Anal. Chem. 1996, 68, 4173-4179
Effect of Diffusion Coefficient Diversity on Steady-State Voltammetry When Homogeneous Equilibria and Migration Are Encountered Keith B. Oldham
Department of Chemistry, Trent University, Peterborough, Ontario K9J 7B8, Canada
An exact treatment is developed to predict the steady-state limiting voltammetric current, IL, for a system in which the reaction O+ + A- / N occurs reversibly in solution, in the presence of supporting electrolyte C+A-. Either or both O+ and N undergo a one-electron reduction at the hemispherical microelectrode. The dependence of IL on the formation constant of the equilibration reaction, the electrolyte concentration, and the support ratio is derived for any trio of values of these parameters and for any combination of diffusivities of the O+, N, and A- species. Restricting attention to the steady state facilitates the modeling of voltammetry, especially at microelectrodes.1 The high current density at microelectrodes makes it possible to extend experimental voltammetric studies toward systems that, because of their high resistivity, would be inaccessible to traditional electrochemical techniques. Such systems include organic solvents of low permittivity, in which electrolytes are very sparingly soluble, and aqueous systems without the usual supporting electrolyte excess. There have been concomitant developments in the theory of steady-state electrochemical processes, which must now incorporate migration, as well as diffusion, in the treatment of transport to the electrode.2-14 However, another complication that sometimes occurs in electrochemical systems is the coupling of species undergoing transport to other solute species through homogeneous equilibria. It is to one example of such a system that this article is addressed. One aspect in which the present study represents an improvement over many of its predecessors is that the diffusivities (diffusion coefficients) of the various species are regarded as distinct. Much prior work has assumed, for mathematical convenience, that all solutes share a common value of (1) Bond, A. M.; Oldham, K. B.; Zoski, C. G. Anal. Chim. Acta 1989, 216, 177. (2) Bond, A. M.; Fleischmann, M.; Robinson, J. J. Electroanal. Chem. 1984, 172, 11. (3) Ciszkowska, M.; Stojek, Z. J. Electroanal. Chem. 1986, 213, 189. (4) Oldham, K. B. J. Electroanal. Chem. 1988, 250, 1. (5) Amatore, C.; Fosset, B.; Bartlett, J.; Deakin, M. R.; Wightman, R. M. J. Electroanal. Chem. 1988, 256, 255. (6) Baker, D. R.; Verbrugge, M. W.; Newman, J. J. Electroanal. Chem. 1991, 314, 23. (7) Norton, J. D.; Benson, W. E.; White, J. S.; Pendley, B. D.; Abruna, H. D. Anal. Chem. 1991, 63, 1909. (8) Norton, J. D.; White, H. S. J. Electroanal. Chem. 1992, 325, 341. (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) Verbrugge, M. W.; Baker, D. R.; Newman, J. J. Electrochem. Soc. 1993, 140, 2530. (12) Oldham, K. B. J. Electroanal. Chem. 1993, 347, 49. (13) Amatore, C.; Montenegro, M. I. Anal. Chem. 1995, 67, 2800. (14) Palys, M. J.; Stojek, Z.; Bos, M.; van den Linden, W. E. J. Electroanal. Chem. 1995, 383, 105. S0003-2700(96)00173-4 CCC: $12.00
© 1996 American Chemical Society
diffusivity. This article quantitatively treats an example of steadystate voltammetry that incorporates all three complications: migration, homogeneous equilibria, and diverse diffusivities. Though they have yet to realize their full potential, voltammetric studies of the class of systems addressed in this article hold promise of providing valuable data, as well as insight into electrochemical fundamentals. Data which are accessible include equilibrium constants, transport parameters, and analyte concentrations. SYSTEM UNDER CONSIDERATION Here, attention is directed to the reduction to a neutral product,
O+(soln) + e- f product of a monovalent cation equilibrium reaction,
O+
(1)
which enters the homogeneous
O+(soln) + A-(soln) / N(soln)
(2)
with the anion A-. This anion arises, in part, from the supporting electrolyte C+A-, which is strong. An example of this behavior occurs when the electroactive cation is a metal ion that forms a neutral ion pair in a solvent such as toluene.15 Another example has O+ representing the solvated proton, A- being the anion of a neutral weak acid. If, as will be assumed, equilibrium 2 is established rapidly, it makes no difference to the objectives of this study whether O+ alone is electroactive or whether the neutral species N is also reducible:
N(soln) + e- f A-(soln) + product
(3)
