On the experimental determinations of ionic activity coefficients

Fluid Phase Equilibria 239 (2006) 120–124

On the experimental determinations of ionic activity coefficients Francesco Malatesta ∗ Dipartimento di Chimica e Chimica Industriale dell’Universit`a di Pisa, Via Risorgimento 35, 56126 Pisa, Italy Received 26 September 2005; received in revised form 4 November 2005; accepted 15 November 2005

Abstract A paper has recently been published by G. Wilczek-Vera and J.H. Vera [G. Wilczek-Vera, J.H. Vera, Fluid Phase Equilbr. 236 (2005) 96–110], to corroborate their belief that ion activity coefficients can be measured, and to rebut opposing arguments. The present analysis, which deduces exact equations valid for any kind of cell in terms of electrolytes instead of constituent ions, reverses their conclusions and confirms that Vera and collaborators have for many years published, as experimental values of ionic activity coefficients, other no-better-identified quantities with no relationship to the unknown values of the real ionic activity coefficients. © 2005 Elsevier B.V. All rights reserved. Keywords: Activity; Activity coefficients; Ionic activity coefficients; Ions; Ion selective electrodes; Junction potential; Potentiometry

1. Introduction

2. General considerations

Wilczek-Vera and Vera have lately published in this Journal a paper [1] to rebut my slashing criticism against Vera school’s determinations of ion activity coefficients [2–4]. Among their arguments, I find one incontrovertible observation, that in a list of papers on the ion activities, I cited inappropriately two papers [5,6], which actually did not deal with this topic and only had a minor role in suggesting to Vera that the ion activity coefficients could be measured [7]. I apologize for the mistake. As for other concerns and conclusions of their paper [1], these are incorrect. Wilczek-Vera and Vera probably misunderstood my arguments. The particularly simple models – a single electrolyte, Nernstian electrodes, and invariant ionic conductivities – had the only aim of permitting a particularly simple demonstration that their method failed, even in the most favourable conditions. Since they claim, on the contrary, that it is only in these particular conditions that their method fails, in this paper complete equations are deduced that are valid for all real cells, thus reaching, by a more laborious route, a general and definitive conclusion. For the reader’s sake, I will use as far as possible the same symbols used by Wilczek-Vera and Vera in [1].

Vera and his co-workers have so far maintained that the emf (Ei,k ) of a cell (Cell 1):

∗

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0378-3812/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2005.11.009

. Reference electrode|solution r ..solution k, containing the ion i|ISE for i contains information about the activity coefficient of i, γ i,k , and that this information can be recovered, using their method for evaluating the liquid junction potentials (EJ,k ) and calibrating the ISEs. Actually, they do not prove the correctness of their approach and greatly underestimate the opposing arguments presented many years ago by Taylor [8] and Guggenheim [9,10]. The approximate equations that Vera and co-workers adopt make an implicit use of arbitrary input values of the individual ionic activities. By examining an electrolyte model whose ion activity coefficients are hypothetically already known, rather than a real electrolyte where no concrete possibility exists of checking whether the values returned are correct or not, I proved that their method does not recover the correct γ i,k , but as many series of arbitrary γ i,k as those preliminarily entered for the arbitrary input values [2,3]. To deny the validity of this proof, Wilczek-Vera and Vera claim that “The scientific path to validate our method for measuring ionic activities is to stop generating imaginary data with faulty equations representing hypothetical experiments carried

F. Malatesta / Fluid Phase Equilibria 239 (2006) 120–124

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Table 1 Ionic activity coefficients reported by Vera and co-workers, references [7,11,12], before (parentheses values) and after their correction for the sign in EJ,k HCl ([11,12]) m

(mol kg−1 )

0.1 0.3 0.5 0.7 1.2 2.0 a

MgBr2 ([7,11]) γCl−

a γHcalc. +

m (mol kg−1 )

γBr−

calc. a γMg +

(0.961) 0.806 (1.153) 0.740 (1.350) 0.713 (1.573) 0.710 (2.262) 0.753 (3.932) 0.913

(0.655) 0.785 (0.507) 0.798 (0.429) 0.821 (0.381) 0.853 (0.316) 0.955 (0.271) 1.166

0.5 1.0 1.5 2 2.5 3

(0.491) 0.583 (0.444) 0.588 (0.459) 0.654 (0.503) 0.757 (0.549) 0.863 (0.601) 0.916

(0.669) 0.480 (2.157) 1.258 (5.682) 2.852 (17.07) 7.672 (58.63) 23.90 (219. 9) 100.3

