Comment on "Negative Deviations from the Debye−Hückel Limiting

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Comment on "Negative Deviations from the Debye-Hückel Limiting Law for High-Charge Polyvalent Electrolytes: Are They Real?" Tarita Biver, and Francesco Malatesta J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b00700 • Publication Date (Web): 05 Sep 2018 Downloaded from http://pubs.acs.org on September 8, 2018

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Journal of Chemical Theory and Computation

Comment on "Negative Deviations from the Debye−Hückel Limiting Law for High-Charge Polyvalent Electrolytes: Are They Real?" Tarita Biver 1 and *Francesco Malatesta 2 1

Dipartimento di Chimica e Chimica Industriale - Università di Pisa - Via Giuseppe Moruzzi 13 - 56124 Pisa - Italy 2 *Retired Professor of the University of Pisa - Via Don Gaetano Boschi 26 - 56126 Pisa - Italy. Email: [email protected]

ABSTRACT A paper (Fraenkel, J. Chem. Theory Comput. 2018, 14, 2609) was recently published in which, starting from data of activity coefficients of electrolytes determined by Malatesta and co-workers in the years 1992 - 2000, an incorrect conclusion is reached, i.e., that the negative deviations from the Debye-Hückel limiting law found at high dilution for high-charge polyvalent electrolytes were presumably not real. The present paper shows the reasons why we cannot share his opinion. The negative deviations are both supported by the theory and experimentally demonstrated.

INTRODUCTION Papers published by Malatesta and co-workers ("our group"; "us") in the years 1992-2000 are quoted many times in the paper "Negative Deviations from the Debye- Hückel Limiting Law for High-Charge Polyvalent Electrolytes: Are They Real?", by Dan Fraenkel.1 He examined the experimental activity coefficients obtained by us for 1-3, 3-1, 2-3 and 3-3 electrolytes down to particularly high dilution levels - in a few cases, 10-5 mol kg-1 and beyond. Data regarding 2-2 and 3-2 electrolytes were not considered and other 3-3 salts showing very evident negative deviations from the Debye-Hückel limiting slope (LL) were also neglected. Fraenkel "corrected" our activity coefficients by repeating extrapolation to infinite dilution with empiric methods; then, by examining the activity coefficients corrected by his methods, concluded that the apparent negative deviations from the LL found by us for 2-3 and 3-3 electrolytes were not real. However, it is not so: his corrected data, indeed, are less accurate than the original ones, and the negative deviations of 2-2, 2-3 and 3-2, and 3-3 electrolytes really exist. We note incidentally that the data for La[Fe(CN)6] he examined were those of 1994,2 which were affected by a systematic error in the dilute region because of a peculiar failure of the electrodes adopted, containing agar-jellified solutions. The problem was identified soon after and new measurements for La[Fe(CN)6] were performed in 1995 with a new kind of cell made entirely of PTFE.3 Presumably, if Fraenkel had considered the revised data instead of those of 1994 his conclusions would have been different. Figure 1 shows the actual behavior of La[Fe(CN)6] and, for comparison, the LL, transformed into a curve because of the logarithmic abscissa used to avoid crowding in the dilute regions.

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Figure 1. Activity coefficients of La[Fe(CN)6].3 For comparison, LL and theoretical curve (numerical integration of the Poisson Boltzmann equation, IPBE) calculated for a hypothetical 3-3 salt made of rigid charged spheres with diameter a = 0.65 nm. The double crosses identify two independent "absolute reference points" derived from additivity thermodynamic relationships, and their range of uncertainty. (The method of the "absolute reference point" is described in the Supporting Information.)

DISCUSSION

Fraenkel's skepticism on the real existence of negative deviations from LL in very dilute solutions of highly charged polyvalent electrolytes derives from his thought that such deviations, if observed at such low concentrations as 10-4 -10-5 mol kg-1, would represent a sort of violation of consolidated electrolyte theory, that predicts the electrolytes to attain necessarily the LL in the limit of I1/2 = 0 (I = the ionic strength). This opinion is clearly expressed by the words he wrote in the Introduction: "Furthermore, the conflict between experimental and the DH theory as mentioned above, is not at high electrolyte concentration, at which the theory is indeed expected a priori to deviate from reality due to its drastic assumptions and approximations; rather, it is at very dilute solutions, for which the theory has long been regarded by most scholars in the art as physically correct".1 This thought seems to arise from a misunderstanding of what the negative deviations really are. The negative deviations exist only at finite (even though low) concentrations, and do not imply a violation of the LL in the limit of infinite dilution; simply, in these cases the LL is attained at dilution levels beyond those accessible to experiments. Figure 1 shows how this is possible. There is no violation of any theory. Negative deviations are predicted by the ion solution theory for such highly charged systems; yet, the Debye-Hückel approximation, that relies on the linear approximation (L-PB) of the Poisson Boltzmann equation (PB), is not able to take into due account the real structure of the ionic atmosphere for highly charged systems. It is perfectly correct to sustain that all theories predict the LL slope as the only valid limit to infinite dilution of d(ln γ±)/d(I1/2) for all electrolytes; however, the ion solution theories do not affirm also that the limit to infinite dilution of the second derivative of ln γ± vs. I1/2 (or m1/2) has to be necessarily positive, i.e., that only positive deviations from the LL are possible. This additional statement derives only from the Debye-Hückel usual approximation (DHT), ln γ± = - AD ǀz+z-ǀ2 I1/2 (1 + a BD I1/2)-1