Nevertheless, in the interest of realism, we shall treat (1) and (3) as reactions occurring in parallel, respectively contributing the fractions (1 - f ) and f, either of which (or neither) could be zero, to the faradaic current I. The system being considered is simpler than many that arise in practice. A three-ion system, such as that treated here, is inherently less elaborate to model than the four-ion system exemplified by studies16,17 of weak acids with added inert electrolyte. A further complication, ignored here but treated elsewhere,18 is the possibility of ion pairing between the supporting cation C+ and A-. Of course, if the product of a reaction otherwise similar to reaction 1 is an ion, there is the potential for ion pairing (15) Santos, J. H.; Cardwell, T. J.; Bond, A. M.; Oldham, K. B. J. Electroanal. Chem., in press. (16) Ciszkowska, M.; Stojek, Z.; Morris, S. E.; Osteryoung, J. G. Anal. Chem. 1992, 64, 2372. (17) Stojek, Z.; Ciszkowska, M.; Osteryoung, J. G. Anal. Chem. 1994, 66, 1507. (18) Oldham, K. B.; Cardwell, T. J.; Santos, J. H.; Bond, A. M. J. Electroanal. Chem., in press.
Analytical Chemistry, Vol. 68, No. 23, December 1, 1996 4173
of the product too, and similar complications can be envisaged even when the product is neutral. The bulk solution contains concentrations cOb, cNb, cAb, cCb of the solutes O+, N, A-, and C+, the reduction product(s) being absent. Though it may have been prepared in another way, the bulk solution is considered to be composed of two components, in addition to the solvent. One of these components, of concentration cE, could be the neutral compound N, or it could be the salt O+A-; in either case, it is the source of the electroactive species, so that
cE ) cOb + cNb
(4)
The second component is the supporting salt C+A-, which contributes all of C+ and part of A- to the solution. Its concentration differs from cE by the factor F, an important quantity called the “support ratio”,
cCb ) FcE
(5)
The support ratio, which may, in principle, take any value between 0 and ∞, is the index generally used to quantify the relative abundances of electroactive species and supporting ions. Note that, though the cation of the supporting electrolyte is electropassive, its anion is identical with one of the two products of reaction 3. Thus, the support ratio here does not have quite the same connotation as in some other studies. Two other interrelationships among the bulk concentrations of the four solute species are provided by the electroneutrality condition,
cOb - cAb + cCb ) 0
(6)
and the equilibrium law,
cNb ) βcObcAb
Figure 1. Fractional association as a function of the support ratio F and the association parameter p. Fractional association is symbolized cNb/cE and V(F,p) in the text. In this figure and Figures 3 and 5, the left-hand border, F ) 0, corresponds to an unsupported solution, whereas the regions shown shaded represent solutions in which the support is “full”, as defined in the text.
(7)
where β is the formation constant for reaction 2. Since, in the subsequent development, the formation constant occurs mostly as its dimensionless product βcE with the bulk electrolyte concentration, it will be convenient to adopt the abbreviation
p ) βcE
(8)
Increasing β or increasing cE favors the formation of the uncharged species N, so p can be considered as a crude measure of how much association occurs in the bulk solution. Accordingly, p will be named the “association parameter”, though the degree of association is strongly influenced also by the support ratio, as Figure 1 illustrates. The working electrode is a shrouded microhemisphere of radius a and area 2πa2, as illustrated in Figure 2. The voltammetric program is either a large negative potential step or a slow negative-going potential ramp, with interest being confined to the final limiting current, IL, that is attained when an enduring steady state has been reached. The limiting current corresponds to total concentration polarization of the electroactive species via reactions 1 and 3. Irrespective of the mechanism of the electrode reaction, the equilibrium law,
cN ) βcOcA
(9)
which is taken to apply instantaneously at all points in the solution, ensures that whenever one of cO or cN is forced to zero, the other follows suit. Accordingly, 4174 Analytical Chemistry, Vol. 68, No. 23, December 1, 1996
Figure 2. Geometry of the electrode, shown in cross section. The shaded zone is demarcated by the electrode surface, a portion of the plane in which the electrode is mounted, and a hemisphere at an arbitrary distance r from the electrode’s center; its volume is 2π × (r3 - a3)/3.