Values that Vera and co-workers calculated from their activity coefficients for Cl− and Br− and the literature values of γ ± .

out in non-existing cells. A possible scientific path would be to verify that the data measured by our group are reproducible, and that unique values for the activities of ions are obtained from these data by proper calibration of the electrodes” [1]. Apart from the remark that the arguments against their method were all based on thermodynamic exact equations and not faulty equations, this is not a valid argument. By the “scientific path” they suggest, indeed, Vera and collaborators reached the conclusion that their data published before 2004 were correct, but they were forced to recognize later [11] a sign error in their calculations and to re-write all previous data tables, which were wrong. They sustain now that the results after the sign correction are nearly the same as before, and present this hypothetical finding as one of the strongest elements of proof for the validity of their method, which, in their opinion, is so self-correcting as to absorb even the effect of a sign error. I submit to the reader a few data published by Vera’s school [7,11,12] before and after the correction of the sign (Table 1): even the order γ cation > γ anion or vice versa can revert, with the correction. However, my objections against the results of Vera’s school does not concern minor questions such as a sign error in the liquid junction potentials, but the basis of the method. Thus, I will not rebut point-by-point the objections they raised against my arguments. Indeed, there is only one important question to be resolved: it is, or is it not possible, for the emf of a real cell to provide any information about the activity of an ionic lone constituent, and not only about the activity of “molecular” electrolytes? If the response is: no, the desired information is not there, then all the arguments of their paper [1] are not relevant. To answer the question, we initially analyze the cells with Nernstian electrodes, and then those with non-Nernstian ISEs. 3. Cells with Nernstian electrodes Many years ago, Taylor [8] demonstrated that the emf of any cell, with or without liquid junctions, “is a function of molecular free energies solely and is not a function of ionic free energies”. Unfortunately, Taylor’s arguments may prove to be obscure because of some terms which are no longer familiar to modern scientists; Vera and collaborators interpreted such arguments as if they were only valid for concentration cells with two solutions of the same electrolyte, differing only infinitesimally in concentration.

However, Taylor’s principle can also be deduced by an independent path, and be expressed in terms more familiar to electrochemists. Let’s start with the general scheme of Cell 1. For simplicity, we will confine ourselves to concrete situations (i.e., naming the different ions K+ , Cl− , Na+ , Br− etc. rather than i, j, k,. . . or 1, 2, 3,. . .). If the two solutions r and k are composed of the same electrolyte, e.g. KCl, or of two electrolytes with an ion in common, e.g. KCl and MgCl2 , we have the more simple situations, the same that Wilczek-Vera and Vera developed to obtain their Eqs. (14) and (22) [1], thus proving (contrary to their intention) that the information about the ionic individual activities is not there.1 A more complex situation is encountered if r and k are composed of two salts with no ions in common, e.g. KCl (r) and NaBr (k), since in this case some divergence problems (actually not real)2 seem to occur because of the divergence to −∞ of the chemical potentials of KCl in k and of NaBr in r. We will develop this kind of situation, which is sufficient for our purposes; indeed, any further addition of electrolytes in the reference solution, salt bridge, or target solutions does not modify the results, except that the equations become more cumbersome. For concreteness, the cells examined are . Ag, AgCl|KCl(solution r)..NaBr(solution k)|ISE for Br− Cell 2 . Ag, AgCl|KCl(solution r)..NaBr(solution k)|ISE for Na+ Cell 3 1 Their extension of Eqs. (14) and (22) [1] to cells with non-Nernstian electrodes is incorrect, because of their Eq. (6), which is only valid for Nernstian electrodes (see Section 4). For Nernstian electrodes, the term that contains a−,k (the activity of the chloride ion) in their Eqs. (14) and (22) [1] is null. It should be noted that the symbols used by Wilczek-Vera and Vera in both equations suggest, misleadingly, the further dependence of the emf on the chloride ion activity, through the term that they indicate as [E−,0 − RTF−1 ln a−,r ]. That is a mistake, however; simply, Wilczek-Vera and Vera have omitted to point out that their E−,0 also depends on a−,r . It is a simple matter to verify that their 0 0 − EAgCl/Ag ) + RTF−1 ln a−,r ], and hence, that the E−,0 means the sum [(EISE

difference [E−,0 − RTF−1 ln a−,r ] does not depend on a−,r . 2 The divergence problems prove not to be real as soon as one recalls that r and k also contain H+ and OH− , which are in common, and that the transport number of an ion whose concentration becomes null, also becomes null, more rapidly than the logarithm of the activity diverges to −∞.

