J. Phys. Chem. B 1999, 103, 1699-1704
1699
Principle of Unchanging Total Concentration and Its Implications for Modeling Unsupported Transient Voltammetry Keith B. Oldham Department of Chemistry, Trent UniVersity, Peterborough ON, Canada K9J 7B8
Stephen W. Feldberg* Department of Applied Science, BrookhaVen National Laboratory, Upton, New York 11973 ReceiVed: September 21, 1998; In Final Form: December 26, 1998
The advantages and disadvantages of electrochemistry with little or no added supporting electrolyte are reviewed. Analysis of such systems is complicated by the increased importance of the migrational component of transport. Some computations and analyses can be simplified by invoking the principle that for any electrochemical experiment where the diffusiVities of all the solute species do not differ significantly from a common Value, the total solute concentration remains uniform and constant. This principle, which applies to both transient and steady-state regimes, holds independently of the cell and electrode geometries and is not compromised by migration and/or convection. The only constraint, additional to the assumption of uniform diffusivities, is the obvious one: that the electron-transfer reaction, and any homogeneous reactions that may participate, do not themselves perturb the number of solute particles (molecules and ions) present. Some consequences of the principle are explored, especially relating to three-ion systems. Using the principle, a framework of procedures is derived which may be followed to obtain a solution to many transient voltammetric problems at a macroelectrode when supporting electrolyte is absent.
Introduction The very earliest voltammetric experiments1,2 employed pure water as solvent, but over the succeeding 75 years it has become customary to use excess supporting electrolyte. Among the reasons for adding copious inert electrolyte are the following: 1. By increasing the cell’s electrical conductivity, the ohmic polarization of the electrode is diminished. 2. When excess supporting electrolyte is present, migration is an unimportant transport mechanism, massively aiding the modeling of the voltammetric experiment. 3. The thickness of the diffuse double layer at the working electrode, and therefore the potential drop across this layer (the φ2 potential), are reduced and become less influenced by variations in the concentrations of the electroactive species, thereby validating background correction. Any “Frumkin effect” is also minimized. 4. Gradients of concentration of electroactive species occur in the vicinity of the working electrode. In the absence of supporting electrolyte, these concentration gradients can give rise to appreciable density gradients and so engender unwanted convection. When other solution components are present in large excess, the dependence of the solution density on the concentration of a minor component is greatly suppressed. 5. One or both of the electroactive species (reactant or product) must be ionic. The activity coefficients of these electroactive ions, in the absence of excess electrolyte, is a function of their concentration and therefore change unwelcomely with time and/or distance during a voltammetric experiment. A high ionic strength enforces a constant activity coefficient. These are persuasive advantages. However, there is also a downside to using a solution that contains large concentrations
of ionic species other than the electroreactant and electroproduct. The disadvantages include the following: 1. It becomes difficult to compare electrochemical evidence from solutions of high ionic strength with that from other studies, such as solution spectroscopies, which are customarily conducted without added electrolyte. 2. Many organic solvents of low permittivity are unable to dissolve large concentrations of electrolytes. Even when special electrolytes are found that do dissolve significantly in such solvents, it is likely that they often do so by forming ion pairs. The desired effect of increasing the ionic strength is not then achieved. 3. Some analytical applications of voltammetry, such as the in situ analysis of river waters, preclude the addition of salts. 4. The speciation of the electroreactant is always in doubt when other solutes are present in excess. In the case of ionic electroreactants, ion pairs or even complex ions may be formed. This prevents or impedes the reliable measurement of data of thermodynamic significance (standard potentials, for example), as well as such other data as diffusivities and rate constants. 5. Impurities are inevitably added along with the supporting electrolyte. 6. The need to have electrolyte present at much higher concentrations imposes an unwelcome upper limit on the concentrations of electroactive species. The advantages of working with little or no added supporting electrolyte have been recognized during the past decade and have led to a spate of theoretical and experimental publications dealing with voltammetry in the absence of excess support, often in the context of microelectrodes 3-11 which can greatly diminish the ohmic polarization, but also at macroelectrodes12,13 where
