Ion Association, Solubilities, and Reduction Potentials in Aqueous Solution Steven 0. Russo and George I. H. Hanania Indiana Unlversity, Bloomington. IN 47405 Aqueous solutions dominate general chemistry teaching. The main reason for this is traditional, reflecting the uhiquitv of water and the ease of conductine.. ex~eriments that . involve inorganicand analytical reactions of electrolyte solutions. More recently, there has been arowinr interest in the biological role of liquid water, and this has enhanced the relevance of electrolytes to chemistry courses for students in biochemical and health-related programs. Certain difficulties are encountered when one deals with the subject of aqueous solutions. In the first place, liquid water has a complex molecular structure and unusual physico-chemical properties, all of which complicate the interpretation of structure-related phenomena. The other difficulty about water is that solutions of electrolytes are far from ideal, largely so because of electrostatic effects. This aspect of nonideality has been treated with much success by the Debye-Hiickel theory, which introduced the concept of ionic activity. Because of its elegant simplicity, the theory has been widely adopted for the calculation of activity coefficients. Indeed, a survey of about 20 undergraduate textbooks shows that physical, analytical, and even some general chemistry texts, deal with the subject. In contrast to the prevalent use of ionic activities, the concept of ion association in aqueous solution has been generally ignored. Even physical chemistry texts tend to omit the subject. Since the dielectric constant of water is so high, and water molecules are so effective in separating ions, i t is commonly assumed that ion-pairing effects in aqueous solutions are negligible. But this cannot be the case, as can be seen from the extensive compilations of stability constants for metal-ligand and other chemical equilibria, including ion association ( I ) . Three exampieb may be cited to illustrate the point. The measured solubilitv of CaSO&in water is erosslv inconsistent with its known soiuhility p;oduct. ~1tCough"thisfact has been amply clarified (2,3),the subject apparently continues to confuse teachers (4, 5). The reduction potential for the iron(II1)-iron(I1) couple has a value that varies depending on the type and concentration of the acid present, hut it may not he clear why this is so (6).Then there is the widespread practice, in chemistry textbooks, of calculating ionic strength and activity coefficients assuming complete dissociation of electrolytes, when such calculations can he as much as 50% in error. As will be demonstrated below, all these uncertainties reflect the unrecognized role of ion association. The treatment of electrolvte solutions is further comnlicared by the concomitance of ion-solvent and aggregation equilibria, hv much ambieuitv about activitv coefficients in concentrated solutions, agd b; hydrophohic&dother structural effects. Despite these limitations. we believe that a combined approach to ionic activity andion association presents to the student a realistic and more auantitative view of electrolyte solutions than is otherwise poisihle. In this article, we develop the above idea and apply i t in two stages. First, by incorporating the combined effects of 148
Journal of Chemical Education
ionic strength and ion association, we show how calculations involving ionic equilibria are carried out. On that basis, we then examine the variahilitv of reduction ~ o t e n t i adata l for two aqueous redox systems: the Fe'--Fe2- couple with anion bindinr and the F ~ ( C N ) ~ - F ~ ( C NcIo; u~ ~- l ewith cation binding. The topic ofredox potenti& has tLe merit of being familiar t o students, and experimental measurements are easy to make. Moreover, since the equations for reduction potentials involve ratios of ionic activities, the errors that accrue from various simplifying assumptions tend to he somewhat lessened. ~
~
Ion Association Considered from an electrostatic viewpoint, the nonideal behavior of electrolyte solutions may he interpreted as due partly to physical and partly to chemical factors. The Dehye-Huckel theory, which assumes complete dissociation of electrolytes into their solvated ions, attributes deviation from ideality to long-range physical forces of interionic attraction. This factor is evaluated in terms of an activity coefficient parameter, which adequately accounts for the observed behavior of strona 1:l electrolvtes in verv dilute ifacti\.aquenussolution. Rut experhentaldete'minations ity coefficients for other electrolvtes, and even for some 1:l electrolytes in more concentrated solutions, show progressive deviation from theory. The discrepancy is attributed to a chemical effect, namely that short-range electrostatic interaction can lead to chemical ion association. In other words, deviation from ideality is due to the combined effects of a physical factor (activity coefficients) and a chemical factor (ion pairing). The latter is treated as a regular chemical equilihrium. Thus, when a symmetrical electrolyte such as a 2 2 salt is dissolved in water, the metal cation M2+ and the (ligand) anion L2- will he in chemical equilihrium with the ion pair M2+L2-: The need for taking ion association into account is illustrated with an example in Figure 1.The upper curve is a plot of the mean ionic activity coefficient f+ for a typical 2:2 electrolyte, varying with concentration up to 1.0 mol/L, based on a Debye-Huckel calculation without ion pairing. (The difference hetween molarity and molality is ignored.) The lower curve is a similar nlot which also incornorates the chemical equilibrium of eq 1. Both curves extrapolate to an f+ of 1.0 in the limit of zero salt concentration. Details of calculations are described in the text below. Figure 1 also shows the experimentally determined mean ionic activity coefficients for ZnS04 (7). I t is evident that, a t least in the case of 2 2 electrolytes, the Dehye-Huckel theory alone does not correctly predict activity coefficients and that incorporating a chemical (ion pairing) equilihrium yields a concordant fit of experimental data. Actually, the fit extends even beyond 1M concentration in this case. The tendency of ions to associate intoion pairs depends on the balance between electrostatic forces and thermal energy.
