9966
J. Phys. Chem. B 1998, 102, 9966-9974
Analysis of Simulated Reversible Cyclic Voltammetric Responses for a Charged Redox Species in the Absence of Added Electrolyte Alan M. Bond† Department of Chemistry, Monash UniVersity, Clayton, Victoria 3168, Australia
Stephen W. Feldberg*,† Department of Applied Science, BrookhaVen National Laboratory, Upton, New York 11973 ReceiVed: July 2, 1998; In Final Form: September 11, 1998
We present an analysis of simulated cyclic voltammetric responses of a dissolved redox salt with no added supporting electrolyte. The redox system of interest is a reversible one electron transfer: AzA + e ) BzA-1. Only the charged, oxidized species AzA and counterion XzX are initially present in solution (analogous systems where the reduced species and counterion are initially present are not explicitly discussed); their relative concentrations are dictated by zA, zX, and the constraint of electroneutrality (systems where zA ) 0 or 1 are not considered). Cyclic voltammetric responses are simulated assuming Nernst-Planck transport, and electroneutrality with perfect IR compensation (easily simulated but experimentally unattainable) and with partial IR compensation. Simulations assuming perfect compensation allow us to elucidate the change in the resistance, Rref, between the working and reference electrodes effected by changes in the depletion layer during the course of the cyclic voltammetric perturbation. Simulation of systems with (more realistic) partial compensation leads to an empirical linear equation which correlates E° with cathodic peak potential, Epc: Epc ) E° + (RT/F)[bz - 1.347((F/RT)Ipc(1 - γ)R0ref)0.929], where Ipc is the cathodic peak current, R0ref is the resistance of the bulk solution (prior to the CV perturbation) between the working and reference electrodes, and γ is the fraction of that resistance compensated (by positive feedback) during the perturbation. The constant bz is a function of zA (range -1000 to +1000) and zX ((1, (2) with weak dependence on DX/DA (range 0.5-2) when DA ) DB (DX, DA, and DB are the diffusion coefficients of species X, A, and B, respectively). The numerical parameters 1.347 and 0.929 remain constant for the ranges of charges and diffusion coefficients studied.
Introduction Traditionally, electroanalytical experiments are carried out using excess inert electrolyte to minimize IR distortion of the response, suppress migration currents, and establish a welldefined double layer as well as minimize the φ2 potential across the diffuse double layer. There are, however, a number of reasons why one might wish to carry out such experiments with little or no added electrolyte (when the species initially present in solution are a charged redox moiety and its counterion): 1. To obtain thermodynamic data that may be validly compared with those obtained from nonelectrochemical analyses where no electrolyte was added. 2. To obtain thermodynamic data under conditions more closely approximating conditions of infinite dilution. 3. To minimize introduction of impurities which might be associated with an inert electrolyte salt. 4. To minimize difficulties possibly associated with the preparation of both components of a redox couple required for potentiometric measurements. 5. To minimize chemical complications associated with added electrolyte, e.g., ion pairing or complexation with redox species, or ion-induced adsorption. In the present work we focus on analyses of cyclic voltammetric responses at a (planar) macroelectrode with semiinfinite * To whom correspondence should be directed. † The authors convey their best wishes to Allen Bard on his 65th birthday.
