Anal. Chem. 1997, 69, 446-453
Limiting Currents for Steady-State Electrolysis of an Equilibrium Mixture, with and without Supporting Inert Electrolyte Keith B. Oldham
Department of Chemistry, Trent University, Peterborough, Ontario K9J 7B8, Canada
An exact treatment derives the steady-state limiting current of a one-electron reduction for the N h O+ + Amixture at a hemispherical microelectrode. Either or both of the neutral N and cationic O+ species may be electroactive. A supporting salt is present at any concentration, including zero or excess; its ions are electropassive and do not interact with the other solutes or each other. The various species are treated as having distinct diffusivities, linked to their mobilities through the Nernst-Einstein relationship. Universal electroneutrality is assumed. The predictions of the model are compared with published experimental data on the reduction of aqueous weak acids; agreement is excellent at intermediate, but poor at low, support ratios. Analysis of the unsupported case shows that the neutral N species dissociates in a narrow zone close to the electrode, and the injection of ions there serves to increase the electric field in the outer region of the transport zone. This enhances cationic migration enormously, leading to an unsupported limiting current that is much more than double the supported value. However, the limiting current is drastically diminished by traces of foreign electrolyte. Curiously, the limiting current with full support adopts the same value when equilibration is fast as when it is very slow, although the mechanisms are totally different. The advent of microelectrodes, with their relative freedom from ohmic polarization, has relieved electrochemists of the requirement to provide excess supporting electrolyte. A penalty attendant to this relief has been the need to model migration, in addition to diffusion, in predicting faradaic currents. Fortunately, the added mathematical difficulties of incorporating migration into voltammetric theory are largely offset if interest is restricted to the steady state, a condition that is rapidly achieved at most microelectrodes. Accordingly, as an incomplete listing1-17 of publications in this field attests, there has been a plethora of research in recent years
aimed at the prediction of steady-state microelectrode currents mediated by diffusion and migration. Almost all of these treatments assume universal electroneutrality and the validity of the Nernst-Einstein link between diffusivity and mobility, as does the present study. When only small amounts of supporting electrolyte are added, or none at all, ohmic polarization, even at a microelectrode, may so degrade voltammetric wave shapes that only the limiting current IL can be measured with precision. Such was the case, for example, with experimental studies on electroactive solutes in toluene by Santos et al.18 Of course, the theoretical interpretation of such experiments need only consider total concentration polarization at the electrode, a mathematical bonus. Because electrode potential is no longer a useful parameter in such studies, its role as the independent variable is taken over, to some extent, by the “support ratio” (F, the concentration of electroactive species divided by that of the electropassive electrolyte, as defined in eq 7). In Santos’ studies, this dependence of IL on F was interpreted in terms of ion pairing in the solution. Though it is not usually so considered, a weak acid may be regarded as an ion pair formed between the solvated proton and the acid’s anion, so experiments such as those conducted in aqueous solution by Ciszkowska et al.19-21 fall in the same category. Earlier work from this laboratory predicted the magnitude of steady-state microelectrode limiting currents as a function of support ratio under conditions in which one22 or two23 ion pairs were formed. However, for mathematical economy, the systems addressed in those studies contained only three ionic species, whereas experiments frequently employ four, two cations and two anions, in addition to neutral solutes. Accordingly, this article seeks to predict the IL versus F relationship for systems containing four ionic species: the electroactive ion, a counterion with which
(1) Bond, A. M.; Fleischmann, M.; Robinson, J. J. Electroanal. Chem. 1984, 172, 11. (2) Ciszkowska, M.; Stojek, Z. J. Electroanal. Chem. 1986, 213, 189. (3) Oldham, K. B. J. Electroanal. Chem. 1988, 250, 1. (4) Amatore, C.; Fosset, B.; Bartlett, J.; Deakin, M. R.; Wightman, R. M. J. Electroanal. Chem. 1988, 256, 255. (5) Kharkats, Yu. I. Electrokhimiya 1988, 24, 539. (6) Kharkats, Yu. I.; Sokirko, A. V. J. Electroanal. Chem. 1991, 303, 17. (7) Kharkats, Yu. I.; Sokirko, A. V. J. Electroanal. Chem. 1991, 303, 27. (8) Baker, D. R.; Verbrugge, M. W.; Newman, J. J. Electroanal. Chem. 1991, 314, 23. (9) Cooper, J. B.; Bond, A. M.; Oldham, K. B. J. Electroanal. Chem. 1992, 331, 877. (10) Oldham, K. B. J. Electroanal. Chem. 1992, 337, 91.
