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DOI: 10.1039/B311624G (Paper) Phys. Chem. Chem. Phys., 2004, 6, 124-128

Activity coefficients in mixed electrolyte solutions†

Francesco Malatesta *, Chiara Fagiolini and Roberto Franceschi
Dipartimento di Chimica e Chimica Industriale dell’Università di Pisa, via Risorgimento 35, 56126, Pisa, Italy. E-mail: franco@dcci.unipi.it; Fax: +39 050 2219260; Tel: +39 050 2219258

Received 23rd September 2003 , Accepted 7th November 2003

First published on 1st December 2003

Abstract

An experimental method is illustrated, which opens up an effective line of attack in the fundamental and practically unexplored field of not-necessarily-dilute, not-necessarily-low-charge, multiple electrolytes. The method is based on the idea that ions of a sufficiently enhanced hydrophobic character (typically, charged metal complexes and organic or partially organic ions) can be studied by means of liquid membrane cells without undergoing any interference by highly hydrophilic ions such as SO42−, OH−, H+, Mg2+, La3+, etc. Proofs are given for hexacyanocobaltate(III) and tris(ethylenediamine) cobalt(III) salts. As a first application, the method is used to find otherwise unknown activity coefficients of trace electrolytes in swamping ionic bulks – the typical situation encountered in equilibrium, kinetic constant, and standard potential determinations. Observed salt effects do not prove in any case to be trivial, and in particular, the activity coefficients of 3:3 salts in 2:2 and 3:2 ionic bulks are found to be completely different from expectations.

Introduction

Notoriously, the pH of the NBS/NIST 0.025 M KH2PO4–0.025 M Na2HPO4 standard buffer which every student would predict at 7.2 actually stands at 6.865.1 The discrepancy, a hydrogen ion concentration of about twice the expected, is entirely due to the activity coefficients. Yet, although the influence of activity coefficients may be dramatic, the level of insight into their behavior in the real reaction bulks, mixed electrolyte solutions often embodying high-charge ions, is still poor today. Only low-charge single electrolytes, in fact, have been faithfully covered for a long time by theory and experiments. Before the recent measurements from our own laboratory, even the activity coefficients of 2:2 single electrolytes were controversial due to the lack of data in the dilute regions, and no data at all were available for 3:3, 3:2 and 2:3 salts. Liquid membrane cells have now led such high-charge single electrolytes to be satisfactorily covered by experiments, although a theoretical analysis as advanced as they would deserve is still awaited.2 However, careful knowledge of single electrolytes and the ability to predict their behavior by theory does not solve the problem which mainly concerns chemists, i.e., how to extend the predictions to multicomponent solutions and test their correctness. Multicomponent solutions have so far been resistant to any significant progress since the situation described in Harned and Robinson's book,3 and the paucity of experimental information frustrates any sound attempt to develop general theoretical models.

When introducing the method of liquid membrane cells, which solved the problem of high-charge single electrolytes, Malatesta and Carrara maintained that their liquid membranes were intrinsically unable to make any selection between ions of the same sign, and therefore, that the measurements were only feasible for single electrolytes.4 However they were wrong, as they had forgotten that aqueous phases, unlike organic membranes, are always ion-selective bulks. As indifferent as the organic phase may be to the substitution of some ions for others of the same sign, the different water–ion interaction energies can lead a membrane to be permeable to only one ion of the aqueous solution, the less hydrophilic one among those of the right sign.

We shall demonstrate that these conditions are actually met in several cases, and therefore, liquid membrane cells make it possible to determine the activity coefficients of a variety of electrolytes in mixed solutions. Results are examined for K3[Co(CN)6] in K2SO4 and KOH, tris(ethylenediamine)cobalt(III) chloride–[Co(en)3]Cl3– in HCl, MgCl2, and LaCl3, and finally for [Co(en)3][Co(CN)6] and [Co(en)3][Fe(CN)6] in MgSO4 and La2(SO4)3, particularly interesting because of the cations and anions all carrying high charges. In this first approach, aimed at introducing the method, the experiments have been limited to identifying, for each mixed solution, the activity coefficients of one electrolyte of nearly null concentration in a varying concentration of the other. However, all desired mixing patterns are equally well allowed, e.g. changing the ratio of concentrations at a constant ionic strength, or the ionic strength at a constant concentration ratio, etc.