linear diffusion. Cyclic voltammetry carried out with excess supporting electrolyte is an excellent tool for confirming the stability of electrode products as well as for evaluating the standard potential of the redox couple of interest.1 However, the resistance between the working electrode and the reference electrode, Rref, can produce a large potential drop, IRref (where I is the current) such that Eeff, the effective potential across the double layer of the working electrode, is significantly different from ECV, the potential specified by the cyclic voltammetric protocol,2,3 i.e.:
Eeff ) ECV - IRref
(1)
Note: a glossary of symbols is at the end of the text. Good electrochemical practice dictates that the reference electrode should be placed as close as possible to the working electrode, thereby reducing Rref (and therefore IRref) as much as possible. With a potentiostat, three-electrode system, and positive feedback,4 the effect of Rref can be further diminished. Positive feedback increments ECV by a voltage component that is proportional to the current, I, and R0ref, the value of Rref measured prior to the CV perturbation and assumed to remain constant during the course of the CV. Thus, instead of eq 1 we can write
Eeff ) ECV - IRref + γIR0ref
10.1021/jp9828437 CCC: $15.00 © 1998 American Chemical Society Published on Web 11/05/1998
(2)
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where 0 e γ e 1. When γ ) 1, one has perfect compensation as long as Rref does not change. Then Eeff ) ECV but the system can also become unstable when γ g 1 and in practice γ < 1. An interesting and important complication arises in systems without added supporting electrolyte: variations in the composition of the depletion layer during the perturbation will effect a change in Rref from the initial value, R0ref, during the course of the CV. The role of that change, ∆Rref, could be critical. If eq 2 is rewritten as
Eeff ) ECV - I(R0ref + ∆Rref) + γIR0ref ) ECV - (1 - γ)IR0ref - I∆Rref (3) and if we define
∆γ ) ∆Rref /R0ref
(4)
Eeff ) ECV - (1 - γ + ∆γ)IR0ref
(5)
eq 3 becomes
and it is clear that a negative ∆γ produced by a negative ∆Reff (eq 4) could effect overcompensation and instability. Thus, determining the sign and magnitude of ∆Rref is an important objective of this work. When we use the term “perfect compensation” we mean that the entire resistance is compensated at all timessthis includes any changes in resistance effected by changes in the composition of the depletion layer. When we say “partial compensation” we mean that γ is fixed and compensation is based on a constant value of IR0ref with no adjustment for changes in resistance during the course of the CV. Note that even when γ ) 1 the system is still only partially compensated since no adjustments are made for the changes in resistance effected by changes in the composition of the depletion layer. In the present analysis we ignore capacitive currents, a reasonable assumption for many cases of interest. Since uncompensated IRref can effect nonlinear scan rates (see eq 1) there is no simple way to execute a background correction. In general, the ratio of the capacitive current to the faradaic current will be proportional to the double layer capacitance of the electrode and the square route of the scan rate and inversely proportional to the concentration of the analyte. When the ratio is too large, e.g., >0.05, the experimental parameters must be appropriately modified. Perhaps a better approach would be to introduce a capacitive current component into the simulation. The systems of interest in the present study initially comprise only the solvent and an ionic redox moiety with its counterion. The electrochemistry is described by the single electron transfer reaction:5
AzA + e ) BzA-1
(6)
where AzA or BzA-1 is the charged redox species initially present in solution. We will consider a wide range of values for zA (a number of cluster complexes of interest to us, e.g. [S2Mo18O62]4-, are highly charged6). However, we do not consider zA ) 0 and zA ) 1: for zA ) 0 no ions would be initially present and the resistance of the bulk solution, in principle, would be infinite;7 for zA ) 1 the product species is uncharged and during reduction the concentration of ions in the depletion layer approaches zero at the electrode surface producing very high local resistance. For oxidations the equivalent excluded charges are zA ) 0 and zA ) -1.
