Negative Deviations from the DebyeâHÃ¼ckel Limiting Law for High

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Negative Deviations from the Debye–Hückel Limiting Law for High-Charge Polyvalent Electrolytes – Are They Real? Dan Fraenkel J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b01260 • Publication Date (Web): 21 Mar 2018 Downloaded from http://pubs.acs.org on March 22, 2018

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Journal of Chemical Theory and Computation

Negative Deviations from the Debye–Hückel Limiting Law for High-Charge Polyvalent Electrolytes – Are They Real?

Dan Fraenkel* Eltron Research & Development Inc., 4600 Nautilus Court South, Boulder, Colorado 80301, United States

* Corresponding author, electronic e-mail: [email protected]

ACS Paragon Plus Environment

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Abstract In the past few decades, emf (E) measurements using improved electrochemical cells have afforded the derivation of mean ionic activity coefficients (γ±’s) of very dilute solutions of binary electrolytes, within the 10–6 – 10–3 molar range (Malatesta et al., J. Sol. Chem. 1994, 23, 11). Such measurements are especially important for highly charged electrolytes whose behavior at very low concentration is not yet fully understood, and whose γ± values, derived from E data, have been claimed by Malatesta and coworkers to exhibit “negative deviations” from the Debye–Hückel (DH) limiting law. Here I examine electrolytes studied by the Malatesta group, which belong to the 3–1, 1–3, 2–3 and 3–3 valence families, and analyze their E and γ± data using the Smaller-ion Shell (SiS) theoretical treatment (“DH–SiS”) of strong electrolyte solutions (Fraenkel, J. Chem. Theory Comput. 2015, 11, 178, 193). The DH–SiS physical model incorporates all three ion-size parameters of a binary ionic solution as the “true” ion–ion collision distances, and leads to an improved DH-like electrostatic theory of ionic activity. Correcting Malatesta’s data by better extrapolation of E to zero concentration results in a more accurate “standard potential”, E°; this affords improved γ± values that (1) fit well with the DH–SiS equations, and (2) agree with the DH limiting law.

Keywords: electrolyte, activity, Debye-Hückel, limiting law, negative deviation


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Introduction The past century has witnessed a continuous intense interest in the thermodynamic behavior of strong electrolyte solutions.1–3 Along with improved methods of measuring excess functions of such solutions, e.g., the (molal, m) mean ionic activity coefficient, γ± (for example, by advanced electrochemical cells), new theories have been devised to more adequately address the essential physical features of ionics in solution. While both measurement and theoretical modeling of simple electrolytes – those of the 1–1, 1–2 and 2–1 valence families – have been accomplished with high degree of accuracy over broad concentration ranges, from ~10–4 to ~10 m, less success was achieved with the more complex systems of highly charged electrolytes, especially at low concentration (< 0.1 m). First, it has been difficult – if not impossible – to derive reliable experimental data for such electrolyte systems, i.e., those with ‫׀‬z+z–‫ ≥ ׀‬3 [z, charge of the positive (+) or negative (–) ion]; and second, researchers have argued that with highly charged ionic systems, the linearization of the Poisson–Boltzmann (PB) equation is physically unjustified. Moreover, apparent negative deviations from the Debye–Hückel (DH) limiting law (DHLL) have been frequently observed with those high-charge systems; that is, initial negative slopes of the log γ±-vs.-m1/2 pseudo-straight lines, which are larger (giving steeper lines) than those of the DHLL lines. These deviations were attributed to deficiencies of the DH theory due to its reliance on the linearized PB (L-PB) equation. Thus, the deviations were explained by a variety of theories that are more complex than the DH theory, but are empirical or semiempirical: the Bjerrum theory of ion pairs,4 the Gronwall–La Mer–Sandved theory,5 the Pitzer theory of ion interaction,6 and other


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theories – all containing adjustable parameters (six different in Pitzer equation!) of no clear physical meaning. While such theories can also address low-charge electrolytes that perfectly adhere to the DHLL, they do not provide a proof that negative deviations from the DHLL should occur with higher ionic charge; nor do they show a correlation between the extent of the deviation and ionic charge. In fitting theory with experiment, the “negative deviation” has been taken care of by merely adjusting the free parameters until a “best fit” was reached. This was done without consideration of the nature of the adjusted parameter(s) and of the effectiveness of the fit (i.e., the standard deviation of the fitted data from the experimental ones). Furthermore, the conflict between experiment 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. 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 attracts counterions to one another more than the coulomb forces? A major progress in the measurement of dilute solutions of strong electrolytes has been achieved through electromotive force (emf) studies by Malatesta and Carrara7 who employed in their electrochemical cells electrodes made of selective permeable liquid membranes. Later, using such cells, Malatesta et al.8–11 centered their effort on highcharge electrolytes, for which they have reported negative deviations from the DHLL in the case of ‫׀‬z+z–‫ > ׀‬3. Such deviations were not observed when cells of the same type


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were used for electrolytes of the 1–1, 1–2, 2–1, 1–3 and 3–1 valence families (i.e., with ‫׀‬z+z–‫ ≤ ׀‬3). An inherent difficulty in electrochemical measurements is the determination of the “standard potential”, E°. One has to make a judicial choice of the method to apply for extrapolating E to E°. Slightly deviated E° values cause relatively large deviations in the γ± values of the highest electrolyte dilutions, since the latter values are derived from very small E° – E differences; yet E values at very low electrolyte concentration are normally less accurate, and if the extrapolation is based on them, the resulting E° may be considerably biased. Malatesta et al. employed extrapolation methods8 in which the fit of a theoretical treatment was optimized for all measured E values; the parameters adjusted for the best fit (lowest standard deviation) were then used for the extrapolation. For example, in those cases fitted to the Pitzer equation,6 the fit provided the parameters α1,