cOs ) cNs ) 0
(10)
at the electrode surface. The symbol r will be used to represent distance measured from the center of the electrode hemisphere, but an alternative spatial coordinate, defined by
x ) (r - a)/r
(11)
will prove useful later. Note that a superscript “s” is being
attached to the c symbol to represent concentrations at the electrode surface (where r ) a and x ) 0), whereas a c symbol with superscript “b” indicates a concentration in the bulk solution (where r f ∞ and x ) 1). Two other superscripts, “h” and “m”, will be introduced later. Unsuperscripted c symbols represent concentrations at any point in the range a e r < ∞ or 0 e x e 1. The objective of this study is to produce an exact expression for IL, which in accordance with IUPAC’s preference will be negative for the reduction that is being treated, in terms of the variables of the system. These latter are the electrode radius a, the bulk electrolyte concentration cE, the formation constant β of reaction 2, the support ratio F, and the diffusivities Di (i ) O, N, A, and C) of the solute species. BULK SOLUTE CONCENTRATIONS By solving eqs 4-7 simultaneously, one may derive the expressions
[x
b
cN
]
(pF + 1)2 + 4p - 1 F 2p 2
cOb ) cE
[
(12)
]
F x(pF + 1) + 4p - 1 ) cE 1 + 2 2p 2
cAb ) cE
[
]
x(pF + 1)2 + 4p - 1 + F 2p
2
∫ cr
(13)
(14)
(15)
Substitution from eqs 12, 14, and 5 into inequality 15 leads to
Fg
94 200p
(x
1+
200p -1 3
)
CONSERVATION EQUATION In this section, a relationship is developed that is obeyed by any solute i (i ) O, N, A or C) during the experiment under discussion. It applies also to the neutral product of the reductive electrode reaction, but we shall have no need to consider this species. Consider the zone, illustrated by shading in Figure 2, lying between the electrode surface and the hemisphere of radius r, where r is finite and greater than a but otherwise arbitrary. Let us formulate an answer to the question, “What is the rate of increase in the amount (moles) of species i in the zone during any experiment with this apparatus and solution?”; i.e., what value has the quantity
∂ 2π ∂t
for three of the four bulk species concentrations, with eq 5 completing the quartet. To illustrate these equations, Figure 1 shows how the fraction cNb/cE of the electrolyte that is associated depends on the support ratio for seven representative values of the association parameter. Notice that, for large values of p, most of the electroactive species is associated irrespective of F, whereas for small p, large values of F are needed to ensure that association is dominant. A question to which we will need an answer later in this article is “What sets of values of the parameters cE, F, and β correspond to solutions that are fully supported?” In other words, “Which solutions contain excess supporting electrolyte?” The answer to this question is not self-evident. Certainly it is not always necessary to have F . 1 for support to be full. The salient feature of a fully supported solution is that the electroactive ions migrate negligibly: therefore, for the system now under consideration, an acceptable criterion might be that full support requires that only a small fraction of the bulk ionic strength be contributed by the electroactive O+ ions. For, if this fraction is small in the bulk, it will be still smaller closer to the electrode. If this criterion is adopted and “a small fraction” is interpreted as “not more than 3%”,19 then support is full when
97cOb e 3(cAb + cCb)
Formula 16 shows that a support ratio of 16 or more will always lead to full support, but that smaller F values will often be adequate. For p larger than unity, inequality 16 is well approximated by F g 3.8/p1/2, showing that, when the association parameter is 16 or more, even a support ratio of unity provides full support!