We will assume that the Debye lengths are very small compared to the dimensions of the depletion layers likely to be involved in the cyclic voltammetric experiment. This will allow us to invoke the electroneutrality constraint, greatly simplifying the analysis. A body of work (see e.g., refs 8 and 9 and references therein) addresses just what happens when that condition does not obtain and is not relevant here. Numerous works have considered the effects of uncompensated resistance on the effect of electrochemical responses. Oldham10 has presented a detailed analysis for the steady-state behavior at a hemispherical electrode of a completely ionized redox salt producing ions AzA and Xzx with the relative concentrations of the ions dictated by electroneutrality constraints and the composition of the salt. Bento et al.11 also investigated steady-state currents at microelectrodes focusing on the reduction or oxidation of a neutral species with low electrolyte/analyte ratios. They also suggest strategies for minimizing the effects of junction potentials which are a source of serious experimental ambiguities in the quest for thermodynamic data. More recently, Bento et al.12 have suggested that hydrodynamic effects play a critical role in microelectrode phenomena. Jaworski et al.13,14 explored the effect of migration on chronamperometric responses at spherical electrodes. Nicholson15 was probably the first to address the problem of uncompensated resistance in cyclic voltammetry, but only small deviations, |IpRref| e ∼0.1 V were considered. In the present paper, we explore effects of uncompensated resistances as large as ∼0.3 V (at T ) 298.2 K). Theory The fundamental assumptions for our computations are that transport is described by the Nernst-Planck equation coupled with the electroneutrality constraint. The Nernst-Planck formalism is
fj ) -Dj
dcj F + ΕDjcjzj dx RT
(7)
where fj, cj, and Ε are dependent upon time (t) and distance (x), and correspond to the flux and concentration of the jth species and electric field, respectively; Dj and zj are the diffusion coefficient and charge of the jth species. Invoking electroneutrality, i.e.
∑j cjzj ) 0
(8)
greatly simplifies the analysis.16 We assume a planar electrode with semiinfinite linear diffusion (no edge effects). The reference electrode is assumed to be well outside the depletion layer.17 Eeff is assumed to be exactly equal to ECV, easy to do in a simulation, not so easy to do in an experiment. However, the simulation does allow us to estimate just what compensation would be required to achieve such a goal. The resistance, Rref, between the working and reference electrodes cannot be constant since the concentrations of all species, and therefore the resistance, will be changing within the depletion layer. It is important to estimate the magnitude of that effect since experimental compensation is almost always based on the presumption that Rref is constant ()R0ref) throughout the time course of the experiment. We have noted that a decrease in Rref (negative ∆Rref) during a CV with nearly full compensation (γ close to unity) could induce instability. However, we will show that I∆Rref for many if not most cyclic voltammetric conditions of interest will be ignorably small. We then explore
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the behavior of systems with varying levels of compensation and propose an approach to estimate the value of E° when compensation is incomplete. Our computations and discussions focus on systems where zA is the charge of the redox species initially present in solution and the reduction product has the charge zA - 1 (see eq 6). The analogous results for oxidations, where the starting species is the reduced redox species, are easily intuited and will not be explicitly computed. We reemphasize here that our analyses presume that there is no complexation and no ion pairing, a condition which becomes increasingly unlikely with increasing |zA|. Some Preliminary Analysis. Some simple analysis will allow us to estimate the magnitude of I∆Rref and to show that when there is perfect compensation it is independent of scan rate, initial concentration, and the diffusion coefficient (once the relative values of the diffusion coefficients have been set). At any given time (i.e., for any value of Eeff)
|I∆Rref| ≈ |i| × area × σbulk(δ/area)
(9)
where I is the current (units A), i is the current density (units A/cm2), δ is the thickness of the depletion layer, area is the area of the electrode, and σbulk (units ohm cm), the resistivity of the bulk solution, is