Experimental

Our cation-exchange liquid membranes are made of organic solutions (in an oil solvent non miscible with water, usually 2-nitrophenyl-octyl-ether, NPOE) of appropriate M[TFPB]z salts, Mz+ being the target cation and [TFPB]− the hydrophobic tetrakis[3,5-bis(trifluoromethyl)phenyl]borate ion (C8H3F6)4B−, which gives salts extremely insoluble in water, soluble and appreciably ionized in organic solvents. The anion-exchange membranes are made of the [TDA]xX salts, Xx− being the target anion and [TDA]+ the tetradodecylammonium ion (C12H25)4N+, which also yields salts extremely insoluble in water. [TDA]3[Co(CN)6] used in this study, like [TDA]3[Fe(CN)6], is a thick, electrically conductive liquid at room temperature and does not need any additional solvent, while other salts, e.g. [TDA]Cl also used in this study, are dissolved in suitable organic oils, here, 3,3,4,4,5,5,6,6,7,7,8,8,8-tridecafluoro-1-octanol.

Structure of cells and membrane electrodes have already been described in papers devoted to single electrolytes,2,4 the only (although fundamental) difference consisting in the solutions examined, which contain two electrolytes instead of one single electrolyte. Additional information concerning salt preparation and apparatus is provided in the ESI sections 1 and 2.†

Fundamentals

Let consider a chain of phases constituting a cell like those used for single electrolytes, but containing two salts, MaXb (Mz+,Xx−) and CcYd (Cs+,Yy−), in the central section,
 
(−) Ag, AgCl |KCl (m0) |K+| KxX (m1) |Xx−| MaXb(m),CcYd(m′) |Mz+| MClz(m2) |AgCl, Ag (+)(cell 1)
where |K+|, |Xx−|, and |Mz+| denote membrane phases only permeable to the corresponding ions. Molal concentrations m0, m1 and m2 are kept constant. If dq charges (dn[thin space (1/6-em)]=[thin space (1/6-em)]F−1dq mol of charges) are allowed to pass throughout the cell at constant T, p, and composition, the overall change of free enthalpy dG is F−1dq{−μ[KCl(m0)][thin space (1/6-em)]+[thin space (1/6-em)]x−1μ[KxX(m1)][thin space (1/6-em)]−[thin space (1/6-em)](az)−1μ[MaXb(m, m′)][thin space (1/6-em)]+[thin space (1/6-em)]z−1μ[MClz(m2)]}, where μ[MaXb(m,m′)] is the chemical potential of MaXb in the mixed solution. The electromotive force, E[thin space (1/6-em)]=[thin space (1/6-em)]−dG/dq, can therefore be written as E[thin space (1/6-em)]=[thin space (1/6-em)](azF)−1μ[MaXb(m,m′)][thin space (1/6-em)]+[thin space (1/6-em)]constant. By expanding μ[MaXb(m,m′)] and then collecting all constant terms in a sort of standard potential, E*, one obtains:
 
E[thin space (1/6-em)]=[thin space (1/6-em)]E*[thin space (1/6-em)]+[thin space (1/6-em)]RT(a[thin space (1/6-em)]+[thin space (1/6-em)]b)(azF)−1ln[thin space (1/6-em)](mγ±)(1)
where γ± is the mean activity coefficient of MaXb in the mixed solution.‡

The problem is whether it is really possible to obtain an |Xx−| membrane so insensitive to Yy−, and an |Mz+| membrane so insensitive to Cs+, for eqn. (1) to hold over a wide range of concentration ratios. (The original considerations that led us to believe this problem to be resolvable, are discussed in the ESI section 3.†) Closely related to this problem, is how to demonstrate that such a condition has really been met. We reached a sound proof for Xx−[thin space (1/6-em)]=[thin space (1/6-em)][Co(CN)6]3− matched with Yy−[thin space (1/6-em)]=[thin space (1/6-em)]SO42− or OH−, and for Mz+[thin space (1/6-em)]=[thin space (1/6-em)][Co(en)3]3+ matched with Cs+[thin space (1/6-em)]=[thin space (1/6-em)]H+, Mg2+, and La3+. (Only one case is discussed in outline hereafter, the others being treated in the supplementary information.)

