Activity coefficients of ions in sodium halide solutions: Critical remarks

Francesco Malatesta

Abstract

The fundamentals of an experimental method proposed by Zhuo et al. [1] to determine activity

coefficients of sodium and halide ions in sodium halide solutions, are critically examined. It is

shown that this method relies on a key hypothesis, which proves to be incorrect, about the liquid

junction potential, whose value is assumed not to change when the concentration of the sample

solution is changed. The direct consequence of this assumption is that results that are interpreted

as the activity coefficients of sodium and halide ions are, instead, conventional values, which only

depend on the mean activity coefficients and transport numbers, and have no connection with the

activity coefficients of the respective ions.

Considerations valid for all papers dealing with the experimental determination of ionic activity

coefficients are derived. Such papers are to be regarded as incorrect.

1. Introduction

A paper was recently published in this Journal, "Activity coefficients of individual ions in aqueous

solutions of sodium halides at 298.15 K", by Zhuo et al. [1], in which conventional values of ionic

activity coefficients are taken for the activity coefficients of the corresponding ionic species. This

confusion is potentially negative for scholars working with electrolyte solutions, owing to the risk

that findings that are a mere artifact of arbitrary assumptions (e.g., the activity coefficients of Na+

found to be lower than those of F- and higher than those of Cl- and Br- in [1]) may be mistaken for

experimental evidence. The purported ionic activity coefficients of [1] rely on an arbitrary

assumption regarding the liquid junction potential (EJ), a quantity that, like the ionic activity

coefficients, can be defined only in conventional terms. Conventions for ionic activity coefficients

(i) and liquid junction potentials are interrelated, and infinite self-consistent combinations exist,

which are perfectly equivalent and interchangeable.

The question of the determination of ionic activities dates back to last century. In late 1921, Lewis

and Randall emphasized that 'it would be of much theoretical interest if we could determine the

actual activity of an ion in a solution of any concentration. This indeed might be accomplished if we

had any general method of calculating the potential at a liquid junction' [2]. As regards the

possibility of calculating the potential at a liquid junction, however, severe reservations were

expressed a few years later by Taylor [3] and Guggenheim [4,5] who proved, respectively, that (i)

also in cells with liquid junctions, the electromotive force depends on the activity of complete

electrolytes, and not on the individual activities of the ions that compose these electrolytes,

exactly as in the case of cells without liquid junctions [3]; and, (ii) the calculation of the liquid

junction potential is impossible without the prior knowledge of the same ion activities that one

would like to derive from the knowledge of the liquid junction potential [4], thus yielding infinite

equivalent possibilities. Even more important, Guggenheim (and to some extent Taylor)

demonstrated that the very concept of a difference of electrical potential between two different

phases has no physical significance; by consequence, also the individual electrode potentials,

chemical potentials of charged constituents of a solution, liquid junction potentials, and activities

and activity coefficients of individual ionic species, are entities with no physical significance and

can only be defined in merely conventional terms [3-5].

A quantity that cannot be defined uniquely cannot be measured a fortiori; therefore, it was to be

expected that after 1930, no scientist would insist on attempting to determine the actual values of

ion activity coefficients. Things evolved otherwise, however. Papers [6-8] may give an idea of the

debate between supporters of the opinion that experimental determinations of ion activity

coefficients are in some way possible, and those who exclude this possibility.

In reality, the discussion had no reason to exist: Guggenheim's arguments that exclude any

possible determination of these quantities have never been invalidated. Yet, ignoring the

conceptual impossibility, Zhuo et al. [1] published their experimental values of ionic activity

coefficients.

To show that their results are not valid, we will not appeal once again to the general principle that

ionic activity coefficients and EJ are concepts with no physical significance; we will act instead (like

the authors of [1]) as if the i and EJ functions were univocally definable and potentially knowable,

and we will examine in depth the effects of the basic assumption paper [1] relies on, i.e., that the

variation of EJ with the concentration of the solution examined "can be safely neglected" [1]. We

will prove in the next sections that, because of this assumption, the quantities taken for i in [1]

are, instead, other functions which only depend on the mean activity coefficients (±) and

transport numbers (ti) and are unrelated to i .