10.1021/jp9837939 CCC: $18.00 © 1999 American Chemical Society Published on Web 02/19/1999
1700 J. Phys. Chem. B, Vol. 103, No. 10, 1999 positive feedback may be required to diminish the distortions introduced by uncompensated resistance.14 When the electroreactant is an ion, there is no inherent reason why supporting electrolyte is required in voltammetric experiments, because the electroactive ion and its counterion will always provide the means whereby current may be carried through the cell, though the increased resistance in the unsupported case may introduce unwelcome distortions. The product of the electrode reaction will also contribute to the solution conductivity, if it is an ion. However, when the electroreactant is neutral, a modicum of added electrolyte is essential; otherwise no significant current can flow through the solutionssuch cases are excluded from consideration here. A special circumstance, rare in practice, occurs when the charges of reactant and product of the electrode reaction have opposite signs. In such “sign reversal” cases, unusual and interesting complications arise8,9,11 which we will discuss briefly. Our primary concern, however, will be with the oxidation or reduction of an anion or a cation to generate another ion of the same sign as the reactant, but with a smaller or a larger charge magnitude. After the generality of the principle of unchanging total concentration is established, attention will be directed to classical voltammetric situations. Thus, in later sections of this article, we shall consider the working electrode to be a planar macroelectrode, large enough that edge effects may be ignored, and fronted by a planar diffusion field of effectively semiinfinite extent. Principle of Unchanging Total Concentration Earlier literature8-11has noted that, under certain steady-state voltammetric conditions with the diffusivities (diffusion coefficients) of all species identical, the total concentration of all (charged and uncharged, electroactive, and electropassive) solute species is uniform throughout the solution in contact with the working electrode, even with little or no added electrolyte. Amatore et al.8 utilized this constraint and recently Jaworski et al.13 simulated chronoamperometric responses under similar conditions, obtaining results which indicated that the total concentration of solute was again time independent. Neither of these research groups commented on the generality of this interesting phenomenon. In the present work it will be established that this principle of unchanging total concentration is considerably more general than might have been inferred from the previous studies; it is valid for any cell geometry and is not compromised by migration or convection, provided only that chemical or electrochemical reactions do not themselves perturb solute numbers. In describing this principle, we use the adjective “unchanging” to emphasize that the total concentration is not only uniform (i.e., unchanging in space), but also constant (i.e., unchanging in time). Aside from the requirement that diffusivities for all species be indistinguishable, the only constraint is that the electron-transfer reaction, as well as any homogeneous reactions that there might be, must conserve the number of dissolved particles (molecules and ions) in the solution phase. This will usually preclude any adsorption, plating, dissolution, aggregation, or disaggregation processes being involved in the electrode reaction. In the following, we shall prove the principle of unchanging total concentration for any geometry when there is transport by diffusion, migration, and convection. When the Nernst-Einstein relationship is used to replace ionic mobilities by their corresponding diffusivities, the general three-dimensional flux equation incorporating diffusion, migra-
Oldham and Feldberg tion and convection is
F fj ) -Dj∇cj - zjcjDj ∇Φ + Wcj RT
(1)