(11) Oldham, K. B. J. Electroanal. Chem. 1993, 347, 49. (12) Norton, J. D.; Benson, W. E.; White, J. S.; Pendley, B. D.; Abruna, H. D. Anal. Chem. 1991, 63, 1909. (13) Norton, J. D.; White, H. S. J. Electroanal. Chem. 1992, 325, 341. (14) Myland, J. C.; Oldham, K. B. J. Electroanal. Chem. 1993, 347, 91. (15) Jaworski, A.; Stojek, Z.; Osteryoung, J. G. Anal. Chem. 1995, 67, 3349. (16) Amatore, C.; Montenegro, M. I. Anal. Chem. 1995, 67, 2800. (17) Palys, M. J.; Stojek, Z.; Bos, M.; van den Linden, W. E. J. Electroanal. Chem. 1995, 383, 105. (18) Santos, J. H.; Cardwell, T. J.; Bond, A. M.; Oldham, K. B. J. Electroanal. Chem., submitted. (19) Ciszkowska, M.; Stojek, Z.; Morris, S. E.; Osteryoung, J. G. Anal. Chem. 1992, 64, 2372. (20) Stojek, Z.; Ciszkowska, M.; Osteryoung, J. G. Anal. Chem. 1994, 66, 1507. (21) Ciszkowska, M.; Stojek, Z; Osteryoung, J. G. J. Electroanal. Chem. 1995, 398, 49. (22) Oldham, K. B. Anal. Chem. 1996, 68, 4173. (23) Oldham, K. B.; Cardwell, T. J.; Santos, J. H.; Bond, A. M. J. Electroanal. Chem., in press.
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© 1997 American Chemical Society
the electroactive ion enters an equilibrium, and two electropassive and nonpairing ions arising from a supporting salt. For simplicity, all ions are regarded as monovalent, though the treatment is tractable for many instances of polyvalence. Diffusivities (diffusion coefficients) are not treated as necessarily sharing a common value. In upcoming sections of this article, a model will be constructed and used to predict the steady-state current for the “four ion plus one neutral” system. Though there are small differences in the formulation, and of course the outcome differs, the philosophy and nomenclature of this derivation closely parallel those in a previous article22 in this journal. Consequently, fewer details of the mathematical procedures are provided than is customary.
Table 1. Properties of the Solute Species role of species
i
zi
νi
Gi
electroactive cation pairing anion neutral molecule or ion pair product, if soluble supporting cation supporting anion
O A N R C X
1 -1 0 0 1 -1
1-f -f f -1 0 0
G G -G 0 0 0
though this assumption is of questionable validity close to the electrode when the support ratio is small.12,24 One or both of the species O+ and N is electroactive. The one-electron reduction of the cation,
DETAILS OF THE MODEL The solution in the cell contains a neutral solute N and ions O+ and A- with which it is in equilibrium
O+(soln) + e- 98 R
N(soln) / O+(soln) + A-(soln)
if it occurs, produces a neutral product R. If the neutral species N is electroactive, the stoichiometry of the electrode reaction is
(1)
This homogeneous equilibration reaction is considered to be established infinitely quickly, so that the law of mass action
cN ) βcOcA
(2)
governs not only the bulk concentrations cbN, cbO, and cbA of the three solutes but also their values everywhere in the cell. β is the formation constant of the neutral species N. Ions O+ and Aare present in equimolar amounts in the bulk solution, so that the total “analytical” concentration cE of N is
cE ) cbN + cbO ) cbN + cbA
(3)
N(soln) + e- 98 A-(soln) + R
(9)
(10)
so that the same products are formed as via the route (1) + (9). In the interest of generality, it will be assumed henceforth, unless stated otherwise, that both of reactions 9 and 10 occur in parallel, with the fraction of the total current carried by reaction 10 being f, which may equal zero, unity, or anything in between. A hemispherical microelectrode of radius a resting on an infinite insulating plane will be treated so that only one distance coordinate, the radius vector r from the hemisphere’s center, is needed. A dimensionless alternative spatial variable,