2. Fundamentals

For any isothermal Voltaic cell, a general equation exists [9],

E = E° - E° - S ss, ln as, + s, ln as, +

s d(ln as)] (1)

in which S is the Nernstian slope RT/F; E° and E° are the standard potentials of the terminal

electrodes; s, and s, are the virtual numbers of moles of any species s (ions or molecules)

formed at the respective electrodes when one virtual faraday of electricity goes through the cell

from left to right in the cell; s (the "transference number of s using the Scatchard definition [9])

are the moles of species s which would be virtually transferred in the direction of the cations for

any virtual faraday, at constant composition (the product s × zs, where zs is the charge of s with its

sign, is the usual transport number of s, ts, null for molecules); and as is the activity of s. The term

- S s

s d(ln as) represents the overall potential introduced by liquid junctions and membranes

encountered along the path which, starting from the metal-solution interface () of the left

electrode (e.g., the internal metallic electrode of an ISE), arrives at the metal/solution interface

() of the right electrode (e.g., the internal metallic electrode of another ISE). Eq. (1) of Ref. [1] is a

corollary of the present Eq. (1). Although the ionic activity coefficients are apparently involved in

Eq. (1) through the as terms, it has been well known since Taylor's paper [3] that these quantities

are connected to one another in such a way that only the activities of neutral sets of ions (e.g., Na+

+ CI-) and molecular species (e.g., H2O) survive. Therefore, the electromotive force of any cell – the

only quantity that the experiments provide – contains no information about the i values [3,6]; any

set of possible i functions that are consistent with the E values is perfectly equivalent to, and

replaceable by, infinite other sets of alternative i functions able to compose the same ± values,

with no physical possibility of distinction.

If one divides a cell into arbitrary sections, then the individual potentials of these sections

conserve, unlike E, a net dependence on individual ionic activities that do not simplify off (e.g., the

individual potential of a section consisting of a silver chloride-silver electrode immersed in a KCI

solution is a function of aCl-). Unfortunately, the potentials of the individual sections are quantities

that cannot be measured, or even defined except in a conventional form. To draw from E the

potential EX of a section X (e.g., an ISE; a liquid junction; a membrane; etc.), which contains in

theory the desired information about one or more i, it would be necessary to know the

complementary quantity EY = E - EX pertaining to the residual sections, which however depends on

other unknown i: and vice versa. If we use an approximate estimation for EY (which is the same as

assigning conventional i values inside all cell sections but X), then we obtain conventional values

also for EX and the i involved in EX. That is exactly the procedure adopted in Ref. [1], where the

convention concerns the connection between the inner solution of the reference electrode and

the sample solution. In Ref. [1], indeed, it is assumed arbitrarily that the double junction

connection allows the liquid junction potential to remain constant when the concentration of the

sample solution varies, 'as the major internal component [i.e., the filling solution of the salt bridge]

stays constant'. This arbitrary assumption (or in other words, this convention) leads us to

determine experimental values of something (we shall name it i for short) which is supposed, in

Ref. [1], to coincide with i / i,R (i,R, the value of i in a solution k indicated as R, selected as a

reference point). Starting from this supposition, the i values are used in connection with an

expanded form of the Debye-Hückel equation in order to identify, by least squares, the value of

the quantity that is assumed to be i,R in i (and would really be i,R if i were i / i,R, which is not

the case). The supposed i,R, and i values used as if these were i / i,R provide conventional i,

which Zhuo et al. [1] take for i. Furthermore, as any error J in EJ introduces opposite errors J/S

and -J/S in In + and In -, these conventional i obey the correct relationship In + + ln- = 2 ln±

as if these were the ion activity coefficients, but this occurrence, of course, has no value to

demonstrate that these are the "correct" i.

A closer examination of the methods used by Zhuo et al. [1] (next section) allows us to better

understand the consequences of the key assumption that the dependence of EJ on the

concentration of sodium halide "can be safely neglected" [1].