where fj is the flux density(a vector) of the jth solute species and W is the solution velocity (also a vector). The symbols Dj, cj, and zj are the diffusivity, concentration, and charge number (all scalars) of the jth species, Φ is the electrostatic potential (also a scalar), F is Faraday’s constant, R is the gas constant, T is the thermodynamic temperature, and ∇ is the operator
∇)
∂ ∂ ∂ + + ∂x ∂y ∂z
(2)
The unit vectors i, j, k associated with coordinates x, y, and z are implicit in our notation. On summing eq 1 over all solute species, we obtain
F
∑j fj ) -∑j Dj∇cj - RT∇Φ∑j zjDjcj + W ∑j cj
(3)
where the summation is for j ) 1 to J, the total number of solute species in the system. Now, if all J species share the same diffusivity D, the electroneutrality principle
∑j zjcj ) 0
(4)
causes the penultimate term to vanish and eq 3 to become
∑j fj ) -D∑j ∇cj + W∑j cj
(5)
Let us now apply the ∇ operator to each term in this equation then, since ∇‚W ) 0 for an incompressible fluid,15 we can write
∂cj
∑j ∂t ) ∑j ∇fj ) -D∑j ∇2cj + W∑∇cj
(6)
where the first equality is a statement of the principle of species conservation. The ∇2 symbol represents
∇2 )
∂2 ∂2 ∂2 + 2+ 2 2 ∂x ∂y ∂z
(7)
and is the Laplacian operator. Equation 6 is valid only in the absence of homogeneous reactions. If, however, a homogeneous reaction involving the jth species occurs, then this reaction will induce a rate of change of sjr in the concentration of this species. Here r is the net rate (forward rate minus backward rate) of the reaction at the location and time in question, while sj is the stoichiometric coefficient of the jth species in the chemical equation for the homogeneous reaction, being positive if the species is a product of the reaction, and negative if it is a reactant. Of course, the expression sjr still applies if the jth species is not involved in the reaction, but sj is then zero. Accordingly, if a homogeneous reaction occurs, eq 6 becomes modified to
∑j cj
∂
∂t
∑j ∇2cj + W∑j ∇cj + r∑j sj
) -D
(8)
If the homogeneous reaction preserves the total number of solute
Principle of Unchanging Total Concentration
J. Phys. Chem. B, Vol. 103, No. 10, 1999 1701
particles, as we henceforth assume, the final term vanishes and there remains
∂
∑j cj ∂t
∑j ∇2cj + W∑j ∇cj
) -D
(9)
We can interpret this equation verbally as an assertion that any change with time in the total solute concentration arises as a result of either a gradient in the concentration of one or more species (∇cj * 0) or from curvature of some concentration profile (∇2cj * 0). Topologically, a voltammetric cell consists of the threedimensional electrolyte solution enclosed by two-dimensional “walls”. In almost all electrochemical experiments, the initial state is one of uniform concentration throughout the electrolyte solution. Therefore, both right-hand terms in eq 9 are zero before the experiment commences, even if the solution is in motion, and therefore the left-hand term is, of course, also zero. Nothing can enter the system to affect the solution, other than through the walls. One of these walls is the working electrode surface. A second is the counter electrode, here assumed to be far enough away16 from the working electrode that reactions at its surface are inconsequential. Other walls, including the gas/ solution interface commonly present, are electrically and chemically insulating. There is often a reference electrode, but this is as innocuous as the insulating walls in the present context since, in principle, the reference electrode senses the potential of the solution without perturbing the solution in any way. The walls cannot perturb the preexisting uniformity of the total concentration through the final term in eq 9. If this is not self-evident, note that all components of a liquid’s velocity are zero at a surface. Thus V ) 0 at the walls, nullifying the final term in eq 9 for all of the solution’s bounding surfaces. If ∑cj is to be perturbed from its initial uniformity, then we must look to the first right-hand term as the cause. Because they are chemically impermeable, there are no fluxes normal to the boundary surfaces, even after the experiment has commenced, other than at the working electrode. Let y be the coordinate locally normal to the surface of the working electrode, with y ) 0 being the surface itself. Electropassive species contribute no flux across the electrode, but every electroactive species experiences a flux across the y ) 0 surface when the electrode is working:
(fj,y)y)0 * 0
j ) electroactive
(10)
where fj,y denotes the flux density of the jth species in the direction of the y-coordinate, being positive if motion is away from the electrode. However, if the number of solute particles (ions and molecules) is conserved by the electrode reaction, these flux densities will all cancel.