α2, β(0), β(1), β(2) and CMX. With these six free numerical factors, when optimized for theory–experiment “best fit” (see, e.g., Table V in Ref. 9), Pitzer equation was extrapolated to zero concentration to provide E°. A risk in such a method of extracting E° is two-fold: (1) The fit considers equally E’s in a wide range of concentration without a solid physical basis of the theoretical expression(s) used, and (2) the extrapolation to E° is based on fit parameters that are fit-range dependent. This is a considerable drawback that may result in inaccurate E° providing not only false γ± values, but – more seriously – false overall trends of the γ±-vs.-I behavior (I, ionic strength). The aim of the present work is to re-examine experimental data of strong electrolyte solutions with ‫׀‬z+z–‫ ≥ ׀‬3 over wide concentration ranges, 10–6 – 1 m, and analyze those data using the theoretical SiS treatment (“DH–SiS”) of strong electrolyte solutions.12 Specifically, the analysis is of electrolytes with trivalent ions in a series that includes


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shared ions, as measured by Malatesta and coworkers. I thus chose for this analysis the 3–1 electrolyte LaCl3;8 the 1–3 electrolytes K3Fe(CN)68 and K3Co(CN)6;10 the 2–3 electrolytes Mg3[Fe(CN)6]2,9 Mg3[Co(CN)6]2,10 Ca3[Fe(CN)6]2,9 and Ba3[Fe(CN)6]2;9 and the 3–3 electrolytes LaFe(CN)68 and LaCo(CN)6.11 The 3–3 family, with ‫׀‬z+z–‫ = ׀‬9, has not been studied theoretically thus far. Therefore, one aim of the current effort is to find out whether 3–3 electrolytes adhere to the DH–SiS theory, and if so, whether theory– experiment fit is achievable using the ion-size parameters (ISPs) of co-ions as employed previously in other systems having the same ions. A further goal of the present study is to reveal whether the extension of experimental measurements to very low concentrations indeed results in a negative deviation from the DHLL. Such a deviation, observed by Malatesta et al. and by others, has been attributed to flaws of the L-PB equation when applied to highly charged ions (see above). However, a recent analysis of the I-range of validity of the linearization of the PB equation13 has shown that with adequate ISP values, γ± data of electrolytes of many valence families can be effectively fitted with the DH–SiS expressions that are based on the L-PB equation, and this is true especially for very dilute electrolyte solutions. Furthermore, an evaluation of the series expansion terms of the PB equation beyond that of the linear form,13 has revealed no substantial contribution of those terms, especially for dilute solutions; i.e., it has confirmed that the L-PB equation is not considerably different from the full, nonlinearized PB equation, and there is no sensitivity to ionic charge and valence. Therefore, if negative deviations from the DHLL do exist, they should originate from factors other than the L-PB equation.


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Before continuing, a brief account on DH–SiS is in order. The “SiS treatment” is an extension of the electrostatic model and theory of Debye and Hückel for the case of sizedissimilar ions.12 For finite-size ions, the DH theory gives the extended DH equation, DHEE, with a single ISP – here denoted å – known as “the distance of closest approach” between any two mutually approaching ions in the solution (but mostly, counterions2). The DHEE reduces to the DHLL at very dilute solution, when the κ factor – the DH reciprocal screening length – approaches zero. In DH–SiS there are two closest approach distances of co-ions: bs of the smaller ion (usually the “+ +” closest distance, so “b+”) and bl of the larger ion (the parallel “– –” distance, thus “b–”); a third closest approach distance is that of counterions, a (i.e., “+ –” that is assumed equal to “– +”; the difference between the two is the chosen reference ion, i.e., whether the positive or negative ion). Even though the DH–SiS physical-mathematical development is more complex than the parallel DH development, the γ±-vs.-I analytic expressions eventually obtained are still rather simple, straightforward and fully transparent.12 The DH–SiS equations include the three ISPs – bs, bl and a – as the only “free parameters,” but when the b’s are known, e.g., when they are chosen as the crystallographic diameters (for example, of Na+ and Cl– in NaCl), the DH–SiS equations remain with a single adjustable parameter, a (in the NaCl case, reflecting the “real” average “contact” distance between Na+ and Cl–). Note, however, that a ≠ å; and unlike å, a possesses a genuine physical meaning, being the average distance between the centers of the cation and anion when the two ions collide; and usually, that distance is of non-hydrated (or, in general, non-solvated) ions. å is a sort of average of a and bs, becoming identical with a when a = bs (see Methods below); this, however, never happens. The DH–SiS equations reduce to the DHEE for the private case


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a = bs = bl (again, a case practically never occurring), and collapse to the DHLL for

κ → 0; that is, at almost infinite dilution. In the context of the present study, one question to ask is whether for all monatomic ions, one can use the crystallographic diameters for the b’s, and whether for the Fe(CN)63– and Co(CN)63– anions, stable – and reasonable – sizes (e.g., diameters) can be employed in the theory–experiment fitting process.