(16)
after considerable algebra. Pairs of F, p values that meet this condition lie in the shaded region in Figure 1. (19) Oldham, K. B.; Myland, J. C. Fundamentals of Electrochemical Science; Academic Press, Inc.: San Diego, 1994; p 264.
r 2 a i
dr
(17)
It is possible to identify four distinct terms that contribute to quantity 17. First, amounts of species may be introduced (or removed) by the electrode reaction itself. The appropriate rate of material increase (mol s-1) is νiI/F, where I is the current and F is Faraday’s constant. The symbol νi denotes the stoichiometric coefficient, which takes the numerical values listed here, i
νi
O N A C
1-f f -f 0
representing the role that each species plays in reactions 1 and 3. Here, f is the fraction of the total current I that is carried by reaction 3. Notice that, because cathodic currents are negative, νiI/F is negative for O+ and N, positive for A-, and zero for C+. Second, the amounts of species O+, N, and A-, but not C+, within the zone will change by virtue of the homogeneous reaction 2. The symbol Gi will be used to represent the rate of generation of species i within the zone from this cause. From the stoichiometry of the homogeneous reaction, it is clear that
GO ) -GN ) GA
(18)
These G terms are functions of the position of the r hemisphere (and of time, except in the steady state). Third, material can diffuse into (or out of) the zone by diffusion across the hemisphere of radius r. The area of this hemisphere is 2πr2, and therefore, by Fick’s first law, the rate of increase of the amount of species i within the zone is 2πr2Di(∂ci/∂r) by this route, Di being the diffusivity (diffusion coefficient) of the species. Fourth, charged species may enter (or leave) the zone by migration across the hemisphere of radius r. The rate of material entry by this cause is 2πr2ziuici(∂φ/∂r), where ui is the mobility of species i and zi is its charge number. The local electrostatic potential is denoted by φ. However, the Nernst-Einstein relationAnalytical Chemistry, Vol. 68, No. 23, December 1, 1996
4175
i
zi
O N A C
1 0 -1 1
ship20 permits the mobility of a monovalent ion to be replaced by DiF/RT, where R and T are the gas constant and absolute temperature. When expression 17 is equated to the sum of its four contributors, the equation
[
]
νiI ∂ci ziFci ∂φ 2 + G + c r dr ) + 2πr D i i i a F ∂r RT ∂r
∫
∂ 2π ∂t
r
2
UNSUPPORTED CASE
(19)
emerges. This is a general material balance relationship, but, when we specify steady-state conditions and extreme concentration polarization, simplification to
0)
[
]
νiIL dci ziFci dφ + Gi + 2πaDi + F dx RT dx
-IL 2πFaDi
In this special case, C+ is absent, and electroneutrality then enforces the equality of cO and cA everywhere. This means that when eqs 23 and 25 are added, the migration terms disappear. The homogeneous generation terms may then be eliminated by adding a suitably weighted version of eq 24 and invoking the equalities in eq 18. The result is
(20)
ensues, after also adopting the new spatial coordinate x, defined in eq 11. Further notational economy arises from the definition
cih )
With some noteworthy exceptions,14,21 problems of this sort are often tackled by making the simplifying assumption that all diffusivities have equal values. There are circumstances, such as when O+ is the hydrogen ion and the solvent is water,16,17 in which this assumption is hopelessly untenable. In less extreme circumstances, it is still of interest to study just how significantly the voltammetry is affected by unequal diffusivities. Before the general case is tackled, consideration will be given in two special cases: when supporting ions are absent, and when they are in excess.
(21)
cOh )
{
cOh + cAh d 2cO + cN dx cNh
(22)
Each cih term may be regarded merely as an abbreviation for a constant. However, a significance that can be attached to each term is that cih is the concentration of species i needed to generate a steady-state limiting current of magnitude |IL| if the species i were electroactive and underwent an uncomplicated one-electron electrode reaction from a fully supported solution. Equation 22 will be called the “conservation equation” of species i. The forms adopted by the conservation equations for each of the four solutes in the present problem are h
(1 - f )cO
FcOhGO dcO cOF dφ + + ) IL dx RT dx
(23)
-cOhx + 2cO +
h
f cN
-f cAh +
FcN GN dcN + ) IL dx
FcAhGA dcA cAF dφ ) IL dx RT dx
0)
dcO cCF dφ + dx RT dx
(24)
(29)
The constant was shown to be zero by setting x ) 0 and invoking boundary condition 10. Thence, setting x ) 1, we arrive at
cOh ) 2cO +
) 2cE
cOh + cAh b cN cNh
[
]
h h x1 + 4p - 1 cO + cA x1 + 4p - 1 - 2p 2p 2p 2cNh
(30) The second step in (30) follows from utilization of the F ) 0 versions of eqs 12 and 13. Finally, the result
IL ) -4πFacE
[ (
)
DN DO 1+ + 2 DA
{
(
DN DO 1+ 2 DA
)}
]
x1 + 4p - 1 (31) 2p
emerges on abandoning the ch abbreviations. This is the steady-state limiting current in the total absence of supporting electrolyte.