(1)

(symbols in the Glossary) which predicts d2(ln γ±)/d(I1/2)2 be always positive; yet, this is a peculiar mistake of DHT and comes up from the L-PB. Unlike L-PB, the complete solution (by numerical methods) of the PB equation (IPBE) 4,5 shows that negative values of d2(ln γ±)/d(I1/2)2 at I1/2 = 0 are also possible, depending of the ion charges, ion sizes, and dielectric constant of the solution. As shown in Figure 2 for simulated 2-2 salts made of charged rigid spheres of diameter a = 0.4 nm and a = 0.8 nm, IPBE predicts different trends 2 ACS Paragon Plus Environment

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depending of the values of a. High values of a give DHT-like trends, with only positive deviations. Low values of a give negative deviations. Anyway, the curves with either positive or negative deviations obey the LL slope in the limit of infinite dilution.

Figure 2. Activity coefficients predicted by IPBE for a 2-2 salt made of rigid charged spheres of diameter a = 0.4 or a = 0.8 nm. For comparison, the LL slope. Like IPBE, also the Bjerrum theory,6 Gronwall - La Mer - Sandwed theory,7 Mayer theory,8-11 HNC,12,13 SMPB,14 and no doubts further theories that overcome the L-PB approximation, all admit the possibility of deviations from the LL in the direction opposite to DHT predictions for ions of high charge and low size, and/or solvents with low dielectric constant. Fraenkel also continued: « A “negative deviation” from the DHLL reflects attraction forces between counterions, which are stronger than the conventional coulomb attraction forces. What can possibly be there at very dilute solutions that attract counterions to one another more than the coulomb forces? ». Yet, the question is slightly different. The negative deviations from the LL do not reflect forces that are stronger than the conventional coulomb attraction forces, since IPBE - for instance - does not consider other forces but coulomb ones, and, despite this fact, it predicts the possibility of negative deviations such as in Figs. 1 and 2. And the same considerations apply also to Bjerrum theory, Gronwall - La Mer - Sandwed theory, Mayer theory, HNC, and SMPB, all applied to the restricted primitive model (i.e., to the same physical model of solution assumed by the DHT, and introducing a as sole parameter). I.e., a negative deviation from the LL reflects only a discrepancy between the coulomb forces that take place in the ionic atmosphere as it is really, and those that would occur in the hypothetical ionic atmosphere that existed if the L-PB were a valid approximation to the PB also for high-charge systems. Furthermore, Fraenkel suggests that the finding of negative deviations in our experiments comes out from the fact that our extrapolations to infinite dilution relied on the semiempirical Pitzer's equation,15-17 and he cites at this regard the best fit parameters of the Pitzer equation reported in Table V of our paper on the alkaline earth hexacyanoferrates(III).18 Actually, he misunderstood the reasons why these parameters were reported. In our papers, indeed, the extrapolations to infinite dilution of the relative activity coefficients did not rely on the Pitzer equation. The latter was used only for interpolation purposes, and its best fit parameters were reported only for reader's utility, since the Pitzer equation is a very compact and useful tool to calculate the values of ln γ± at any wished concentration, obviously within the experimental range, with approximately the same precision of the experimental data. Extrapolations to infinite dilution were instead performed with the aid of theoretical treatments based on the restricted primitive model, by adjusting their unique parameter, the distance of closest approach a between ion centers. Specifically, we used the DHT, Bjerrum theory, 3 ACS Paragon Plus Environment