Methods The DH–SiS equations used here, as in previous studies, have been provided in a number of recent publications, in which they were proven over and over again to be very effective in analyzing experimental data of many electrolytes; the studies have covered electrolytes of different valence families, broad concentration ranges, and wide ranges of temperature and permittivity.12–21 Here, following a brief description of the essence of DH–SiS, I shall concentrate on the methods and mathematical expressions directly relating to the current analysis. DH–SiS at a glance. DH–SiS is an extension of the electrostatic DH theory to sizedissimilar ions.12 The extra electrostatic potential energy of ionic interaction is sizedependent so it is different for the smaller (s) and larger (l) ions. The activity coefficient of the chosen i ion, γi, at sufficiently dilute solution, is approximated as only its electrical energy part. A “charging process” correlates γi with the field-average extra electrostatic energy per unit charge, Ψi, through the expression

 µ i ,el ln γ i  =  kT

 1 2  = z i Ψi  2

(1)


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µi,el being the electrical part of the chemical potential of ion i carrying a zi charge, k Boltzmann factor and T absolute temperature. Denoting the charge of the positive ion as z+ and that of the negative ion as z– (see above), the expression for γ± in a binary electrolyte system is

ln γ ± =

1 z + z − Ψ± 2

(2)

in which Ψ± is defined (in analogy to ln γ±) as

Ψ± ≡

1 zi ∑ z i Ψi ; z = i∑ z i =+ , − =+ ,−

(3)

The s- and l-ions can each be positive or negative, so an (s,l) set is either (+,–) or (–,+). Accounting for the electroneutrality condition,

∑zν i

i

= 0 , eq 3 becomes

i =+, −

Ψ± =

1 1 z i Ψi = (ν l Ψs + ν s Ψl ) ∑ z i = s ,l ν

(4)

with νs and νl being, respectively, the multiplicities of the s- and l-ions in the electrolyte “molecular” formula, and νs + νl = ν (= ν+ + ν–). Combining eq 4 with eq 2, one gets

ln γ ± =

1 z + z − (ν l Ψs + ν s Ψl ) 2ν

(5)

An equation for the function γ± vs. I is obtained by mathematically deriving Ψs and Ψl as functions of the reciprocal screening length, κ (= BI1/2) and the ISPs.


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Assuming bs = b+, bl = b–, zs = z+, and zl = z–, one eventually obtains

log γ + = − z +2

κ  2 exp[κ (a − b+ )] − κ (a − b+ ) − 2  A × 1 −  B 1 + κa  1 + κb + 

(6)

log γ − = − z −2

κ  2 exp[κ (b− − a)] − 2κ (b− − a ) − 2  A × 1 +  B 1 + κa  1 + κb − 

(7)

and using the definition ν log γ± = ν+ log γ+ + ν– log γ–,

κ A × × B 1 + κa (8)  ν − 2 exp[κ (a − b+ )] − κ (a − b+ ) − 2 ν + 2 exp[κ (b− − a )] − 2κ (b− − a ) − 2  × + × 1 −  1 + κb+ ν 1 + κb−  ν 

log γ ± = − z + z −

In eqs 6 – 8, A, B and κ are the conventional parameters of the DH theory, so in water at 25 °C,

A = 0.51002 (kilogram-solvent/mole-solute)1/2

and

B = 0.32848 (kilogram-

solvent/mole-solute)1/2Å–1. Recalling that the ISPs a, b+ and b– are of the “+ –”, “+ +” and “– –” closest ion distances, respectively, ISP additivity requires that a = (b+ + b–)/2. A nonadditivity factor, d, is defined as d ≡ 2a – b+ – b–, so the condition d = 0 (or d/2 = 0) means full ISP additivity, and d (d/2) has otherwise either a positive or negative value. Computations of activity coefficients, based on eqs 6 – 8, are performed as detailed previously.12

Estimation of Eo. A major task of the current study is the determination, using Nernst equation, of the electrochemical potential, E, at zero concentration (i.e., Eo). Two well-known literature methods3 were employed previously in combination with DH–SiS, to show (for HCl at low permittivity,17 and for aqueous sulfuric acid19) that literature claims3 of negative deviations from DHLL are incorrect or at least unsubstantiated. Here


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I again use those two methods: In Method 1, one plots E + (νRT/NF) ln m± against m1/2, where R is the gas constant, N the number of equivalents (so N = z+ν+ = z–ν–), and F Faraday constant. Eo is obtained as the intercept of the pseudo-straight line at very high dilution (limiting line), at m = 0. Due to compromised measurement accuracy and systematic experimental errors in measuring E’s at very low concentration, extrapolated lines may not be entirely straight. The extrapolation to m = 0 may then sometimes be better done with slightly curved lines such as those obtained by polynomial regression.17 Method 2 is based on incorporating the DH extended equation, DHEE, into Nernst equation, thus obtaining the combined equation

= +

ℱ

ln − ‫׀ ׀‬

⁄ ℬå ⁄

(9)

Plotting E0 vs. m should yield ideally a straight line with slope zero. In eq 9, applicable for very dilute solutions, A and B are again the usual DH parameters and å is the ion-size parameter of DHEE. The problem is that one has to use a “correct” å value, not just assume a “wild-card” value for å (such as a flat 4.0 Å, as done in many literature analyses). I have employed eq 9 successfully17,19 when using å values calculated from DH–SiS by a method developed previously (see Appendix 3 of Ref. 12). This method can be generalized to any electrolyte valence family, as shown next. At low ionic strength (I < 0.1), the single-ion extra-electrostatic interaction energies, given in reciprocal length unit, are12 "