Consider a variant of (cell 1),

 
Ag, AgCl | KCl(m0) |K+| K3[Co(CN)6](m1) |[Co(CN)6]3−|

…K3[Co(CN)6](m), K2SO4(m′)…

|K+| KCl (m0) | AgCl, Ag

(cell 2)
consisting of two membrane electrodes, for [Co(CN)6]3− and K+ respectively, dipping in the mixed solution. No problems are encountered on the right-side electrode since only one kind of cation, K+ leaving the water's H+ aside, is involved. As to the left-side electrode, the terminal membrane |[Co(CN)6]3−| is possibly affected by the exchange reaction 2 [Co(CN)6]3− (org)[thin space (1/6-em)]+[thin space (1/6-em)]3 SO42− (aq)[thin space (1/6-em)]⇆[thin space (1/6-em)]3 SO42− (org)[thin space (1/6-em)]+[thin space (1/6-em)]2 [Co(CN)6]3− (aq). Such a situation is the typical one to which the Nicolsky equation of the ion selective electrodes applies,5 yielding the left electrode potential ϕleft to depend on the ion activity of both [Co(CN)6]3− (mγ) and SO42− (m′γ′) and the relevant charges,
 
ϕleft[thin space (1/6-em)]−[thin space (1/6-em)]const.[thin space (1/6-em)]=[thin space (1/6-em)]−(RT/3F)ln[mγ[thin space (1/6-em)]+[thin space (1/6-em)]Kij(m′γ′)3/2](2)
with Kij[thin space (1/6-em)]=[thin space (1/6-em)]constant. Eqn. (2) is not liable to direct verification since the individual ion activities are intrinsically unknowable. However, keeping m′ constant and m[thin space (1/6-em)]≪[thin space (1/6-em)]m′ (e.g., m[thin space (1/6-em)]<[thin space (1/6-em)]10−2m′) the activity coefficients become practically independent from m, and constant. Therefore, −(RT/3F)ln[mγ[thin space (1/6-em)]+[thin space (1/6-em)]Kij(m′γ′)3/2] transforms into P[thin space (1/6-em)]−[thin space (1/6-em)](RT/3F)ln(m[thin space (1/6-em)]+[thin space (1/6-em)]Q), P and Q being constants. The right-side electrode potential, ϕright, also becomes constant. Thus, the cell's emf E[thin space (1/6-em)]=[thin space (1/6-em)]ϕright[thin space (1/6-em)]−[thin space (1/6-em)]ϕleft takes the form:
 
E[thin space (1/6-em)]=[thin space (1/6-em)]E*′[thin space (1/6-em)]+[thin space (1/6-em)](RT/3F) ln(m[thin space (1/6-em)]+[thin space (1/6-em)]Q)(3)
Eqn. (3) corresponds to a concave curve if reported vs. ln[thin space (1/6-em)]m. The Nernst-slope-bearing straight line of the K3[Co(CN)6]–K2SO4 mixtures for m/m′[thin space (1/6-em)]>[thin space (1/6-em)]10−4 [Fig. 1, (a)] means that Q is negligibly small compared with m, i.e., the anion-exchange electrode is responsive to [Co(CN)6]3− only, exactly as if it was dipped in a pure solution of K3[Co(CN)6] free of K2SO4. (A fortiori, the same profitable situation is also fulfilled for higher m/m′ ratios, where E is no longer a linear function of ln[thin space (1/6-em)]m because of the activity coefficients and of the potassium concentration which are no longer constant.)


