemistry of ions in IL solvent systems on the basis of their activities, providing a rigorous thermodynamic framework.

Introduction

Solutions of electrolytes are ubiquitous in nature and play vitn class="Chemical">al roles in chemical, biological, and ecological processes. They are formally defined as a liquid or solid phase containing a substrate that bears ions (an ionophore, e.g., an ionic crystal such as NaCl) or one that is capable of producing ions (an ionogen, e.g., an organic acid that can dissociate into ions) and a substance that dissolves these species (often a neutral molecular solvent, such as water). The physical chemistry of such ionically conducting solutions underlies the entire field of electrochemistry and has been termed “ionics”.[1] Although studies on ionics have mainly been performed in aqueous solutions, which limits the types of substances we can handle, over the last 100 years nonaqueous solvent systems (e.g., molten salts, organic molecular liquids, solid oxides, etc.) have also been studied, enabling us to handle often unstable species that have many practical applications, such as Li ion transport in batteries,[2] electrodeposition of elements (e.g., Al),[3,4] and hydride (H–) transfer[5,6] or conduction.[7,8]

Ionic liquids (ILs) are one of the most recent classes of nonn class="Chemical">aqueous solvent systems to have been established. An IL is defined as a solvent-free combination of cations and anions that displays at least one liquid-like phase below 350 °C, whereas a room-temperature ionic liquid (RTIL) is an IL that exists in a liquid-like phase at 25 °C.[9,10] In this article, the ILs that we discuss are all RTILs. ILs can be described by apparently contradicting concepts: they are electrolytic solutions in the limiting case of zero molecular solvent, like an ionic crystal, but being liquid at room-temperature implies the possibility that they may play the role of a virtually nonvolatile[11] electrolytic “solvent”. Accordingly, ILs are often thought of as the missing link between neutral molecular solvents and molten salts.[12] This concept has been drawing increasing interest over the past few decades,[13−18] with ILs having been found to exhibit interesting and useful properties of dissolution and solvation not only as solvents for reactions,[19−29] but also as reactive solutes,[30−32] in which an IL molecule is embedded (or tagged)[33−37] as a part of the solute in neutral molecular solvents. More importantly, these properties can be finely tailored to meet specific needs by designing structures of the IL molecule, facilitating a wide range of chemical transformations and amplifying the importance of the chemistry of ILs.

In terms of thermodynamics, as ILs are composed of only an ionophore as the solvent, representing the limiting case of zero molecular solvent as a reaction medium, it is possible to define unique sn class="Chemical">tandard states for solutes. These standard states are distinct from those of solutes in molecular solvents, wherein ideal behavior for the physical properties of ions can be observed at infinite dilution of an electrolytic solution, which eliminates long-range and strong electrostatic interactions. Thus, in terms of describing the kinetic and thermodynamic properties of (electro)chemical reactions or extractions, what factors must be considered when an IL is used to dissolve solutes (neutral molecules or electrolytes) compared to those for a conventional molecular solvent? This question concerns the solvation of the solutes (solute–solvent interactions), which is thermodynamically quantified by the Gibbs energy change upon transfer (Gibbs transfer energy) of a given solute from a molecular solvent to an IL. A theoretical framework to answer this question, called the protoelectric potential map, has previously been constructed by Radtke, Himmel, and Krossing et al.[38−40] We will therefore focus briefly on introducing the physical meaning of the Gibbs transfer energy of a single ion and the current experimental difficulties associated with the determination of this quantity, especially when involving an IL as a solvent.

In the case of neutral solute molecules, the quantification of solvation can, in principle, be made in a straightforward manner by measuring the solubility of the solute in two solvents and then n class="Chemical">taking the difference. Thus, a number of thermodynamic properties have been measured in ILs, such as the solubility of gases[41] or neutral solutes,[42−47] starting from fundamentals like activity and expanding to applications like the transformation of organic substrates.

However, in the case of an electrolyte as a solute that produces ions, the quantification of solvation is accompanied by a complication, that is, the difference between the electrical potentials of the two phases has to be considered upon transfer of single ions. The importance of this is illustrated, for example, by the permselective transfer of ions across neuronal membranes, eventually invoking an electrical signal in nerve cells,[48] or by the release of ions from a hydrophobic self-assembled monolayer mimicking biological molecular recognition on protein surfaces.[49]

There are two ways to define the Gibbs transfer energy of a single ion: “real” and “intrinsic” energies. This is caused by the formn class="Chemical">alism of the electrochemical potential for an ionic species.[50−52,38] Thus, when an electrochemical potential difference (Δμ̃) is expressed by three potential differences, i.e., chemical (Δμ), surface (Δχ), and outer (Δψ) (eq , where z is the charge of the species, and F is the Faraday constant), two new types of potential differences can be defined:[50−52,38] (1) the real chemical potential difference,[53−55] Δα = Δμ + zFΔχ (eq ), and (2) the inner potential difference, Δϕ = Δχ + Δψ (eq ), where Δμ is generally called the intrinsic chemical potential difference.[56−58] Note that both the real (Δα) and the intrinsic (Δμ) chemical potential differences are differences of absolute quantities. The former can be directly measured through a determination of the outer potential difference (Δψ), whereas the latter cannot because the inner potential difference (Δϕ) cannot be directly determined. For example, an attempt to compare the potentials for the electrode reaction in which Ag+ is reduced to Ag metal in two different solvents would fail when determining the intrinsic Gibbs transfer energy of Ag+ from one solvent to the other. This is because, upon physical contact of the two different solvents, the difference in chemical potentials of the two solvents yields an inner potential difference[50] that manifests itself on a practical level as a conditional liquid junction potential difference,[59,58] involving the presence of a spectator electrolyte. However, whereas the real chemical potential difference (Δα) results from a mixture of two physically quite different effects (solvation in the bulk and surface properties of the bulk received by an ion transferring from just outside of the bulk to the interior part as measured in, for example, a jet (Kenrick) cell[60]), the intrinsic chemical potential difference (Δμ) exclusively characterizes differences in ion–solvent interactions in the two bulk solutions.For ionics in molecular solvents, as the solubility of a given electrolyte has to be measured in a manner similar to that for neutral molecules, which affords the intrinsic§ Gibbs transfer energy of the whole salt, methods have been devised to divide the intrinsic Gibbs transfer energy of the electrolyte into contributions from its two constituent ions, that is, extrathermodynamic assumptions have been sought[61−66] and established over the last several decades. Only the so-called reference electrolyte assumption[67,68] using tetraphenylarsonium tetraphenylborate (TA+TB–) or tetraphenylphosphonium tetraphenylborate (TP+TB–), otherwise known as the “TATB assumption” or “TPTB assumption”, respectively, and discussed in more detail later (see Scheme for structures), has been found to afford reasonable intrinsic Gibbs transfer energies of single ions, which are supported by recent theoretical ab initio calculations using the cluster-pair approximation.⊥

Scheme 1

Chemical Structures of TA+, TP+, and TB–

Meanwhile, although some progress has been made in the field of n class="Chemical">ionics in ILs, it is still not fully developed. For example, the electrode potential of the redox couple between protons and dihydrogen gas,[69−71] acidity,[72−80] correlations between acidities in ILs and those in molecular solvents,[79,81,82] studies of reference electrodes (wherein negligible liquid junction potential (NLJP) difference is frequently assumed),[83−87] interfaces between two immiscible electrolyte solutions (ITIES) for charge-transfer processes,[88−91] and three-phase electrodes using droplets[92] were reported. However, almost all of these measurements are formal concentration-based and not activity-based. Although thermodynamic measurements on mixing behaviors[93−98,47] and vapor pressures[11,99−102] of ILs have been reported, no thermodynamic behavior for any electroactive electrolyte has been reported. Comparing acidities in ILs and molecular solvents may yield an opportunity to estimate the intrinsic transfer energy of H+ to an IL from a molecular solvent (e.g., water,[81] acetonitrile,[79] or dimethylsulfoxide[82]). Unfortunately though, there is a critical problem, as it first requires an estimation of the intrinsic transfer energy of the conjugate acid or base on the basis of the acidities of these charged species. These issues indicate that there is no basis to even attempt to quantify the intrinsic transfer energy of a single ion to an IL from a molecular solvent by applying the TATB or TPTB assumption. Although ITIES studies are activity-based, the nonpolarizable character of the interface makes interpretation of the acquired quantities quite complex, except in a few cases in which quite hydrophobic ILs are used.[103−106]

In this article, we initially discuss the mass action law and sn class="Chemical">tandard states for ionic solutes in ILs. We then confirm our postulations by potential difference measurements of electrochemical cells using silver and ferrocenium salts as typical probe electrolytes in 1-ethyl-3-methylimidazolium bis(trifluoromethanesulfonyl)imide ([emim+][NTf2–] or [emim+][TFSI–]; Scheme ) at 25 ± 1 °C as a representative IL, which was previously found not to exhibit nanoscale ordering or aggregation of the alkyl chains in nonpolar domains.[107−109] We also measured the potential differences at higher temperatures for cells containing the NTf2– salts of 1-butyl-3-methylimidazolium and 1-methyl-3-octylimidazolium (bmim+ and omim+, respectively; Scheme ). These measurements resulted in a successful determination of the reaction stoichiometry of electrode reactions involving these salts, their activity coefficients, and the standard electrode potential for the reduction of Ag+ to Ag metal in [emim+][NTf2–]. We have also assembled a continuum explanation for solvation by conventional molecular solvents and ILs by means of solubility measurements and the introduction of the new concept of “pseudo-single ions” under the scheme of the TATB or TPTB assumption to analyze intrinsic solvation parameters for several interesting electrolytes, thermodynamically highlighting solvation by [emim+][NTf2–]. On the basis of these two observations, we now have a thermodynamic basis for ionics in ILs. For example, on the basis of activity, we show the possibility to connect a pH scale in an IL to a unified pH scale[110] previously developed for all phases. This was achieved by taking down thermodynamic and nonthermodynamic fences (i.e., conditional liquid junction potential differences) between conventional molecular solvents and [emim+][NTf2–]. Our experimental results not only provide a thermodynamic basis for ionics in ILs but also, for the first time, a thermodynamic “bridge” to ionics in conventional molecular solvents on the basis of evidence obtained from activity measurements.