Ψ∗ ≈ "($%&

(10)

')


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for the condition b+ < a, and

Ψ∗ ≈

"

(11)

"%

for the condition b– > a. The mean ionic Ψ∗ is12

Ψ±∗ =

*

Ψ∗ +

'

Ψ∗

(12)

The ISPs a and b+ are of the particular electrolyte examined, as obtained by the DH–SiS fit with experiment over a broad electrolyte concentration range (see below). Substituting eqs 10 and 11 into eq 12 provides

Ψ±∗ ≈

" , "+% * (%&' )-

(13)

,

and, therefore, å in eq 9 is calculated using the expression

å=.+

*

(. − / )

(14)

For symmetrical electrolytes,

å = . + $ (. − / )

(14a)

Methodology of data analysis. The theoretical analysis applied in the current work is done as follows. The original literature experimental data of γ± vs. m are fitted with theory (DH–SiS) as before (see, for example, description in Ref. 12). The “best fit”


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provides the initial set of the three ISPs. The b’s are usually kept as those of the ions in other electrolyte systems (for example, b of K+ is always the crystallographic diameter, 2.66 Å). From the optimized a and b+, å is calculated (for Method 2, see above). From the experimental E’s, E° is now obtained by both methods, Method 1 [giving E°(1)] and Method 2 [E°(2)], by graphical extrapolation to zero concentration. An arithmetic averaging of E°(1) and E°(2) gives E°ave. Using E°ave, E°ave – E is recalculated and, from Nernst equation, γ± is also recalculated. With the new set of recalculated γ± values, the theoretical fit with DH–SiS is repeated to provide more refined ISPs (usually, affording somewhat different a), and å is recalculated. å is used again in Method 2 to provide a more refined E°(2). From the new E°s, a more accurate E°ave is obtained. This regressive calculation loop is repeated until no improvement in E°ave is achieved anymore. The set of E°ave – E and γ± data is now fixed, and so is the set of the fitted theoretical γ± data. The regression analysis just described is, as expected, sensitive to the inaccuracies in the measurements at very low concentration, and, due to its nature, it becomes gradually less accurate for experimental data at increased concentration. Therefore, it is imperative to choose for the analysis a concentration range that can be reasonably believed to best represent the system at small enough m, where Methods 1 and 2 are both applicable. The theory–experiment fit of γ± is thus done within the concentration range judged to be the most reliable.

Results Table 1 lists the electrolytes examined in the current study with their ISPs and corrected Eo’s, the number of experimental data points used for the analysis, and the concentration range; with reference to the specific literature sources of the various


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electrolytes. All data are of aqueous solutions at 25 °C. In two cases, those of K3Fe(CN)6 and Mg3[Fe(CN)6]2, two different analyses are summarized and compared; they are based on different sets of experimental data and reflect the variability of the analysis results, and the resultant Eo’s and the ISP a (the only adjustable parameter of theory–experiment fit). In all cases, the b’s were chosen as those of the corresponding ions, either the crystallographic values or when otherwise, the values obtained previously using the DH– SiS equations for theory–experiment fit of γ±. This provides full consistency with the previously analyzed electrolytes, or the same ions in other electrolytes. The parameter d/2 is shown here in all cases to be very small (in most cases, 3; and if it does exist, it probably reflects factors other than the charge of the ions of a binary electrolyte.


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Figure 7. The mean ionic activity coefficient (molal), γ±, and the individual ion activity coefficients of the smaller (γs) and larger (γl) ions, as functions of the square root of ionic strength, I1/2, for LaFe(CN)6 (I/m = 9) in water at 25 °C. Experimental γ±’s (diamond symbols) are the corrected data of Malatesta et al. (Ref. 8). Lines represent γ’s computed by DH–SiS: The full line is that of γ±; upper dashed line is that of the cation’s γs; lower dashed-dotted line is that of the anion’s γl. (a) Full I range. (b) γ±’s at very low I range, 0 – 0.0015. (c) γ±’s at low I range, 0 – 0.05; dotted line is the limiting law.


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Discussion Low-charge electrolytes, those with ǀz+z–ǀ ≤ 2, hence of the 1–1, 2–1 and 1–2 valence families, are by far more common and important in nature than high-charge electrolytes, those with ǀz+z–ǀ ≥ 3. For example, in biological and physiological functions of the living cell and in living body fluids, the most common ions are H+, Na+, K+, Mg2+, Ca2+, Cl–, HCO3– and single-charge organic anions. Indeed, most electrolyte theories developed over the years, including the well-known and thoroughly tested hypernetted chain (HNC) and Mean Spherical Approximation (MSA) theories,1,12 have been examined mostly against experimental data of the low-charge electrolyte systems; this, however, was primarily because γ± data of low-valence electrolytes were found to fit better with theory. Whenever systems with ǀz+z–ǀ ≥ 3 were examined (e.g., the 3–1 LaCl3 and 2–2 sulfates), the theory–experiment fit was usually poor (or poorer) and the lack of agreement was commonly explained by partial ion association or other factors ignored in the theoretical development, such as those related to ion charge. One argument, as mentioned above, was that the linearization of the PB equation (as in the DH theory and in DH–SiS) is not justified in the case of multi-charged ions.3,5,7,9,10 Another problem with the high-charge electrolytes has been the lack of experimental data at low concentration, below 0.1 m. This is the reason why the experimental work of Malatesta et al. is so important. Even if systems with ǀz+z–ǀ ≥ 3 are less prevalent in nature than those with ǀz+z–ǀ ≤ 2, they are of great theoretical interest. For example, is the limiting law correct for cases with ǀz+z–ǀ ≥ 3? Is it applicable for ǀz+z–ǀ = 9? The concern over the adequacy of employing the plain (unmodified) DH theory for high-charge electrolytes has been raised occasionally in dealing with the theory–experiment fit of