σbulk )
1 F2
(10)
∑j cjDjzj
2
RT
Note that area cancels out in eq 9. We can define a current function, χpc, as defined by Nicholson and Shain:18
χpc )
ipc 1/2 F D |ν| FcA RT A
[
]
≈
1 2
(11)
where cA is the bulk concentration of the redox species initially present in solution, DA is its diffusion coefficient, and F, R, and T have their usual significance. With perfect compensation, the value of ipc, the cathodic peak current density, will be similar to values obtained for systems with supporting electrolyte, hence approximating χpc ≈ 1/2. With the electroneutrality constraint (eq 8) and the presumption that the bulk solution contains a redox ion and its counterion eq 10 becomes
σbulk )
1 2
F c (D z 2 - DXzAzX) RT A A A
(12)
When the diffusion coefficients are the same for all species, σbulk (eq 10) becomes
σbulk )
1 2
F D c (z 2 - zAzX) RT A A A
(identical D values) (13)
The thickness of the depletion layer, δ, can be approximated as
δ≈ Combining eqs 9-14 gives
x
RT DA F |ν|
(14)
F |I ∆R | ≈ RT pc ref 2(z
2 A
1 - zAzX)
(15)
Equation 15 says nothing about the actual sign. It does, however, approximate the magnitude of the maximum value of |i∆Rref|; it also tells us19 I∆Rref will be independent of cA, DA and ν. It is also useful to analyze the behavior of the uncompensated IRref. A primary goal of this work is the evaluation of E° from a voltammogram that may involve only partial compensation. We define the uncompensated resistance, Ru, by
Ru ≈ (1 - γ)R0ref )
(1 - γ)xrefσbulk area
(16)
For experimental purposes, R0ref is an easily and directly measured parameter (much easier to measure than xref or σbulk). For theoretical calculations we define an uncompensated resistance parameter, Pu:
Pu )
1 2
(1 - γ)xref zA2
[ ] F|ν| RTDA
- zAzx
1/2
≈
F I R RT pc u
(17)
Equation 17 will serve as the basis for characterizing the effect of uncompensated resistance on the shape of the CV. In the following analyses we use an explicit finite difference algorithm to simulate CVs in two different ways: with perfect compensation (i.e., Eeff ) ECV) and with partial compensation (i.e., Eeff ) ECV - I(1 - γ)R0ref). The programs can model a broad class of systems with any number of sequential oneelectron transfers and independently assigned E° values (a simple modification allows introduction of Butler-Volmer heterogeneous kinetics). Any number of species may be considered, each with its specified initial concentration and charge (consistent with constraints of charge neutrality); the distance between working electrode and reference along with level of compensation can be specified. The QuickBASIC computer programs for executing these calculations are available on request. A more refined implicit finite difference approach for simulating systems with Nernst-Planck transport has been described in detail by Rudolph;20 however, the explicit finite difference code is considerably simpler to program and is adequate as long as there are no homogeneous kinetics and as long as the diffusion coefficients of the different species are not dramatically different. Analysis: Simulations with Perfect Compensation Simulation of I∆Rref during the Course of a CV. The objective of these simulations, executed with “perfect compensation” (i.e., Eeff ) ECV at all times), is to ascertain the magnitude of I∆Rref, the change in the compensating potential effected by the change in resistance within the depletion layer. Figure 1 shows the computed values of (F/RT)I∆Rref during the course of a cyclic voltammogram as a function of (F/RT)(ECV - E°). The curves in Figure 1a were computed for zA ) -1, -2, -3, and -4 with zX ) 1 and diffusion coefficients the same for all species; curves in Figure 1b are for zA ) 2, 3, and 4 with zX ) -1. As suggested by eq 15, |i∆Rref| decreases dramatically with increasing values of zA; the results are independent of scan rate, diffusion coefficients (assuming relative values are fixed), and the initial concentration of the redox salt. When zA e -1 (see Figure 1a), the reduction product, BzA-1, produced in the depletion layer is more highly
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Figure 1. Plot of (F/RT)I∆Rref vs (F/RT)(ECV - E°) (a, top) for zX ) 1 and, for curves from highest to lowest: zA ) -1, -2, -3, and -4, respectively; (b, bottom) for zX ) -1 and for curves from highest to lowest: zA ) 2, 3, and 4, respectively.