(25)
Equation 31 is a very credible outcome. When p is small, corresponding to negligible association between the O+ and Aions, the equation reduces to
(26)
IL ) -4πFacEDO
The task is to solve these four differential equations simultaneously, subject to boundary conditions 10, 12-14, and 5, and with assistance from relationships 9, 18, and
cO - cA + cC ) 0
cOh + cAh cN ) constant ) 0 cNh
DO -
h
(28)
and indefinite integration now gives
which permits the conversion of eq 20 into
FcihGi dci ziciF dφ + νicih + ) IL dx RT dx
}
(27)
this last being the general electroneutrality condition. (20) Reference 19, p 239.
4176 Analytical Chemistry, Vol. 68, No. 23, December 1, 1996
(32)
when p , 1 and F ) 0 which is the well-known result for the reductive limiting current of an unsupported monovalent cation. On the other hand, when the association parameter is large, so that free ions are sparse, the form adopted by (31) is (21) Kharkats, Yu. I.; Sokirko, A. V. J. Electroanal. Chem. 1991, 303, 17.
DN(DO + DA) IL ) -4πFacE 2DA
(33)
when p . 1 and F ) 0 and reflects the diffusivities of all the species present. Notice that, as might be expected, increasing DN or DO enhances the steadystate limiting current, whereas increasing DA diminished it. While theoretically interesting as a limit, it should be stressed that accurate experimental results conforming to the predictions of this section are unlikely to be achieved. Under total concentration polarization, the concentration of O+, and thereby also that of A- in the absence of support, is predicted to be zero at the electrode surface. Extreme ohmic polarization (iR drop) will ensue, even with a microelectrode, and there is a strong likelihood of extraneous effects vitiating the voltammetry. Furthermore, conditions of zero support are notoriously difficult, if not impossible, to achieve under the conditions of an electrochemical experiment, especially in ionophilic solvents such as water. The unsuspected presence of minute concentrations of electroinert ions can have a dramatic effect. For example, the final diagram of this article reveals that the presence of as little as 1 µM of adventitious electrolyte would (with DA ) DN ) DO/ 10) diminish the limiting current of an unsupported 10 mM aqueous solution of a weak acid (pK ) 5) by a factor of about 4. FULLY SUPPORTED CASE Here, the concentrations of the supporting ions C+ and A- are large in comparison with cO, and this has a number of simplifying consequences. The migration of O+ is thereby inhibited, so that diffusion becomes the only important transport mechanism. Moreover, the concentrations of C+ and A- become virtually uniform throughout the cell and, therefore, negligibly different from their bulk values cCb and cAb. The uniformity of cA means, as a corollary to eq 9, that the ratio cN/cO of the concentrations of the two diffusing species, N and O+, is the constant βcAb everywhere in the cell. There is, therefore, no driving force for either direction of equilibration reaction 2 to occur, so that species O+ and N diffuse independently of each other. Hence, each contributes a limiting current given by the standard formula -2πFacibDi for a fully supported one-electron steady-state reductive limiting current.22 Accordingly,
IL ) -2πFa(DOcOb + DNcNb)
[
]
cNb cE
) -2πFacE DO - (DO - DN)
full support
(34)
Figure 3. Quantity U(F,p,p) graphed as a function of the support ratio F and the association parameter p. This quantity is the normalized steady-state limiting current IL/(-4πFacEDO) in cases where DA ) DN ) DO.