Scheme 2

Chemical Structures of emim+, bmim+, omim+, and NTf2– (or TFSI–)

Results and Discussion

Energetics upon Dissolution of an Electrolyte into an IL: Application of the Mass Action Law

As an IL represents the limiting case of zero neutral molecular solvent, when the dissolution of a n class="Chemical">salt, A+X–, into an IL, [B+][X–], which has an anion in common with the salt, is considered, the mass action law as it applies to neutral molecular solvents (eq ) is not applicable because each ion is still adjacent to oppositely charged ions, even after infinite dilution of the solution by the IL. This allows us to deduce that the dissolution has to be described by A+X– becoming a component of the IL (eq , where μ is the intrinsic chemical potential, and x is the mole fraction of [A+][X–]) rather than by solvation of the individual constituent ions of A+X– (i.e., A+ and X–).Alternatively, it is still possible to describe the thermodynamics of the dissolution by a dissociation energy that determines the concentration of “free” ions in the IL as an aggregate of ion-pairs (the so-called constituent-based approach). This energy is defined by treating the IL as a traditional dilute electrolyte solution[111−113] or a concentrated electrolyte solution,[114−116] both of which models have been discussed as a central matter in the ionics of ILs.[117] Although these solution models are attractive to explain our results described later, the dissociation energy is still under debate, and the results in our study cannot determine the energy because, provided it is observable, it would be expected to be constant, resulting in a parallel shift of solubility product constants and standard electrode potentials.

Therefore, a component-based approach, which we have adon class="Chemical">pted, is important when a thermodynamic quantity is considered, as previously discussed by Krossing et al.[39] For example, for the determination of the acidity of a given compound (A+X– is H+X– in this case) on an absolute scale in an IL, the acidity of the proton has to be based on the compound being a component of the IL in the IL solution. Rogers et al.,[98] Welton et al.,[97] and Doherty et al.[73] also pointed this issue out. Krossing et al.[39] discussed the effects of the formation of the homoconjugate (HX2–) on acidity in IL solutions. Even for a salt, A+Y–, that does not have an anion in common with the IL, [B+][X–], the chemical potential of A+Y– in [B+][X–] will be the sum of that of [A+][X–] and [B+][Y–] at infinite dilution (eq , where x is the mole fraction of [A+][Y–]), assuming the absence of a preferential interaction between A+ and Y–. This assumption is not just applicable to a single, special case. For example, the absorptivity of the Kosower salt (which is an iodide salt of a pyridinium that exhibits absorption only when its ions are ion-paired) in ILs consisting of 1-butyl-3-methylimidazolium (bmim+) as a common cation indicates that ILs and the salt apparently form only statistically random ion contacts.[118] pKa measurements of a series of ylide salts with different counteranions also confirmed no effect of the anions on the respective pKa’s.[82] Meanwhile, the chemical potential of [B+][X–] is defined as the sum of the chemical potentials of [B+] and [X–] because [B+] and [X–] are constituents of the solvent. At infinite dilution (or pure [B+][X–]) this will yield eq , which was successfully used to assess the Gibbs energy of fusion for ILs from the solid to the liquid state, which is relevant to the melting point of an IL,[119] and is evaluated in relation to vaporization, lattice formation, and dissociation processes of ILs.[120]This results in a significant difference between the solvation of an electrolyte by an IL and by a molecular solvent, in terms of a Born–Fajans–Haber cycle as proposed in Figure . In the thermodynamic cycle for solvation by a molecular solvent (Figure a), a solid electrolyte forms two separate ions in the gas phase, which are then solvated by solvent molecules. When the dielectric permittivity of the solvent is low, these exist largely as ion pairs in the form of an associated salt, with an association constant symbolized by KA. The solubility product constant, Ksp, of the electrolyte is defined as the solubility product when solute–solute interactions are eliminated by infinite dilution.

Figure 1

Born–Fajans–Haber cycles for the assessment of the solubility product constant, Ksp, of an electrolyte (denoted as “filled circled plus, filled circled minus”) to two different types of solvents: (a) a neutral molecular solvent; and (b) an IL, wherein the ionic solvent molecule is denoted as “open circled plus, open circled minus”. Subscripts (s), (g), (sol), and (IL) for each ion or pair of ions indicate the electrolyte or its ionic constituents in the solid phase, gas phase, solvated by neutral solvent molecules, and solvated by the IL, respectively.

Born–Fajans–Haber cycles for the assessment of the solubility product consn class="Chemical">tant, Ksp, of an electrolyte (denoted as “filled circled plus, filled circled minus”) to two different types of solvents: (a) a neutral molecular solvent; and (b) an IL, wherein the ionic solvent molecule is denoted as “open circled plus, open circled minus”. Subscripts (s), (g), (sol), and (IL) for each ion or pair of ions indicate the electrolyte or its ionic constituents in the solid phase, gas phase, solvated by neutral solvent molecules, and solvated by the IL, respectively.

Meanwhile, for solvation in an IL (Figure b), the two constituent ions of the n class="Chemical">electrolyte become an integral part of the IL to yield two additional IL components. Note that the stoichiometric number (n) of ionic constituents of an IL involved in the reaction depends on the nature of the electrolyte, which has to be determined experimentally. For example, in the case of salts of transition metal cations forming complexes with multiple anions of ILs upon dissolution,[121−123] the corresponding residual cations are supposed to be in the stoichiometry required for electroneutrality. This situation is completely different from that in a molecular solvent because the electrolyte exists as two IL components (not two individually solvated constituents), even at infinite dilution in the IL (eq , where “s” and “IL” in subscript denote a substance in the solid phase and solvated by an IL, respectively, and the activities are in the molarity scale unless otherwise noted by a subscript “x”, indicating them being in the mole fraction scale). On the basis of the mass action law as shown in Figure b, the solubility product constant, Ksp, is defined as the solubility product when interactions between the two solute ions (not solute and solvent ions) are eliminated by infinite dilution as shown in eq , where γ, γs, and γIL are the activity coefficients of solutes, the solid electrolyte, and the IL, respectively, C is the concentration [mol/L], xs and xIL are the mole fractions of the solid electrolyte and the IL, respectively, μ°,IL is the standard chemical potential of the solute, i (e.g., ), in the IL, μ°,s is the standard chemical potential of the solid electrolyte, j (), in the solid phase, R is the gas constant, and T is the temperature of the solution. In the case of a salt that has an ion in common with one of those of the IL as a solvent, eqs and 6′ define the solubility constant, Ks, instead (the notation used is the same as that for eqs and 6).The solvation scheme for an electrolyte in an IL as proposed by us in Figure b might be too simple to describe thermodynamic quantities relevant to a particular phenomenon, for example, the dissolution of an electrolyte in this study, as it is based on a limited, although still increasing, number of experimental observations in ILs. However, as described below, we found experimentally that this scheme works quite effectively to explain potential differences at various temperatures developed by the dissolution of both silver and ferrocenium salts to ILs that have different lengths of alkyl chain in the cationic constituent of the ILs, forming nonpolar domains with nanoscale ordering when the chain is longer that propyl.[107−109] Therefore, our observations allowed us to determine the stoichiometric number (n in eqs and 6) of ionic constituents of ILs involved in the dissolution, as well as activity coefficients (γ in eq ) of each component under this solvation scheme, resulting in the determination of solubility product constants, pKsp, in an IL, which are used for the experimental assessment of the absolute intrinsic standard chemical potential of H+ in an IL in the last section of the paper.