The cell's emf (E′ = E values shifted by a constant), varying the molal concentration of (a) K3[Co(CN)6] in 0.0904 mol kg−1 K2SO4; (b) [Co(en)3]Cl3 in 0.1054 mol kg−1 MgCl2; (c) [Co(en)3]Cl3 in 0.0646 mol kg−1 LaCl3; (d) K3[Co(CN)6] in 0.1113 mol kg−1 KOH; (e) [Co(en)3]Cl3 in 0.1000 mol kg−1 HCl. Full lines, eqn. (3), Q = 0; dotted lines, eqn. (3), Q = 8 × 10−7 (a), 5 × 10−7 (b, c), 4 × 10−6 (d), 1.1 × 10−5 (e).
Fig. 1 The cell's emf (E′[thin space (1/6-em)]=[thin space (1/6-em)]E values shifted by a constant), varying the molal concentration of (a) K3[Co(CN)6] in 0.0904 mol kg−1 K2SO4; (b) [Co(en)3]Cl3 in 0.1054 mol kg−1 MgCl2; (c) [Co(en)3]Cl3 in 0.0646 mol kg−1 LaCl3; (d) K3[Co(CN)6] in 0.1113 mol kg−1 KOH; (e) [Co(en)3]Cl3 in 0.1000 mol kg−1 HCl. Full lines, eqn. (3), Q[thin space (1/6-em)]=[thin space (1/6-em)]0; dotted lines, eqn. (3), Q[thin space (1/6-em)]=[thin space (1/6-em)]8[thin space (1/6-em)]×[thin space (1/6-em)]10−7 (a), 5[thin space (1/6-em)]×[thin space (1/6-em)]10−7 (b, c), 4[thin space (1/6-em)]×[thin space (1/6-em)]10−6 (d), 1.1[thin space (1/6-em)]×[thin space (1/6-em)]10−5 (e).

By developing the cell equation at m′ no longer constant and sufficiently high m/m′ ratios for Q to be neglected, one has:

 
E[thin space (1/6-em)]=[thin space (1/6-em)]E‡[thin space (1/6-em)]+[thin space (1/6-em)]RT/3F ln[m(3m[thin space (1/6-em)]+[thin space (1/6-em)]2m′)3γ4±](4)
where the quantity E‡[thin space (1/6-em)]+[thin space (1/6-em)](RT/3F)ln[(3m[thin space (1/6-em)]+[thin space (1/6-em)]2m′)3γ4±], constant for m′ constant and m[thin space (1/6-em)]≪[thin space (1/6-em)]m′, corresponds to E*′ of eqn. (3). (The general form that eqn. (4) relies on is reported in ESI section 6.) On the other hand, eqn. (4) is also related to the equation pertaining to the particular cell 2 that does not contain K2SO4 (m′[thin space (1/6-em)]=[thin space (1/6-em)]0), i.e. the same cell already used to determine the activity coefficients of pure K3[Co(CN)6] in water,6
 
E[thin space (1/6-em)]=[thin space (1/6-em)]E‡[thin space (1/6-em)]+[thin space (1/6-em)](RT/F)ln[thin space (1/6-em)]3[thin space (1/6-em)]+[thin space (1/6-em)](4RT/3F)ln(mγ±)(5)
where E‡[thin space (1/6-em)]+[thin space (1/6-em)](RT/F)ln[thin space (1/6-em)]3 (for 1:3 or 3:1 electrolytes) is eqn. (1)’s E*. Hence, it is possible to link E‡ to the emf values measured in the solutions of pure K3[Co(CN)6] (whose activity coefficients are already known),6 thus identifying γ± in the K3[Co(CN)6]–K2SO4 mixed solutions whose E values are measured with the same cell. The same considerations apply to the other systems shown in Fig. 1. The actual values of the activity coefficients of the solutions of Fig. 1 can therefore be determined. Practical details of the procedure adopted to minimize errors, are reported in ESI, section 8. Activity coefficients of [Co(en)3][Co(CN)6] and [Co(en)3][Fe(CN)6] in the solutions of MgSO4 and La2(SO4)3 are determined in the same way.