∑j fj,y)y)0 ) 0
(
(11)
In this case, the first right-hand term, like the second, lacks the capability of changing the total concentrations from its initial value. We conclude that when (1) all solute species share the same diffusivity, (2) the electrode reaction conserves the number of solute particles, and (3) homogeneous reactions, if any, also conserve solute particle numbers, then no mechanism exists by which a change may be effected in the total solute concentration. This unchanging total concentration is, of course, the uniform
total concentration of all solutes in the original cell solution. This proves the principle of unchanging total concentration. It will be derived again below, in a more restrictive setting, en route to the proof of another useful principle. Two Principles in Planar Semiinfinite Geometry Consider an electrode reaction17 in which the charge number of some chemical entity X changes
Xz1(soln) + (z1 - z2)e- a Xz2(soln)
(12)
without any disruption of the chemical integrity of X. This is the simplest example of conservation of solute particle number. The product species may, or may not, be present in solution prior to the experiment. We shall invariably associate subscripts 1 and 2 with the reactant and product, respectively. Thus c1 will denote the concentration of the electroreactant, a variable in both space and time, and z2 will denote the charge number of the electroproduct, an invariant integer. In addition to the two electroactive species, the solution may contain any number of other solutes, to which we allocate the subscripts 3, 4, ..., J. These may be uncharged or charged with no restriction on the sign or magnitude of the charge, save for the constraint of electroneutrality (eq 4). We define the initial concentration of the jth solute as cjb and the total initial solute concentration as cb. In this section, the electrode is assumed planar and of large enough area that edge effects can be ignored. Thus, transport will occur only along a linear dimension that may be identified with the x-axis. Because we ignore convection here, the transport equation
∂cj zjFDcj ∂φ ∂x RT ∂x
fj ) -D
(13)
contains only a diffusive and a migratory term. This is a simplified version of eq 3, with the diffusivity D assumed common to all J species. Summation over all solutes leads to
∂
∑j fj ) -D∂x∑j cj
(14)
because electroneutrality causes the final term to disappear. Next, differentiate eq 14 with respect to x and apply the conservation relationship ∂fj/∂x ) -∂cj/∂t which applies generally in planar geometry. This leads to
∂
∑j
∂x
∂2 cj ) D ∂x2
∑j cj
(15)
which shows that Fick’s second law applies to the total solute concentration. To solve this differential equation, three boundary conditions are needed. In the vast majority of voltammetric experiments, the concentrations cj of each of the J solutes is uniform initially, as therefore is their sum
∑j cj ) ∑j cjb ) cb
at t ) 0 and all x
(16)
This identity becomes the initial boundary condition. Moreover, the solution uniformity is generally maintained in the bulk of the solution, which provides a second boundary condition.
∑j cj ) ∑j cjb ) cb
as x f ∞ and all t
(17)
1702 J. Phys. Chem. B, Vol. 103, No. 10, 1999
Oldham and Feldberg
The third boundary condition comes from taking the x ) 0 case of eq 14. Because, at the electrode surface, f1 ) -f2 and fj ) 0 for 3 e j e J, summation of eq 13 over all J species leads to
∂
∑j cj ) 0
at x ) 0 and all t
∂x
(18)
A solution that satisfies all of eqs 16-19 is
∑j cj ) ∑j cjb ) cb
at all x and all t
(19)
Therefore, by the uniqueness property of differential equations, result 19 is the only valid solution. This paragraph has established the principle of unchanging total concentration for the special case of planar semiinfinite diffusion/migration. However, a second valuable principle can be established for this circumstance. Again sum eq 14 over all J species, but multiply each term by zj prior to the summation. Electroneutrality now causes the second term to vanish, so that
∑j zjfj )
-FD ∂φ RT ∂x
∑j zj2cj
(20)
three solutes is uncharged are rather trivial and the voltammetry in such cases has been thoroughly explored in the literature.8,9,11 The remainder of this article will therefore be confined to discussion of cases in which the three solutes are all ions. Within the three-ion class there are two broad subclasses18 according as the reactant and product share the same sign (z1z2 > 0) or have opposite signs (z1z2 < 0). These have been termed11 the “sign retention” and “sign reversal” cases. The latter case, exemplified in eq 24 with an additional electropassive ion, is rare or nonexistent in experimental practice. Voltammetry in Three-ion Systems The electroneutrality principle (eq 4) and the principle of unchanging total concentration (eq 19) place severe constraints on the individual ion concentrations when only three ions are present. Thus, if the concentration of any one of the three ions is specified, those of the other two are automatically prescribed. For example, if the reactant ion is given a zero concentration at the electrode surface, c1s ) 0, as might occur under extreme concentration polarization, the other two ions are predicted to acquire the concentrations