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For simplicity of notation, we omit the electric charges in superscripts and subscripts. The respective cell equations, in the usual form that Wilczek-Vera and Vera [1] also adopt, which involve the individual ion activities, are:

where the abbreviation Qk,r is substituted for the cumbersome k term RTF−1 r {2tK , . . . , aw }, which only depends on the activity of electrically neutral electrolytes, not ionic lone species. Likewise, Eq. (2) yields:

0 0 EBr,k = EISE(Br) − EAgCl/Ag + RTF−1 ln aCl,r
k −1 −1 − RTF ln aBr,k − RTF ti z−1 i d(ln ai )

0 0 − EAgCl/Ag + 2RTF−1 ENa,k = EISE(Na)

(1)

0 0 ENa,k = EISE(Na) − EAgCl/Ag + RTF−1 ln aCl,r
k + RTF−1 ln aNa,k − RTF−1 ti z−1 i d(ln ai )

(2)

r

r

i

i

 We develop now the term i ti z−1 i d(ln ai ) present in both equations, recalling that the solutions also contain H+ and OH− and that the transport number of any ion (here, arbitrarily, H+ ) can always be expressed as 1 minus the sum of all other transport numbers:  ti z−1 i d(ln ai ) i

= tK d(ln aK ) − tNa d(ln aNa ) − tCl d(ln aCl ) − tBr d(ln aBr ) − tOH d(ln aOH ) + (1 − tK − tNa − tCl −tBr −tOH ) d(ln aH ) (3) We now collect the terms pertaining to any ti , recalling that products such as aK × aCl can be expressed as the activity of the corresponding electrolyte, (a±KCl )2 , and ratios such as aK /aH or aNa /aH can be rewritten as (aK × aCl )/(aH × aCl ) or (aNa × aBr )/(aH × aBr ), i.e., (a±KCl /a±HCl )2 or (a±NaBr /a±HBr )2 , respectively. Hence, we obtain  ti z−1 i d(ln ai ) i

= 2tK [d(ln a±KCl ) − d(ln a±HCl )] + 2tNa [d(ln a±NaBr ) −d(ln a±HBr )] − 2tCl d(ln a±HCl ) − 2tBr d(ln a±HBr ) − tOH d(ln aw ) + d(ln aH )

(4)

aw is the water activity, arises from aH × aOH . Substituting the right-hand side member of Eq. (4) for i ti z−1 i d(ln ai ) in Eq. (1), we obtain 0 0 EBr,k = EISE(Br) −EAgCl/Ag +2RTF−1 [ln a±HBr,k − ln a±HCl,r ]
k {2tK [d(ln a±KCl ) − d(ln a±HCl )] − RTF−1 r

+ 2tNa [d(ln a±NaBr ) − d(ln a±HBr )] − 2tCl d(ln a±HCl ) − 2tBr d(ln a±HBr ) − tOH d(ln aw )}

(5)

or EBr,k =

0 EISE(Br)

0 − EAgCl/Ag

− 2RTF

(6)

(7)

Eqs. (6) and (7) prove definitively that no term exists in the emf of these cells which may depend on the activity of an ionic constituent, but only on the activity of “molecular” electrolytes. The same kind of result is obtained for higher numbers of electrolyte components and/or more highly charged ions in the reference solution, salt bridge and target solution. If, e.g. LiF is introduced in the left or right or both solutions of the cells, or a LiF salt bridge is interposed between the two solutions, the only change consists in one more term, 2tLi [d(ln a±LiF ) − d(ln a±HF )] − 2tF d(ln a±HF ), inside the integrand of Qk,r . The kind and geometry of the liquid junctions have no relevance, since the equations are completely general. We must conclude, therefore, that no cell of any kind, with Nernstian electrodes, can provide information about ion activity coefficients. 4. Cells with non-Nernstian ISEs Wilczek-Vera and Vera assume that the same equations valid for cells with Nernstian electrodes may be extended to cells with non-Nernstian ISEs by the mere substitution of an empirical slope Si for Si0 (the Nernst slope) in Eq. (8): Ei,k = Ei,0 + Si0 ln ai,k + EJ,k