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DHLL+B2 approximation of Mayer's theory,11 and IPBE. Each theory was used independently from others, and the results were compared for mutual control. The results were in general nearly coincident, with the exception of the DHT that cannot be used confidently for extrapolation with 2-2, 2-3 and 3-2, and 3-3 electrolytes, since the relative activity coefficients of these electrolytes in very dilute regions generally present negative deviations that DHT is not able to reproduce. In the case of 2-2, 2-3, 3-2, and 3-3 electrolytes, a complementary, completely independent method was also used by us to identify the real values of the activity coefficients. We denoted this method as the method of the absolute reference point. It relies on thermodynamic relations of additivity, and, for a salt MnXy (Mr+, Xx-, both r and x >1), it grounds on emf values measured in three auxiliary cells, for MClr, KxX, and KCl, all built using the same set of four electrodes. Thus, the possibly problematic extrapolation to infinite dilution of the relative activity coefficients of a 3-3 salt, for instance, transforms into the much simpler extrapolation of the relative activity coefficients of a 1-3 salt, a 3-1 salt, and a 1-1 salt, which usually do not present any particular problem. For brevity, the rationale and details of the absolute reference point method are not described here but in the Supporting Information, with an example of application. The independent results of the absolute reference point method and of the different theory-assisted extrapolations to zero were generally in excellent agreement, thus providing a mutual corroboration of the results. In a few cases - electrolytes displaying particularly intense negative deviations (3-3 salts and La2(SO4)3)19,20 - the theory-assisted extrapolations to zero were unsatisfactory in spite of the very high dilutions reached; as to say, the Bjerrum theory, DHLL+B2, and IPBE suggested non coincident, broad extrapolation values, with large levels of uncertainty. In these cases, the values of the activity coefficients were fixed using the sole method of the absolute reference point. The extrapolation methods with which Fraenkel corrects our results, on the contrary, are fundamentally empiric. He chooses to extrapolate the value of the "standard potential" E° of the cell, rather than the value of ln γ±ref of a reference solution as we prefer. In theory, both methods (finding either E° or ln γ±ref) are perfectly equivalent in the case of a true cell without transport, but Fraenkel did not consider the specific characteristics of the liquid-membrane cells, for which it is impossible to define once and for all an invariable value of E° (which is the reason why in our papers we preferred always the symbol E* rather than E° for the liquid-membrane cells), thus making preferable to extrapolate to zero the value of ln γ±ref. For compactness, we refer once again readers to the Supporting Information. Briefly, the use of a single, average E° for all groups of measurements carried out with a determinate cell in different days causes the errors to increase, sometimes very much, since any group of measurements actually would require a different E°. But this is only a secondary aspect of the problem. To deduce E°, Fraenkel uses two slightly different methods. In the first method, the function E°', E°' = E - [(ν+ + ν-)/(ν+z+)] RTF-1 ln m±

(2)

E°' = E° + [(ν+ + ν-)/(ν+z+)] RTF-1 ln γ±

(3)

i.e.,

is extrapolated down to m1/2 = 0 where the intercept of E°' is equal to E°. To extrapolate E°' down to m1/2 = 0, he relies on empirical straight lines that, however, have no theoretical reason for the correct extrapolation be impacted. The only "authorized" straight line, indeed, would be the LL, but often, despite the high dilution levels, the trend of the experimental points does not yet obey the LL when we are dealing with salts with both z+ and ǀz-ǀ >1. Furthermore, the range of concentrations in which the empiric straight line has to be fitted to the data is quite undefined. Sometimes, Fraenkel used slightly curved lines obtained by polynomial regression instead of straight lines, or also "graphical extrapolation to zero concentration".1 Such empiric extrapolations have no reason to be considered more correct than the original ones of our papers, which usually relied on three independent theories at least, and were corroborated whenever possible (i.e., with 2-2, 2-3 or 3-2, and 3-3 salts) by the absolute reference point method. 4 ACS Paragon Plus Environment

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As for the second method used by Fraenkel, it resembles the theory-assisted extrapolation also adopted in our papers in so far as concerns DHT and excluding the IPBE, DHLL+B2, and Bjerrum theories used for additional control by us. The function E°", E°" = E°' - [(ν+ + ν-)/(ν+z+)] RTF-1 ln γ±(DHT, a )

(4)

E°" = E° + [(ν+ + ν-)/(ν+z+)] RTF-1[ ln γ± - ln γ±(DHT, a)]