3. The quantities that the method of Zhuo et al. [1] identifies

Let us examine what really happens in a cell, (I'), very similar to cell (I) of Zhuo et al. [1],

Cu|REF-E ⁞ solution b (salt bridge) ⁞ solution k |Na-ISE|Cu' (I')

For E to represent the difference between the potential of the right side metal Cu' and the left side

metal Cu of the cell, in (I') we have exchanged left and right electrodes, compared with cell (I) of

Ref. [1]. As in cell (I) of [1], the ISE is Nernstian. The only difference between cell (I) of [1] and this

cell (I') is the salt bridge b, which in cell (I) is filled by a different electrolyte (10% sodium nitrate,

[1]), while in cell (I') it contains NaX (concentration mb) like k (concentration m). This condition is

required for exact mathematical development to be feasible (with a heteroionic salt bridge as in

cell (I), EJ depends on the individual profiles of ion concentrations in the transition zone between b

and k, which are time-dependent until they reach some no better identified stationary

equilibrium). By indicating as nw+ and nw- the hydration numbers of the cation and anion, the

transference number of water holds w = nw+t+ – nw-t-,

The electromotive force of cell (I') has the same conventional form reported in [1] for cell (I):

E = E° + S ln ai + EJ (2)

(i = Na+; EJ, the liquid junction potential between b and k; E° = E°ISE - EREF-E + E'J, where E'J is the

liquid junction potential on the left side of b, constant). Following Zhuo et al. [1], we select one

particular solution k, named R, as our reference point, and define E as the difference between

the E values measured in each solution k and in R,

E = S ln (ai /ai,R) + EJ (3)

According to the general equation of voltaic cells, we have

EJ = - S s

b

ks d(ln as) (4)

(the sum and subscript s cover all ionic and molecular species present). In cell (I'), Eq. (4) becomes:

EJ = S {ln[m-/(mb-b)] – 2b

kt+ d(ln m±) –

b

k(nw+t+ – nw-t-) d(ln aw)} (5)

(Since we are dealing with one single electrolyte, t+, t-, ±, +, -, and aw are all functions or the

same single variable m, and the integrals depend only on the extreme values of m and not on the

intermediate values encountered.) The function EJ is, therefore,

EJ = EJ - EJ(R) = S {ln[m-/(mR-,R)] – 2R

kt+ d(ln m±) –

R

k(nw+t+ – nw-t-) d(ln aw)} (6)

It is clear from Eq. (6) that a higher or lower concentration or NaX in the salt bridge solution b has

no effect on the value of EJ, and this latter cannot be 'safely neglected', in contrast with the key

hypothesis or Zhuo et al. in [1]. Of course, it is not so easy to prove that the same thing happens

when the electrolyte of the salt bridge is not the same as in k, which is the case in cell (I) used by

Zhuo et al. [1]; but it is the duty of Zhuo et al. to prove how and why a salt bridge made up of a

different electrolyte should cause EJ to be negligible, as they hypothesize in [1].

To estimate the magnitude of EJ in cell (I'), let us consider for a moment that t+ is roughly

constant, thus obtaining:

EJ S {t- ln[m-/(mR-,R)] – t+ ln[m+/(mR+,R)] – (nw+t+ – nw-t-) ln(aw,k /aw,R)} (7)

Regarding possible values for EJ in real situations, an instructive exercise based on past

experimental data of high accuracy [10] is reported in Appendix A. It shows how the activity

coefficients of the cation and the anion which are found to be consistent with EJ = 0 turn out to

be instead the activity coefficients of the anion and the cation — i.e., interchanged – if EJ varies

by only 5 mV along the concentration range covered in [1].

In the specific situation of t+ = t- and nw+ = nw- , we have

EJ (S/2) ln[(- +,R)/(+ -,R)] (8)

proving that the opinion that in these conditions EJ is null is as arbitrary as assuming that +

remains equal to - at all concentrations.