(8)

thus obtaining their Eq. (6) [1], Ei,k = Ei,0 + Si ln ai,k + EJ,k . Their assumption is arbitrary and incorrect. Nernst’s equation and Si0 have a precise thermodynamic derivation, and Nernstian electrodes are electrodes in which, due to the intrinsic nature of the equilibria and kinetic processes involved, all the correct reasons subsist for the exact application of the Nernst equation. Conversely, non-Nernstian ISEs are such that no theoretical reasons exist for the Nernst equation to hold, and no theoretical relationships between the electrode potential and ai,k can any longer be deduced, starting from a stringent analysis of the equilibria and kinetic processes involved. With a non-Nernstian ISE, connected to an undoubtedly Nernstian electrode for the same ion, so as to obtain a cell without any liquid junctions, Cell 4: Nernstian electrode for i|i in the solution k(mi,k ) |non-nernstian ISE for i one can only take note of the occurrence of an empirical relationship between the concentration of the target ion, mi,k , and the experimental emf of the cell without transport, EISE(i,k) : EISE(i,k) = const + SISE(i) ln mi,k + (dev)i,k

−1

× [ln a±HBr,k − ln a±HCl,r ] − Qk,r

× [ln a±NaBr,k − ln a±HBr,k + ln a±HCl,r ] − Qk,r

(9)

where SISE(i) and (dev)i,k are the empirical slope of EISE(i,k) and the empirical function that allows for the deviations of

F. Malatesta / Fluid Phase Equilibria 239 (2006) 120–124

EISE(i,k) from the linear trend. The absolute potential ψi,k of the left electrode obeys the Nernst law for absolute potentials: ψi,k = ψi0 + Si0 ln ai,k

(10)

and hence ψISE(i,k) , the absolute potential of the ISE, obeys Eq. (11): ψISE(i,k) = ψi0 + Si0 ln ai,k + const + SISE(i) ln mi,k + (dev)i,k (11) 0 the sum ψi0 = const, Si the sum Si0 + By naming ψISE(i) 0 SISE(i) , and Si ln δi,k the empirical function (dev)i,k (δi,k , another empirical function), one obtains 0 + Si ln mi,k + Si0 ln (γi,k δi,k ) ψISE(i,k) = ψISE(i)

(12)

Hence, the correct equation to be used for Eq. (8) whenever the cell contains a non-Nernstian ISE, is Ei,k = Ei,0 + Si ln mi,k + Si0 ln (γi,k δi,k ) + EJ,k

(13)

which is incompatible with the arbitrary assumption of WilczekVera and Vera, Si0 ln(γ i,k δi,k ) = Si ln(γ i,k ), implicitly adopted in their Eq. (6) [1], with the only exception of the case in which (dev)i,k = 0 and Si = Si0 (in other words, the case of a Nernstian ISE). We are now able to correctly extend Eqs. (6) and (7) of the previous section, valid for Nernstian electrodes only, to the case of non-Nernstian ISEs for Br− and Na+ respectively. Eq. (6) becomes 0 0 − EAgCl/Ag − (RTF−1 + SBr ) ln mHBr,k EBr,k = EISE(Br)

− RTF−1 ln[(γ±HBr,k )2 δBr,k ] + 2RTF−1 ln a±HCl,r − Qk,r

(14)

i.e. (RTF−1 + SBr ) ln mHBr,k substitutes for 2RTF−1 ln mHBr,k , and RTF−1 ln[(γ HBr,k )2 δBr,k ] substitutes for 2RTF−1 ln γ HBr,k . No terms that contain the activity or activity coefficient of any lone ion have arisen with the substitution of a non-Nernstian ISE for the Nernstian electrode. We thus conclude that the use of a non-Nernstian Br− -ISE in Cell 2 does not produce any improvement as regards the lack of information about single ion activity coefficients. As for Cell 3, using a non-Nernstian ISE for Na+ Eq. (7) becomes: 0 0 ENa,k = EISE(Na) −EAgCl/Ag +SNa ln mNa,k − (RTF−1 ) ln mH,k

+ RTF−1 ln[δNa,k (γ±NaBr,k )2 ] − 2RTF−1 ln γ±HBr,k + 2RTF−1 ln a±HCl,r − Qk,r

(15)

and in turn, Eq. (15), like Eqs. (6), (7), and (14), does not contain any information about single ion activity coefficients. The same reasoning can easily be applied to any real or imaginary possible cells.