charged; the net result is a decrease in the resistance within the depletion layer thus effecting a negative ∆Rref throughout the course of the CV. Since the reductive currents during the initial half-cycle of the CV are negative (IUPAC convention), I∆Rref will be positive during this phase of the CV. With oxidation during the second half-cycle, I becomes positive, ∆Rref remains negative, and therefore I∆Rref becomes negative. When zA g 2, ∆Rref will be 0 or positive (Figure 1b) throughout the course of the CV and I∆Rref will be negative during the initial (reductive) half-cycle of the CV and positive during the second half-cycle. Note also that at potentials well negative of the reduction peak, I∆Rref is virtually constant: a result of the change in the thickness of the depletion layer; |∆Rref| increases with t1/2 while the |I| increases with t-1/2.21 The data in Figure 1 suggest that as long as zA e -3 or g 3 (as suggested by eq 15) the maximum values of |I∆Rref| at room temperature (∼300 K) will be less than 0.001 V. Even when ∆Rref is negative (Figure 1a), this should not threaten the stability of the compensation. As indicated by eq 15, an increase in |zX/zA| will further decrease |I∆Rref|. Shape of CVs Simulated with Perfect Compensation. We focus here on the shapes of the cyclic voltammograms with perfect compensation. In Figure 2a the CVs for zA ) -1, -2, -3, -4, -1000, 4, 3, and 2 with zX ) (1 as required are presented as the dimensionless current function, χ (as defined by Nicholson and Shain18), plotted as a function of the normalized potential (F/RT) (ECV - E°). The shape of the CV for zA ) -1000 is indistinguishable from the classical CV computed for a system with excess inert electrolyte.18 The difference (F/RT)(Epa - Epc) does not change significantly from 2.22, the value when there is excess electrolyte; there are, of course, small changes induced by changing the reversal potential. The effect of migration on normalized peak potentials ((F/RT)(Ep - E°)) for the forward and reverse peaks is shown in Figure 2b as a function of 1/zA. Note that there is a smooth
Figure 2. With zX ) (1, (a, top) Plot of the current function, χ (eq 11), vs (F/RT)(ECV - E°) for increasing zA ) -1, -2, -3, -4, -1000, 4, 3, and 2 corresponding to increasing peak currents; (b, middle) plot of (F/RT)(Ep - E°) vs 1/zA; (c, bottom) Plot of the cathodic and anodic peak current functions (relative to the cathodic peak current function with excess electrolyte) vs 1/zA.
transition from negative to positive values of 1/zA and the interpolated value of (F/RT)(Ep - E°) corresponding to 1/zA ) 0 is within 0.004 of 1.109 ()0.0285 V at T ) 298.2) the value predicted for a reversible CV with excess electrolyte.18 The effect of migration on the normalized current function, χp/χpc,se, as a function of 1/zA is shown in Figure 2c (χpc,se is the current function for the cathodic peak when there is excess electrolyte,
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Figure 4. Selected plots of (F/RT)(Epc - E°) vs Pw0.929 (see eq 18) for DA ) DB ) DX and zA ) -1, -3, -8, 1000, 8, 4, 2 for highest to lowest lines, respectively. See Table 1.
Figure 3. (a, top) Plot of χ vs (F/RT)(ECV - E°) for zA ) -1, zX ) 1 with Pu (eq 17) ) 1.973, 4.932, 9.86, 14.80, and 19.73. All curves were initiated at (F/RT)(ECV - E°) ) 0.3(F/RT) and all CVs, save one, were reversed at (F/RT)(ECV - E°) ) -0.3(F/RT). The CV reversed at (F/RT)(ECV - E°) ) -0.5(F/RT) corresponds to Pu ) 19.73. (b, middle) CV for Pu ) 0 with ∆Estep ) 0.005 V. (c, bottom) Same as (b) but with ∆Estep ) 0.002 V.