a (eq 2); a corresponding stoichiometric equilibrium constant is defined in terms of concentrations (eq 3): K = [hIL]I[M . [L] = xlC(1- x)=
(3)
where square brackets indicate equilibrium molar concentrations, C i s the total (analytical) salt concentration, and x is the extent (fraction) of ion pairing. Ionic activities are related to molar concentrations by the activity coefficient parameters, f:
I t is assumed, though not rigorously, that individual ion activity coefficients can be calculated by use of the extended Debye-Huckel expression: log f, = -O.SlOZ?IJf/(l
.25
.50
.75
i1.o
CONCENTRATION Figure 1. Varlatlon of mean ionic activity caefficlent wah concernration(moll L) for a typical 2:2 eiecbolyte in water at 25 ' C . Upper curve: f+ calculated from eq 5 on the assumption of complete dissociation. Lower curve: f* calculated from eq 5 but including ion association, f l = 200. Experimental points are shown for ZnSO,.
The determining- parameters include ionic charge, concen. tration, temperature, and dielectric constant. SGong solvation and high dielectric constant, as in water, favor dissociation into free ions. Poor solvation and low dielectric constant promote ion association, as occurs in most nonaqueous media. Of course. other factors beine.. eoual. . . a hieher ionic charge and a higher concentration also promote ion association. Ion oairs nlav an imnortant role in holding the certiarv structurkofpr&ei;ls and enzymes (where they are called sait bridees). in maintaining micelle stabilitv, in lowerine the pola&of biological mo~eculesto facilitate their crossGg of membranes. and in generallv attenuating- chemical reactivity in organic solven&. What is the experimental evidence for ion association? Classically, the evidence comes from measurements of ionic conductivities, hut also from a variety of thermodynamic methods, notably potentiometry and spectrophotometry, and more recently from specific spectroscopic techniques hama an and NMR). Kinetic studies of reaction mechanisms ion pair may be: (1) solvent separated, have shown that (2) solvent shared. that is. nartiallv desolvated. or (3) a contact ion pair, in'which thitwo ions are not separated by anv solvent molecule. Furthermore, the binding of ions mar not be purely electrostatic. If orbital overlap isinvolved, the resulting complex ion may be held by (directed) covalent bonds or by partially ionic bonding. The important point to note here is that ion association is formally equivalent to metal-ligand binding, that is, to complex-ion formation and to acid-base protonation. In this discussion, we consider only the equilibrium aspect of eq 1, regardless of the mechanism of ion pairing or the struiture ofthe ion nair. Charees mav be omitted to eeneralize the treatment. thermodinamie stability cons&nt, KO, for formation of the ion pair, is defined in terms of activities, ~~~~
~~
~
~~~~
~~
~
~
~
~
+ 0.329R;q)f)J
(5)
which applies a t 25 O C to an aqueous ionic species with concentration C; (mol/L), charge Zi, and ion-size parameter R; (A) a t ionic strength I = (1/2) xC;Z?. Equations 2,3, and 4 relate the stoichiometric to the thermodynamic ion-association constant, through the influence of ionic strength, also called the neutral salt effect, such that
K = PVM.~L#ML) (6) It follows that, whereas the equilibrium constant K determines the extent of ion pairing (eq 3), it is also a function of the ionic strength of the solution, but ionic strength is itself dependent upon the extent of ion pairing. Because of this interplay of the two variables, calculations involving ionic equilibria entail making a series of approximations, a situation ideally suited for iterative computations. The following examples illustrate the method proposed for treatment of data. Example-CaSO.