Let us now return to non-approximated equations; we want to understand the implications of the

assumption EJ = 0 for the "i" values obtained. To distinguish these activity coefficients from the

"real ones", let us use the symbol instead of . Assuming EJ = 0 automatically transforms Eq. (3)

(and Eq. (3) of [1]) into:

E = S ln[m+/(mR+,R)] (9)

(+/+,R = the function of the previous section). We have now two equations, Eq. (9) which is the

result of the assumption EJ = 0, and Eq. (10), which is the exact equation for cell (I')

E = 2S {ln[m±/(mR ±,R)] – R

kt+ d(ln m±) –

R

k(nw+t+ – nw-t-) d(ln aw)} (10)

By connecting these two equations, we obtain

ln[m+/(mR+,R)] = 2 {ln[m±/(mR ±,R)] – R

kt+ d(ln m±) –

R

k(nw+t+ – nw-t-) d(ln aw)} (11)

or, with the approximation that t+ is roughly constant, to better estimate the magnitude of this

quantity,

ln[m+/(mR+,R)] 2 t- ln[m±/(mR ±,R)] – (nw+t+ – nw-t-) ln(aw,k /aw,R) (12)

The exact Eq.(11) and approximate Eq.(12) prove that the quantities Zhuo et al. [1) took for

ln[(m+)/(mR+,R)], from which they derived their supposed values of +, are instead a different

thing, i.e., a function of the mean activity coefficients and transport numbers (approximately, a

function of t- and ± in the case of +; in the similar development for X-, not reported for brevity, a

function of t+ and ±), which has no relation with ln[(m+)/(mR+,R)].

Of course, these considerations, which are strictly valid for cell (I'), cannot be extended accurately

to cell (I) used by Zhuo et al. [1] because of the presence of the heteroionic salt bridge; but the

conclusions cannot be appreciably different, leaving no doubt that the values reported as

experimental ion activity coefficients in Ref. [1] are mere conventional values, with no plus point

among the infinite alternative sets of conventional values, including those in which + and -

correspond to - and + of [1].

4. Conclusions

The paper of Zhuo et al. [1] proposed a method aiming to measure the ion activity coefficients,

whose results for sodium halide solutions were indeed reported as the activity coefficients of Na+,

F-, Cl-, and Br-. Yet ever since the Taylor and Guggenheim papers [3-5] it has been known that the

activity coefficients of individual ionic species have no physical meaning and can be defined only in

conventional terms. The supposition that experimental methods can identify the "real" values of

quantities of itself without physical significance and conventional is nonsense; the "experimental"

results are necessarily conventional and not experimental. We therefore examined the basic

assumptions and structure of the method used to determine - and + of [1], and concluded that

these - and + correspond to some no better identified functions – using heteroionic salt bridges –

or identified functions – using homoionic salt bridges – of the mean activity coefficients and

transport numbers, which do not seem to be of any practical or theoretical utility. A warning is

necessary, not to accept as true the orders + > - or vice-versa as if these were experimental

results, in [1] and other papers that pretend to determine activity coefficients of individual ionic

species [7, 11-29]; these orders indeed are a mere effect of the conventions adopted, and can

always be reverted by adopting a different convention, as exemplified in Appendix A. Papers that

report experimental values of ionic activity coefficients are all to be regarded as intrinsically

incorrect.

List of symbols

a dimensionless activity

ai; as; aw dimensionless individual activity of i ; s ; water

b inner solution of the salt bridge

C dimensionless concentration (molar scale)

E electromotive force

EJ liquid junction potential

F Faraday constant

h inner solution of the reference electrode

i any ionic species

k sample solution

m dimensionless concentration (molal scale)

nw+, nw- hydration numbers of the cation and anion

R gas constant

s any chemical species

S Nernst's slope

ts (= zs s) transport number of s (null for uncharged species)

T absolute temperature

y+ ; y- ; y± activity coefficients stated in terms of C

zs charge of s, with its sign

left side terminal electrode

activity coefficient

i ; s individual activity coefficient of i; s

± mean activity coefficient

i,R individual activity coefficient of i in a reference solution R

s, s mol of any species s formed at the respective electrodes when one faraday of

electricity goes through the cell (from left to right in the cell)

conventional value for

i / i,R (hypothetically, i / i,R )

s transference number of s (virtual moles of s transferred in the direction of the cations

for one virtual Faraday)

right side terminal electrode

Acknowledgments

This work has been supported by the University of Pisa. The author wishes to thank Dr. Dmitri P.