Potential Difference Measurements Using Silver and Ferrocenium Salts as Probes To Investigate Activity Coefficients and Reaction Stoichiometries

To obtain insight into the reaction stoichiometry and the activity of a n class="Chemical">salt dissolved in [emim+][NTf2–] as a representative IL, we measured the potential differences of electrochemical cells (Cells 1a and 1b∥ where the liquid junction was formed by a Vycor glass frit,¶ and c1 or c2 are the concentrations of solutes that were varied) using the Ag+ and ferrocenium (Fc+) salts of NTf2– as probes in the IL and found an interesting medium effect.AgNTf2 is found to form emim+Ag(NTf2)2– upon dissolution in [emim+][NTf2–],[123] and therefore the electrode reaction of interest and the potential (E) can be described by eqs and 8 as an application of eqs and 6′, respectively, according to the Nernst equation, where R is the gas constant, T is the temperature (298.15 K), F is the Faraday constant, xIL is the mole fraction of the IL, for which the activity coefficient was assumed to be unity, and n is the stoichiometric number of ionic constituents of the IL involved in the electrode reaction. Figure a shows the dependence of the electrochemical potential of a Ag+/Ag electrode configured as in Cell 1a on a function of the concentration of [emim+][Ag(NTf2)2–] as a solute and the mole fraction of the IL as a solvent, as indicated in eq . The observed potentials have been converted so as to refer to E°(Fc+/Fc) in [emim+][NTf2–] determined by cyclic voltammetry (see the Experimental Section for details). The observation of a Nernstian response of the potential to the logarithmic term in eq (the slope was found to be 100% of the theoretical coefficient only when n is set to 2) clearly indicates that the activity coefficient (γ) of [emim+][Ag(NTf2)2–] is constant over a wide range (10–5–10–1 M) of analytical concentrations of Ag+NTf2– in [emim+][NTf2–]. From the observed linear relationship over this concentration range, we extrapolate a value of unity at the limit of a dilute solution (eq ) for the constant activity coefficient of the silver cation in [emim+][NTf2–]. Accordingly, the standard electrode potential (E°) of eq was determined for the first time to be 515 ± 8 mV versus E°(Fc+/Fc) in [emim+][NTf2–] at 25 ± 1 °C by eq , formally defining the activity of the silver cation on the basis of eq . Importantly, eq indicates that the potential difference is determined by the ratio of analytical concentrations of the silver salt to the IL, strongly supporting our solvation scheme. Equation also implies that, in a region of low analytical concentration of the silver salt, the activity of Ag+ can be approximated to the analytical concentration. In fact, Abbott et al.[124] observed such a relationship for the Ag/AgCl electrode in [emim+][Cl–]. Although the range of concentrations tested was half that in our investigations, their observation strongly supports the generality of ours.Although the observation of an activity coefficient of unity for the silver salt in [emim+][NTf2–] supports our solvation scheme, it is not sufficient proof because even a reaction not involving [emim+] in the stoichiometry (eq ) could yield an equation of the type shown in eq , and the stoichiometric number might happen to be 2. We therefore also measured the electrode potential developed by the ferrocenium/ferrocene (Fc+/Fc) redox couple to test whether Fc+ is coupled with [NTf2–] in the electrode reaction, as stated in eq , for which the potential is defined by eq , where “(IL)” and “(IL, x)” in subscripts denote a substance solvated by the IL, and the activities are in molarity units and mole fraction units, respectively. Note that the mole fraction with the stoichiometric number of the IL is involved in the denominator in the logarithmic term. Figure b shows potentials of the Fc+/Fc redox couple measured by using Cell 1b when the ratio of C([Fc+][NTf2–]) to C(Fc) was varied at 25 ± 1 °C and then converted so as to refer to E°(Fc+/Fc) in [emim+][NTf2–] determined by cyclic voltammetry (see the Experimental Section for details). In this measurement, we observed an ideal Nernstian slope of the potential to the logarithmic term in eq (we found that the slope is 100% of the theoretical coefficient only when setting the stoichiometric number to 1), indicating that the activity coefficient of [Fc+][NTf2–] in [emim+][NTf2–] is a constant that is set to unity over a wide range (10–4–100 M) of analytical concentrations of Fc+NTf2– on the basis of the same argument as above for the silver cation (eq ). The intercept being close to zero (1 ± 1 mV) is consistent with E°(Fc+/Fc) in [emim+][NTf2–] determined by cyclic voltammetry. We also found such an ideal Nernstian response passing through the origin at 39 ± 1 °C (see the Supporting Information). As Fc+ is a more bulky cation than emim+ and the observed stoichiometric number is again the same as it is expected to be according to eqs and 10, this observation justifies our solvation scheme, and the solubility product constant, Ksp, and the solubility constant, Ks, are thus defined by eqs and 6′, respectively, wherein all of the activity coefficients can be set to unity.An activity coefficient of unity for the mole fraction of the IL indicates that the IL solution behaves as an ideal solution, the properties of which are determined by the number of particles in the solution. This is unlikely to be applicable for other ILs having longer alkyl chains in the imidazolium ring, for example, the NTf2– salts of bmim+ and omim+ (Scheme ), which are found to exhibit nanoscale ordering of the alkyl chains in nonpolar domains.[107−109] In fact, we found only about 97 and 93% of the theoretical Nernstian responses in a similar experiment to Figure b except using [bmim+][NTf2–] and [omim+][NTf2–], respectively, as the IL at 25 ± 1 °C (see the Supporting Information). However, it has been shown that the extent of this ordering is strongly dependent on temperature,[108] and indeed we found that we were able to recover ideal Nernstian responses (100% of the theoretical coefficient) at 39 ± 1 and 67 ± 1 °C for [bmim+][NTf2–] and [omim+][NTf2–], respectively (see the Supporting Information). It is noteworthy that these deviations are not caused by nonideal excess molar volumes of mixtures of these ILs and ferrocenium, similar to those observed by Lopes and Rebelo et al.[93] who tested mixtures of 1-methyl-3-alkylimidazoliums, because values of the functions of concentrations in the abscissa axes do not depend on the actual volumes of the solutions. Therefore, these observations merit future studies to investigate the colligative properties of various IL solutions.

Figure 2

Dependence of the potentials of (a) an Ag+/Ag electrode and (b) an Fc+/Fc redox couple at a Pt electrode in an IL ([emim+][NTf2–]) on functions of the concentrations of (a) Ag+NTf2– and (b) Fc+NTf2– in the IL at 25 ± 1 °C under an N2 atmosphere in the dark. The dependence when a sintered glass frit with a porosity of 5 μm was used in the cell instead of a Vycor frit is shown by white squares (□). The red lines indicate linear correlations between the potentials and the functions, that is, (a) ln(C([emim+][Ag(NTf2)2–])/xIL2) and (b) ln(C([Fc+][NTf2–])/(C(Fc) × xIL), where C = concentration [mol/L], acquired by a linear least-squares fitting method. The potentials were measured in two different batches of the IL vs an Ag+/Ag electrode in the IL (C0(Ag+NTf2–) is ca. 1 mM) and then converted to values vs the standard electrode potential of Fc+/Fc in the IL, which was determined by the method described in the Experimental Section.

Dependence of the potentials of (a) an Ag+/Ag electrode and (b) an n class="Chemical">Fc+/Fc redox couple at a Pt electrode in an IL ([emim+][NTf2–]) on functions of the concentrations of (a) Ag+NTf2– and (b) Fc+NTf2– in the IL at 25 ± 1 °C under an N2 atmosphere in the dark. The dependence when a sintered glass frit with a porosity of 5 μm was used in the cell instead of a Vycor frit is shown by white squares (□). The red lines indicate linear correlations between the potentials and the functions, that is, (a) ln(C([emim+][Ag(NTf2)2–])/xIL2) and (b) ln(C([Fc+][NTf2–])/(C(Fc) × xIL), where C = concentration [mol/L], acquired by a linear least-squares fitting method. The potentials were measured in two different batches of the IL vs an Ag+/Ag electrode in the IL (C0(Ag+NTf2–) is ca. 1 mM) and then converted to values vs the standard electrode potential of Fc+/Fc in the IL, which was determined by the method described in the Experimental Section.

Evaluation of the Gibbs Transfer Energies of “Pseudo-Single Ions” from an IL

Our confirmation that the activity n class="Chemical">coefficients of the silver salt and the ferrocenium salt in [emim+][NTf2–] are unity prompted us to evaluate the Gibbs transfer energies of ions on the basis of the reference electrolyte assumption,[61−66] using TP+TB–. As discussed in the Introduction, it is thermodynamically impossible to access single-ion properties such as the transfer energy of a single ion. However, the salt TP+TB– (or TA+TB–) possesses two important properties for measurements of the Gibbs transfer energy of ions: (1) oppositely charged ions having the same symmetrical shape, and (2) a large steric hindrance that buries the charge of the ion.[125] On the basis of these properties, the degree of (de)stabilization experienced by TP+ (or TA+) upon solvation by a solvent is assumed to be identical to that of TB–.The Gibbs energy change for the transfer of TP+TB– from one solvent (sol 1 denoted in subscript) to another solvent (sol 2 denoted in subscript) (eq ) can be thermodynamically determined from the difference between the logarithms of the solubility product constants measured in the two solvents, as shown in eq , where pKsp = −log(Ksp). When the Gibbs transfer energies of the two single ions are defined as shown in eqs and 13d, the transfer energy of the salt can be expressed as the sum of the transfer energies of the two single ions (eq ). Herein, the TPTB (and also the TATB) extrathermodynamic assumption states that the Gibbs transfer energy of the salt, TP+TB– (or TA+TB–), from sol 1 to sol 2 receives equal contributions from the transfer energies of the TP+ (or TA+) and TB– constituent ions (eq ). This enables us to estimate the transfer energies of single TP+ (or TA+) and TB– ions, which has been tested and found to be valid quantitatively, with an uncertainty of ±3 kJ/mol in the energy.[67]Once the Gibbs transfer energies of the TP+ and TB– constituents are determined by eq , the transfer energy of any given ion can be derived by subtracting the Gibbs transfer energy of TP+ (or TB–) from the Gibbs transfer energy of a TP+ (or TB–) salt of that ion. Although a difficulty one might encounter is the low solubility of this bulky salt (TP+TB–) in molecular solvents, fortunately, we found that the solubility of the salt is 3.4 mM in [emim+][NTf2–] at 25 °C, which is bracketed by 1.4 mM in acetonitrile[63] and 7 mM (for TA+TB–) in DMSO.[62] We also found that the solubility of a neutral bulky molecule, tetraphenylmethane (TM) in [emim+][NTf2–] is 0.4 mM, which is bracketed by 0.1 mM in methanol[126] and 0.6 mM in acetonitrile,[126] indicating the interesting dissolving ability of [emim+][NTf2–].