At 25 O C , the solubility of calcium sulfate in water is 0.0155 M, the thermodynamic solubility product KspOis 2.4 X lOW, and the ion association constant KO is 200 (1).The relevant equilibria correspond to eq 1, except that the solid salt is in equilibrium with solvated ions in the saturated solution. We now apply eq 6 to get K, and eq 3 to get x, using a computer algorithm. In the first iteration, Co = 0.0155 M , I = 4C, and f~ and f~ are calculated using eq 5. HenceK = 37.8 and x = 0.293 (extent of ion-pair formation). Next, the value of C is recalculated, C = Co(1 - x) = 0.0110 M , and the second iteration produces a new set of numbers yielding yet another value of C, and so on. The computations end when the change in x becomes less than, say, 0.1%. which usually requires four iterations. Table 1lists the numbers obtained. Inspection of the data in Table 1shows that 27.2% (0.0042 M) of the ions are ion-paired and 72.8% (0.0113 M each) are free ions, which accounts for the observed total solubility of 0.0155 M. If the solution were assumed to be perfectly ideal, the solubility would be equal t o the square root of the soluTable 1.
Extent of ion Pairing in 0.0155 M CaSO,(aq) at 25
iteration
[MI
I
f~
1'
K
1 2
0.0155 0.0110 0.0113 0.0113
0.0620 0.0436 0.0453 0.0452
0.456 0.496 0.494 0.495
0.414 0.462 0.458
37.6 46.1 45.3 45.3
3 4
IMI = concentration01 tea Caz+a SO1'lmollLl
0.458
OCa
x
0.293
0.269 0.272
0.272
In saturated aaueoussolutlon: I =
~
Volume 66
Number 2
February 1989
149
bility product, namely 0.0049 M, which isless than one-third of the actualvalue. Likewise, if we take the experimental solubility of 0.0155 M and assume ideality (a co&non simplification in textbooks), the soluhili~yproduct would be the quantity that is 10square of the solubility or 2.4 X fold too high. Of course, an electrolyte solution is not ideal, and the thermodynamic (zero ionic strength) solubility aM. aL = (0.0113)2(0.495)(0.458) = product should be K,," 2.9 X in agreement with the standard value. There has been some confusion in the literature over the solubilitv and solubilitv product for CaSOd (4. 5): it was therefor; reassuring to !&ow that the relationships that had been explained in an old review (2) have now been amply clarifiedin a recent review (3).Unfortunately, the error has cropped up again in a more recently published school experiment (5) that first employs ion-exchange chromatography for the determination of solubility, a correct procedure, but then incorrectly uses this quantity, which includes the concentration of free ions and ion pairs, to obtain the (wrong) K., The error here demonstrates the need for taking a more consistent approach to ionic equilibria. Table 1(last row) also shows that, although the thermodynamic ion-association constant for CaSO4 is 200, the concentration equilibrium constant is only 45, clearly so because the activity coefficients are less than 0.5 instead of being 1.0 as in an ideal solution (ea 6). A smaller K means less ion pairing.And so, for agiven electrolyte, thedecrease inactivity coeificients works in a direction opposite to that of ion association. Hence, it is the balance of these two factors that determines the positions of the resulting equilibria in sulu-
a
(expressed in mol %) as a function of the total (analytical) salt concentration in each of the three cases. These calculations ignore the formation of ion triplets, hydrolysis, and other equilibria; they also assume the applicability of the extended Debye-Huckel expression (eq 5) to individual ions within the concentration range covered. Despite these limitations, some general features emerge: 1. Figure 2 shows that, except for 1:l salts, ion pairing occurs to the extent of 2040%; it could be even higher in cases where the ionassociation constants are greater than the average values taken here. This is not a negligible effect. In the light of these results, it is regrettable that so many textbooks that deal with the topic of ionic activities provide the student with examples of ionic strength calculations, almost invariably and unrealistically ignoring ion pairing in aqueous solution. 2. 3:3 salts exhibit an exceptional behavior in that ion-pair formation rapidly rises (to a 45% maximum at 0.01 M) and then decreases. This is an elegant demonstration of the powerful effectof high charges on (lowering) activity coefficients, an effect that turns out to be opposite to and greater than the mass action (chemical) effect of increasing concentration. Davies (8)discusses this and other aspects of ion association in detail.