Zarubin of Moscow University of Technology for enlightening discussion and kind criticism.

Appendix A.

The present exercise evidences how arbitrary is any assumption regarding liquid junction

potentials and individual ionic activities, proving the existence of infinite ways in which the same

experimental data can be interpreted in terms of equally plausible, self-consistent sets of EJ and

ionic activity coefficients. It is an implicit reply to papers [1, 7, 11-29], whose authors have

deplorably insisted on ignoring the fundamental, straightforward arguments with which, eighty

years ago, Guggenheim explained why activities and activity coefficients of individual ionic species

cannot be physically defined [4].

It was impossible to use the experimental E of [1] for this exercise, because of the presence in the

cell of two or more electrolytes, which prevent any possibility of calculating EJ (in the case of more

than two species of ions the integrand in Eq. (1) is not a perfect differential, and so the integration

cannot be performed without further knowledge of the variation of the concentrations along the

transition layer [30]). The experimental data we consider are therefore those measured for KCI by

Shedlovsky and Macinnes [10], which are expressed in terms of molar concentrations C rather

than molal concentrations m; activity coefficients (y) are also stated in terms of C (ai = Ci yi = mi i).

The cell examined by Shedlovsky and Macinnes was

Ag, AgCl|KCl(CR) ⁞ KCl(Ck) |AgCl, Ag (II)

where CR was 0.1 mol dm-3. The left section is equivalent to the reference electrode and salt

bridge of cells (l) and (I'), and the right side AgCI/Ag electrode is the same thing as a Nernstian ISE

for Cl-. The range of experimental concentrations was 0.005 - 3 mol dm-3; we select for the

exercise the sub-range 0.005-1 mol dm-3, the same explored for sodium halides in [1]. Over this

range, KCI presents the advantage that the transport numbers are practically independent of the

concentration (t+ varies from 0.4905 at m = 0 to 0.4898 at m = 0.1 and 0.4882 at m = 1 mol kg-1

[31]), and therefore, the integral R

k t+ d(ln m ±) of Eqs. (5) and (6) can be safely equated to 0.490 ×

ln[mk±k /(mR±R)]. A further advantage is that the very small difference between t+ and t-

guarantees that the minor term R

k (nw+ t+ – nw- t-) d(ln aw) of Eqs. (5) and (6) is safely negligible (nw+

= 6 ± 1, nw- = 6 ± 1 [32]). Hence, for any possible series of conventional y+ and y- we are able to

calculate the corresponding values or EJ or EJ (there is no difference, both the reference

concentration and left concentration are 0.1 mol dm-3) via Eq. (6), which for cell (II) reduces to:

EJ= S {t- ln[Cky-,k /(CR y-,R)] – t+ ln[Cky,k /(CRy+,R)]} (A1)

The exercise (whose results are summarized in Table A1 and Fig. A1) starts from the convention

that EJ is constantly null, as in [1]. From these null EJ (EJ(1)) and experimental E, conventional

values of ln y-(1) = In (y-,k/y-,R) consistent with EJ(1) are derived from the cell equation written in

conventional form,

E – EJ = – S ln[Cky-,k /(CRy-,R)] (A2)

Experimental ln y± matched with conventional ln y-(1), allow conventional ln y+(1)= 2 ln y± –

ln y-(1) to be deduced, as well. Afterwards, extrapolation to ionic strength zero of Y+ = LL – ln y+

and Y- = LL – ln y- (LL, the theoretical limiting law for ln y+ and ln y-) provides the required ln y+,R

and ln y-,R necessary in order to deduce conventional values or y+ and y- consistent with EJ(1)

(columns y+(1) and y-(1) of Table A1).

Table A1

Effect of two different conventions on the liquid junction potentials and ion activity coefficients.