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electrolyte solutions; observing negative deviations from the limiting law for those ionic systems has further exacerbated the problem. Because the DH–SiS treatment is an improved electrostatic ionic theory going beyond the old DH theory, and since the effectiveness of an “all-electrostatic” treatment should be especially manifested at very low ionic concentration, analyzing Malatesta’s experimental data by the DH–SiS equations is of special interest. In the overall DH–SiS equation (eq 8), a charge factor appears three times: (i) in ǀz+z–ǀ, (ii) in the conversion of m to I, and (iii) in the ratio νi/ν that gives the relative contribution of the positive and negative ions as reference ions. Malatesta and coworkers were able to measure Es of electrolytes with high-charge ions at very low concentration. However, for converting their E data to γ± values, they needed the standard potential, Eo. Unfortunately, obtaining Eo from measured Es was compromised by using inaccurate methods in extrapolating E to zero concentration. Applying Methods 1 and 2 of the present paper for obtaining more accurate Eo values of the Malatesta measurements, is now shown to give improved experimentally-derived γ±’s. Such values, for nine different binary electrolytes with ǀz+z–ǀ ≥ 3, are examined in this paper against the prediction of the DH–SiS theoretical expressions through the usual theory–experiment fitting procedure. I have chosen to examine electrolyte systems that, according to Malatesta and coworkers, adhere to the DH theory, and other systems that were claimed by those authors to not obey that theory but, instead, deviate negatively from the limiting law: The electrolytes LaCl3 and K3M(III)(CN)6 with M(III) = Fe, Co, were found by Malatesta et al. to follow the DHLL. The M(II)3[M(III)(CN)6]2 electrolytes with M(II) = Mg, Ca, Ba and M(III) = Fe, Co, and the electrolytes


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LaM(III)(CN)6 with M(III) = Fe, Co, were claimed to deviate negatively from the DHLL. In all cases, exactly the same procedure has been applied here for the evaluation of Eo from Nernst equation. A reviewer has raised the point that the issue at stake and the agreement with theory (e.g., DH–SiS) should be checked also for γ± values of the relevant electrolytes, as obtained by methods other than emf measurements. However, unfortunately, owing to technical limitations, other methods (such as vapor pressure, isopiestic point, boiling point elevation, freezing point depression, etc.) cannot be applied successfully to solutions of very low ionic concentration; those methods are usually effective only down to 0.1 m or at most, 0.001 m. Thus, Malatesta’s emf method employing his special electrodes appears as the only existing method by which sufficiently accurate γ± data can be generated at concentrations well below 0.001 m. The correspondence of Malatesta’s data as corrected in the present work with data of other groups, available especially at higher concentration, can be done (as demonstrated for LaCl3 in Fig. 3, and for K3Co(CN)6 in Fig. 4) by showing the agreement of data from the different sources with the entire computed behavior based on DH–SiS expressions, using a fixed optimized set of ISPs. (For the above examples, see the chosen ISPs in Table 1). The main result of the current analysis is that all binary ionic systems are shown to essentially follow the DHLL. This outcome should be pleasing to theorists of electrolyte solutions. Were this not the case, one would have had to assume that there is a discontinuity in the theoretical effect of the ionic charge on the activity of electrolytes as a function of concentration, such that below ǀz+z–ǀ = 4 (i.e., for electrolytes with ǀz+z–ǀ being 1, 2 or 3, since ǀz+z–ǀ is always an integer), the charge effect displays one physical


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behavior, and above ǀz+z–ǀ = 3 (i.e., for electrolytes with ǀz+z–ǀ being 4, 6, 9, etc.), it unexplainably adopts another, different physical behavior. From a theoretical viewpoint, there seems to be no justification for a “discontinuity” of the charge effect or for otherwise assuming a continuous shift of this effect, as a function of its extent, from obeying to disobeying the DHLL. Thus, to my great satisfaction, the recalculated γ± data of all systems examined in this work, except LaCo(CN)6 (see above), fit very well with the theoretically predicted values based on the DH–SiS equations; this happens when employing the ISPs b+ and b– as the known diameters of the cation and anion, respectively, and when only slightly adjusting a with respect to the sum of the ionic radii. Furthermore, since the DH–SiS theory computes the activity of individual ions24 and obtains the mean ionic activity by combining the contributions of the positive and negative ions in a binary system, the general “constancy” of the single-ion activity of an ion at low concentration, in the presence of different counterions, can establish the effectiveness of the theoretical treatment as offered in the present work. Clearly, as concluded by visually examining eqs 6 and 7, for a given ion, the same function of activity vs. ion strength is obtained for a constant a. Thus, the activity of the ion is dependent on the counterion only to the extent that a varies. According to Table 1, a changes moderately between systems having a common ion, as shown for K+, La3+, Fe(CN)63– and Co(CN)63–. Figure 8 depicts the quantitative change of γ+ of La3+ (Fig. 8a) and γ– of Fe(CN)63– (Fig. 8b), as a function of ion strength, for electrolytes of the present analysis. In both cases, the behavior is initially the same, regardless of the counter ion, up to a certain ion strength (or concentration) beyond which the single-ion activities start to diverge, reflecting the effect of the