Table 2.
.
Example-Ca(lO& Another common experiment in general chemistry involves the determination of the solubility and solubility product for calcium iodate. At 25 OC, the solubility (S) of Ca(103)2 6H20 in water is 0.00784 mol/L, the solubility product (K,,O) is 7.0 X 10-7, and the ion-association constant (KO) for the ion pair Ca2+IOx- is 7.8 (1). Since K' is small, onlf. little ion pairing is expected. The problem is similar to that of CaS04,but here we have a 21 unsymmetrical electrolyte, the ion pair carries a net charge, and there is an extra free iodate anion in solution. Iterative calculation yields: [Ca2+]= 0.00735, [103-] = 0.0152, x = 0.063 (6.3%ion pairing), I = 0.0225, which enables the ionic activity coefficients a~~ = (0.00735) to be calculated. Hence, KsPo = a~ (0.0152)2(0.580)(0.865)2 = 7.4 X lo-? in agreement with the standard value. However. if we take the experimental solubility but assume the solkion to be ideal, K,, = (0.00784)(2 X 0.00784)2 = 1.9 X 10-6. which is 2.7 times the correct value. The error is smaller than in the case of CaSO+ It is instructive to see how the soluhilitv of calcium iodate isaccounted for. First, if we assume complete ideality, K,, = (S)(2S)' = 7.0 X 10." privinpr S = 5.6 X lo-:' M (71% of the total solubility). Next, h e introduce activity coefficients but not ion pairing. Here I = 3S = 0.0168, and hence fc$+ = 0.616, fro,- = 0.876. Since 7.0 X = (S)(2S)2(0.616) (0.876)2we get S = 7.18 X 10-3 M (92% of the total solubilitv). . The remainine 8% is clearlv the contribution of ion pairing. Table 2 summarizes and contrasts the data for Ca(IOq)1 . ".- and CaSOa. The first. havine asmaller ionic charae. shows 29% nonidekity. The second; a 2.3 salt, shows 68% nonideality. Likewise, ion pairing accounts for only 8% in Ca(I03)2 but 27% in CaS04. Even larger ion-pairing effects would be expected for electrolytes of higher ionic charge.
Total
Contrlbutlons to Solubllltya
.
100
100
Percenteoa conwibutlons to ms t m l lmeasuredl wlublllh In water at 25 'C. Solid phases:~a(l6&.6~$3andcsso,. PH&.'I~~-assmiation constants insolution: P = 7.8 (calolum Iodate); 200 (caldum sulfate). ~etells d calculatims am In hetext.
.
.
.
Other Examples In order to explore the above ideas further, we extended the calculations to cover a range of salt concentrations, from zero to 0.4 M, taking typical examples of symmetrical 1:1, 22, and 3 3 salts. Figure 2 is aplot of the extent of ion pairing
150
Journal of Chemical Education
CONCENTRATION Figure 2. Extent of ion pairing (mot %) In aqueous solutions of electrolytes. concentration ranme from zero to 0.4 M. Detalls of calcuiations are In the text. Typcal ,an-pair stablllty constant$ are ass~med(a) I I salt. wllh = 1, lo) 2 2 salt. with KD = 200 lc) 3 3 salt. wllh KD = 6000
3. Impairing also occurs in solutions of unsymmetricalelectrolytes, as in the above case of calcium iodate, and in complex mixtures of electrolytes such as the familiar example of sea water. The main constituents of sea water are Na+ and Cl- ions, with appreciable amounts of Mg2+,Ca2+,Kt, HC03-, and other minor species, which together form a medium of approximately constant icnie streneth - (.I = 0.70 M and oH 7.M.0). These conditions are sufficientlyprecise for making reliable calculations on ionic equilibria. Indeed, it has been determined that, in normal sea water, about 70%of the bicarbonate is present as free HCOJ-, 50%of the sulfate as free Sod2-,and only 10%of the carbonate as free C03Z-, the rest in each case being ion-paired to the cations in solution (9). These numbers indicate a significant geochemical role for ion association. Other applications are presented in the section on redox potentials that follows. ~~~~
~
~~
~
.