see eq 11). Here, too, there is a smooth transition from negative to positive values of 1/zA and the interpolated value of χp/χpc,se corresponding to 1/zA ) 0 is 1.000. While the simulations of systems with perfect compensation are interesting and perhaps helpful for our understanding of migration effects, executing an experiment with perfect compensation is, at least now, not possible. Thus it is essential to examine the simulations of CV with partial compensation to develop an aproach for evaluating E°. Analysis: Simulations with Partial Compensation Effect of Uncompensated Resistance on the Shape of a CV. A sense of how uncompensated resistance modifies the shapes of the CVs can be gleaned from the CVs in Figure 3a. Figure 3a shows normalized CVs (χ vs (F/RT)(ECV - E°)) for zA ) -1 and zX ) 1 with Pu (eq 17) ) 1.973, 4.932, 9.86, 14.80, and 19.73. All curves were initiated at (F/RT)(ECV E°) ) 0.3(F/RT) and all CVs, save one, were reversed at (F/ RT)(ECV - E°) ) -0.3(F/RT). The CV reversed at (F/RT) (ECV - E°) ) -0.5(F/RT) corresponds to Pu ) 19.73. Note
that the anodic peak potential is less positive than anodic peak of the corresponding CV reversed at -0.3(F/RT) for Pu ) 19.73; that is because the current is significantly lower and therefore IRu is smaller. This is a major reason for the difficulty in simply using the average (Epc + Epa)/2 as an estimate of E°. Instability Caused by Negative ∆Rref. When Pu ) 0 (i.e., when γ ) 1) zA ) -1 and zX ) 1, the simulation exhibits instability (Figure 3, b and c). This instability is expected since ∆Rref is negative when zA < 0 and the system becomes overcompensated (see eqs 2-5 and accompanying discussion). We are a bit surprised that the instability does not appear until well into the anodic wave. We confirmed that changing Estep, the potential step size in the simulation, from 0.005 V (Figure 3b) to 0.002 V (Figure 3c) did not greatly change the onset of the instability, although it did affect the magnitudesthe simulation in Figure 3c crashed. We did not explore this phenomenon further. Determining E° from CVs with Partial Compensation. The best way to determine the E° from an experimental CV is by fitting the simulated CV to the experimental CV: Set the values of the known input parameters for the simulation and adjust the unknown parameters until the simulated and experimental CVs match. Another approach is to examine the relationship between Epc and E° . We noted earlier (see Figure 3a) that both Epa and Ipa depend on the potential at which the CV scan is reversed; this presents yet another parameter to be considered in the analysis. Running the full CV (i.e., scanning forward and back) is still useful since it can confirm that there are no obvious complications and also, by averaging Epc and Epa, provide a rough estimate of the value of E°. Figure 4 shows some selected plots (for zA ) 1000, 8, 2, -1, -3, and -8) of (F/RT)(Epc - E°) as a function of Pw0.929, where Pw is defined
Pw )
F F |I |(1 - γ)R0ref ) |i |(1 - γ)xrefσbulk (18) RT pc RT pc