(5)

i.e.,

(where ln γ±(DHT, a) is the value predicted by eq 1 for a suitable a) is plotted vs. m (although it would be better a plot vs. log m to expand the dilute regions). It is expected that E°" tends to E° more quickly than E°'; an horizontal trend with E°" = const = E° should be reached at low concentrations, so long as ln γ±(DHT, a) is able to reproduce the actual trend of ln γ±. However, this is not the case for La[Fe(CN)6], La[Co(CN)6], and other salts that deviate negatively from the LL, because of the curved trend - with a positive second derivative of E°" vs. m - which in these cases is observed for the expected flat trend. The low-size scale adopted in Fraenkel plots (his Figure 1 b) makes this fact scarcely discernible, yet it occurs. The extrapolation method based on E°" can work correctly only with salts with a DHT-like behavior, such as K3[Co(CN)6] (with which, indeed, E°" displays a nearly flat trend). Conversely, it is unfit for salts with negative deviations in the diluted regions, since the trends of ln γ± (negative deviations) and of ln γ±(DHT, a) (positive deviations) are incompatible. It would be preferable, rather, to substitute ln γ±(LL) for ln γ±(DHT, a) in the expression for E°". As for Fraenkel's E°, additional discussion is required. He wrote "A major task of the current study is the determination, using Nernst equation, of the electrochemical potential, E, at zero concentration (i.e., E°) "; which, however, is an incorrect sentence since (i) an electrochemical potential is a completely different thing than an electromotive force E, and (ii) E° does not represent the value of E at zero concentration (which is -∞) but the value of E when m±γ± = 1. However, those are no more than mistakes coming from inattention, and do not affect Fraenkel's procedure to determine E°, which is conceptually correct apart for the disputable extrapolation methods adopted (see above). The problem is another. The E° of a liquid membrane cell is not a constant quantity to determine once and for all, but a quantity that depends on the concentration of the electrolyte solutions that fill the electrodes. For a determinate salt, different values rather than a single value of E° are usually required, since a slow osmotic flux of water through the membranes slowly modifies the concentrations of the internal solutions of the electrodes, thus modifying from day to day the E° value to use; and in addition, in some cases it is necessary to refill the electrodes, or even to use electrodes with different internal solutions and membrane compositions to confirm the general soundness of the results. On the contrary, Fraenkel uses his E°(1), E°(2) and E°ave as invariable and specific to the salts examined, LaCl3, K3[Fe(CN)6], K3[Co(CN)6], Mg3[Fe(CN)6]2, Mg3[Co(CN)6]2, Ca3[Fe(CN)6]2, Ba3[Co(CN)6]2, La[Fe(CN)6], and La[Co(CN)6] (Table I of Ref. 1). Actually, such values are non-exportable and cannot be used by other scientists for new measurements. Every E°(1), E°(2) and E°ave of Table I 1 apply only to a restricted sub-set of measurements among those performed by our group of research for the corresponding salt. For instance, the E°ave = 0.118712 V reported for La[Co(CN)6] applies only (and approximately) to the first five series of measurements we performed for La[Co(CN)6],19 and not to series 6. According to our present recalculations, the six series of determinations for La[Co(CN)6] required individual E°'s, 0.11988 V, 0.11952 V, 0.11944 V, 0.11963 V, 0.11944 V, and 0.07345 V (series 6 employed a different pair of electrodes). It is worth to note that a much preferable possibility consisted in storing, instead of E°(1), E°(2) and E°ave, the activity coefficients γ±ref for suitable reference solutions, as γ±ref is an exportable quantity. We have recalculated the activity coefficients of La[Co(CN)6] by applying the Fraenkel E°ave = 0.118712 V to the "raw" values of E reported in our original paper,19 series 1 - 5. The resulting corrected ln γ± are compared with the original ones in Figure 3. Although the Fraenkel procedure increases the random error because of the use of a single E°ave for five different E° 's, this effect is so small that cannot be detected in the plot. Our ln γ± rely on the determination of ln γ±ref by means of the absolute reference point method, since our 5 ACS Paragon Plus Environment


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theory-assisted extrapolation to zero with the aid of IPBE, Mayer theory and Bjerrum theory did not provide a univocal result. The corrected data on the contrary rely on Fraenkel's extrapolations. It is evident that there exist a bias between Fraenkel's values and ours. However, it is worth to note that not only our original values, but also his "corrected" values lie under the LL line, i.e. deviate negatively from the LL.

Figure 3. The activity coefficients of La[Co(CN)6] according to the original paper 19 (filled circles), and corrected by Fraenkel (open diamonds). The trend of the original values recalls the IPBE curve predicted, in water at 298.15 K, for a population of +3 and 3 charged rigid spheres with diameter a = 0.66 nm.