~~
Redox Potentials
Before we consider the combination of ion association and oxidation-reduction equilibria, it is helpful to review the relevant equations. Discussion is here confined to reversible e = red). An inert electrode one-electron transfer (ox inserted into the mixture acquires an electrical potential, E, given by the Nernst equation:
+
At fixed temperature, this potential is determined by FAQ,the standard reduction potential for thecouple, and by the ionic activities of oxidant and reductant species. Activity coefficients relate ionic activities to molar concentrations as already defined (eq 4). For equimolar concentrations of oxidant and reductant, in water at 26 "C, the measured reduction potential. E" (in millivolts), becomes"
witha parallelset of relations for the reductant species (red). The ligand L is a counterion, and ox-L and red-L are the ion pairs or complexes. For ligand binding t o oxidant, KoXo(eq 10, which corresponds to eq 2) is the thermodynamic ion association or stability constant, and K, (eq 11, corresponding to eq 6) is the equilibrium constant expressed in molar concentrations at the appropriate ionic strength. The terms and relations for the reductant species are strictly comparable to these. [L] is the concentration of free ligand a t equilihrium. By suitable combination and rearrangement of the above set of equations, i t can be deduced that 1+ [L] .K,, (12)
Equation 12 summarizes the relevant facts and shows how the observed potential (WIdiffers from thestandard pocenTwo factors are involved: the activity coefficient tial (En). correction for long-range electrostatic interaction and the ion-association eauilibrium for short-ranee chemical binding. The chemicai factor also indirectly includes an activity correction, since the eauilibrium constants..K..A and K,. are functions of ionic strength (eq 11). The relative coniiibutions of the two factors depend on the ionic charges and on electrolyte concentration. Two limiting, and simplifying, conditions may be noted: (1) When ion sssociation is very weak, that is, whenKis sufficiently small, eq 12 reverts to eq 8. Under those conditions, ionic streneth is the onlv consideration. " (2) When ion association is very strong, the term in K becomes dominant, and eq 12 can he simplified to the approximate relation
@ = E? + 59.2 log (KrdIKo,)
In the limit of extreme dilution, activity coefficients approach unity, and EO' attains the ideal value EO. In real solutions, a t finite ionic strength, f, and f,,d will be unequal because the oxidant and reductant species carry different charges, and so fl will shift away from EO. In fact, as eq 8 shows, i t is the ratio of the two activity coefficients that determines the extent of the shift. Thus, on this basis, for a solution a t 0.10 M ionicstrength, a 3+/2+ couple would have EO' about 20 mV less positive than Eo, whereas a 3-14couple would be 40 mV more oositive than its EO. The deviation is a measure of the elect;ostatic nonideality in electrolyte solutions, exclusive of chemical interactions. Actually, redox measurements show that the nonideality in such systems is considerably greater, a fact that again demonstrates the need for taking chemical effects into account. The role of ion association can be assessed as would be done for any side reaction. Omitting charges, consider each of the oxidant and reductant ions to be independently involved in a single ligand-binding equilibrium: ox
11
ox-L
+
red
ItL
(9)
red-L
On the oxidant (ox) side of the equilibrium, we have KO: = [ox-Ll . f d o x l .fa.. [Ll ./L' K, = K2V, .fdf0,.d
'
(10)
(11)
The symbol Eo' was originally defined (10) as the experimentally determined "formal potential" in a solution containing unlt formula weights oer liter, 1 F each. of oxidant and reductant. A s used here lea 8). @ is'the observed Dotential for anv eouimolarmixture of oxidini and reductant, and es6ecia11yfor very diute solutions where ionic strengths and activity coefficientscan be more reliably calculated.