x ) (r - a)/r
(11)
Combination of this with the equilibrium law 2 shows that
x1 + 4p - 1 cbO ) cbA ) cE 2p
(4)
2p + 1 - x1 + 4p cbN ) cE 2p
(5)
and
will also be used later; it takes values between zero and unity as the solution phase is transitted from the surface of the electrode into the bulk. Thus, since the superscripts “s” and “b” are being employed respectively to specify conditions at the electrode surface and in the bulk, xs ) 0 and xb ) 1. Consideration is restricted to the limiting current, which corresponds to the concentrations of the electroreactants being reduced to zero at the electrode surface
csO ) 0 and csN ) 0
where the quantity p, given by
p ) βcE
(6)
is a dimensionless parameter reflecting the degree of association. Two supporting ions, C+ and X-, may also be present in the bulk solution at concentrations related to cE by
cbC ) cbX ) FcE
(7)
(12)
Note that even if one of the N, O+ pair is electropassive, its concentration will nevertheless be forced to zero at the electrode by virtue of law 2. At the electrode surface, the total flux Jsi (mol s-1) of each of the species (i ) O, N or A, as well as the product R, if it dissolves in the solution) involved in the electrode reaction is proportional to the current IL, the constant of proportionality being νi/F
Jsi ) νiIL/F
(13)
where F is the support ratio. The electroneutrality condition
0)
∑ zc ) c i i
O
- cA + cC - cX
(8)
is taken to apply both in the bulk and throughout the solution,
where F is Faraday’s constant and νi is a stoichiometric coefficient which takes values as listed in Table 1. Please note the sign (24) Smith, C. P.; White, H. S. Anal. Chem. 1993, 65, 3343.
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447
conventions being used: IL is negative for the cathodic reaction under discussion, and J is positive for fluxes away from the electrode.
DOA 1 ) 2 (1/DO) + (1/DA)
CONSERVATION EQUATIONS With i ) O, A, N, C, or X, and φ representing the local electrical potential, the conservation equation
On addition of the five thus-modified conservation equations, the useful result
dci 2πaDiziciF dφ νiIL + + + Gi 0 ) 2πaDi dx RT dx F
(14)
applies at any point x (0 e x e 1) in the solution phase. If it is a soluble species, this equation also applies to the neutral product R of the electron-transfer reaction; however there is no call to consider that species further. A similar conservation equation has twice been derived rigorously in the recent literature,22,23 and it will suffice here to identify the source of the each of the four terms. The first and second terms originate respectively in the diffusive and migratory fluxes of the solute i. The third term represents the rate at which i is being injected into the solution from the electrode; in fact this term is just Jsi , as given in eq 13. The fourth term, Gi, is defined as the rate of generation (mol s-1) of species i, in the zone between the electrode surface and x, by virtue of the equilibration reaction 1. It is evident that
GO ) GA ) -GN
(15)
from the stoichiometry of the reaction. These equalities are already incorporated into Table 1, where the unsubscripted G is used to represent the total rate (mol s-1) of dissociation of the species N into ions, i.e., the total rate of reaction 1 in the zone. When the conservation equation is written successively for each of the five important species, inserting the appropriate terms from Table 1, one finds
0 ) 2πaDO
dcO 2πaDOcOF dφ (1 - f )IL + + +G dx RT dx F