The membrane electrodes for [Co(CN)6]3− or [Fe(CN)6]3− and [Co(en)3]3+, practically unresponsive to SO42−, OH−, H+, Mg2+, and La3+ respectively, make a first group of interesting mixed solutions available. Like [Co(CN)6]3− or [Fe(CN)6]3− and [Co(en)3]3+, all ions whose charge is shielded by a hydrophobic shell-or counterbalanced by a hydrophobic tail-are predictably accessible to examination in the presence of more hydrophilic species. Besides, no particular difference in hydrophilic character is expected between Mg2+ and Ca2+ (or Sr2+ or Ba2+), or between sulfate and carbonate or phosphate, or between H+ and Li+ or Na+, thus expanding the set of allowable supporting electrolytes. Not all electrolyte combinations thus obtained will probably enjoy selectivity properties as favorable as those examined in the present paper, but sufficient for less extreme m/m′ ratios. (The criteria to be adopted in such cases are discussed in the ESI section 5).

Application: activity coefficients of a trace electrolyte in a second electrolyte

The present application is only aimed at showing an example of the possibilities that the method offers; nevertheless, it touches on a topic of quite wide interest. Kinetic and equilibrium parameters like rate constants, acidity and formation constants, redox potentials, etc., generally need to be determined in a supporting electrolyte of much greater concentration than the reacting species, to preserve the constant value of the activity coefficients and accessory properties (diffusion potentials, absorptivity values, etc.). There are no doubts, so doing, that the activity coefficients of the trace species remain unchanged, as desired, yet their values are masked and remain unknown; thus, no predictions can be made about what will occur in a different supporting medium, or up to the infinite dilution limit. Measuring these constant-but-unknown activity coefficients may be therefore very interesting.

Consider a mixed stock solution where m[thin space (1/6-em)]≪[thin space (1/6-em)]m′ and where the activity coefficients of K3[Co(CN)6] no longer vary on further diminishing m–thus proving indistinguishable from the limit for m[thin space (1/6-em)]=[thin space (1/6-em)]0 in m′ (the Fig. 1 solutions obey this condition). On diluting such a solution, one obtains the relative activity coefficients for vanishing concentrations of K3[Co(CN)6] at decreasing concentrations of K2SO4, down to the extreme limits allowed by the liquid membrane cell. Extrapolation to ionic strength zero-to transform the relative activity coefficients into real activity coefficients-is unnecessary, since the activity coefficients found are linked to that of the stock solution, which is linked in turn (viaE‡, eqns. (4) and (5)) to pure K3[Co(CN)6] in water, whose activity coefficients are already known. Likewise, the activity coefficients of trace amounts of K3[Co(CN)6] in KOH, [Co(en)3]Cl3 in HCl, [Co(en)3]Cl3 in MgCl2, [Co(en)3]Cl3 in LaCl3, [Co(en)3][Co(CN)6] in MgSO4, [Co(en)3][Co(CN)6] in La2(SO4)3, and finally, [Co(en)3][Fe(CN)6] in MgSO4 and [Co(en)3][Fe(CN)6] in La2(SO4)3, are determined. The results (Figs. 2, 3, and 4; the full details and source data are available in the ESI) deserve some discussion, since they are not so obvious. In particular, for [Co(en)3][Co(CN)6] in MgSO4 the results appeared to be so discordant from expectations, as to suggest repeating a second series of measurements for [Co(en)3][Fe(CN)6]. (The results fully corroborated those of [Co(en)3][Co(CN)6], thus proving that the experiments were right and that the theory has to adapt oneself to allow for such effects.)


Activity coefficients of trace amounts of K3[Co(CN)6] in solutions of KOH (mole ratio 1∶110) and K2SO4 (mole ratio 1∶103) compared with the corresponding smoothed values in water (bold line) and IPBE predictions for an infinitely diluted 1:3 salt in a 1:1 salt solution (a = 0.43 nm) or 1:2 salt solution (a = 0.37 nm).
Fig. 2 Activity coefficients of trace amounts of K3[Co(CN)6] in solutions of KOH (mole ratio 1∶110) and K2SO4 (mole ratio 1∶103) compared with the corresponding smoothed values in water (bold line) and IPBE predictions for an infinitely diluted 1:3 salt in a 1:1 salt solution (a[thin space (1/6-em)]=[thin space (1/6-em)]0.43 nm) or 1:2 salt solution (a[thin space (1/6-em)]=[thin space (1/6-em)]0.37 nm).