We note that our original values for La[Co(CN)6] are more appropriate than those corrected in order to attain the limiting slope in the limit of infinite dilution. For m < 10-5 mol kg-1 also our values get over the LL line, a symptom that the cell was no more working properly. That is a very logical thing, considering the interference by H+ ions (that compete with La3+ at the surface of the permselective membrane for lanthanum), which is no longer negligible when the concentration of La3+ is lower than 10-5 mol kg-1. The symmetrical interference by OH- at the surface of the permselective membrane for [Co(CN)6]3-, on the contrary, is presumably unimportant owing to the results obtained in the determinations of the activity coefficients of K3[Co(CN)6] in the presence of KOH,21 proving that the hydrophilic ions OH- are not able to displace the hydrophobic ions [Co(CN)6]3- from the surface of the corresponding membrane. Our original measurements for alkaline-earth hexacyanoferrates(III)18 did not benefit yet of the improved method that relies on two measurements of Eref in the reference solution - before and after any daily series of determinations - and therefore the results were slightly less precise than those for, e.g., La[Co(CN)6] 19 or Mg3[Co(CN)6]2.22 According to our original determinations, the 2-3 salts Mg3[Fe(CN)6]2,18 Ca3[Fe(CN)6]2,18 Sr3[Fe(CN)6]2,18 Ba3[Fe(CN)6]2,18 Mg3[Co(CN)6]2,22 and Ca3[Co(CN)6]2 22 all exhibit only moderate negative deviations from the LL. Therefore, to increase the sensitivity of the test, we report the results in the expanded form ln γ± - ln γ±(LL) (Figure 4). Our original ln γ± were anchored to their definitive position: as for Mg3[Co(CN)6]2 and Ca3[Co(CN)6]2,22 by means of both the theory-assisted extrapolation of ln γ' to infinite dilution and the determination of ln γ±ref by the method of the absolute reference point; as for Mg3[Fe(CN)6]2, Ca3[Fe(CN)6]2, Sr3[Fe(CN)6]2, and Ba3[Fe(CN)6]2,18 only by the method of the absolute reference point. The absolute reference points, with their ranges of uncertainty, are shown in Figure 4 in the form of double crosses (note: for Ba3[Fe(CN)6]2 the double cross has arbitrarily been reported around the point at m = 8.85×10-4 mol kg-1 since the true reference point, at m = 9.808×10-3 mol kg-1,18 was outside the range of the figure).


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Figure 4. Negative deviations from the LL of 2-3 salts in dilute solutions. Filled circles, Ca3[Fe(CN)6]2; open triangles, Ca3[Co(CN)6]2; open squares, Mg3[Fe(CN)6]2; open circles, Mg3[Co(CN)6]2; open diamonds, Sr3[Fe(CN)6]2; stars, Ba3[Fe(CN)6]2 The double crosses identify the reference values determined by the "absolute reference point" method and their range of uncertainty. Dashed line: the LL. Full lines = IPBE predictions for hypothetical 2-3 salts made of hard charged spheres, for two empirical, indicative values of a, 0.64 nm (upper) and 0.56 nm (lower). Fraenkel's corrected values for Mg3[Fe(CN)6]2, Mg3[Co(CN)6]2, Ca3[Fe(CN)6]2, and Ba3[Fe(CN)6]2, based on his re-extrapolations (E°ave reported in Table I of Ref. 1; for Mg3[Fe(CN)6]2 we have selected his second option, E°ave = 0.1125755 V, and not E°ave = 0.112448 V) are shown in Figure 5.

Figure 5. The trend of the corrected activity coefficients of Fraenkel in the dilute regions, for Ca3[Fe(CN)6]2, Mg3[Fe(CN)6]2, Mg3[Co(CN)6]2, and Ba3[Fe(CN)6]2 (same symbols as in Figure 4), and two indicative DHT curves, a = 0.5 nm and a = 0.3 nm.

The points of Figure 5 are appreciably more scattered than the original ones of Figure 4 because of the use of E - E°ave (with an E°ave value invariable) instead of E - Eref (with at least one "fresh" Eref for any day of work). The indicative DHT curves show how, disregarding the trend of the points with m < 10-4 mol kg-1, these "corrected" ln γ± should go to zero in the limit of infinite dilution. The information obtained from the absolute reference points that anchor each ln γ±ref to a precise position - with a margin of uncertainty between 7 ACS Paragon Plus Environment