(13) Equation 13 is very useful for predicting the effect of ligand binding on reduction potentials, and consequently on the relative stabilities of oxidation states of complex ions where electrostatic effects are dominant. These ideas will now be discussed in a number of specific cases. F&,
F&
The standard reduction potential for this well known ferric-ferrous couple is 770 mV (in water, a t 25 OC and zero ionic strength). However, solutions of iron salts usually require the presence of a strong acid to repress acidic dissociation of the aquo cations, or contain a suitable ligand that stabilizes (forms a more stable comolex with) one of the two oxidation itates relative to the other. If the ligand is anionic, i t tends to form ion pairs withFe3+ and Fez+,but the oxidant ion pair will have a bigger stability constant because of its higher electrostatic charge. The result is a neeative shift in the reduction potentia~bfthe system, whicK is precisely what is observed in practice. Consider, for instance, an aqueous solution of FeC13 and FeC12,0.0020 M each, in the presence of 0.10 M HC1. The ion association constants are: K,O = 30 (for Fe3+C1-) and K,do = 2.5 (for Fe2+C1-), indicating weak complexation (I). The described iterative procedure, or even simple guessing, gives an estimate of the ionic strength and activity coefficients, hence the eauilihrium constant and extent of ion oairine.. (ea . . 11 and 91, f i r the oxidant and also for the reductant species. Then, usina ea 12. we find the exoected reduction ootential EO' = 133 &, a b o k 40 mV more negative than thestandard value. The ex~erimentalvalue under these conditions is 736 mV (11). Analytical textbooks frequently quote the "formal" potential as 700 mV, but that refers to l F solutions of oxidant and reductant in 1 F HC1. Can the potential a t such high concentration be reliably derived? Fortunately, it is the ratio of two activity coefficients Vo./fred) that is involved. When applied to a solution of FeCI3 and FeC12, 0.01 M each, in 1.0 M HC1, under the simplified assumptions of Cl- binding, Volume 66 Number 2 ~ e b r u a t y1989
151
and a total ionic strength of 1.0 M, the above type of calculation yields EO' = 707 mV. If activity coefficients are obtained using the Davies equation, which includes a salting out term and is more applicable a t high concentrations than the DebyeHuckelequation (a), the result is found t o he 703 mV. A formal potential a t about 700 mV does not seem unreasonable. Interestingly, the calculations show that the shift in potential is due equally t o the activity coefficients and to ion ~ a i r i n (about e 50% each). ~h~i~on(111)-iron(11)'couple has a somewhat less positive formal notential of 680 mV for 1F solutions in the nresence of 1 F-HPSO~.Here we have stronger ion bindiig to the HSOa- anion: KmO= 1.1X lo4.and K7.d0 = 1.6 X lo2(I).We can perform a simplified calcuiation byconsidering 0:h0 M solutions of ferrous and feric sulfates in 1.0 M HoSO.,. bv assuming that I = 1.0 M and free ligand conce&&& [HS04-] = 1.0 M, and by ignoring all other side reactions. The above treatment then yields EO' of 635 mV, not in very good agreement with the experimental formal potential but possibly reflecting the morecomplex mixture or ions in solution. In this case, the chemical effect accounts for 80% of the shift in reduction notential. If notential measurements are made in the presence of a mixtire of HzSO4 and H3POb an even lower formal reduction notential annears. In view of the expected additional ligand Linding to yhospbate ions, it is not difficult to see whv thisshould he so.' and thereis no need to attempt an unreliable calculation. Fe(C%*, Fe(chl)$Another iron(II1)-iron(I1) redox system is one containing an equimolar mixture of potassium ferricyanide and potassf: um ferrocyanide (hexacyanoferrates). This couple has a number of features that make it narticularlv suitable for the present discussion. The two sal& are readiiy available, and the mixture is stable in dilute aqueous solution; the redox potential is easily measured; the ions are anionic and therefore provide a contrast to the cationic ferric-ferrous counle: and,-having high charges, the ions are clearly expectei td participate in ion pairing. Indeed, the redox potential of this system has been determined over a wide range of conditions (12, I3), enabling precise comparison of calculated with experimental valuis; We consider first an aqueous solution of K3Fe(CN)~and KaFe(CN)6,1.00 X M each, noting that the ion-pairing ligand is the K+ cation. The standard quantities for this system are: En= 355 mV; KOx0= 30 (for K+Fe(CN)e3-); KrdO = 230 (for K+Fe(CN)&); a n d 1 = 0.00156 M. We now ohtain the activity coefficients (eq 5) and consequently find that KO, = 23 and K,d = 163 (eq 6). What is the extent of deviation due to nonidealitv? Supnose that we neelected ion pairing. Under these condi;ione,ihe nonideality term = 59.2 log (0.680/0.510) = 7.7 mV (ea 8). a small shift as would be expected for a very dilute solution. If ion pairing is included, the corresponding term from eq 12 yields a 10.1 mV shift, making the predicted reduction potential EW = 355 10 = 365 mV. The experimental value is 366 mV. Agreement is excellent. In order t o test this model further, we ask what if the above solution also contained 0.1 M KCl? As a neutral salt. KC1 should increase the ionic strength and, as a ligand, KG should enhance ion ~airina.As the above calculations have just shown, both efrects should make the redox potential more positive. This turns out to he the case. Details of the calcul&ons are not given here, hut the results are summarized in Tahle 3, together with data for the same svstem in the presence of a calcium salt (with a bigger role for ion pairing due to its higher charge). The last column gives the corresponding redoxdata in the presence of an acid,that is, a situation in which the ligand is H+. Although protonation is not a purely electrostatic phenomenon, thequantitiative treatment of all these equilibria is strictly comparable, and indeed the concordant results confirm expectations.