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different a parameters of the different electrolytes. As common to cations,12 La3+ in LaCl3 passes a minimum in the γ+-vs.-I1/2 curve (Fig. 8a), and the trend in LaFe(CN)6 suggests that a similar behavior would happen if γ± values at higher I were available. In contrast, the anion Fe(CN)63– exhibits a monotonic γ– decrease with increasing I (Fig. 8b), as observed with other anions.12

Figure 8. Individual-ion activity coefficients (molal), γi’s, of ions of the present analysis, as functions of the square root of ionic strength, I1/2: (a) γ+ of La3+ in LaCl3 (dashed line) and in LaFe(CN)6 (dashed-dotted line). (b) γ– of Fe(CN)63– in K3Fe(CN)6 (dashed line), in Mg3[Fe(CN)6]2 (dashed-dotted line) and in LaFe(CN)6 (dotted line). Finally, in view of the very good theory–experiment fit of the high-charge electrolytes examined in the present study, and the lack of negative deviations from the DHLL, there seems to be no support by Malatesta’s own low-concentration γ± data, as corrected here, for his claim (and that of others cited by him) that the linearization of the PB equation is unjustified and ineffective in the case of high-charge ions. Indeed, as mentioned in the Introduction section, my recent analysis13 of the ion strength range of


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the effectiveness of the L-PB equation, has revealed no sensitivity to ion charge, especially at very low ionic strength. At such a low I, even the 3–1 electrolytes (e.g., LaCl3) – which exhibit theory–experiment disagreement at higher ion strength13 – are in full agreement with the linearization (through the DH–SiS theoretical treatment).

Conclusion The thermodynamic behavior of high-charge electrolytes with ǀz+z–ǀ ≥ 4 has been a puzzle for many decades. Measuring their ionic activity at very low concentration ( 3 (i.e., of the valence families 2–2, 2–3/3–2, 1–4/4–1, 3– 3, etc.). Two main conclusions immediately emerge: First, without anomaly with respect to the limiting law, there is no basis for claiming that the L-PB equation is unsuitable for high-charge electrolytes. Second, by strictly adhering to the DHLL, the high-charge systems examined here extend the use of an all-electrostatic theory, such as DH–SiS, to very dilute solutions of electrolytes with ǀz+z–ǀ > 3; this suggests that the theory should apply to dilute solutions of electrolytes of essentially all valence families. Furthermore, the fit at very high ionic dilution is better with DH–SiS than with DHLL because the DH–SiS mathematical expressions do not merge with DHLL at very low concentration; they only approach the limiting straight line asymptotically from above this line, thus providing, as appropriate, a positive – not negative – “deviation”; the latter is fully consistent with the physics of the electrostatic model and theory. Stated otherwise, the “deviation” is, in fact, that of the limiting law – as straight line – from the more accurate upward-curved line of DH–SiS. DHLL is the tangent of the physically more correct non-linear log γ±-vs.-I1/2 curve (of DH–SiS), and this is true even at ion concentration as low as 10–6 m. The fit of DH–SiS with the corrected experimental data of Malatesta and coworkers is done with co-ion ISPs as the respective ionic diameters, so no ion hydration is involved in the physical process leading to the excess thermodynamic behavior. Also, adhering to the theory means that electrolytes even as highly charged as LaFe(CN)6 are not ionassociated in aqueous solution within the m-ranges of the Malatesta measurements.


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Appendix: Can negative deviations from the limiting law be detected in ion electrical conductance studies?

A reviewer has pointed out that if negative deviations from the limiting law exist in activity measurements, they should also exist in electrical conductance measurements of the same electrolytes. If electrolytes with |z+z–| > 3 do not deviate from the limiting law in conductance studies, they should not deviate from the limiting law of activity (DHLL) either. The reviewer further emphasized that studies of Apelblat (e.g., Ref. 25) show no apparent difference in behavior between electrolytes with |z+z–| ≤ 3, and with |z+z–| > 3. A note on the Apelblat study and the relevance of conductance measurements to the present work, appears, therefore, necessary. Apelblat’s aim was to improve the determination of limiting conductivities of electrolytes in solution through employing the Quint–Viallard (QV) conductivity equation for the extrapolation of the conductivity to zero concentration. The QV equation is a Justice-type expression (see Ref. 1, p. 523) incorporating a (speculative) ion association constant as an adjustable parameter. An ion-size factor enters the constants of the QV equation and appears also in the DH expression (DHEE); the latter is used for deriving activity coefficients for the mass-action expressions. Apelblat chose the (questionable) Kielland values for the ion-size parameters. The purpose of this note is not to critique the Apelblat study or his very approach to the problem of extrapolation of conductance data; that is a separate issue. What is relevant in Apelblat’s work to the present study, is not the theory–experiment fit and the extrapolation yielding the limiting conductivities, but rather the very rigorous compilation of literature conductance data of many electrolytes of various valence families (including “exotic” ones, e.g., 1–6 and 6–