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± 0.008 (Mg3[Fe(CN)6]2) and ± 0.019 (Ca3[Fe(CN)6]2) - is completely ignored in this case. It is not clear why these "corrected" values of ln γ± should be preferred to the original ones, except for a prejudicial postulation that negative deviations from the LL are inadmissible. The reason why Fraenkel insists in believing that the negative deviations should not exist - excepting very particular situations in which strong interactions other than the usual coulomb forces are occurring - is because he reached the conviction that L-PB is not considerably different from the full, nonlinearized PB equation, independently of the charges involved. He wrote at this regard a paper, reporting an evaluation of the series expansion terms of the PB equation beyond that of the linear form, which revealed no substantial contribution of those terms for, in particular, dilute solutions.27 It is not easy to carefully inspect all statements in his work to identify where, in his analysis, he introduced the item/items that make his conclusions non-applicable to, or non-comparable with, the situations studied many years ago by Guggenheim,4, 28 who for 2-2 electrolytes found considerable differences between the L-PB and complete numerical integration of PB, in the charge density distributions around the central ion. Guggenheim's analyses proved that the L-PB is unsuitable for situations equivalent to 2-2 electrolytes in water (or to 1-1 electrolytes in a solvent in which the dielectric constant is a quarter than in water). One of the authors of the present paper (F. Malatesta) repeated in the years 1970 the Guggenheim computations using his improved method of numerical integration of the Poisson Boltzmann equation (he was developing and checking the IPBE algorithm),5 thus confirming the same conclusions for 2-2 electrolytes and also covering the case of 33 electrolytes (unpublished results). Fraenkel did not make mention of the Guggenheim papers (Refs. 4 and 28) in his reexamination of the limits of applicability of the L-PB approximation to PB. CONCLUSIONS Fraenkel's DH-SiS theory 29 - the reason why he re-examined our experimental data on 2-3 and 3-3 salts - is an extension of the DH to size-dissimilar ions, which relies on the L-PB approximation and therefore, like DHT, is not able to reproduce the behavior of salts that deviate negatively from the LL in the dilute region. Fraenkel solved this problem by denying the real existence of the negative deviations; he re-extrapolates the E°'s of the liquid membrane cells with fundamentally empiric criteria that ignore the experimental trend between 10-4 and 10-5 mol kg-1 and the values of ln γ±ref obtained with the method of the absolute reference point, thus obtaining for 2-3 salts (not, however, for 3-3 salts) alternative values of ln γ± with no negative deviations, with which the DH-SiS equation works satisfactorily. Yet, even if his recalculated ln γ± were really the correct ones for Mg3[Fe(CN)6]2, Mg3[Co(CN)6]2, Ca3[Co(CN)6]2, and Ba3[Fe(CN)6]2, the problem would remain for [Co(en)3][Fe(CN)6] and [Co(en)3][Co(CN)6]. Indeed, for La[Co(CN)6] his "corrected" activity coefficients confirm - instead of denying - the presence of the negative deviations, and for La[Fe(CN)6] he did not use the correct experimental data,3 but earlier data 2 that in the dilute regions were affected by a severe systematic error and were recognized incorrect by us for a long time.2, 3 The real data of La[Fe(CN)6], like those of La[Co(CN)6], confirm the presence of the negative deviations. Furthermore, other salts that Fraenkel did not consider in his paper1 display negative deviations more enhanced (and less disputable) than those of the 2-3 salts. Studied by our group of research, the sulfates of Mg, Ca, Sr, Zn, Cd, Co, Ni, and Mn among 2-2 salts;23, 24, 25 La2(SO4)320 and [Co(en)3]2(SO4)3 (en = ethylenediamine)26 among 32 salts; [Co(en)3][Fe(CN)6] and [Co(en)3][Co(CN)6] in addition to La[Fe(CN)6] and La[Co(CN)6] among 3-3 salts.19 Particularly large negative deviations are those observed for the 3-2 salts La2(SO4)3 and [Co(en)3]2(SO4)3 (Figure 3 of Ref. 20) and the 3-3 salts [Co(en)3][Fe(CN)6] and [Co(en)3][Co(CN)6] (Figs. 2 and 3 of Ref. 19). It is impossible to deny the real existence of the negative deviations from the LL. It is worth to stress that the negative deviations from the limiting law observed in dilute solutions of highcharge electrolytes do not contrast with the principle that all electrolytes have to obey the limiting slope of the Debye-Hückel theory in the limit of infinite dilution, and do not represent a violation of the general theory of the electrolytes. Indeed, similar deviations are predicted, in the same conditions of charges 8 ACS Paragon Plus Environment

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involved and dielectric constant of the solvent, also by a mere electrostatic model of charged rigid spheres, provided this model is developed to a level that goes beyond the linearized Poisson-Boltzmann equation. Theories that are not able to rationalize this fact should require revision.