+
152
Journal of Chemical Education
Table 3.
Effects d Actlvlly CoeHlclents and Ion Arsoclatlon on Ferrlcvanlde-Ferrocvanlde Reduction Potential p' (mVV
Ideal P (standard value) wim sctlvity correction wlth activity and ion pairing Measured potentlal P' Nanideality contribution: activity ( % ) ion pair (%)
Table 4.
Effect
355 362 365 366
355 395 423 424
355 395 430 442
355 393 570 560
76 24
58 42
46 54
18 82
ol Llgands on Redox Potenllals of
Ferrlc-Ferrous
Couolesa Llgand
@'(V)
water HCI
EDTA
0.77 0.70 0.56 0.25 0.15
limlt at I = 0. 25 OC Cl- binding (text) acetate binding citrate binding chelate
B.
phen blPY
1.16 1.10
Iris complexws. covalent bonding Iris ~mplexws.m a l e n t bonding
C.
hemoglobin mvwlobln ,
0.15 0.10
Fe In porphyrin and protein Fe in DorDhvrin and orotein
A.
Acetic acid Citrio aoid
-
ccmment
. .,
m row o- ronll ) r w x poton! d m e q ~ o o ~ s ~ on o l a! n 25-C
(A1 nmspoasnseot anlor I I Q O ~ ~ S@ cu~atedur~nll. 13 e~rsemom~ n exmmenta n data IBI r m nevtral N-heterocysllc:Ilgands:phen = 1.K-phenanthrollne, blpy = 2.2'-blpyrldyl:experimental data: (C) hemoproteins: experlmewal data.
-
Tahle 3 provides some insieht into the relative roles of the physical and chemical factor; that shift the redox potential of a well defined inoreanicsvstem. The consistent aareement between calculated and m&ured quantities indicates that ion pairing cannot be ignored even in dilute solutions. Moreover, we can rationalize the shifts in redox potential quite successfully in electrostatic terms and as consequences of the laws of chemical eauilibrium. For instance. add in^ KC1 shifts the equilibrium &wards more ion pairing. We predict a areater effect hv the divalent calcium ions. as is observed (Fable 3, columnc). In the case of HC1, it tuins out that the reductant (ferrocvanide ion) forms two successive nrotonated species, whereas the oxidant (ferricyanide) does not protonate (13).The result, as the last column of Table 3 shows. is quite a large pH effect, the redox potential becoming mord positive by ahout 200 mV as the pH of the solution aDbroaches 1.And, not surprisingly, more than 80%of the shift is due to chemical binding. Finally, it is noteworthy that these variations in the ferricyanide-ferrocyanide redox potential make a good class or laboratory demonstration: the experiment is easy to perform, impressive to watch, and challenging to interpret. Other Appllcatlons
The above treatment of ionic equilibria can be extended to incorporate the formation of more than one complex (ion triplets, multiple equilibria, polynuclear species). For instance, if there are two successive ligand-binding steps, as in the protonation of ferrocyanide a t low pH, the term (1 K[L]) in eq 12 becomes (1 K,[L] + K1K2[LI2)whereKl and KPare the stahility constants for the first and second ligandbinding steps, which applies to either or both of the redox ions (numerator and denominator). And this can be further extended to cases involviue more steowise eouilihria. Since stability-constant data a r e readily hailable'(l), predicted values of reduction potential can be obtained for avariety of
+
+
~
~
~~
well characterized redox systems. In most cases involving anionic lieands, the calculated ootentials agree with experis . 