0 ) 2πaDA
(16)
dcA 2πaDAcAF dφ f IL +G dx RT dx F
(17)
dcN f IL + -G dx F
(18)
0 ) 2πaDN
dcC 2πaDCcCF dφ 0 ) 2πaDC + dx RT dx
(19)
(20)
Notice that, by virtue of the definition of G, this term is indissolubly linked with f in the grouping G - (f IL/F). It is judicious to combine the five conservation equations in various ways that eliminate the unknown dφ/dx, f, and G terms. To achieve the first of these combinations, divide each of the five conservation equations by the corresponding diffusivity, except that (18) is divided instead by half the harmonic mean DOA of the diffusivities of the O+ and A- ions, i.e., by 448
Analytical Chemistry, Vol. 69, No. 3, February 1, 1997
IL d[cO + cA + (2DNcN/DOA) + cC + cX] + (22) dx FDO
appears after eq 8 has been exploited. A second valuable combination is obtained by dividing eqs 17 and 18 by DA and eq 19 by DCcC/cA. On then summing the three, one secures
0 ) dcA +
DN cA dcN + dcC DA cC
(23)
A third useful combination leads directly to
0 ) d(cC/cC) + (dcX/cX)
(24)
from eqs 19 and 20. From these various differential equations, from boundary conditions 4, 5, 7, and 12, and from the ancillary relations 8 and 2, we shall ultimately determine how the limiting current IL depends on the support ratio F. SOLUTION FOR THE LIMITING CURRENT Integration of eq 24 between the limits x and xb gives
cXcC ) cbXcbC ) F2c2E
(25)
eq 7 having provided the second equality. Integration of eq 22 between the limits of xs and xb leads to
[
0 ) 2πa cbO - csO + cbA - csA +
2DN b (c - csN) + cbC - csC + DOA N
]
cbX - csX +
IL (26) FDO
initially and to
and
dcX 2πaDXcXF dφ 0 ) 2πaDX dx RT dx
0 ) 2πa
(21)
[
]
csC DN DN x1 + 4p - 1 -IL +F) + 1(27) 4πaFDOcE DOA DOA 2p cE after incorporation of the electroneutrality equality csA + csX ) csO + csC, the boundary conditions 4, 5, 7, and 12, and subsequent rearrangement. Only the presence of the csC/cE term in eq 27 now impedes straightforward access to the limiting current. To proceed further, it is felicitous to undimensionalize all concentration terms by dividing them by cE. Thus let Ci ) ci/cE. In this new notation, eq 2 is CN ) pCOCA, which becomes
dCN ) pCO dCA + pCA dCO
(28)
on differentiation. This expression may be inserted into eq 23 to produce
0 ) CC(1 + pˆ CO) dCA + CA(dCC + pˆ CC dCO)
(29)
where pˆ is being used as an abbreviation for pDN/DA. Next, consider the equation
CA ) CO + CC - CX ) CO + CC - F2/CC
(30)
(which comes from combining the electroneutrality law 8 with result 25) and its derivative
dCA ) dCO + [1 + (F2/C2C)] dCC
(31)
These two equations may now be combined with eq 29 so as to eliminate all terms involving CA. The result is
dCC -CC[CC(1 + 2pˆ CO + pˆ CC) - pˆ F2] ) dCO C (C + 2C + pˆ C C ) + pˆ F2C C
O
C
O C
(32)
O
after rearrangement. This Darboux differential equation interrelates the variables CC and CO, both of which range over the interval xs e x e xb. It cannot be solved analytically, but it can be integrated numerically by following the simple strategy detailed earlier22 and illustrated in Figure 1. This integration procedure makes use of our knowledge of the values of both variables at one end of the range and of CO at the other,
CbO )
x1 + 4p - 1 ; CbC ) F; C sO ) 0 2p
(33)
to determine C sC, i.e., csC/cE. This is all that is needed to complete the determination of IL via eq 27. The final result is summarized by the equation
DN 1 + 2p - x1 + 4p -IL x1 + 4p - 1 + + ) 4πaFDOcE 2p DOA 2p F - C sC{CbO;CbC;pˆ } (34)
Figure 1. Darboux algorithm used to determine the concentration of the supporting cation at the electrode surface. The initial value of M is immaterial; it was often 8.