[Co(en)3]Cl3 in HCl (concentration ratio 1∶117), in MgCl2 (concentration ratio 1∶204), and in LaCl3 (concentration ratio 1∶130) solutions, compared with the smoothed activity coefficients of [Co(en)3]Cl3 in water (bold line) and the IPBE predictions for a 3:1 salt at zero concentration in a 1:1 (a = 0.34 nm), 2:1 (a = 0.35 nm), and 3:1 (a = 0.37 nm) salt solution.
Fig. 3 [Co(en)3]Cl3 in HCl (concentration ratio 1∶117), in MgCl2 (concentration ratio 1∶204), and in LaCl3 (concentration ratio 1∶130) solutions, compared with the smoothed activity coefficients of [Co(en)3]Cl3 in water (bold line) and the IPBE predictions for a 3:1 salt at zero concentration in a 1:1 (a[thin space (1/6-em)]=[thin space (1/6-em)]0.34 nm), 2:1 (a[thin space (1/6-em)]=[thin space (1/6-em)]0.35 nm), and 3:1 (a[thin space (1/6-em)]=[thin space (1/6-em)]0.37 nm) salt solution.

[Co(en)3][Co(CN)6] and [Co(en)3][Fe(CN)6] in MgSO4 and in La2(SO4)3 compared with pure [Co(en)3][Co(CN)6] in water and the IPBE predictions. The IPBE calculations refer to a 3:3 single electrolyte (a = 0.46 nm), to an infinitely diluted 3:3 electrolyte in a 2:2 electrolyte solution (a = 0.41 nm), and to a mixture of a 3:3 and 3:2 electrolyte with a concentration ratio 1∶26.5 (a = 0.54 nm).
Fig. 4 [Co(en)3][Co(CN)6] and [Co(en)3][Fe(CN)6] in MgSO4 and in La2(SO4)3 compared with pure [Co(en)3][Co(CN)6] in water and the IPBE predictions. The IPBE calculations refer to a 3:3 single electrolyte (a[thin space (1/6-em)]=[thin space (1/6-em)]0.46 nm), to an infinitely diluted 3:3 electrolyte in a 2:2 electrolyte solution (a[thin space (1/6-em)]=[thin space (1/6-em)]0.41 nm), and to a mixture of a 3:3 and 3:2 electrolyte with a concentration ratio 1∶26.5 (a[thin space (1/6-em)]=[thin space (1/6-em)]0.54 nm).

The activity coefficients of K3[Co(CN)6] in K2SO4 and KOH are slightly but perceptibly lower than those of pure K3[Co(CN)6] of an equal ionic strength (I, mol kg−1), and very similar to one another. The results are compared with simulation curves obtained by solving the unmodified exponential form of the Poisson–Boltzmann equation (numerical integration method IPBE)7 for trace amounts of 1:3 electrolytes dissolved in 1:2 or 1:1 electrolytes (in the ESI a copy of the IPBE algorithm is available). We assumed a single value for the distance of closest approach a of the ions, since IPBE, like the Debye–Hückel theory, does not consistently admit the use of different values of a for different ions of the same solution.3 Lower values of a correspond to stronger interactions and vice-versa. According to Brønsted's principle of specific interaction, the principal interactions that condition the excess free energy of infinitely diluted K3[Co(CN)6] in K2SO4 are those between the target cations K+ and bulk anions SO42− (as encountered in pure K2SO4, a[thin space (1/6-em)]=[thin space (1/6-em)]0.34 nm) and target anions [Co(CN)6]3− and bulk cations K+ (as in pure K3[Co(CN)6]; a[thin space (1/6-em)]=[thin space (1/6-em)]0.42 nm). Therefore, one expects that the most appropriate average value for a has to be between 0.34 and 0.42 nm. As a matter of fact, a is actually found to hold ca. 0.37 nm (Fig. 2). Likewise, the expected value for K3[Co(CN)6] in KOH is between 0.42 nm (as in pure K3[Co(CN)6]) and 0.47 nm (pure KOH; deduced by applying IPBE to the KOH data from ref. 8) and the empirical value that best reproduces the experimental data is really 0.43–0.44 nm.