4 gi\ r s sornr examples: A wide range of mental ~ ~ l u e able ~otentialsis covered, reflecting differences in the strength of chemical binding ofligand toiron in the ferric and ferrous states. The reference point in Table 4 isEOfor Fe3+(aq),Fe2+(aq) in the ideal (zero ionic strength) state. Anionic ligands lower ~otential. the reduction ~-~ . a fact which mav be attributed to stronger electrostatic interaction with Fe". And so, negativelv chareed lieands tend to stabilize the hieher oxidation stateof a ckion& redox couple (Table 4, A). This conclusion would apply not only t o iron but also to other metal redox systems, such as Co3+-Co2+and its complex ions. In contrast, neutral N-heterocyclic organic ligands with delocalized pi-electron rings bind covalently to Fe3+ and to Fez+,more strongly t o the latter, thus stabilizing the lower, ferrous oxidation state and raising the redox potential to 1V or more (Table 4, B). And yet, if the same type of ligand also contains a negatively charged side group, as in the case of 4,7-dihydroxyphenanthrolinea t high pH, the redox potential drops markedly, once again reflecting the importance of charge considerations. In the biochemically important hemonroteins (cvtochrome. hemoglobin,. mvoplobin, peroxi. da;e, catalase); the iron atom is coordinated in a mac~ocyclic ii-lieand with two ~ o s i t i v charges e neutralized, making it an ~ e * ~ ~ e " c o u p he l e .redox potential is understandahly very low,and the ironW) state unstable with respect tooxidation. Since hemoglobin and myoglohin carry and store 0, only in the ferrous state, blood has a constant supply of reducing enzymes, to convert any oxidized hemoglobin into its physiologically active (ferrous) state. ~~~~
Conclusion In the above discussion, the subject of ion association and redox potentials was treated within the wider context of ligand-binding equilibria, and i t was applied to specific cases
where electrostatic interactions are chemically more s i p i f i cant than electronic effectsor structural considerations. The nonideality of aqueous electrolyte solutions was viewed as having physical (activity coefficient) and chemical (ion-pairing) components, whose variable contrihutions depend on the magnitudes of ionic charges, concentrations, and stability constants. This approach;it seems to us, has the merit of alerting the learner to regard ion pairing as a natural concept, be i t in a simple aqueous solution, a complex mixture (as in sea water or biological fluids), a nonaqueous medium, or even a fused salt. On this basis. textbooks and laboratorv manuals can deal with the topic of ionic equilibria in a more com~rehensive.albeit sim~lified.manner. The common assumption of ideality then 6ecomes one of disregarding physical as well as chemical interactions. This is a balanced and chemically more satisfying notion for student and teacher alike. Acknowledgment The authors wish to thank Dennis G. Peters of Indiana University for very helpful discussion and advice. Literature Clted 1. 8mith.R. M.;Ma~ll.A.E.CrifieolStobilifyConstants. Val.4,lnorgonicComplezea; Plenum: New York, 1976. 2. Meitea, L.; P d e , J. S. F.;Thomas, H. C. J. Chom. Edue. 1966.43.661. 3. Martin, R. B. J. Chom. Edue. 1986.63.471. 4. Sawyer, A. K. J. Chem. Edue. 1985.62.362; 1983,60,416. 5. Masterman. D. J. Chem. Educ. 1987.64.408. 6. Anslytied chemistrytextbooka:slsoHondbook ofChemistry and Physics:CRC. 7. Harnod, H. S.; Owen, B. B. The Physical Chemistry of Eleefmlyfa Solutions; Reinhold: New York, 1943;p426. 8. Dsvies, C. W.Ion Associotion:Butbmrtha: London, 1962: Chapter 9. 9. Butler, J. N. Corbon Dioxide Equilibn'o and Thcir Applieofions; Addisan-Wesley New York. 1982;pp 119.124. LO. Bram, R. A ; Swift, E. H. J. Am. Chem. Soc. 1949,71,2719. 11. Moiler. M. J. Am. Chem. Soc. 1931,41.1123. . 1935.39.945. 12. Kolthoff, I. M.; Tomsiak, W. J. J. P h y ~Chem. 13. Hanania.O. I. H.: 1rvine.D.H.;Eston,W.A.;Georp, P. J.Phys. Chom. l967,71,2022. 14. Kolthoff, I. M.; Elving, P. J., Eds. Treatise on Analytical Chemistry. 2nd ed. Wiley: New York. 1978: Part I, Vol. 1, p 539. Alao in some analytical chemistrytextbooks.
Volume 86 Number 2
February 1989
153