In the case of an IL as a solvent, for example, [emim+][NTf2–], the above assumn class="Chemical">ption using TP+TB– cannot be applied directly for the determination of the transfer energy of an ion from the IL to a molecular solvent because, under our component-based scheme for solvation by the IL, dissolution of the salt, TP+TB–, into the IL will yield two new IL components, that is, [TP+][NTf2–] and [emim+][TB–] solvated by the IL (eq ), also supported by a pure entropic mixing behavior of TP+ and TB– in the IL (see the Supporting Information for calorimetric measurements), where pKsp was calculated to be 4.94, as shown in Table .

Table 1

Total Solubilities (C), Mean Activity Coefficients (γ±), and Solubility Product Constants (pKsp) or Solubility Constants (pKs) of Electrolytes and TM in Four Solvents, Including [emim+][NTf2–], on the Molarity Scale at 25 ± 1 °Ca

	[emim+][NTf2–]		MeOH				water–EtOH (80:20 mol/mol)			water
electrolyte or molecule	C/mM	pKsp	C/mM	γ±	αb	pKsp	C/mM	γ±	pKsp	C/mM	γ±	pKsp
TP+TB–	3.4 ± 0.4	4.94j,k				9.04c,d			12.30c,d			17.13c,d
TM	0.4 ± 0.1	3.4 ± 0.2g				3.8h
emim+TB–			3.9 ± 0.1	0.79	0.89	5.12
emim+TB–i	43.3 ± 0.8	1.36g
TP+NTf2–			200 ± 10	0.39	0.60	2.67	3.2 ± 0.2	0.90	5.08
TP+NTf2–i	92.1 ± 0.6	1.04g
Et4N+NTf2–							191 ± 2	0.52	2.00	17.5 ± 0.2	0.88	3.63
emim+Pic–			205 ± 10	0.38	0.60	2.65				57 ± 1	0.82	2.67
emim+NTf2–										46.4 ± 0.4e	0.83	2.83j
Ag+TB–		12.4 ± 0.2j,k				14.4f						17.2f
Bu4N+NTf2–							4.5 ± 0.5	0.88	4.80
Bu4N+NTf2–i	554 ± 20	0.26g
Bu4N+Bu4B–	100 ± 10	2.00	3.1 ± 0.3	0.81	0.91	5.29

Data from this work unless otherwise noted; solubility product constants (±0.10 in pKsp = −log(Ksp in mol2 L–2) unless noted) have been corrected for mean activity coefficients (γ±) calculated by (1) using the Davies equations in molecular solvents or (2) assuming unity in [emim+][NTf2–]; solubility constants (±0.10 in pKs = −log(Ks in mol L–1)) have not been corrected for activity coefficients.

The degree of dissociation of an electrolyte estimated by using a method described in the Supporting Information, where the association constant was assumed to be the average value (63 M–1) of association constants for TP+pic– (54 M–1)[128] and Bu4N+TB– (72 M–1).[61]

Taken from ref (63).

Estimated from the transfer energies of relevant ions and confirmed to be consistent with pKsp for TA+TB–.[61,62]

Calculated using the solubility[127] in the mole fraction scale with an assumption that the density of the solution is the same as that of water (997 g/L).

Taken from ref (129).

As a solubility constant.

Taken from ref (126).

In the cases when an electrolyte yields a single salt upon its dissolution (i.e., eqs S9a and S9b).

The standard state for [emim+][NTf2–] as a solvent was taken to be unit mole fraction.

In the case when an electrolyte yields two salts upon dissolution (e.g., eqs and S1).

Data from this work unless otherwise noted; solubility product n class="Chemical">constants (±0.10 in pKsp = −log(Ksp in mol2 L–2) unless noted) have been corrected for mean activity coefficients (γ±) calculated by (1) using the Davies equations in molecular solvents or (2) assuming unity in [emim+][NTf2–]; solubility constants (±0.10 in pKs = −log(Ks in mol L–1)) have not been corrected for activity coefficients.

The degree of dissociation of an electrolyte estimated by using a method described in the Supporting Information, where the association n class="Chemical">constant was assumed to be the average value (63 M–1) of association constants for TP+pic– (54 M–1)[128] and Bu4N+TB– (72 M–1).[61]

Estimated from the transfer energies of relevant ions and confirmed to be n class="Chemical">consistent with pKsp for TA+TB–.[61,62]

Calculated using the solubility[127] in the mole fraction scn class="Chemical">ale with an assumption that the density of the solution is the same as that of water (997 g/L).

As a solubility consn class="Chemical">tant.

In the cases when an electrolyte yields a single n class="Chemical">salt upon its dissolution (i.e., eqs S9a and S9b).

The standard sn class="Chemical">tate for [emim+][NTf2–] as a solvent was taken to be unit mole fraction.

In the case when an electrolyte yields two n class="Chemical">salts upon dissolution (e.g., eqs and S1).

However, an estimation of the Gibbs transfer energies of single ions is possible if the transfer reactions of so-called pseudo-single ions are defined as shown in eqs and 15c,which are ann class="Chemical">alogous to eqs and 13d, but bridge component- and constituent-based energetics. This definition is based on an assumption that the difference in the solvation energies of TP+NTf2– and emim+TB– by the IL is invoked by the presence of counterions, that is, NTf2– and emim+, respectively. Therefore, elimination of the presence of NTf2– and emim+ from TP+NTf2– and emim+TB–, respectively, affords the net solvation energies of TP+ and TB–, the magnitudes of which are assumed to be the same. Here, we define a combination of ions, for example, in the left-hand side of eq , as a “pseudo-single ion” because this combination represents TP+ stoichiometrically (see the Supporting Information for a rigorous derivation of the transfer energies of pseudo-single ions). Thus, the transfer energies of these two pseudo-single ions are calculated to be 11.7 kJ/mol for eqs and 15c when “sol” is methanol by using the pKsp of TP+TB– in methanol (Table ). In a similar manner, the transfer energies from the IL to water are calculated to both be 34.8 kJ/mol (Table ). Only a few transfer energies of TP+ and TB– have been reported so far,[103−106] for example, in ITIES studies of ion transfer, at an interface between water and tetraalkylammonium (or tetraalkylphosphonium) tetrakis(pentafluorophenyl)borates, which are quite hydrophobic (their pKsp to water was estimated to be <14). Our values compare well with these reported values. More importantly, our classical method does not depend on whether a given IL is miscible with a molecular solvent or not.

Table 2

Estimated Standard Molar Gibbs Energies of Transfer of Ions, Salts, and TM at 25 ± 1 °C between [emim+][NTf2–] (IL, Bold Font) and Relevant Molecular Solvents Involved in Table a

	ΔtrG°/kJ mol–1
species transferred	IL → MeOH	MeOH → water	MeOH → water–EtOH	water–EtOH → water	IL → water	water → CH3CN	IL → CH3CN	IL → water–EtOH	MeOH → CH3CN
TP+	11.7	23.1	9.3		34.8		2.1b	21.0	–9.0b
TB–	11.7	23.1	9.3		34.8		2.1b	21.0	–9.0b
TM	2.3
emim+	9.8	–3.0			6.8
NTf2–	–2.4		4.5	6.7	8.8			2.1
Ag+	−0.3	−7.1			–7.4	–24.1c	–31.5
H+					–36.9 ± 3.3d	45.7 ± 3.3e, 46f	8.8 ± 3.3g
Et4N+				2.6c
Pic–		3.1h
Bu4N+	8.9	21.0i		6.0c		–29.7j		23.9	–8.7
Bu4B–	9.9								–9.3k

Data on the intrinsic chemical potential scale from this work unless otherwise noted; energies (±3.0 kJ/mol)[67] have been calculated by using values shown in Table .

Calculated from the pKsp (5.68)[63] of TP+TB– in acetonitrile.

Ref (67).

Calculated using eq .

Calculated from values for energies of transfer of H+ from the IL to water and CH3CN.

Taken from ref (62) where the value was confirmed by two independent methods: (1) a potential difference between two hydrogen electrodes in water and acetonitrile; (2) a comparison of acidities and solubilities of relevant compounds.

Calculated by using eq .

Ref (130).

Calculated from the pKsp of Bu4N+TB– in both solvents.[131]

Ref (132).

Calculated from the total solubility (304 ± 10 mM) of Bu4N+Bu4B– in acetonitrile with a correction for the degree of dissociation of the salt in the same manner as described in footnote b in Table , where the association constant was assumed to be 20 M–1, as reported[133] for similar electrolytes.

Data on the intrinsic chemicn class="Chemical">al potential scale from this work unless otherwise noted; energies (±3.0 kJ/mol)[67] have been calculated by using values shown in Table .

Calculated from the pn class="Chemical">Ksp (5.68)[63] of TP+TB– in acetonitrile.

Calculated from vn class="Chemical">alues for energies of transfer of H+ from the IL to water and CH3CN.

Taken from ref (62) where the vn class="Chemical">alue was confirmed by two independent methods: (1) a potential difference between two hydrogen electrodes in water and acetonitrile; (2) a comparison of acidities and solubilities of relevant compounds.

Calculated from the pn class="Chemical">Ksp of Bu4N+TB– in both solvents.[131]

Calculated from the totn class="Chemical">al solubility (304 ± 10 mM) of Bu4N+Bu4B– in acetonitrile with a correction for the degree of dissociation of the salt in the same manner as described in footnote b in Table , where the association constant was assumed to be 20 M–1, as reported[133] for similar electrolytes.