GENERAL CASE The situation in which each of F and p is free to adopt any positive value will now be addressed. Of course, the result is more complicated than for the two limiting cases considered earlier. It will be demonstrated that the limiting current has the form
[
{
IL ) -4πFacE DOU(F,pˆ ,p) - DO -
(
)} ]
DN DO 1+ 2 DA
V(F,p)
(35) in this general case. The function V depends on the support ratio F and the association parameter p; in fact, it equals cNb/cE, as graphed in Figure 1. The function U depends not only on these same two variables but also on a modified association parameter, defined by
where the second step invoked eq 4.
Equation 34 shows that the operative diffusivity is DO when association is prevalent, so that cNb/cE is close to unity, but DN when there is little association, so that cNb/cE ≈ 0. Recalling that in this section we are restricted to the shaded zone, Figure 1 can be used to assess the value of the cNb/cE fraction and thereby evaluate eq 34 semiquantitatively. For example, when F ) 100 and βcE ) p ) 0.03, this diagram shows that cNb/cE ≈ 0.7, so that the effective diffusivity is a weighted combination of DO and DN, namely 0.3DO + 0.7DN. Of course, eq 13 can provide more accurate values of the complementary weights. (22) Reference 19, p 279.
DN DA
pˆ ) p
(36)
Figure 3 illustrates these dependences when pˆ ) p. The form of eq 35 enables us to regard the V term as a coefficient expressing how seriously IL is affected by inequalities among the various diffusivities. When O+, N, and A- share a common diffusivity D, IL equals -4πFacEDU(F,p,p), the V “correction term” having vanished. To derive eq 35, start with the four conservation equations. Add eqs 23, 25, and 26 as they stand, and eq 24 after first multiplying by (cOh + cAh)/cNh. This addition gives Analytical Chemistry, Vol. 68, No. 23, December 1, 1996
4177
cOh )
{
}
cOh + cAh d 2cO + cN + 2cC dx cNh
(37)
after relationships 18 and 27 are exploited. Because cOh is independent of x, integration gives
2cO +
cOh + cAh cN + 2cC ) cOhx + constant ) cOhx + 2cCs cNh (38)
where “constant” was identified as 2cCs by setting x to zero and invoking boundary condition 10. On now setting x to unity, we find
(
cOh ) 2 cOb +
)
cOh + cAh b cN + cCb - cCs 2cNh
(39)
Figure 4. Procedure used to solve the Darboux differential equation to find cCb. Starting with point b, the coordinates of sucessive points are calculated until point s is reached.
Expressions for the bulk concentrations cOb, cNb, and cCb are known from eqs 12, 13, and 5, but the surface concentration, cCs, of the supporting cation is more difficult to access. Multiply eq 24 by cAhcC, eq 25 by cNhcC, and eq 26 by cNhcA. Cancellations occur on subsequent addition and lead to
cOm-1 ) cOm -
(45) b
cAhcC dcN + cNhcC dcA + cNhcA dcC ) 0
(40)
From eqs 27 and 9,
cN ) βcOcA ) βcO(cO + cC)
(41)
and therefore, on differentiation,
dcN ) (2βcO + βcC) dcO + βcO dcC
(42)
and when these two equalities are substituted into eq 40, one finds
dcC -cC(2βˆ cO + βˆ cC + 1) ) dcO 2cC + cO + βˆ cC
(43)
cOb M
cCm-1 ) cCm +
m
m
m
cO cC 2βˆ cO + βˆ cC + 1 M 2c m + c m + βˆ c m C O C
which constitutes a discrete equivalent of differential eq 43. After M applications of this formula, one sets cCs ) cC0. Figure 4 shows M ) 24, but in practice M is chosen large enough that, to the precision sought, further increase in M has no effect on the calculated value of cC0. The above description of the solution of the Darboux equation was couched in terms of concentrations, but in practice dimensionless equivalents are used with cO and cC replaced by cO/cE and cC/cE. Similarly, βˆ is replaced by pˆ ) βˆ cE. After the calculation, one may then set
after rearrangement. Here,
cAh
DN βˆ ) β h ) β DA cN
(44)
Equation 43 is the equation from which a value of cCs is to be sought. Except when βˆ ) 0 (in which case cN ) [(cO2/4) + constant]1/2 - cO/2), eq 43 cannot be solved to give cC as an explicit function of cO. However, numerical solutions are easily found for Darboux differential equations23 such as this. The integration is illustrated in Figure 4. The curve in this diagram represents the solution to eq 43. Its slope at each point is given by (43) in terms of the coordinates of that point, but those coordinates are initially unknown, except at the terminus labeled “b”. This terminus corresponds to the bulk solution, so its coordinates are cOb and cCb, given by eqs 12 and 5. The second terminus, labeled “s”, represents the electrode surface where cO ) cOs ) 0. It is the cC coordinate, cCs, of this terminus that is sought. The numerical procedure involves interpolating M - 1 points, equally spaced in cO, along the curve. The points are numbered 0, 1, ..., m, ..., M, with m ) 0 being the s terminus and m ) M the b terminus. Starting at point b, one calculates the coordinates (cOm-1, cCm-1) of the (m - 1)th point from those of the mth by the fomulation (23) Murphy, G. M. Ordinary Differential Equations and Their Solutions; Van Nostrand: Princeton, NJ, 1960; p 39.