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1). The data are presented in the figures.25 An inspection of the figures reveals that most experimental data perfectly (or almost perfectly) adhere to the Kohlrausch relation [i.e., the equivalent conductivity, Λ, being a linear function of c1/2 (c, molar concentration) with a negative slope], and therefore also to the simple Debye–Hückel–Onsager (DHO) limiting law,1–3

Λ = Λ − 12 ∗ ( + | |) + 4 ∗ 5Λ 67

(A1)

α*, β* and ω being the well-known constants of the DHO theory, and Λo the limiting Λ. As apparent from Apelblat’s Figs. 1 – 6 and 10 – 13, electrolytes with both |z+z–| ≤ 3 and |z+z–| > 3, all follow the DHO limiting law. On this, the reviewer is absolutely correct. While this is satisfying and in full accord with the current study on activity coefficients, a word of caution is in order here. The claimed negative deviations of electrolytes from the DHLL, as found in the literature (see main text), are not deviations from the straight-line behavior, but rather unexplainable steeper negative slopes of the limiting straight lines. The slope of the DHLL line incorporates the charge of the ions by the factor |z+z–|. In the DHO limiting law, eq (A1), the slope of the straight line is a more complex structure, in which there are two contributions – that of electrophoresis and that of relaxation. The ion charge appears only in the former contribution and the charge factor is (z+ + |z–|). The latter contribution (of relaxation) includes the limiting conductivity that tends to be larger for complex ions (those frequently present in high-charge electrolytes), thus further “diluting” the chargerelated electrophoretic contribution to the slope of the limiting law. As a result, deviations of electrolytes from the DHO limiting law – if occurring – are apparently


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impossible to detect. Such deviations would only somewhat affect the value of Λo, thereby also somewhat affecting the slope of the limiting law. This can be easily realized by converting eq (A1) to a form analogous to the DHLL,

8

89 − 1 = − +2 ∗

(;' |;* |) 8

(A2)

The left-hand term in eq (A2) is equivalent to lg γ± in DHLL, both factors being small negative numerical fractions approaching zero at infinite dilution, (i.e., when Λ ≈ Λo and γ± ≈ 1, respectively). The term in the brackets on the right-hand side of eq (A2) is equivalent to |z+z–|A of DHLL. A negative deviation from the DHLL would be equivalent to the factor |z+z–| being larger than expected, say 4.2 instead of 4 in the case of 1–4 electrolytes. In the case of the DHO limiting law, a larger (z+ + |z–|) factor due to that deviation (say, 5.2 instead of 5) would be compensated by a larger Λo value obtained by the extrapolation to zero concentration, thus diminishing, or eliminating, the effect on the slope [term in the brackets of eq (A2)], i.e., the factor (z+ + |z–|)/Λo may not vary significantly. Some minor change could occur on the left-hand side, due to the small change in Λo. The conclusion here is that a deviation (negative or positive) from the conductivity limiting law will not be observed, or it will be too small to be detected convincingly. Furthermore, Λo data of electrolytes of complex ions are usually not very accurate, and, therefore, so are the ionic equivalent conductivities of the complex ions. Thus, even if – in principle – deviations from the DHO limiting law could be detected, in practice they would be undetectable since we do not have a stable value for the limiting


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conductivity of the complex ion. A deviation will be manifested by a somewhat different value of Λo without a standard value to compare with. The overall conclusion from all the above is that deviations from the limiting law may occur in activity studies but not show in conductivity studies. As a first choice, sans rock-solid evidence to the contrary, one should assume that all electrolytes of all valence families adhere to the limiting laws of both activity and conductivity. On this, I fully agree with the reviewer.

Glossary Abbreviations/acronyms DH

Debye–Hückel (ion activity theory)

DHEE

DH extended equation

DHLL

DH limiting law

DHO

Debye–Hückel–Onsager (ion conductivity theory)

DH–SiS

SiS treatment of the DH theory

emf

Electromotive force

ISP

Ion-Size Parameter; distance of closest approach between two ions

L-PB

Linearized Poisson–Boltzmann (equation)

PB

Poisson–Boltzmann (equation)

SiS

Smaller-ion Shell

QV

Quint–Viallard (equation of conductivity)

Mathematical/physical notations Å

Ångstrom (unit)

A

Constant in the DHLL equation

å

Distance of closest ion-ion approach in DHEE equation

a

An ISP; distance of closest approach between counterions

B

Constant of the DHEE equation; equals κ/I1/2

b

An ISP; distance of closest approach between co-ions


35


c

Concentration in Molar units, moles per liter of solution

CMX

Parameter in Pitzer equation

clc

Calculated (see figure frames)

d

Nonadditivity factor; equals 2a – b+ – b–

E

Electrochemical potential

exp

Experimental (see figure frames)

F

Faraday constant

I

Ionic strength

i

Chosen ion

k

Boltzmann factor

M

Molar, moles per liter of solution

m

Molal, moles per kilogram of solvent

N

Number of equivalents (= z+ν+ = z–ν–)

T

Absolute temperature

R

Gas constant

V

Volt (unit)

z

Ion charge number, or valence

(1), (2)

Of Method 1, 2 (see text)