For completeness, we can now cn class="Chemical">alculate Gibbs transfer energies of emim+ and NTf2– as single ions from the IL to methanol (denoted as “M”), as shown in eqs and 15e, respectively.For example, the transfer energy for emim+ is calculated to be +9.8 kJ/mol from eq , where values were taken from Tables and 2. The energy for NTf2– is calculated to be −2.4 kJ/mol in the same manner. The value for emim+ may at first appear to be unusual because, in the Born equation in which an ion is more stabilized by a solvent as a dielectric body of which the permittivity is higher, emim+ can be expected to be stabilized more by methanol (εr = 32.6) than by [emim+][NTf2–] (εr = 12.3). However, note that the Gibbs energy of formation of [emim+] in [emim+][NTf2–] contains a contribution from the IL.To verify the above procedure, we tested the transfer energies of emim+ and NTf2– from the IL to methanol by calculation of the Gibbs energy change of dissolution of the IL to water, that is, the solubility of the IL in water (pKsp). This can be considered to be a transfer energy of the salt, emim+NTf2–, from the IL, where [emim+][NTf2–] exists in the IL, to water, where emim+ and NTf2– exist as individual solvated ions (eq ).On the basis of the thermodynamic cycle shown in Scheme with Gibbs transfer energies of emim+ from the IL to methanol, followed by water, and NTf2– from the IL to methanol, followed by aqueous ethanol and then water, as shown in Table , the Gibbs energy change is calculated to be 15.5 ± 3.0 kJ/mol, which is in perfect agreement with the value (16.2 ± 0.5 kJ/mol)[127] obtained from the pKsp value in Table directly determined by solubility measurements. This confirms that our solvation scheme in Figure b is appropriate to describe the medium effect by [emim+][NTf2–], and it enables us to thermodynamically compare solvation by various solvents, now including ILs. In a similar manner, we also estimated the Gibbs transfer energy of Ag+ from the IL to various neutral molecular solvents through the solubility product constants of Ag+TB– determined by potentiometric titrations, as shown in Tables and 2 (see the Supporting Information for details).

Scheme 3

Thermodynamic Cycle Relevant to the Calculation of the Solubility Product Constant of [emim+][NTf2–] (IL) in Water, pKsp

“(IL, x)”, “(IL)”, “(M)”, “(aq)”, and “(aq. ethanol)” in subscripts denote a substance solvated by the IL, the IL, methanol (M), water (aq), and aqueous ethanol (aq. ethanol), respectively. The activities are in molarity units, except where noted by a subscript “x” symbol, in which case the activity is in mole fraction units.

“(IL, x)”, “(IL)”, “(M)”, “(aq)”, and “(n class="Chemical">aq. ethanol)” in subscripts denote a substance solvated by the IL, the IL, methanol (M), water (aq), and aqueous ethanol (aq. ethanol), respectively. The activities are in molarity units, except where noted by a subscript “x” symbol, in which case the activity is in mole fraction units.

Error in the Gibbs Transfer Energy of a “Pseudo-Single Ion” Caused by the TATB Assumption

In molecular solvents, the uncertainty (in both precision and accuracy) of the Gibbs transfer energy of a single ion is thought to be ±3 kJ/mol[67] as long as the vn class="Chemical">alue is derived from reliable experimental data of solubilities and a reliable reference electrolyte assumption, that is, the TATB assumption.

As the TAn class="Chemical">TB assumption in an IL has been applied in this study, for the first time, with a pseudo-single ion, we have evaluated both its precision (self-consistency) and its accuracy (correctness). Testing the transfer energy of emim+ and NTf2– by the experimental solubility of the IL to water is a good method to show the high precision of the TATB assumption in the IL with the pseudo-single ion, treating a series of solubilities of relevant salts. Another good example is, as evaluated later, an excellent agreement of the transfer energy of H+ from water to acetonitrile (45.6 kJ/mol; see Table ), which was derived by using the TATB assumption in the IL with the pseudo-single ion, with that determined previously (46 kJ/mol;[62] see Table ). Unfortunately though, these tests cannot determine the accuracy of the transfer energy of a pseudo-single ion because the former is only related to the transfers of salts (not single ions), and the latter just shows consistency in a series of calculations of conditional liquid junction potential differences.

Theoretically, the absolute accuracy of the n class="Chemical">TATB assumption cannot be clarified by any experimental thermodynamic method. Thus, we chemically deduced whether the concept of the TATB assumption, that is, that the magnitude of solvation for the bulky and symmetrically shaped cation and anion (i.e., TA+, TP+, and TB–) is the same, can prevail in [emim+][NTf2–] by making a comparison between TATB and another pair of bulky and symmetrically shaped ions (Scheme ), that is, tetrabutylammonium (Bu4N+) and tetrabutylborate (Bu4B–), in terms of the relative accuracy. We found that Bu4N+Bu4B– can work as a reference electrolyte because the Gibbs transfer energies of both Bu4N+ and Bu4B– from methanol to acetonitrile have almost the same magnitudes as those for TP+ and TB–, as shown in Table . This result is not surprising as Bu4N+ is found to sometimes show a similar magnitude of solvation to that for TA+. The IL consists of two different bulky ionic molecules, that is, emim+ and NTf2–, and these ionic constituents have the potential to interact with the alkyl group-containing Bu4N+Bu4B– salt in a chemically different manner from the interaction with the TATB salt that comprises only aryl groups. Nevertheless, quite interestingly, we found that the Gibbs transfer energies of Bu4N+ and Bu4B– from the IL to methanol show similar magnitudes to those for TP+ and TB–. This strongly indicates that the TATB assumption can be applied for [emim+][NTf2–] as long as the TATB assumption in conventional molecular solvents is effective. Therefore, we conclude that the uncertainty of the transfer energy of a pseudo-single ion in an IL is the same as that for a single ion in molecular solvents, that is, ±3 kJ/mol.[67]

Scheme 4

Chemical Structures of Bu4N+ and Bu4B–

Estimation of Conditional Liquid Junction Potential Differences

On the basis of the observations described above, we can now evaluate n class="Chemical">conditional or practical liquid junction potential differences (ΔLJE) at interfaces between [emim+][NTf2–] and conventional molecular solvents, including water. In principle, there are two methods to evaluate ΔLJE: (1) the Izutsu three-component method,[59,134] which enables us to quantify each component (diffusion and Gibbs transfer energies of charged species, and solvent–solvent interactions); and (2) a method[135] that takes the difference between an experimental potential difference of two Ag+/Ag electrodes and a potential difference estimated by a Gibbs transfer energy of Ag+. Although the latter method has a disadvantage in that it yields only conditional values, this is still useful as long as the same conditions are used throughout the experiments. In this study, we used the latter method to evaluate ΔLJE caused by diffusion of Et4N+ClO4–, which is used as a typical supporting electrolyte for molecular solvents in electrochemistry.

To estimate ΔLJE, the electrochemical cells shown in n class="Gene">Cells 2a and 2b (“IL” denotes [emim+][NTf2–], “sol” denotes methanol or acetonitrile as a solvent in Cell 2a, and “aq” denotes water as a solvent in Cell 2b)∥ were constructed, where the liquid junction was formed by a Vycor glass frit.# The potential differences of the cells (ECell 2a and ECell 2b) were measured as −152 ± 10 and −431 ± 5 mV at 25 ± 1 °C for Cell 2a where “sol” was methanol and acetonitrile, respectively, and −596 ± 5 mV at 25 ± 1 °C for Cell 2b.These potential differences are composed of three terms: the difference in activities of Ag+ in the IL and molecular solvent, the Gibbs transfer energy of Ag+ from the IL to the molecular solvent, and ΔLJE that is in question. Thus, as the first and second terms are known, the conditional liquid junction potential differences were calculated to be −123 ± 33 and −78 ± 32 mV from the IL to the methanol and acetonitrile solutions used in Cell 2a, respectively, and −74 ± 32 mV from the IL to the aqueous solution used in Cell 2b (see the Supporting Information for details). These liquid junction potential differences clearly indicate that the assumption of an NLJP difference between [emim+][NTf2–] and a molecular solvent cannot be recommended to be used with methanol, acetonitrile, or water as a solvent. In other words, these potential differences have to be taken into account.

Assessment of (Electrode) Potentials in an IL in Comparison with Molecular Solvents

The elimination of conditionn class="Chemical">al liquid junction potential differences formed at interfaces between an IL and a conventional molecular solvent now enables us to assess the thermodynamic basis of electrochemistry using ILs and to bridge the realms of conventional molecular solvents and ILs. Here we discuss (1) the electrode potential of the ferrocenium/ferrocene (Fc+/Fc) redox couple in [emim+][NTf2–], and (2) the absolute intrinsic standard chemical potential of H+ in [emim+][NTf2–] derived from the standard Gibbs transfer energy of a proton from [emim+][NTf2–] to molecular solvents including water.

The Ferrocenium/Ferrocene (Fc+/Fc) Redox Couple

The electrode potential of the n class="Chemical">Fc+/Fc couple is recommended to be used in nonaqueous solvents for the purpose of an internal potential standard. This electrode potential has also been used for making comparisons of potentials measured in different solvents under the assumption that the electrode potential of the Fc+/Fc couple is invariant with solvent type, as long as the solvent dielectric permittivities are similar. This is often referred to as the “Fc assumption”, or more generally this is one of the “reference couple assumptions”.[67] Recent electrocatalysis studies[136−138] using ILs have also used the Fc+/Fc couple to assess their potentials in comparison to those in conventional molecular solvents. However, no estimation of conditional liquid junction potential differences has been made so far. Therefore, we have tested the Fc assumption in [emim+][NTf2–] as a typical IL.