Activity coefficients of [Co(en)3]Cl3 in MgCl2 and LaCl3 do not differ appreciably from those of the pure electrolyte at equal ionic strength, while HCl lowers the values. The different interaction strengths can be evaluated from the IPBE values of a to be used for the corresponding single electrolytes. HCl (principal interactions, H+–Cl−) requires a[thin space (1/6-em)]=[thin space (1/6-em)]0.54 or 0.55 nm;9 MgCl2 (Mg2+–Cl−), 0.55 nm; LaCl3 (La3+–Cl−), 0.51 nm; [Co(en)3]Cl3 ([Co(en)3]3+–Cl−), 0.37 nm (for dilute regions) or 0.35 nm (an empiric adjustment that covers a wider range of ionic strengths).2 These values suggest that the distance of closest approach needed to simulate the activity coefficients of [Co(en)3]Cl3 should be intermediate between 0.55 and 0.35 nm both in HCl and MgCl2, and between 0.51 and 0.35 nm in LaCl3. On the contrary, the values actually requested are all lower than expected, ca. 0.37 nm in LaCl3, 0.35 nm in MgCl2, and still less (0.33–0.34 nm) in HCl, thus diagnosing stronger than expected interactions not easy to rationalize.

A much more complex and unexpected behavior is observed for [Co(en)3][Co(CN)6] and [Co(en)3][Fe(CN)6], two salts characterized by strong ionic interactions. Thus, unlike e.g. K3[Co(CN)6] or [Co(en)3]Cl3 whose single solutions do not display negative deviations from the limiting law, [Co(en)3][Co(CN)6] and [Co(en)3][Fe(CN)6] do so. The behavior of these 3:3 salts in water is consistent with the IPBE approximation for a population of 3+ and 3− charged spheres with a value of a of ca. 0.46 nm.10 If the corresponding ions are dispersed in a more concentrated supporting electrolyte whose ions are not so highly charged, e.g. MgSO4 or La2(SO4)3, the negative deviations are expected to decrease, and actually do. However, lanthanum sulfate would also be expected, owing to the higher charge, to cause larger negative deviations than magnesium sulfate at equal ionic strength. On the contrary, the experiments said no, and no mistakes are possible in the experiments since the same results have been independently obtained for both [Co(en)3][Co(CN)6] and [Co(en)3][Fe(CN)6].

The discrepancy is surprising, since the experimental results do not seem to be the logical extension of the known behaviour of the component single salts. In both La2(SO4)3 and MgSO4 supporting electrolytes the interactions trace cation – bulk anion are the same, [Co(en)3]3+–SO42−, while the interactions trace anion – bulk cation are different, [Co(CN)6]3−–La3+ or [Co(CN)6]3−–Mg2+. The interactions between [Co(CN)6]3− and La3+ involved in the first case are the same also present in La[Co(CN)6], which displays considerable negative deviations, while conversely, the interactions between [Co(CN)6]3− and Mg2+ are the same as in Mg3[Co(CN)6]2 – which only displays very moderate negative deviations. Hence, the larger negative deviations are reasonably predicted for [Co(en)3][Co(CN)6] in La2(SO4)3 and not in MgSO4. However, the prediction is wrong. Alternatively, one can observe that IPBE, when applied to a simulated mixture of a 3:3 salt (trace amounts) and 2:2 salt, or 3:3 salt and 3:2 salt (the same concentration ratios as in the experiments), is able to reproduce the experimental results fairly well, provided a is properly adjusted as an empirical parameter. The proper values are 0.54 nm in La2(SO4)3 and 0.41 nm in MgSO4. Comparison of such values with those referring to the pure electrolytes in which the same cation-anion interactions occur, reveals that 0.54 nm is not far from the mean value between 0.38 nm and 0.66 nm found in [Co(en)3]2(SO4)3 and La[Co(CN)6] respectively. However, the value of a for Mg3[Co(CN)6]2 is 0.67 nm, nearly the same as for La[Co(CN)6]; thus, one would expect that the proper value of a for [Co(en)3][Co(CN)6] in MgSO4 should again be ca. 0.54 nm, and not 0.41 nm. Identical considerations apply to the corresponding solutions of the hexacyanoferrate (III) salt.