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Greek notations

α

Pitzer parameter

α∗

Parameter in electrophoretic part of DHO conductivity equation

β

Pitzer parameter

β∗

Parameter in relaxation part of DHO conductivity equation

γ

Molal ionic activity coefficient

κ

Reciprocal screening length in DH theory

Λ

Equivalent conductivity

µ

Chemical potential

ν

Number of ions in the “molecular formula” of an electrolyte

ω

Random-walk parameter in DHO equation

Ψ

Extra electrostatic (field-averaged) potential energy per unit charge


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Superscripts Asterisk that represents dividing Ψ by δ, (= q2/εkT, with q being the

*

fundamental charge, and ε solvent permittivity) o

limiting (value)

(0), (1), (2) Order of Pitzer β parameter new

Of new

Subscripts ±

Mean ionic

+, –

Of positive (+), negative (–) ion

ave

Average

el

Electric (part)

i

Of ion i

s, l

Of smaller (s), larger (l) ion

1, 2

Order of Pitzer α parameter

References (1) Bockris, J. O’M.; Reddy, A. K. N. Modern Electrochemistry; 2nd ed.; Plenum Press: New York, 1998; Vol. 1. (2) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions; 2nd ed.; Butterworths: London, 1959. (3) Harned, H. S.; Owen, B. B. The Physical Chemistry of Electrolytic Solutions; 3rd ed.; Reinhold Publishing Corp.: New York, 1958. (4) Davies, C. W. Ion Association; Butterworths: London, 1962. (5) 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. (6) 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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(7) Malatesta, F.; Carrara, G. Activity Coefficients of Electrolytes from the E.M.F. of Liquid Membrane Cells. I. The Method – Test Measurements on KCl. J. Sol. Chem. 1992, 21, 1251–1270. (8) 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. Sol. Chem. 1994, 23, 11–36. (9) 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. Sol. Chem. 1996, 25, 61–73. (10) 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. Sol. Chem. 2000, 29, 449– 461. (11) Malatesta, F.; Trombella, S.; Giacomelli, A.; Onor, M. Activity coefficients of 3:3 electrolytes in aqueous solutions. Polyhedron 2000, 19, 2493–2500. (12) Fraenkel, D. Simplified electrostatic model for the thermodynamic excess potentials of binary strong electrolyte solutions with size-dissimilar ions. Mol. Phys. 2010, 108, 1435–1466. (13) Fraenkel, D. Ion Strength Limit of Computed Excess Functions Based on the Linearized Poisson–Boltzmann Equation. J. Comput. Chem. 2015, 36, 2302– 2316. (14) Fraenkel, D. Computing Excess Functions of Ionic Solutions: The Smaller-Ion Shell Model versus the Primitive Model. 1. Activity Coefficients. J. Chem. Theory Comput. 2015, 11, 178–192. (15) Fraenkel, D. Computing Excess Functions of Ionic Solutions: The Smaller-Ion Shell Model versus the Primitive Model. 2. Ion-Size Parameters.

J. Chem.

Theory Comput. 2015, 11, 193–204. (16) Fraenkel, D. Monoprotic Mineral Acids Analyzed by the Smaller-Ion Shell Model of Strong Electrolyte Solutions. J. Phys. Chem. B 2011, 115, 557–568.


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(17) Fraenkel, D. Effect of Solvent Permittivity on the Thermodynamic Behavior of HCl Solutions: Analysis Using the Smaller-Ion Shell Model of Strong Electrolytes. J. Phys. Chem. B 2011, 115, 14634–14647. (18) Fraenkel, D. Single-Ion Activity: Experiment versus Theory. J. Phys. Chem. B

2012, 116, 3603–3612. (19) Fraenkel, D. Electrolytic Nature of Aqueous Sulfuric Acid. 1. Activity. J. Phys. Chem. B 2012, 116, 11662–11677. (20) Fraenkel, D. Agreement of electrolyte models with activity coefficient data of sulfuric acid in water. J. Chem. Thermodynamics 2014, 78, 215–224. (21) Fraenkel, D. Theoretical analysis of aqueous solutions of mixed strong electrolytes by a smaller-ion shell electrostatic model. J. Chem. Phys. 2014, 140, 054513-1–054513-16. (22) Spedding, H. F.; Atkinson, G. In The Structure of Electrolyte Solutions; Hamer, W. J., Ed.; Wiley: New York, 1959; Chapter 22, pp. 319–339. (23) Wynveen, R. H.; Dye, J. L.; Brubaker Jr., C. H. Activity Coefficient and Conductivity Measurements of High-charge (3-1, 1-3, 3-2) Electrolytes. I. J. Am. Chem. Soc. 1960, 82, 4441–4445. (24) Vera, J. H.; Wilczek-Vera, G. Classical Thermodynamics of Fluid Systems. Principles and Application; CRC Press: Boca Raton, 2016; Chapter 27. (25) Apelblat, A.; The Representation of Electrical Conductances for Polyvalent Electrolytes by the Quint–Viallard Conductivity Equation. Part 3. Unsymmetrical 3:1, 1:3, 3:2, 4:1, 1:4, 4:2, 2:4, 1:5, 1:6 and 6:1 Type Electrolytes. Dilute Aqueous Solutions of Rare Earth Salts, Various Cyanides and other Salts. J. Sol. Chem. 2011, 40, 1291–1316.


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For Table of Contents use only

Title: Negative Deviations from the Debye–Hückel Limiting Law for High-Charge Polyvalent Electrolytes – Are They Real?

Author: Dan Fraenkel

TOC/Abs Graphic:


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