c2s
The right-hand term in eq 20 is proportional to the ionic strength µ defined as
∑j zj cj
µ ) /2 1
2
(21)
while the left-hand term in eq 20 is the molar flux density of charge, equal to i/F, where i is the current density. Thus, the summation ultimately leads to
i)
-2F2Dµ ∂φ RT ∂x
(22)
Recognize that the ionic strength and the potential gradient are each functions of x and t, whereas the current density depends on t alone. Thus the spatial variations of µ and ∂φ/∂x must be such that the x-dependence vanishes in their product:
∂φ -RTi * f(x) ) ∂x 2F2D
µ
(23)
This simple principlesthe uniformity of the product of the ionic strength and the field strengthswill prove useful later in this article. It holds for systems that contain solutes with a common diffusivity and where the electrode reaction preserves particle number, but only for planar geometries. Special Cases Up to this point in the article, our findings have been applicable to electrolyte solutions of any composition. We now direct attention to applications of the principle of unchanging total concentration in specific simple instances. Apart from the bizarre circumstance in which a chemical entity X can exist as both a cation and an anion and undergo a reaction such as
X2+(soln) + 3e- f X-(soln)
(24)
at the junction between an electrode and a solution of the salt XX2, the simplest meaningful applications of the principle of unchanging total concentration are to systems of three solutes, at least two of which are ionic. The cases in which one of the
z3cb ) z3 - z2
(25)
-z2cb z3 - z2
(26)
c3s )
at the electrode surface. Note our use of the superscript s to denote conditions at the electrode surface. Now, for the concentrations given by eqs 25 and 26 to be both positive and finite, as they must be in practice, the terms z3, (z3 - z2) and -z2 must all have the same sign, which can occur only if the product and electropassive ions have opposite signs (z2z3 < 0). Unless the product is initially present at unusually large concentrations, the electropassive species will be the reactant’s counterion with a charge number of sign opposite to that of z1. Thus, usually, extreme concentration polarization of the reactant is possible only if the charge number z2 of the product species has the same sign as z1, the charge number of the reactant. That is, concentration polarization of the reactant in a three-ion system is experienced only when there is sign retention, whenever the electropassive ion is a counterion of the reactant. What limit is encountered in sign-reversal processes? In such cases, it is concentration polarization with respect to the counterion that is limiting. For, when c3s ) 0, both of the other surface concentrations
z2cb z2 - z1
(27)
-z1cb ) z2 - z1
(28)
c1s )
c2s
are acceptably positive only when z1 and z2 differ in sign (z1z2 < 0). Despite the surface concentrations of the reactant and product being “locked into” the constant values given in eqs 27 and 28, this type of concentration polarization does not correspond to a potential-insensitive limiting current. See ref 11 for further details. Ionic Strength as a Dependent Variable Applied to the three-ion system, the principle of unchanging total concentration, the electroneutrality principle, and the
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J. Phys. Chem. B, Vol. 103, No. 10, 1999 1703
definition of ionic strength
b
3
cj ) cb ∑ j)1
(29)
3
zjcj ) 0 ∑ j)1
(30)
3
zj cj ) 2µ ∑ j)1 2
(31)
constitute a trio of similar linear equations that can be solved simultaneously. For each concentration one finds
2µ + zk(zj - zk)(zj - zm)zmc
µ-
b
cj )
(zj - zk)(zj - zm)
zkzmµb z1z3
)2 (32) (zj - zk)(zj - zm)
where subscripts k and m identify the two ions other than j. As before, the superscripts b and s respectively indicate conditions in the bulk solution and at the electrode surface, x ) 0. The second equality above is valid if the product ion is absent from the bulk solution, which we henceforth assume. Equation 32 establishes that each ion’s concentration is linearly related to the ionic strength. For example, the counterion’s concentration is b
µ-
z2µ z3