The failure of IPBE is probably the consequence of intrinsic limitations of the Poisson–Boltzmann original equation form; thus it is possible that other, more advanced theories solve the problem of the strange behavior observed.§ However, as far as we know, no electrolyte theory has so far predicted similar facts. On the other hand, such occurrences have so far been unknown, since it has been utterly impossible to detect them by other experimental methods.

Conclusions

The present introductory study proves that the method of liquid membrane cells, originally devised for only single electrolytes, also extends reliably to many kinds of mixed electrolyte solutions, and in particular, to many mixed solutions of high charge electrolytes whose behavior has so far been a mystery. Like the electrolytes examined here, salts of many organic ions or charged metal complexes are expected to be liable to examination when mixed with electrolytes made of considerably more hydrophilic ions such as H+ (and presumably, Li+, Na+,…), Mg2+ (Ca2+,…), La3+ (Al3+,…), OH− (F−, Cl−,…), SO42− (CO32−, SO32−,…), PO43−, etc.. Supplementary hydrophilic electrolytes (e.g., NaOH added to a bulk of Na2CO3 or Na3PO4 to repress hydrolysis) are freely allowed.

The method can further be improved by taking advantage of ion selective reliable electrodes such as the glass membrane electrode for H+, lanthanum fluoride membrane electrode for F−, etc., to trace also hydrophilic ions. Thus, for instance, the activity coefficients of both HCl and [Co(en)3]Cl3 can be determined in the respective mixed solutions. Although such a possibility was unimportant in the present study (the extreme concentration ratios examined lead the activity coefficients of HCl to be the same as those, already known, of the corresponding pure solutions), in the usual mixed solutions, the ability to determine activity coefficients of different components is highly desirable.

The experimental determinations presented, though restricted to a limited set of mixed electrolyte solutions all of the same type (a target electrolyte of negligible concentration inside a supporting electrolyte of variable concentration) have already revealed some unexpected peculiarities of the multicomponent systems containing high-charge ions, which deserve thorough theoretical examination, and will probably lead to a critical revision of current ideas. It is easy to prophesy that the extended opportunity of determining activity coefficients in mixed solutions will produce a much higher level of insight into real ionic medium effects, with a significant impact both on the field of electrolyte theory, and on the study of equilibrium and kinetic aspects of chemical reactions involving ions.

Acknowledgements

The research was supported by MURST (COFIN 2002).

References

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Footnotes

† Electronic supplementary information (ESI) available: (Part A) 1. Apparatus and reagents. 2. Modus operandi. 3. Ion selectivity and water-induced ion selectivity. 4. Tests for ion selectivity (Tables S1–S5). 5. Less demanding checks. 6. Mixed electrolytes with (and without) ions in common. 7. Revised activity coefficients of [Co(en)3][Co(CN)6] in water at 298.15 K (Table S6). 8. Activity coefficients of reference mixed solutions (Tables S7–S15). 9. Activity coefficients of trace electrolytes in supporting electrolytes of variable concentration. (Tables S16–S24). 10. Ion hydration effects. (Part B). The IPBE algorithm: how to use the computation program; the source program IPBE. See http://www.rsc.org/suppdata/cp/b3/b311624g/
‡ Particular forms to be used for electrolytes with ions in common, are reported in the ESI section 6.† Corrections needed to allow for the change in the free energy arising from the water molecules that escort the ions that enter the membranes, are neglected in eqn. (1) for simplicity (for the exact forms see ESI section 10; the differences are negligible in dilute solutions).
§ C. W. Outhwaite has informed the authors of a theoretical paper on electrolyte mixtures which he, B. Hribar Lee, V. Vlachy, L. B. Bhuiyan, and M. Molero, have recently published.11 It is possible that their theoretical models are able to rationalize our results.
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