ASSOCIATED CONTENT Supporting Information Content: (i) Thermodynamics of the cells with permselective liquid membranes. (ii) Difference between E* and the standard potential E° of the typical cells without transport, and determination of the activity coefficients with a liquid membrane cell. (iii) The "Absolute Reference Point" method. (iv) Theoretical IPBE curves for ln γ± of 2-3 or 3-2 and 3-3 electrolytes. This information is available free of charge via the Internet at http://pubs.acs.org

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]. Notes The authors declare no competing financial interest. GLOSSARY Abbreviations/Acronyms DH, Debye-Hückel theory; DHLL, or LL, the DH limiting law; DHT, Debye-Hückel equation for real concentrations; DHLL+B2, the Mayer theory in its usual form, with drastic truncation of the cluster series; PB = Poisson-Boltzmann equation; L-PB = linearized approximation of PB; IPBE = accurate numerical integration of PB. Mathematical / Physical notations a = diameter of the rigid charged spheres used to simulate the ions in the so called primitive model of the electrolytes (= distance of closest approach between the centers of the spheres). AD = constant of the DHLL equation (-AD ǀz+z-ǀ = the LL slope of ln γ± vs. I1/2). Its value in water at 298.15 K is 1.172 mol-1/2 dm3/2 (or mol-1/2 kg1/2 considering molal and molar concentrations as indistinguishable at high dilutions). BD = constant of the DHT equation; BD I1/2 = the reciprocal screening length in the DH theory. Its value in water at 298.15 K is 3.281 nm-1 mol-1/2 dm3/2 (or nm-1 mol-1/2 kg1/2 considering molal and molar concentrations as indistinguishable at high dilutions). E = electromotive force (emf) of a cell. E° = standard potential of a cell without transport; by extension, according to Fraenkel,1 the value of E when m±γ± = 1 in a liquid membrane cell. E* = utility emf that we used for E° in the cells with permselective liquid membranes. As in our papers we used always m and not m±, E* is the value of E when mγ± = 1. F = Faraday constant (the charge of one mol of protons). γ± = mean activity coefficient (on the molal scale). γ±' or γ' = relative mean activity coefficient = γ± / γ± ref ; more general, γ' = k · γ± where k is an unknown constant, needing identification for the experimental γ' be transformed into γ±. i = generic ion. 9 ACS Paragon Plus Environment


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I = ionic strength / mol kg-1 = (1/2) Σi m i z i2 , where the sum extends over all ions of the solution. m = molal concentration / mol kg-1. m± = m ν± ν+, ν- = number of cations A and anions B of a generic salt AaBb, ν+ = a, ν- = b. ν = ν+ + νν± = (ν+ν+ ν-ν-)1/ν R = gas constant. T = absolute temperature. z+, z-, zi = charge of cation, anion, generic ion i. REFERENCES (1) Fraenkel, D. Negative Deviations from the Debye-Hückel Limiting Law for High-Charge Polyvalent Electrolytes: Are They Real? J. Chem. Theory Comput. 2018, 14, 2609-2620. (2) Malatesta, F.; Giacomelli, A.; Zamboni, R. Activity Coefficients of Electrolytes from the Emf of Liquid Membrane Cells. III: LaCl3, K3Fe(CN)6, and LaFe(CN)6. J. Solution Chem. 1994, 23, 11−36. (3) Malatesta, F. Activity Coefficients from the Emf of Liquid Membrane Cells IV: Revised Activity Coefficients of Lanthanum Hexacyanoferrate(III). J. Solution Chem. 1995, 24, 241−252. (4) Guggenheim, E. A. The accurate numerical solution of the Poisson-Boltzmann equation, Trans. Faraday Soc. 1959, 55, 1714-1724. (5) Malatesta, F; Rotunno, T. An Advanced Algorithm for Theoretical Evaluation of Electrolyte Properties according to the Unlinearized Poisson-Boltzmann Equation. Gazz. Chim. Ital. 1983, 113, 783-787. (6) Bjerrum, N. Untersuchungen uber Ionenassoziation. K. Dansk. Videnskab. Selskab. Math.-Fys. Medd. 1926, 7, 1–48 (7) Gronwall, T. H.; La Mer, V. K.; Sandved, K. Über den Einfluss der sogenannten höheren Glieder in der Debye−Hückelschen Theorie der Lösungen starker Elektrolyte. Phys. Z. 1928, 29, 358−393. (8) Mayer, J. E. The Theory of Ionic Solutions. J. Chem. Phys. 1950, 18, 1426-1436. (9) Poirier, J. C. Thermodynamic functions from Mayer's theory of ionic solutions. I. Equations for thermodynamic functions J. Chem. Phys. 1953, 21, 965-972. (10) Poirier, J. C. Thermodynamic functions from Mayer's theory of ionic solutions. II. The stoichiometric mean ionic molar activity coefficient. J. of Chem. Phys. 1953, 21, 972-985. (11) Malatesta, F.; Rotunno, T. A Fast Numerical Algorithm for Evaluation of Electrolyte Properties According to the Mayer Theory. Gazz. Chim. Ital. 1983, 113, 789-792. (12) Rasaiah, J.C. The hypernetted chain (HNC) equation for higher valence electrolytes. Chem. Phys. Lett. 1970, 7, 260–264. (13) Rasaiah, J.C. Computations for higher valence electrolytes in the restricted primitive model. J. Chem. Phys. 1972, 56, 3071–3085. (14) Outhwaite, C.W.; Molero, M.; Bhuiyan, L.B. Primitive model electrolytes in the modified PoissonBoltzmann theory. J. Chem. Soc. Faraday Trans. 1993, 89, 1315–1320. (15) Pitzer, K. S.; Mayorga, G. Thermodynamics of Electrolytes. II. Activity and Osmotic Coefficients for Strong Electrolytes with One or Both Ions Univalent. J. Phys. Chem. 1973, 77, 2300−2308.