When an electrochemical cell as shown in Cell 3a was n class="Chemical">constructed, in which the liquid junction was formed by a Vycor glass frit□ and the left-hand side of the cell worked as a Ag+/Ag reference electrode in acetonitrile for cyclic voltammetry, the half-wave electrode potential of Fc+/Fc in [emim+][NTf2–] was measured as +36 ± 5 mV at 25 ± 1 °C versus the Ag+/Ag reference electrode. Under our scheme for solvation by an IL ([emim+][NTf2–] here), the electrode reaction occurring on a glassy carbon disc as the working electrode (denoted as “GC”) is represented by [Fc+][NTf2–](IL) + e– → Fc(IL) + [NTf2–](IL,  as depicted in Figure b and eq , and the half-wave potential, E1/2, of Fc+/Fc measured by cyclic voltammetry is equal to eq , where the first term is the standard electrode potential. As the mole fraction can be approximated to unityunder the conditions of cyclic voltammetry, the standard electrode potential (but with the junction potential difference) is then calculated to be +32 ± 5 mV by subtraction of the diffusion term in the IL, (RT/2F) ln(DFc/DFc+), where DFc/DFc+ is found to be 1.37.[139] According to eq and Scheme S3 (see the Supporting Information), the addition of ΔLJE to this half-wave potential yields −46 ± 33 mV as the standard electrode potential of the Fc+/Fc couple in [emim+][NTf2–] without being biased by ΔLJE at the liquid junction as depicted in Cell 3b. Meanwhile, the half-wave potential of Fc+/Fc in acetonitrile was measured as +78 ± 5 mV at 25 ± 1 °C versus the same Ag+/Ag reference electrode by using Cell 4 in which no liquid junction is formed. The formal electrode potential (at I = 0.10 M) is calculated to be +78 ± 5 mV by subtraction of the diffusion term in acetonitrile, (RT/2F) ln(DFc/DFc+), where DFc/DFc+ is found to be 1.03.[140] This potential is clearly different by +124 ± 33 mV from −46 ± 33 mV that was calculated above in [emim+][NTf2–] without bias from ΔLJE. Therefore, this indicates that the Fc assumption can be used by taking this difference into account.

Absolute Intrinsic Standard Chemical Potential of the Proton

The determination of the standard electrode potentin class="Chemical">al of Fc+/Fc in Cell 3b enabled us to calculate the standard electrode potential of Fc+/Fc in [emim+][NTf2–] versus SHE in water. Previously,[141] we discussed electrode potentials of Fc+/Fc in acetonitrile versus SHE, which is contacted with the acetonitrile phase through various types of liquid junctions, and suggested +509 ± 32 mV as a formal electrode potential of Fc+/Fc at I = 0.02 M in acetonitrile versus SHE, in which the conditional liquid junction potential difference is eliminated. This potential is converted to +499 ± 32 mV for a formal potential of Fc+/Fc at I = 0.10 M on the basis of Kolthoff and Thomas’s observation[142] and supported by Roberts’ recent observation.[143] As the standard electrode potential and the formal electrode potential (I = 0.10 M) of Fc+/Fc in Cells 3b and 4 were determined to be −46 ± 33 and +78 ± 5 mV, respectively, we suggest +375 ± 33 mV as the standard electrode potential of Fc+/Fc in [emim+][NTf2–] versus SHE in water, in which the conditional liquid junction potential difference is eliminated (Cell 5). This value is in good agreement with the value of +361 ± 32 mV for this potential, calculated from the standard potential of the Ag+/Ag electrode as shown in eq (E° = +515 ± 8 mV vs E°(Fc+/Fc) in the IL), the standard electrode potential of Ag+/Ag in water (E° = +799 mV vs SHE[144]), and the standard molar Gibbs energy of transfer of Ag+ from water to the IL (+7.4 ± 3.0 kJ/mol, Table ). Thus, it will be useful for comparing the overpotentials of given electrocatalytic reactions in water to those in [emim+][NTf2–] IL.This electrode potential affords the standard transfer energy of a proton as shown in eq , which contains a species similar to [emim+][Ag(NTf2)2–], formed upon dissolution of a silver salt because the proton is also known to form a homoconjugate with two conjugate bases (i.e., anions) in ILs.[145,146,72,75] This energy is an important value to thermodynamically relate a Brønsted acidity scale in an IL, that is, [emim+][NTf2–] here, to a unified pH scale for all phases.[110]Therefore, we can estimate the standard electrode potential of a hydrogen electrode in [emim+][NTf2–] (eq ) from the formal potential, which was reported to be −65 ± 5 mV versus Fc+/Fc at the standard states of 1 M of H2 and H(NTf2)2–.[69],○ In this case, the activity coefficients of H2 and H(NTf2)2– are reasonably assumed to both be unity because (1) unlike for CO2 gas, the standard solubility (or the Henry constant) of H2 gas in [emim+][NTf2–] was found to be almost independent of the H2 pressure;[147,148] and (2) H+ is sterically shielded by two NTf2– anions, which would be more crowded than in the case of Ag(NTf2)2–, which itself exhibited an activity coefficient of unity as shown in Figure a. Then, using a saturating concentration of H2 at 25 °C, which is estimated to be 3.6 mM as an average of two reported values (2.6[147] and 4.6[69] mM), the standard electrode potential is calculated to be +7 ± 5 mV versus Fc+/Fc in the IL at the standard states of 1 bar of H2 and 1 M of H(NTf2)2– (see the Supporting Information for details of this calculation).△ Thus, the standard electrode potential of eq is calculated to be +382 ± 34 mV versus SHE (in which the conditional liquid junction potential difference is eliminated). From this, the standard Gibbs energy change for the reaction shown in eq is +36.9 ± 3.3 kJ/mol.Thus, the intrinsic standard Gibbs transfer energy of a proton from water to the IL can be estimated to be +36.9 ± 3.3 kJ/mol (eq ) because the Gibbs energy changes of formation of dihydrogen in the IL and water at the standard state of 1 bar are defined to be equal. In a similar manner, the intrinsic standard Gibbs transfer energy of a proton from acetonitrile to the IL (eq ) can be estimated to be −8.8 ± 3.3 kJ/mol (see the Supporting Information for details).It is quite noteworthy that the difference (+45.7 ± 3.3 kJ/mol) between the two transfer energies of a proton from the IL to water and acetonitrile is in excellent agreement with the transfer energy of a proton from water to acetonitrile (+46 kJ/mol;[62] see also Table ). Table shows the standard Gibbs transfer energies of a proton from the IL to methanol and acetonitrile, in addition to water on the intrinsic chemical potential scale. Accordingly, the intrinsic standard transfer energy of a proton from the IL to acetonitrile is a small positive value, whereas that from the IL to water is a large negative value, that is, the intrinsic standard chemical potential in the IL is similar to that in acetonitrile and much more positive than that in water. This indicates that a proton is much less stabilized by the IL compared to that in water, and the magnitude is similar to that in acetonitrile. This is consistent with previous observations,[73,76] wherein values of pKa in acetonitrile and [bmim+][NTf2–] were found to be similar.

Considering the intrinsic sn class="Chemical">tandard chemical potential of a proton in the IL enables us to firmly anchor the acidity of a proton in the IL to the unified pH scale for all of the phases (pHabs or pHabsH).[110,134] pHabs and pHabsH define the standard Gibbs energy of formation of H+ in the gas phase[110] and in water,[134] respectively, to be zero, and the intrinsic single-ion solvation free energy of a proton in water is −1105 ± 8 kJ/mol,[56] for the dissolution of 1 mole of H+ in the gas phase at a pressure of 1 bar into water to a concentration of 1 M. We can thus estimate the intrinsic standard chemical potential of a proton in [emim+][NTf2–] to be −1068 ± 9 kJ/mol (pHabs = 187 ± 2 or pHabsH = −6.5 ± 0.6). Interestingly, this value is comparable to the theoretically estimated intrinsic value of −1071 kJ/mol[80] for a proton in 1-butyl-3-methylimidazolium NTf2–.

Although the intrinsic transfer energies and n class="Chemical">conditional liquid junction potential differences we acquired in this study are values for one particular IL ([emim+][NTf2–]), our methods and observations are general and enable the experimental thermodynamic investigation of solution chemistry and electrochemistry of all ILs and permit, for the first time, accurate comparison on the basis of activity with observations in conventional molecular solvents under the scheme of a reference electrolyte assumption, that is, the TATB assumption.

Conclusions

Modern ILs, which arose a few decades ago, have been found to exhibit attractive properties for extraction, separation, synthesis, catalysis, electrodics, and so on. Therefore, ionics (where “ionics” refers to the physical chemistry of ions in solution and in liquids arising from molten solids composed of ions)[1] has also been studied in ILs with the goal of understanding, prescribing, and engineering their properties. However, these studies are either semiempirical, concentration-based (i.e., not activity-based), or require a specific IL. This situation originates from the absence of a good exemplar for defining the thermodynamic standard states of ionic solutes in ILs. Until now, this has prevented us from applying thermodynamic principles to ionics in ILs.