c3 ) 2 (z3 - z1)(z3 - z2)
(33)
The existence of these linear relationships makes µ an ideal candidate for the working variable, dependent on both x and t, in voltammetric studies without support, playing the same role that the concentration of the electroreactant does in fully supported voltammetry. Not only can the spatial and temporal variation of each of the three concentrations be expressed in terms of the ionic strength profile, but so also can the three flux densities. Of course, this is only possible when there are no more than three ions. If the cj terms in eq 14 are replaced via eq 32 and the potential gradient ∂φ/∂x is likewise replaced via eq 23, the formula b zji ∂µ z2µ i - 2D F ∂x Fµ fj ) (zj - zk)(zj - zm)
(34)
results. Thus the local flux of each species reflects the local values of both the gradient of the ionic strength ∂µ/∂x and the ratio µb/µ. Choosing x ) 0 and j ) 3 leads to the formula s b i z3µ - z2µ ∂µ s ) ∂x 2DF µs
( )
∂2µ z2µ i ∂µ ∂c2 ∂x2 Fµ2 ∂x ) ∂t (z2 - z1)(z2 - z3) 2D
(35)
because, for the electropassive ion, f3s ) 0. This provides a relationship between the ionic strength at the surface of the electrode, and its gradient there, which is valid for any threeion system under the conditions assumed. Partial Differential Equation Obeyed by the Three-ion System The conservation identity ∂c2/∂t ) -∂f2/∂x may be combined with the x-derivative of the j ) 2 instance of eq 34 to produce
(36)
This result can now by converted, by use of the j ) 2 case of eq 32, into the partial differential equation b ∂µ ∂2µ z2µ i ∂µ )D 2∂t ∂x 2Fµ2 ∂x
(37)
which relates the dependent variable µ to the independent variables x and t. Note the presence in this equation of the current density i, which is a function of t, but not of x. This undesirable complication can be avoided by using relationship 35 to remove the dependence on current. Unfortunately, this is at the expense of further complicating the partial differential equation, which then becomes s b
z2µ µ ∂µ s ∂µ 1 ∂µ ∂2µ ) D ∂t ∂x2 (z µs - z µb)µ2 ∂x ∂x 3 2
( )
(38)
The time dependence has been transferred into the µs and (∂µ/ ∂x)s terms. Comparison with Supported Voltammetry The foregoing section has revealed, for the unsupported threeion case, certain mathematical relationships between the variables that are analogous to relationships in the classical, fully supported, versions of voltammetry. Here we shall review the standard paradigm19 used in predicting the outcome of a fully supported voltammetric experiment, and cite the corresponding relationships when the experiment is unsupported. No particular voltammetric technique (chronopotentiometry or cyclic voltammetry, etc.) is addressed. Instead, the principles common to all are discussed. The comparison is made by way of Table 1. The starting point of most derivations is a partial differential equation relating a dependent variable, usually the concentration c1 of the electroreactant in a supported experiment, to the independent spatial and temporal variables, x and t. As shown in Table 1, this differential equation is Fick’s second law classically. In the unsupported three-solute case, this becomes replaced by eq 37, with the ionic strength as the dependent variable. However, with minor changes, any of the three concentrations could be used, instead of µ, as the dependent variable, because of the linear relationships between µ and each of the cj concentrations. Note that the two differential equations in Table 1 are equivalent when the product is uncharged, i.e., when z2 ) 0, but otherwise the unsupported case obeys a considerably more complicated differential equation. Especially troublesome is the denominatorial µ2 factor, because this destroys the equation’s linearity. Nonlinear differential equations, even of the “ordinary” kind, are notoriously difficult to solve20 and it seems unlikely that any useful fully analytic solution will be found to eq 16 (except in the degenerate z2 ) 0 case, which will not be discussed further) even when the boundary conditions are simple. Numerical solution, however, is feasible. Three conditions must be specified in order to solve either of the tabulated differential equations. Two of these, the initial condition and the remote boundary condition, are essentially identical in the fully supported and unsupported cases, as Table 1 shows. The third boundary condition is provided by the
1704 J. Phys. Chem. B, Vol. 103, No. 10, 1999
Oldham and Feldberg
TABLE 1: Comparison of Equations Describing Supported and Unsupported Voltammetry full support partial differential equation
no support (three-ion case)
2
b ∂ µ z2µ i ∂µ ∂µ )D 2∂t ∂x 2Fµ2 ∂x
∂ c1 ∂c1 )D 2 ∂t ∂x
2