Pw is closely related to Pu (eq 17) but is explicitly defined in terms of the experimentally measured parameters R0ref, γ, and Ipc (Ipc is corrected for background and capacitive currents). In Figure 4 the marked intercept (F/RT)(Epc,se - E°) ) -1.109 (equivalent to -0.0285 V at T ) 298.2 K18) denotes the peak potential for a reversible electron transfer with excess supporting electrolyte. The maximum shift of (F/RT)(Epc - Epc,se) is ∼-11 (or equivalent to ∼- 0.28 V at T ) 298.2 K); the corresponding shift in (F/RT)(Epa - Epa,se) will be a bit less than 11 and will depend on the reversal potential (see Figure 3a). The unusual power dependence on Pw (0.929) was deduced empirically from
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TABLE 1: Least-Squares Evaluation of bz and m for p ) 0.929 and DA ) DB ) DX (See Eq 19) zX ) (1
zX ) (2
Figure for zX ) (1
zA
bz
m
sd
bz
m
sd
Figure 4 Figure 4
1000 8 7 6 5 4 3 2 -1 -2 -3 -4 -5 -6 -7 -8 -1000
-1.1067 -1.1779 -1.1877 -1.2000 -1.2222 -1.2497 -1.3014 -1.4023 -0.5824 -0.8514 -0.9246 -0.9630 -0.9838 -1.0015 -1.0137 -1.0205 -1.1053
-1.346 -1.345 -1.346 -1.347 -1.346 -1.346 -1.345 -1.349 -1.345 -1.348 -1.348 -1.347 -1.349 -1.348 -1.348 -1.349 -1.346
1.0 × 10-2 4.3 × 10-3 4.8 × 10-3 6.7 × 10-3 4.2 × 10-3 5.1 × 10-3 5.7 × 10-3 8.9 × 10-3 8.4 × 10-3 6.1 × 10-3 5.7 × 10-3 5.9 × 10-3 5.2 × 10-3 4.9 × 10-3 4.7 × 10-3 7.3 × 10-3 1.0 × 10-2
-1.1067 -1.1656 -1.1779 -1.1878 -1.2003 -1.2235 -1.2530 -1.3184 -0.7384 -0.9052 -0.9567 -0.9813 -1.0001 -1.0128 -1.0201 -1.0283 -1.1054
-1.346 -1.348 -1.346 -1.346 -1.347 -1.346 -1.347 -1.347 -1.345 -1.348 -1.347 -1.348 -1.348 -1.348 -1.348 -1.348 -1.346
1.0 × 10-2 8.0 × 10-3 4.3 × 10-3 4.9 × 10-3 6.7 × 10-3 4.0 × 10-3 5.4 × 10-3 3.8 × 10-3 5.8 × 10-3 5.4 × 10-3 5.9 × 10-3 5.3 × 10-3 5.0 × 10-3 4.7 × 10-3 7.3 × 10-3 6.2 × 10-3 1.0 × 10-2
Figure 4 Figure 4 Figure 4 Figure 4
Figure 4
Figure 5. Plot of bz vs 1/zA for zX ) (1 (b) and (2 (O) (see Table 1).
the minimization of the standard deviation (sd) for 34 data sets (for zA ) 1000, 8, 7, ... 2, -1, -2, ... -8, -1000 with zX ) (1 and (2) by optimization of the parameters m, bz, and p:
sd )
[
n
[
∑1
]
F
(Epc - E°) - bz - mPpw RT
]
2 1/2
n-1
(19)
The fully optimized values of p ranged from 0.926 to 0.932. For ease of analysis we set p ) 0.929 and optimized only bz and m (sd values were not quite as good but well within the limit of expected experimental errors). The results, as seen in Figure 4, are linear working curves with virtually identical slopes m ()1.347 ( 1.2 × 10-3)ssee legend for details. Values of bz, m, and sd (eq 19) for all indicated values of zA and zX are given in Table 1. The operative equation, then, is
(20) (F/RT)(Epc - E°) ) bz + mPpw ) bz - 1.347P0.929 w The value of bz can be determined from Table 1 for given values of zA and zX. Alternatively, the plots of bz vs 1/zA for zX ) (1 and (2 (Figure 5) are nearly linear so that we can write
bz(zX ) (1) ) -1.109 - 0.547/zA
(21)
bz(zX ) (2) ) -1.109 - 0.410/zA
(22)
These approximations for bz reproduce the intercepts in Table 1 to better than 0.04 or within 10-3 V at T ) 298.2 K. Note that the intercepts of eqs 21 and 22 are both -1.109, the value