C a

/mol dm-3

E a

/mV

y± a

EJ(1)

/mV

y-(1)b

y+(2)c

y+(1) b

y-(2) c

EJ(2) d

/mV

y-(2, check) e

y+(2, check)

e

y±(check) f

0.0050080 70.765 0.927 0 0.916 0.936 -2.892 0.936 0.916 0.926

0.010003 54.025 0.902 0 0.879 0.923 -2.214 0. 923 0.879 0.901

0.010053 53.895 0.902 0 0.880 0.923 -2.213 0.923 0.880 0.901

0.020002 37.489 0.870 0 0.837 0.902 -1.542 0.902 0.837 0.869

0.020003 37.483 0.870 0 0.837 0.902 -1.547 0.902 0.837 0.869

0.030000 27.890 0.849 0 0.811 0.887 -1.152 0.887 0.811 0.848

0.030285 27.688 0.848 0 0.810 0.886 -1.143 0.886 0.810 0.847

0.040000 21.170 0.832 0 0.790 0.874 -0.875 0.874 0.790 0.831

0.050106 15.905 0.819 0 0.774 0.863 -0.659 0.863 0.774 0.817

0.059995 11.731 0.807 0 0.761 0.854 -0.486 0.854 0.761 0.806

0.080004 5.113 0.787 0 0.738 0.837 -0.216 0.837 0.738 0.786

0.10000 0 0.772 0 0.720 0.824 0 0.824 0.720 0.771

0.20000 -15.757 0.722 0 0.665 0.782 0.664 0.882 0.665 0.721

0.50000 -36.453 0.658 0 0.595 0.725 1.549 0.725 0595 0.657

0.9974 -52.442 0.624 0 0.556 0.698 2.258 0.698 0.556 0.623

1.0000 -52.508 0.624 0 0.556 0.698 2.256 0.698 0.556 0.623

a Ref. [10]. b From E and EJ(1), via Eq. (A2). c Alternative convention. d Consistent with y+(2) and y-(2), via Eq.

(A1). e From E and EJ(2), via Eq. (A2). f y±(check) = (y+(1) y-(1))1/2 = (y+(2) y-(2))

1/2 = (y+(2,check) y-(2,check))1/2.

The values are consistent with y± (last column of Table A1; the very minor systematic difference is

because of the two independent extrapolation procedures). As a check, y+(1) and y-(1) are

introduced in Eq. (A1) to recalculate EJ(1) and are found to reproduce correctly the original null

values. Therefore, we can conclude that C, E, y±, EJ(1), y+(1), and y-(1) are consistent with each

other, and EJ(1), y+(1), and y-(1) are plausible.

At this point, however, we change our convention and assume that the ionic activity coefficients

are instead y+(2) = y-(1) and y-(2) = y+(1), and calculate the corresponding EJ via Eq. (A1) (column

EJ(2))· By repeating for EJ(2), as a check, the identical procedure that we performed for EJ(1),

these new conventional EJ are used to back-deduce the corresponding y- and y+ (y-(2,check) and

y+(2,check)), and these are correctly found to be equal to the original y-(2) and y+(2) or, in other words,

equal to y+(1) and y-(1) respectively.

Fig. A1. Two opposite possibilities, among the infinite ways of interpreting the experimental E of Ref.[10] in

terms of ionic activity coefficients of K+ and Cl-: y+(1) and y-(1), results consistent with EJ(1) = 0 values; y+(2) and

y-(2), results consistent with EJ(2) values of Table A1.

We must conclude, therefore, that also C, E, y±, EJ(2), y+(2), and y-(2) are consistent with each other,

and EJ(2), y+(2), and y-(2) are plausible, like EJ(1), y+(1), and y-(1), although the activity coefficient

values that pertained to cations now pertain to anions and vice-versa. Note that this interchange

between y+ and y- corresponds to a very moderate variation in the trend of EJ (always zero in one

case, varying from -2.89 to +2.26 mV in the other case). Since the concept or the "true value" of

EJ has no physical significance, and in no case is it possible to measure, or even calculate, this

quantity except in conventional terms, any choice between the opposite possibilities, EJ(1), y+(1),

y-(1) or EJ(2), y+(2), y-(2), is arbitrary. And likewise, infinite other self-consistent combinations can be

obtained by varying, in infinite arbitrary ways, the convention for the liquid junction potentials or

ionic activity coefficients.

These considerations extend to all papers [1, 7, 11-29] in which "experimental evidence" is given

of activity coefficients of cations higher or lower than those of anions.

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