theory just presented, the limiting currents in those reported experiments in which uniunivalent supporting electrolytes (LiClO4, KCl, LiCl) were added to 10.0 mM ethanoic acid. Assuming25 a pK of 4.76 for the acid, species N, then β ) 57.5 m3 mol-1, cE ) 10.0 mol m-3, p ) 575, and CbO ) 0.0408. Furthermore, using the value20 DA ) 0.97 × 10-9 m2 s-1 and calculating from the known mobility26 of the hydrated proton that DO ) 9.16 × 10-9 m2 s-1 at 293 K, then DOA ) 1.75 × 10-9 m2 s-1. In the absence of alternative information, it is assumed that DN ) DA, so that DN/ DOA and pˆ are assigned values of 0.553 and 575, respectively. After incorporation of these various numbers, eq 34 becomes
-IL ) 0.571 + 4πaFDOcE F - C sC{CbO ) 0.0408; CbC ) F; pˆ ) 575} (35)
The format of the last term in this equation is intended to convey the identities of the three data that need to be input to the Darboux algorithm shown in Figure 1 in order to calculate C sC. Equation 34 is complicated enough that its implication is unclear without deeper analysis. One minor conclusion that can be drawn immediately is that, since the quantity f is nowhere involved in eq 34, the limiting current is uninfluenced by whether the electroreactant is the cation O+, the neutral species N, or both. Before analyzing the theoretical predictions further, their numerical import will be compared with some published experimental results. That experimental system will then provide a basis for further discussion of the insight that the model brings to an understanding of the role of the equilibration reaction. APPLICATION TO PUBLISHED EXPERIMENTS In three journal articles,19-21 Ciszkowska et al. reported and discussed experiments in which steady-state microelectrode voltammetry was performed on aqueous solutions of a number of weak acids with various added amounts of a range of supporting electrolytes. Here an attempt will be made to model, via the
To completely duplicate the conditions of Ciszkowska’s experiments, we set a ) 8.28 µm (the hemispherical equivalent27 of the 13 µm radius disk actually used). Thereby, using also the DO and cE values cited above
IL ) (-919 nA)[0.571 + F - C sC{CbO ) 0.0408; CbC ) F; pˆ ) 575}] (36) Values calculated using this equation are listed in the second column of Table 2 for a variety of support ratios. The most remarkable features of the tabulated theoretical data are the very marked dependence that the steady-state limiting current exhibits on support ratio and the massive effect that (25) Harris, D. C. Quantitative Chemical Analysis; Freeman: San Francisco, CA, 1982; p 663. (26) Oldham, K. B.; Myland, J. C. Fundamentals of Electrochemical Science; Academic Press, Inc.: San Diego, CA, 1994; p 12. (27) Oldham, K. B.; Zoski, C. G. J. Electroanal. Chem. 1988, 256, 11.
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449
Table 2. Steady-State Limiting Currents Calculateda by the Present Treatment and Measuredb Experimentally for a 10.0 mM Solution of Ethanoic Acid Containing Various Concentrations of Uniunivalent Supporting Electrolyte at Support Ratio G
cbO DNcbN DOAH -IL ) + ) 4πaFDOcE cE DOAcE 2pDN
(37)
where the abbreviation
-IL/nA
a
F
theor
0 10-8 10-7 10-6 10-5 0.0011 0.010 0.048 0.13 0.26 0.51 1.0 1.9 5.0 ∞
525 302 272 240 207 132 97.4 78.9 72.0 69.1 67.5 66.6 66.1 65.8 65.4
exptl
127 109 89 74 70 68 67 66 65
Column 2, from eq 36. b Column 3, from Figure 2 of ref 20.
minute additions of electrolyte have. Thus, the current is predicted to suffer an 8-fold diminution in magnitude on passing from no support to excess. Moreover, a concentration of supporting electrolyte as small as one-millionth of the electroactive concentration is calculated to more than halve the limiting current when added to an unsupported solution! One can confidently expect that these predictions will never be verified experimentally, because it will be impossible to prepare an ethanoic acid solution with the requisite low inert electrolyte level. Even the autoprotolysis of water will interfere. Ciszkowska and her co-workers did not tabulate their experimental results, so it was necessary to measure the coordinates of points plotted on rather a small graph (Figure 2 of ref 20) in order to recreate the data listed in the third column in Table 2. Recognize that the theory/experiment comparison in Table 2 is absolute; there has been no fitting of any kind. Thus the agreement at the larger support ratios is amazingly good and probably fortuitously so. At support ratios lower than ∼0.05, however, the discrepancy becomes significant and gets progressively worse. It may be speculated that the solutions in question contained somewhat more electrolytic impurities than the experimenters recognized. In this context, it should be pointed out that errors can arise not only from “foreign” electrolytes but from the acid reagent itself. The calculations crucially depend on exactly the same numbers of hydrogen and ethanoate ions being present in the bulk solution. Thus, even a minute trace of contamination by strong acids or metal ethanoates will lower the limiting current at low support ratios. Theory already in the literature22 permits a calculation of how serious this effect is for the case of excess A- ion, and minor changes can adapt that treatment to cover cases of excess O+ ion. ANALYSIS OF THE UNSUPPORTED CASE An expression for the limiting current when support is totally absent comes directly from eq 26 by setting cC and cX to zero, and is 450