initial condition remote boundary condition link to the current density
c1 ) c1b at t ) 0 c1 f c1b as x f ∞
µ ) µb at t ) 0 µ f µb as x f ∞
∂c1 i ) (z2 - z1)FD ∂x
i)
link to the electrode potential
exp{(z1 - z2)F(E - E°′)/RT} ) c1s/c2s ) c1s/(c1b - c1s)
( )
electrical program, which usually relates either the current or the potential to time. The last step in a standard voltammetric derivation uses the link that has not been employed as a boundary condition to predict the voltammetric response. For example, in a controlledcurrent experiment, the solution to the differential equation is found and used to predict the concentration of the reactant (or the ionic strength of the solution) at the electrode surface, which in turn is used to predict how the electrode potential changes with time, as in the final line of Table 1. In principle, the information contained in the third column of Table 1 is adequate to predict the outcome of any reversible voltammetric experiment that satisfies the assumptions incorporated in the present model. In practice, however, it is likely that numerical, rather than analytical, implementation will be needed to make quantitative predictions. The equations in the final row in Table 1 list the electrode potential E assuming accurate correction for ohmic polarization (“IR drop”). In the absence of any correction for this polarization, E will differ from the applied potential by a term of magnitude ∆φu, representing the “uncompensated resistance” ∆φu/Ai. The magnitude of this correction can be found once the ionic strength is known as a function of distance. The ohmic polarization ∆φ equals the potential difference between the solution layer adjacent to the electrode and that at a point occupied by the inlet to the reference electrode. If xref represents the location of this inlet, then
∆φu )
∫0x
ref
∂φ -RTi dx ) 2 ∂x FD
∫0x
ref
dx µ
(39)
Because the ionic strength profile µ(x) is itself dependent on the current, this “ohmic” polarization will not obey Ohm’s law and ∆φu may be either greater or less in magnitude than ∆φu,static, the value of ∆φu when µ ) µb then given by eq 39 as
∆φu,static )
-RTixref F2Dµb
(40)
2FDµs ∂µ z3µs - z2µb ∂x
s
( )
exp{(z1 - z2)F(E - E°′)/RT} ) c1s/c2s ) [(z2 - z3)(z1µs - z2µb)]/[z1(z1 - z3)(µb - µs)]
Note that ∆φu > ∆φu,static when |z2| < |z1| whereas ∆φu < ∆φu,static when |z2| > |z1|. See Reference 14 for a more complete discussion of this phenomenon. Acknowledgment. We thank Professor Alan Bond for motivating this research and for providing an opportunity for us to work together on this topic at Monash University. K.B.O. gratefully acknowledges the financial support of the Natural Sciences and Engineering Research Council of Canada; S.W.F. thanks the U.S. Department of Energy, Contract No. DE-AC0298CH10886, for its support. References and Notes (1) Heyrovsky´, J. Chem. Listy 1922, 16, 256. (2) Heyrovsky´, J. Philos. Mag. 1923, 45, 303. (3) Norton, J. D.; Benson, W. E.; White, H. S.; Pendley, B. D.; Abruna, H. D. Anal. Chem. 1991, 63, 1909. (4) Amatore, C.; Bartelt, J.; Deakin, M. R.; Wightman, R. M. J. Electroanal. Chem. 1988, 256, 255. (5) Cooper, J. B.; Bond, A. M.; Oldham, K. B. J. Electroanal. Chem. 1992, 331, 877. (6) Norton, J. D.; White, H. S.; Feldberg, S. W. J. Phys. Chem. 1990, 94, 6772. (7) Daniele, S.; Corbetta, M.; Baldo, M. A.; Bragato, C. J. Electroanal. Chem. 1996, 407, 146, and additional citations therein. (8) Amatore, C.; Deakin, M. R.; Wightman, R. M. J. Electroanal. Chem. 1987, 225, 49. (9) Myland, J. C.; Oldham, K. B. J. Electroanal. Chem. 1993, 347, 49. (10) Palys, M. J.; Stojek, Z.; Bos, M.; van der Linden, W. E. J. Electroanal. Chem. 1995, 381, 105. (11) Oldham, K. B. J. Electroanal. Chem. 337 1992 91. (12) Nicholson, R. S. Anal. Chem. 1965, 37, 667; ibid 1965, 37, 1351. (13) (a) Jaworski, A.; Donten, M.; Stojek, Z. J. Electroanal. Chem. 1996, 407, 75. (b) Jaworski, A.; Donten, M.; Stojek, Z. J. Electroanal. Chem. 1997, 420, 307 (a correction to 13a). (14) Bond, A. M.; Feldberg, S. W. J. Phys. Chem. B 1998, 102, 9966. (15) Levich, V. G. Physicochemical Hydrodynamics; Prentice-Hall: Englewood Cliffs, NJ, 1962; p1 ff. (16) If the counter electrode reaction satisfies the criteria of common diffusivities and conservation of particle number, then its location is irrelevant. (17) When |z1 - z2| > 1 we assume that virtually no intermediate species are produced by the sequential one electron transfers. (18) More refined classifications have been proposed in refs 9 and 10. (19) For many examples see: Macdonald, D. D. Transient Techniques in Electrochemistry; Plenum: New York, 1977. (20) Murphy, G. M. Ordinary differential equations and their solutions; Van Nostrand: Princeton, NJ, 1960.