4178 Analytical Chemistry, Vol. 68, No. 23, December 1, 1996
U(F,pˆ ,p) )
cE + cCb - cCs cCs )1+FcE cE
(46)
Notice that a dependence on F enters the U function not only via the F term in (46) but also, via eqs 12 and 5, as a component of the coordinates of the b terminus. A dependence on the unmodified association parameter p enters by the same route. The significance attaching to the U function is that, in the equidiffusivity case, this is the multiplicative factor by which IL is smaller than the unsupported limiting current. As Figure 3 shows, this factor is in the range 1/2 e U e 1 when DA ) DN. When the ch abbreviations are abandoned, eq 39 becomes
[
]
DN(DO + DA) b IL ) -2 (cE - cNb) + cN + cCb - cCs 2πFaDO 2DODA
(47) when relation 4 is also incorporated. Combination with eq 46 now leads to
[
{
IL ) -4πFacE DOU(F,pˆ ,p) - DO -
(
)} ]
DN DO 1+ 2 DA
cNb cE
(48) Since the function V(F,p) was chosen to represent the cNb/cE fraction,
Because it plays only a passive role (i.e., it undergoes no net transport in the steady state), the supporting cation’s diffusivity, DC, does not affect IL. There are three special circumstances in which eq 35 adopts simpler forms: (i) When the diffusivity DN of the neutral species is the harmonic mean of the diffusivities of the two ions O+ and A- from which it forms, i.e.,
(
)
1 1 1 1 ) + DN 2 DO DA
(50)
IL ) -4πFacEDOU(F,pˆ ,p)
(51)
Then
with no contribution from the V term. (ii) When species N diffuses twice as fast as the ion A-, DN ) 2DA. Then
IL ) -4πFacE[DOU(F,2p,p) - DAV(F,p)]
(52)
(iii) When the diffusivities of N and A- are equal. Then
[
IL ) -4πFacE DOU(F,p,p) -
]
DO - D V(F,p) 2
(53)
where the unsubscripted D represents the common diffusivity of the neutral species and the anion. Figure 5 shows the predictions of eq 53 when DO exceeds D 10-fold, as could be appropriate when O+ is the hydrated proton. Figure 5. Quantity U(F,p,p) - 0.45V(F,p), plotted as a function of the support ratio F and the association parameter p, equal to βcE. This quantity is the normalized steady-state limiting current IL/ (-4πFacEDO) in cases where DA ) DN ) 0.1DO. As in Figures 1 and 3, the abscissa is scaled semilogarithmically, except at the lefthand margin.
V(F,p) )
2 cNb F x(pF + 1) + 4p - 1 )1+ cE 2 2p
(49)
eqs 35 and 48 are identical. This result shows that the diffusivities of the three species O+, N, and A- affect the limiting current, each in a unique way.
ACKNOWLEDGMENT The help of Jan Myland and the financial support of the Natural Sciences and Engineering Research Council of Canada are appreciated. The hospitality of Monash University, where the manuscript of this article was drafted, and Professor Alan Bond is gratefully acknowledged. Received for review September 13, 1996.X
February
22,
1996.
Accepted
AC9601730 X
Abstract published in Advance ACS Abstracts, October 15, 1996.
Analytical Chemistry, Vol. 68, No. 23, December 1, 1996
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