Analytical Chemistry, Vol. 69, No. 3, February 1, 1997
2 H ) (DN/DOA )[2DNp + (DOA - DN)(x1 + 4p - 1)] (38)
represents a constant for a given identity and concentration of species N and which equals 364 for the experiments analyzed above. Equation 37 then leads to the -525 nA value given in Table 2 for the unsupported limiting current. Ciszkowska et al. claimed that weak acids share with strong acids (and other strong uniunivalent electroactive electrolytes) an enhancement factor of 2, but the present findings are in line with the predictions of Kharkats.5-7 What causes the large enhancement in the limiting current when support is withdrawn? Evidently this has something to do with the presence of the neutral species, but how it achieves this amplification is not self-evident. It should be noted that if N, the weak ethanoic acid in the experimental case being discussed, were a passive spectator in the experiment, simple theory9 shows that the unsupported limiting current would be only
x1 + 4p - 1 ) -37.6 nA -4πaFDOcE 2p
(39)
twice the diffusion current. Conversely, if the equilibration reaction did not interfere, and the neutral species alone were electroactive, its diffusion current would amount to
2p + 1 - x1 + 4p -2πaFDNcE ) -47.7 nA 2p
(40)
The unsupported limiting current that the present theory predicts exceeds either of these figures by an order of magnitude. Some radical explanation must account for such an unprecedented enhancement. The opaque mathematics of the previous sections is not needed to appreciate what the conditions are like when the solution is devoid of supporting electrolyte. Electroneutrality then insists that the two ions, O+ and A-, will have identical concentrationsscall the common value cIsat any point in the solution. At that point, the law of equilibrium then decrees that the concentration of the neutral species be cN ) βcI2. At the electrode surface, x ) 0, both cI and cN must be zero, and they are required to sum to cE in the bulk, x ) 1. This is not quite enough information to tie down the concentration profiles, but indefinite integration of the cO ) cA ) cI, cC ) cX ) 0 version of expression (22), combined with the cN ) βcI2 law, shows that these profiles are as described by the equations
cI ) (DOA/2βDN)[x1 + 2Hx - 1]
(41)
2 cN ) (DOA /2βD2N)[1 - x1 - 2Hx + Hx]
(42)
and
Figure 2. Theoretical concentration and potential profiles for limiting conditions in the case of 10.0 mM ethanoic acid without support. The ions O+ and A- are respectively the hydronium H3O+ cation and the ethanoate CH3COO- anion; they share a common profile and a common bulk concentration of 0.408 mM. The bulk concentration of species N, the undissociated CH3COOH, has the much greater value 9.592 mM. Data for the profiles are provided by eqs 41, 42, and 44. The inset shows the concentration profiles very close to the electrode surface.
when support is absent. These profiles are portrayed in Figure 2 for the 10.0 mM ethanoic acid case discussed earlier. In this type of graph, an uncharged species such as N, the CH3COOH(aq) molecule in our example, subject only to diffusive transport, should display a linear concentration vs x relationship. Indeed, Figure 2 shows that N’s profile is virtually linear except in the immediate vicinity of the electrode, where a sharp bend leads to a horizontal intersection with the axis. This curvature of the N graph, conspicuous only close to x ) 0, is attributable to the neutral species’ dissociation, which evidently occurs close to the electrode primarily. The role of this process will be quantified in the next section of this article. The plot in Figure 2 for the ions is not so readily interpreted, because these are subject to migratory, as well as diffusive, transport. To investigate the role of migration requires that the electric field first be delineated. The magnitude of dφ/dx can be evaluated as
DO - DA H F dφ H ) + (43) RT dx x1 + 2Hx - 1 DO + DA x1 + 2Hx after suitably combining eqs 16-18, aided by the previous results of this section. (Note that the actual field, -dφ/dr, equals (1 - x)4(dφ/dx)/a.) On integration, the local electric potential φ in the solution, with respect to the bulk, is found to be given by
Fφ DOA ) [x1 + 2Hx - x1 + 2H] + RT DA
{
}
[x1 + 2Hx - 1][x1 + 2H + 1]x 1 ln 2 [x1 + 2Hx + 1][x1 + 2H - 1]
Figure 3. Inventory of material transfers predicted for unsupported 10.0 mM ethanoic acid under limiting steady-state conditions at a hemisphere of radius 8.28 µm. The values at the left relate to total fluxes at the electrode surface; those at the right correspond to the bulk. The dissociative transfer shown in the center of the diagram occurs throughout the transport zone, but most of it happens very close to the electrode.