In this study, we have proposed a standard component-based solvation scheme for an ionic solute in an IL, in which the solute becomes an integral part of the IL to yield two additional components (not two constituents) of the IL phase. This is strikingly different from solvation in molecular solvents of high permittivity, in which each ionic constituent is completely solvated by the solvent molecules. Thus, an IL represents the limiting case of zero molecular solvent. The effectiveness of this solvation scheme was justified by four observations: (1) an ideal Nernstian response of the electrochemical potential of a Ag+/Ag electrode in [emim+][NTf2–]; (2) ideal Nernstian responses (at 25 ± 1 and 39 ± 1 °C) of the Fc+/Fc redox couple to a ratio of concentrations of Fc+, Fc, and importantly the IL; (3) observations of ideal Nernstian responses at higher temperatures for the Fc+/Fc couple in ILs having even longer alkyl chains in the imidazolium ring; and (4) pure entropic mixing of two solutions of IL each containing TP+ and TB– salts, respectively. This confirmation allowed us to determine the stoichiometric number of ionic constituents of the IL involved in the dissolution of electroactive electrolytes, the activity coefficients of which can be set to unity in the case of the IL, [emim+][NTf2–], at 25 ± 1 °C or higher temperatures, from which the standard electrode potential of a Ag+/Ag electrode in [emim+][NTf2–] (eq ) was also determined, for the first time, to be 515 ± 8 mV versus E°(Fc+/Fc). An activity-based determination of the solubility product constants of electrolytes resulted in the assessment of intrinsic Gibbs transfer energies of emim+ and NTf2– from an IL to water (or actually the intrinsic Gibbs transfer energy of relevant salts), completely describing the solubility of the IL in water.

Meanwhile, the concen class="Chemical">pt of the Gibbs transfer energy of a “pseudo-single ion” from an IL to a molecular solvent, which bridges component- and constituent-based energetics, has been established by considering two constituent ions of [emim+][NTf2–], wherein the solute cation and anion are accompanied by [NTf2–] and [emim+], respectively, in terms of electroneutrality and stoichiometry. This type of consideration is an extension of the concept of the hydration or solvation number for a single ion in the case of a molecular solvent, wherein such a number is not necessarily required to experimentally determine the Gibbs transfer energy of a single ion in terms of electroneutrality and stoichiometry. We then found, for the first time, that a commonly used reference electrolyte assumption, that is, the TATB assumption, which assumes the same magnitudes of solvation for the bulky and symmetrically shaped TA+ and TB– ions in molecular solvents, can also prevail in the IL by a comparison of the intrinsic Gibbs transfer energies of H+.

Using the TAn class="Chemical">TB assumption with this concept under our model of solvation, we are now able to assess ionics in ILs with the usual thermodynamic conventions to compare with ionics in molecular solvents. For example, a thermodynamic comparison of a given reaction (e.g., the one-electron-reduction of ferrocenium to ferrocene) in different solvents (an IL and other types of solvents in this case) was achieved by eliminating the conditional liquid junction potential difference, which is composed of the inner potential difference (solvent–solvent interactions) and electrolyte-dependent terms. On the basis of this comparison, we also discussed the intrinsic standard chemical potential of H+ in [emim+][NTf2–] as a representative IL and clarified that the proton is more acidic than in water by 6.5 ± 0.6 units on the unified[110,134] pH scale. We believe that this work has provided a key to the realm of the thermodynamics of ionics in ILs and strengthens progress on activity-based ionics in ILs with a rigorous thermodynamic basis.

Experimental Section

General

Electrochemical measurements were n class="Chemical">conducted with a BioLogic SP-200 potentiostat. Cyclic voltammograms and Osteryoung square-wave voltammograms were recorded in deaerated IL and acetonitrile solutions of ferrocene under N2 atmospheres at 25 ± 1 °C. A standard three-electrode cell was used, which consisted of a disc electrode (glassy carbon, 1 mm diameter) as a working electrode, a platinum wire as a counter electrode, and a reference electrode as indicated in Cells 3a and 4, in which liquid junctions were formed by a Vycor frit on the bottom of a glass tube retaining a silver wire and an electrolyte solution. Half-wave potentials of Fc+/Fc were confirmed to not vary with scanning rate in the voltammetric measurements. NMR spectra were measured on a JEOL ECA600 NMR spectrometer. Isothermal titration calorimetry was performed on a TA Instruments Nano ITC standard volume 601000 instrument equipped with two symmetrical cells for the titrand and the reference (0.95 mL each) and an automatic injection syringe for the titrant. The instrument was first calibrated by measuring the heat of ionization of 2-amino-2-hydroxymethyl-propane-1,3-diol (Tris; Sigma-Aldrich Trizma Base, T1503) with hydrochloric acid in aqueous solution and by the mixing of water to water. In each experimental run, an IL solution (typically 2.0 mM) as a titrand (prepared in a glove box) was loaded into the reaction cell, which was thoroughly purged with N2 until thermally stabilized in advance. The injection syringe was filled with an IL solution (typically 20 mM) as a titrant and then inserted into the cell under N2 flow, after which the cell was allowed to reach equilibrium at 25.00 °C for typically 30 min. After that, 40 μL of the titrant was automatically injected once every 5 min a total of five times. After each injection, the heat evolved resulting from the injection was recorded once per second and integrated. Some solutions were analyzed by using an HPLC system with a Tosoh TSKgel SP-2SW cation exchange column at 35 °C, a Shimadzu LC-20ADsp pump with a Rheodyne 7725i injector, and a Shimadzu CDD-10Asp conductivity detector. A mixed solution of acetonitrile/NaH2PO4 (0.1 M) buffer (7:3 v/v) was used as the eluent.

Materials

A silver wire (>99.99% from n class="Chemical">Aldrich) was coiled and washed with dilute nitric acid before use. Tetraphenylphosphonium tetraphenylborate (TP+TB–),[63] 1-ethyl-3-methylimidazolium tetraphenylborate (emim+TB–),[149,150] tetraphenylphosphonium bis(trifluoromethanesulfonyl)imide (TP+NTf2–),[151] tetraethylammonium bis(trifluoromethanesulfonyl)imide (Et4N+NTf2–),[152] tetrabutylammonium bis(trifluoromethanesulfonyl)imide (Bu4N+NTf2–),[153] tetrabutylammonium tetrabutylborate (Bu4N+Bu4B–),[154] ferrocenium bis(trifluoromethanesulfonyl)imide (Fc+NTf2–),[155−157] and [Ru(bpy)2(dmb)]2+(PF6–)2[158] were prepared according to the reported methods. 1-Ethyl-3-methylimidazolium picrate (emim+pic–) was prepared in a similar manner to the reported procedure.[159] 1-Methylimidazole was distilled in vacuo before use. Methanol (Aldrich, anhydrous) and ethanol (Kanto, anhydrous) were vacuum-transferred before use. Tetraethylammonium perchlorate (electrochemical grade in assay of >99.0%) was used as received from Nacalai Tesque. Lithium bis(trifluoromethanesulfonyl)imide (LiNTf2, >99.7%) was purchased from Kanto Chemical Co., Inc. Water was purified by an Advantec (RFD240NA) Water Distillation Apparatus (Toyo Roshi Kaisha, Ltd.). Other chemicals were obtained from commercial sources and used without purification.

1-Ethyl-3-methylimidazolium Bis(trifluoromethanesulfonyl)imide ([emim+][NTf2–])

This IL was prepared several times by a modification of the reported procedure.[160] The tyn class="Chemical">pical procedure is as follows. 1-Methylimidazole (19.8 g, 0.241 mol) and bromoethane (35.0 g, 0.321 mol) were added to acetonitrile (10 mL), and the solution was heated at 50 °C for 30 h under a N2 atmosphere. After confirmation of the disappearance of 1-methylimidazole, acetonitrile was removed, resulting in solidification of the white bromide salt of emim+. 1.25 g of activated charcoal (grade for column chromatography from Nacalai Tesque) was added to a 125 g aqueous solution of the bromide salt. The mixture was heated up to 80 °C for 30 min and then allowed to cool down to room temperature overnight on a benchtop. The mixture was first passed through a membrane filter made of polypropylene and then through a membrane filter made of Teflon with a porosity of 0.1 μm, to remove the charcoal powder. To the resulting colorless and transparent solution in the flask, an aqueous LiNTf2 solution (ca. 50 mL prepared by dissolution of 69.2 g of LiNTf2 (0.241 mol) in water, followed by filtration through a membrane filter made of polypropylene with a porosity of 0.22 μm) was added. The mixture was shaken well and allowed to stand for a while until it formed a binary phase consisting of aqueous and IL layers. After observation of the binary phase in the flask, the top aqueous layer was removed, and deionized water (ca. 50 mL) was added to the flask, and then the mixture was shaken well again to remove lithium bromide from the IL phase. This procedure for the removal of lithium bromide was repeated (typically 10 times) until a negative AgNO3 test of the top aqueous layer indicated that the bromide was absent from the IL phase. It was then repeated five more times to absolutely ensure the complete absence of bromide. The residual water on and in the resulting IL was removed in vacuo at 60 °C for 72 h and thereafter quantified as less than 10 ppm by a Karl-Fischer titration. Yield: 53 g (56%). Density: 1.52 g/mL at 25 °C (by pycnometry); lit.:[161] 1.517 g/mL. UV–vis absorption (l = 1 cm; vs water): 0.35 at 350 nm, 0.01 at 400 nm, and 0.005 at 450 nm. Electrochemical window (Pt foil used as the working electrode at j < 1 mA/cm2 under an argon atmosphere): 4.2 V; lit.[162] 4.2 V.