123

5. Conclusions A correct development of the starting-points also accepted by Wilczek-Vera and Vera [1] proves that their arguments in favor of Vera school’s determinations of ion activity coefficients and rebutting my adverse criticism [2,3], are inconsistent. Indeed, the exact equations deduced for the electromotive force of their cells – and more generally, any cells – prove that in no case can the emf contain terms that depend on the activity coefficient of a ionic lone constituent, irrespective of the use of traditional electrodes or of Nernstian or non-Nernstian ISEs, of a salt bridge or a direct contact between the solution, of a single electrolyte or as many different electrolytes as one wishes, etc. If, all the same, the Vera method seems to provide convincing values of the ion activity coefficients, this finding can only prove that the method is misleading, irrespective of the suggestions of the similar values deduced by different researchers, or referring to different approximations for EJ,k . The assertion of Wilczeck-Vera and Vera that Taylor’s arguments were valid only for “two reversible electrodes immersed in solutions of the same electrolyte, differing only infinitesimally in concentration” [1], is incorrect. Their “Approaches for the new millennium” [1] are inconsistent. The prediction that accurate experimental measurements of the liquid junction true potentials will become feasible in future [1,11] will not come true; such measurements are conceptually impossible [4,9,10], and no technical progress can change this fact. To conclude, the ion activity coefficients that Vera and his co-workers have published for many years, before or after the correction of the sign in their approximate expressions for EJ,k , are all to be rejected; such values have no relationship with the true values of the ionic activity coefficients, which remain unknown. This fact should be carefully considered by scientists who use “experimental” values of ion activity coefficients to check their theoretical computations, see e.g. [13]. One should realize that no experimental data about ion activity coefficients can exist; the values published in literature are based on wrong assumptions. List of symbols ai activity of an ionic species i (a−,r : of the anion, in the solution r) aw water activity a± mean activity of an electrolyte (a±MX,k : of MX, in the solution k) (dev)i,k empirical deviation term defined by Eq. (9) E potential 0 , E0 E0 standard potential (e.g., EISE AgCl/Ag : of the ISE, of the Ag/AgCl electrode) 0 Ei,0 the difference between EISE and the potential of the reference electrode in Cell 1 Ei,k emf of Cell 1 EJ,k liquid junction potential between the reference solution r and target solution k EISE(i,k) emf of Cell 4 F Faraday constant i a generic ion

124

m Qk,r

R Si Si0 SISE(i) ti T zi

F. Malatesta / Fluid Phase Equilibria 239 (2006) 120–124

molality or dimensionless molality, as appropriate (mi,k : of a species i, in a solution k) k RTF−1 r {2tK [d(ln a±KCl ) − d(ln a±HCl )] + 2tNa [d(ln a±NaBr ) − d(ln a±HBr )] − 2tCl d(ln a±HCl ) − 2tBr d(ln a±HBr ) − tOH d(ln aw )} universal gas constant practical slope defined as Si0 + SISE(i) Nernst’s slope RT(zi F)−1 slope of EISE(i,k) versus ln mi,k transport number of i absolute temperature dimensionless charge of i

Greek letters γi activity coefficient of i γ± mean activity coefficient δi,k empirical function defined by the relationship (dev)ik = Si0 ln δi,k ψi,k absolute potential of a Nernstian electrode for i in the solution k

ψi0

standard absolute potential of a Nernstian electrode for i ψISE(i,k) absolute potential of an ISE for i in the solution k 0 ψISE(i,k) standard absolute potential of the ISE for i References

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

G. Wilczek-Vera, J.H. Vera, Fluid Phase Equilbr. 236 (2005) 96–110. F. Malatesta, J. Solution Chem. 29 (2000) 771–779. F. Malatesta, Fluid Phase Equilbr. 233 (2005) 103–109. F. Malatesta, AIChE J. 52 (2) (2006), doi: 19.1002/aic.10651, in press. A. Haghtalab, J.H. Vera, J. Solution Chem. 20 (1991) 479–493. A. Haghtalab, J.H. Vera, J. Chem. Eng. Data 36 (1991) 332–340. E. Rodil, J.H. Vera, Fluid Phase Equilbr. 205 (2003) 115–132. P.B. Taylor, J. Phys. Chem. 31 (1927) 1478–1500. E.A. Guggenheim, J. Phys. Chem. 33 (1929) 842–849. E.A. Guggenheim, J. Phys. Chem. 34 (1930) 1540–1543. G. Wilczek-Vera, E. Rodil, J.H. Vera, AIChE J. 50 (2004) 445– 462. [12] E. Rodil, K. Persson, J.H. Vera, G. Wilczek-Vera, AIChE J. 47 (2001) 2807–2818. [13] H.-y. Lin, L.-s. Lee, Fluid Phase Equilbr. 237 (2005) 1–8.