achieve truly limiting conditions when supporting electrolyte is totally absent, the potential needs to be negative infinite at the working electrode interface. This comes as no surprise, because there are no ions there. The need for a theoretically infinite cell voltage provides yet a further barrier to any practical realization of support-free voltammetry. Easily overlooked in Figure 2 is the large magnitude that dφ/dx has far from the electrode. Even at x ) 1, eq 43 gives dφ/dx a value of 24.9RT/F, whereas in the corresponding unsupported steady-state electrolysis, free of homogeneous reaction complications, it is known10 to be only RT/ F. This proves to be the key to the puzzling enhancement. Knowing the concentration and potential profiles, it is now possible to decipher the entire material inventory accompanying the unsupported steady-state voltammetry of 10.0 mM ethanoic acid under conditions of total concentration polarization. This is shown in Figure 3 for the case f ) 0, i.e., when the neutral species is electropassive. The numbers in this diagram correspond to the magnitudes of the various terms in eqs 16-18 for the model experiment being considered. Considering that its bulk concentration exceeds that of the ions 25-fold, species N is seen to play a very small numerical role in Figure 3. Nevertheless its effects are profound. My explanation of the huge limiting current predicted for the unsupported case is as follows. The rapid diminution in ion content as the electrode is approached forces the neutral species to dissociate. That dissociation injects ions into a region in which many fewer ions would otherwise be present. This weakens the electric field close to the electrode, permitting more of the applied potential to be developed in the outer regions, as commented upon above. In turn, this fosters a massive migratory flux of O+ into the transport zone, a flux that Figure 3 shows to account for 89% of the current. In contrast, the uncharged species contributes only 9% directly.
(44)
Figure 2 includes a plot of this result and shows that, in order to
DISSOCIATIVE REACTION The G term is analyzed in this section. Recall that this radiusdependent function represents the net forward rate at which Analytical Chemistry, Vol. 69, No. 3, February 1, 1997
451
to deal with. However, this situation is not without its complications and surprises. As a result of ion redistribution caused by the steady-state electrolysis, there may be small differences between the concentrations of the supporting ions in the bulk and at locations in the transport zone; i.e., the excursions (cC - cbC) and (cX - cbX) will not be zero away from the bulk region and may have values commensurate with the concentrations of the minor solution components O+, A-, and N. Nevertheless, the relative differences (cC - cbC)/cbC and (cX - cbX)/cbX will be very small indeed when supporting electrolyte is in excess and this has important consequences. Let (cC - cbC)/cbC, or equivalently [(cC/FcE) - 1], have the very small value δ at some point in the solution. Then, Figure 4. The rate of ion-pair dissociation close to the electrode according to eq 46, plotted as a function of both spatial variables: r and x.
cC ) FcE(1 + δ)
(47)
and via eq 25, reaction 1 proceeds within the total volume 2π(r3 - a3)/3 between the electrode and the location in question. Of greater chemical familiarity is a rate function that reflects the rate of creation of product in unit time and unit volume, i.e., the conventional chemical reaction rate (with units mol m-3 s-1). This will be denoted g and is given by
(48) It follows that
cC + cX ) FcE(2 + δ2 - δ3 + ...)
4
1 dG (1 - x) dG ) g) 2πr2 dr 2πa3 dx
cX ) F2c2E/cC ) FcE(1 + δ)-1 ) FcE(1 - δ + δ2 - δ3 + ...)