Silver Bis(trifluoromethanesulfonyl)imide (AgNTf2)

This compound was prepared by the reported procedure[163,164] from n class="Chemical">silver carbonate (>95% from Wako Pure Chemical Industries Ltd.) and bis(trifluoromethanesulfonyl)amine (>98% from Wako Pure Chemical Industries Ltd.) and precipitated from the ethereal solution with anhydrous hexane under an argon atmosphere with a yield of 50%. Anal. Calcd for C2AgF6NO4S2: C, 6.19%; N, 3.61%. Found: C, 5.93%; H, 0.06%; N, 3.88%. This is a deliquescent fine white powder, and, therefore, it was stored under an argon atmosphere at room temperature in the dark.

Potentiometric Titrations

The titrations were performed by using a combination of a syringe holding a titrant and a custom-made, air-tight, jacketed electrochemicn class="Chemical">al cell holding a magnetic stirring bar and a titrand at 25 ± 1 °C (temperature maintained by a water circulator). The main compartment of the cell has five top ports, which are sealed with Teflon–silicone rubber gaskets in GL14-threaded fittings when unused. The main compartment held a neat IL or an IL solution (typically 1.515 g determined by weight). The five ports were usually fitted with: (1) a silver coil, (2) a platinum coil, (3) a glass tube retaining a Vycor frit to isolate an IL solution of [Ru(bpy)2(dmb)]2+(PF6–)2 (denoted as “Ru2+”), into which a glassy carbon electrode was immersed to act as an internal redox standard, (4) a glass tube retaining a Vycor frit (or a sintered glass frit with a porosity of 5 μm) to isolate an IL solution of AgNTf2 (ca. 1 mM), into which a silver coil was immersed to act as a reference electrode, and (5) two Teflon tubes: one supplying a stream of N2 in and out of the cell and the other attached to the syringe needle. After addition of a certain volume of the titrant, the solution in the main compartment was stirred well for at least 15 min, and the potential difference between ports 1 and 4 was measured, wherein port 4 was treated as the left-hand side of Cells 1a, 1b, and S1. During this measurement, the electrochemical potential of port 4 was calibrated occasionally against the half-wave potential[165] of the redox couple of the ruthenium complex, E1/2(Ru3+/Ru2+), in port 3 measured by cyclic voltammetry using ports 2, 3, and 4 as counter, working, and reference electrodes, respectively. Finally, this potential versus E1/2(Ru3+/Ru2+) in the IL was first converted to a value versus the half-wave potential of the ferrocenium/ferrocene (Fc+/Fc) couple[139] in the IL and then to a value versus the standard electrode potential of the Fc+/Fc couple according to eq , wherein the term for the ratio of diffusion coefficients (DR/DO) is reported as 1.37, and the terms for the mole fraction can be approximated to unity.[139] The ratio of diffusion coefficients (DR/DO) at 39 ± 1 °C was determined to be 1.3 by the Shoup–Szabo method[166] using Fc and Fc+NTf2– solutions. For the experiment as depicted in Figure b, only ports 2, 4, and 5 were used. The actual concentration [mol/L] of Ag(NTf2)2– and the mole fraction of NTf2–, with regard to all of the anionic species in the IL solution in the main compartment to which the titrant was added dropwise were calculated by the method described below.

Measurements of Potential Differences of Cells 2a and 2b

These potential differences were measured by two methods: (1) direct measurement; and (2) indirect measurement through the hn class="Chemical">alf-wave potential of the ferrocenium/ferrocene couple in the IL as follows. In the case of Cell 2a in which “sol” is methanol or acetonitrile, the half-wave potentials were determined in reference to a Ag+/Ag electrode in the IL (Cell 6a) and a Ag+/Ag electrode in acetonitrile (Cell 6b), and then the potential difference of Cell 2a was calculated by subtraction of the latter half-wave potential from the former half-wave potential. In the case of Cell 2b, the potential difference was calculated in the same way, except with the use of Cells 6a and 6c. This indirect method afforded a value in agreement with that measured directly, within the experimental error of the measurements as denoted in each potential difference.

Calculation of Concentration and Mole Fraction in Potentiometric Titrations

To interconvert between the n class="Chemical">concentration [mol/L] of Ag(NTf2)2– and the mole fraction of NTf2– with regard to all of the anionic species in the IL solution in the main compartment, to which the titrant was added dropwise, the molar volume of [emim+][Ag(NTf2)2–], which is formed upon dissolution of AgNTf2 in [emim+][NTf2–],[121,123] was first measured as 0.15 L/mol at a molality of 0.660 mol/kg at 25 ± 1 °C. The molar volume of [emim+][TB–] was not measured because we prepared ca. 1 mM of emim+TB– solution, at which the mole fraction of [emim+][NTf2–] can still be approximated to unity. The molar volume of [Fc+][NTf2–] was measured as 0.30 L/mol at a molality of 0.660 mol/kg at 25 ± 1 °C. For the experiment as depicted in Figure a in the text, an emim+Ag(NTf2)2– solution and neat [emim+][NTf2–] were placed in the syringe and the main compartment, respectively. For the experiment as depicted in Figure S2, an emim+TB– solution and an emim+Ag(NTf2)2– solution were placed in the syringe and the main compartment, respectively. To calculate the concentration, c [mol/L], and the mole fraction, xIL, eqs and 23 were used, respectively, wherein the molar volume of emim+Ag(NTf2)2– or emim+TB– was assumed to be constant upon dilution of a titrant (vt [L] in the range of 100–400 μL of ct [mol/L] with density of dt [g/L]) with a titrand, the initial weight, wd [g] (initial moles of NTf2– anion, md [mol]), of which was typically 1.515 g (0.003520 mol). In eq , d is the density of pure [emim+][NTf2–] (1517 g/L).[161] In eq , mt′ is the number of moles of [emim+][NTf2–] in the titrant, MIL is the molar mass constant of [emim+][NTf2–] (391.31 g/mol), and MAgNTf is the molar mass constant of AgNTf2 (388.02 g/mol). For the experiment as depicted in Figure b, instead of using the syringe, solid Fc+NTf2– (mFc+ [mol]) was added to a solution of the IL (md [mol]) containing Fc (mFc [mol]) in the main compartment under nitrogen flow. In this case, the ratio of C([Fc+][NTf2–]) to C(Fc) and the mole fraction, xIL, were calculated by eqs and 25, respectively.

Solubility Measurements

When the solvent was [emim+][NTf2–], the totn class="Chemical">al solubility of a given salt was determined by measuring a 1H NMR spectrum of the solution containing an internal standard. A sealed glass tube containing D2O was placed in an NMR tube equipped with a J-Young valve, and, in a glove box, the salt, [emim+][NTf2–], and an [emim+][NTf2–] solution of dimethylsulfone as an internal standard were weighed and sealed in the NMR tube under an argon atmosphere. The amount of dimethylsulfone solution added was adjusted in order not to exceed one-tenth of the final concentration of the salt. The tube was allowed to stand at 25 ± 1 °C overnight. After observation that a small amount of solid salt was still left inside the tube, the tube was loaded into the spectrometer at room temperature, and the NMR spectrum was measured as soon as possible. A change of the density upon the dissolution was corrected for when the final concentration exceeded 0.1 M. This experiment was repeated at least twice using different final concentrations of dimethylsulfone (including none) to estimate the experimental error and confirm that the presence of dimethylsulfone does not affect the solubility more than the experimental error. When the solvent was methanol, water–ethanol, or water, the total solubility of a given salt was determined by measuring a UV–vis absorption spectrum of the solution. A solution saturated with the salt of interest at 25 ± 1 °C was diluted by a certain ratio, and the spectrum was recorded. The concentrations were calculated from molar absorption coefficients of TB– (2.12 × 103 M–1 cm–1 at 274 nm in methanol[128]) and pic– (1.44 and 1.56 × 104 M–1 cm–1 at 355 nm in methanol and water, respectively[128]) and corrected to the solubility. The molar absorption coefficient of TP+ was unavailable, and so its value was determined to be 3850 ± 40 M–1 cm–1 at 275 nm in water–ethanol (80:20 mol/mol) by using a solution of TP+NTf2– of a known concentration. Et4N+NTf2– in water and aqueous ethanol were analyzed using an HPLC system. The total solubilities of Bu4N+NTf2– and Bu4N+Bu4B– were determined volumetrically.

Calculation of Solubility Product Constant

The solubility product consn class="Chemical">tant was calculated from the mean activity, which is a product of the mean activity coefficient, γ±, the total solubility, C [mol/L], and the degree of dissociation, α (taken to be unity,[167] except in methanol as discussed below), as shown in eq , wherein “sol” denotes the solvent used. In eq , values of γ± were estimated by using the Davies equation[61,168] (eq ), which is one of the extended Debye–Hückel equations, where εr is the relative permittivity (32.7, 35.9, 78.3, and 54.5 in methanol,[169] acetonitrile,[170] water,[171] and water–ethanol (80:20 mol/mol)[172] as “sol”, respectively), and T is the temperature (298.15 K). In the case of methanol, γ± and α were iteratively calculated by using eqs and 28(167) where KA is the association constant, which was assumed to be the averaged value (63 M–1) of association constants for TP+pic– (54 M–1)[128] and Bu4N+TB– (72 M–1).[61]