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(16) Pitzer, K.S.; Mayorga, G. Thermodynamics of electrolytes. III. Activity and osmotic coefficients for 2– 2 electrolytes. J. Solution Chem. 1974, 3, 539–545. (17) Pitzer, K.S.; Silvester, L. F. Thermodynamics of electrolytes. 11. Properties of 3:2, 4:2, and other highvalence types. J. Phys. Chem., 1978, 82, 1239–1242. (18) Malatesta, F.; Giacomelli, A.; Zamboni, R. Activity Coefficients from the Emf of Liquid Membrane Cells V. Alkaline Earth Hexacyanoferrates (III) in Aqueous Solutions at 25 °C. J. Solution Chem. 1996, 25, 61−73. (19) Malatesta, F.; Trombella, S.; Giacomelli, A.; Onor, M. Activity coefficients of 3:3 electrolytes in aqueous solutions. Polyhedron 2000, 19, 2493−2500. (20) Malatesta, F.; Bruni, F.; Fanelli, N. Activity coefficients of lanthanum salts at 298.15 K. Phys. Chem. Chem. Phys. 2002, 4, 121-126. (21) Malatesta, F.; Fagiolini, C.; Franceschi, R. Activity coefficients in mixed electrolyte solutions. Phys. Chem. Chem. Phys. 2004, 6, 124-128. (22) Malatesta, F.; Bruni, F.; Fanelli, N.; Trombella, S.; Zamboni, R. Activity and Osmotic Coefficients from the Emf of Liquid Membrane Cells. VIII. K3[Co(CN)6], Mg3[Co(CN)6]2, and Ca3[Co(CN)6]2. J. Solution Chem. 2000, 29, 449−461. (23) Malatesta, F.; Zamboni, R. Activity and osmotic coefficients from the Emf of liquid membrane cells. VI – ZnSO4, MgSO4, CaSO4 and SrSO4 in water at 25 °C. J. Solution Chem. 1997, 26, 791–815. (24) Malatesta, F.; Carbonaro, L.; Fanelli, N.; Ferrini, S.; Giacomelli, A. Activity and osmotic coefficients from the Emf of liquid-membrane cells. VII: Co(ClO4)2, Ni(ClO4)2, K2SO4, CdSO4, CoSO4, and NiSO4. J. Solution Chem. 1999, 28, 593–619. (25) Malatesta, F.; Trombella, S.; Fanelli, N. Activity and osmotic coefficients from the Emf of liquid membrane cells. IX: Mn(ClO4)2 and MnSO4 in water at 25 °C. J. Solution Chem. 2000, 29, 685–697. (26) Malatesta, F.; Carrara, G.; Colombini, M.P.; Giacomelli, A. Activity coefficients of electrolytes from the e.m.f. of liquid membrane cells. II—Multicharged electrolyte solutions. J. Solution Chem. 1993, 22, 733– 749. (27) Fraenkel, D. Ion Strength Limit of Computed Excess Functions Based on the Linearized PoissonBoltzmann Equation. J. Comput. Chem. 2015, 36, 2302-2316. (28) Guggenheim, E. A. Activity coefficients and osmotic coefficients for 2-2 electrolytes, Trans. Faraday Soc. 1960, 56, 1152-1158 (29) Fraenkel, D. Simplified electrostatic model for the thermodynamic excess potential of binary strong electrolyte solutions with size-dissimilar ions. Mol. Phys. 2010, 108, 1435-1466



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