(49)
(45) and will differ negligibly from the constant 2FcE. Accordingly, eq 22 may be replaced by
To better understand the role of the dissociative reaction, the magnitude of g at locations throughout the cell will be evaluated for the electrolysis of unsupported 10.0 mM ethanoic acid under steady-state limiting conditions. Continuing to pursue the case in which the species N is electropassive, so that f ) 0, eq 18 shows that G equals 2πaDN (dcN/dx) and hence
g)
2 DN DOA d 2c N H2 (1 - x)4 4 (1 x) ) ) a2 dx2 2βa2DN (1 + 2Hx)3/2
(1 - x)4 (3.66 µmol m-3 s-1) (46) (1 + 727x)3/2
-IL dx/2πFaDO ) d[cO + cA + (2DNcN/DOA)]
(50)
when supporting electrolyte is in excess. A second consequence of the extreme smallness of any relative difference in the magnitude of the supporting cation concentration is that the term dcC/cC may be replaced by zero in eq 23;
0 ) dcA + (DN/DA) dcN
(51)
remains. This result shows that the concentrations of the anion and the neutral species are linearly related; in fact
cA + (DN/DA)cN ) constant ) cbA + (DN/DA)cbN ) csA (52) where the second equality follows from two differentiations of eq 42, and the third from substituting the known constants. Thus the net dissociative reaction rate is 3.66 × 10-3 mol L-1 s-1 at the x ) 0 surface but falls off extremely rapidly, as depicted in Figure 4, becoming half its surface value at x ) 0.000 80, i.e., only 6.6 nm from the surface. Of course, the first-order rate constant for the dissociation of ethanoic acid must well exceed g/cN, in order for the assumption of equilibrium to be plausible. And since we are treating the surface concentration csN as zero, there is a clear conflict between the model and experimental reality. In this respect, at least, the treatment represents a limiting case. ANALYSIS OF FULLY SUPPORTED CASE The last two sections have concerned the unsupported case exclusively. Now attention is turned to the other end of the F scale, where supporting electrolyte is present in excess. One would expect the case of a large support ratio to be more easily treated because there is now no migration of the O+ or A- ions 452
Analytical Chemistry, Vol. 69, No. 3, February 1, 1997
The two alternative expressions for “constant” follow by evaluating the left-hand side of eq 52 at xb and at xs respectively, recognizing that, by arguments that have been exercised before, csN ) 0. When eq 50 is integrated between the limits of x ) xs ) 0 and x ) xb ) 1, we obtain
-IL/2πFaDO ) cbO + cbA - csA + (2DNcbN/DOA)
(53)
because csO and csN are both zero. The difference cbA - csA may be found from eq 52, whence
[
]
DNcbN 2DN DN b -IL ) cbO + cN ) cbO + 2πFaDO DOA DA DO Equation 54 may be reorganized to
(54)
IL ) (-2πFa)(DNcbN + DOcbO)
(55)
This result might have been predicted heuristically and has been so predicted17,20,21 for the case of full support. It corresponds to the independent diffusion of the neutral species and the electroactive cation, uninfluenced by the homogeneous reaction. It is satisfying to see such predictions confirmed rigorously here. Substitution into eq 55 of data appropriate to the Ciszkowska experiment modeled previously leads to a value of -65.4 nA for the limiting current, as reported in Table 2 for F ) ∞. There is one very strange aspect of the derivation of eq 55. Nowhere has use been made of cN ) βcOcA, the equilibrium law! That statement needs qualification. The law was, of course, used to determine the bulk concentrations of the O+, A-, and N species. Furthermore, qualitative use of the mass action law was made in arguing that csN must be zero because csO is, or vice versa. But even that qualitative use is rendered unnecessary if we regard (28) Myland, J. C.; Oldham, K. B., in preparation.
both species as electroactive. The conclusion to be drawn is that eq 55 applies whether equilibration occurs in the transport zone or not, provided that both O+ and N are electroactive. Expressed differently, it appears that eq 55 is valid whether the speed of the equilibration reaction is infinitely fast, as we have previously assumed, or even if it is so slow that it affects only the composition of the bulk solution. The repercussions of this finding, which surprisingly also applies when supporting electrolyte is absent, have been investigated and will be published28 in a separate communication. ACKNOWLEDGMENT The help of Jan Myland and the financial support of the Natural Sciences and Engineering Research Council of Canada are greatly appreciated. Received for review July 19, 1996. Accepted October 9, 1996.X AC960719I X
Abstract published in Advance ACS Abstracts, December 15, 1996.
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