expected for a reversible electron transfer with excess supporting electrolyte. Note also that the plot of bz vs 1/zA is similar to but not the same as Figure 2b which shows the value of (F/ RT)(Epc - E°) vs 1/zA obtained under condtions of perfect compensation. Recall that bz is the intercept term obtained when γ ) 1 and Pw w 0 (see eq 18); setting γ ) 1 is not equivalent to perfect compensation: there is no correction for changes in resistance caused by changes in the composition of the depletion layer. The generality of eq 20 extends to systems where the diffusion coefficients are not identical. While it is reasonable to expect that DA = DB for most redox couples, the value of DX could well be different. Figure 6a shows (F/RT)(Epc - E°) as a function of Pw0.929 for zA ) -1, zX ) 1, DA ) DB, and DX/DA ) 0.5, 1, and 2. The curves for the different DX/DA ratios produce slightly displaced curves with the maximum difference in intercept of 0.43 or about 0.011 V at T ) 298.2 K. However, when zA ) -2 and zX ) 1 (Figure 6b) or zA ) 2 and zX ) -1 (Figure 6c) the curves for the different DX/DA values are virtually on top of each other; the maximum difference in intercept is 0.16 or about 0.004 V at T ) 298.2 K. For |zA| > 2 that difference becomes less than 0.06, i.e., less than 0.0015 V at T ) 298.2 K. The parameter and sd values for these plots are summarized in Table 2. Note, however, that the intercept values, bz, for DX/DA ) 2 and 0.5 are not symmetrically displaced above the value for DX/DA ) 1. For zA ) 2 and zX ) -1, bz values for DX/DA ) 2 and 0.5 are both more negative than for DX/DA ) 1. This may seem inconsistent with the other data presented in Table 2; however, note that for any given value of zA reported in Table 2 the average of the bz values for DX/DA ) 2 and 0.5 are all about 0.04 more negative than the bz value for DX/DA ) 1. This curiosity deserves further study. The linearity of these plots is also sustained when DA * DB. We explore a limited set of examples here with the constraint that DA ) DX and DB/DA ) 0.5, 1, and 2. Figure 7a shows (F/RT)(Epc - E°) as a function of Pw0.929 for zA ) -1, zX ) 1, DA ) DX, and DB/DA ) 0.5, 1, and 2. The differences in bz values diminish but do not disappear with larger values of zA, e.g., zA ) -8 (Figure 7b) and zA ) -1000 (Figure 7c). Why do we not see the ln(DB/DA ) dependence predicted for systems with excess electrolyte? The reason, we believe, is that the transport of species A and B is also affected by the diffusion coefficient of species X.
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a
b
c
Figure 6. Plot of (F/RT)(Epc - E°) vs Pw0.929 (a, top) for zA ) -1, zX ) 1, DA ) DB and DX/DA ) 0.5 (highest line), 1 (middle line), and 2 (lowest line); (b, middle) for zA ) -2, zX ) 1, DA ) DB, and DX/DA ) 0.5, 1, and 2; (c, bottom) for zA ) 2, zX ) -1, DA ) DB, and DX/DA ) 0.5, 1, and 2. See Table 2.
TABLE 2: Least-Squares Evaluation of bz and m for p ) 0.929, DA ) DB, and Variable DX (See Eq 19) Figure
zA
zX
DX/DA
bz
m
sd
Figure 6a
-1 -1 -1 -2 -2 -2 2 2 2
1 1 1 1 1 1 -1 -1 -1
2 1 0.5 2 1 0.5 2 1 0.5
-0.8371 -0.5824 -0.4094 -0.9649 -0.8514 -0.8047 -1.4654 -1.4023 -1.4315
-1.345 -1.345 -1.343 -1.344 -1.348 -1.348 -1.343 -1.349 -1.347
1.2 × 10-2 8.4 × 10-3 6.1 × 10-3 5.5 × 10-3 6.1 × 10-3 6.6 × 10-3 4.8 × 10-3 8.9 × 10-3 3.7 × 10-3
Figure 6b Figure 6c
Interpretation of Experimental Data A trivial rearrangement of eq 20 produces the working equation:
Epc ) E° + (RT/F)(bz - 1.347P0.929 ) w
(23)
A plot of Epc vs Pw0.929 will have an intercept of E° + (RT/F)bz where bz can be adequately approximated by eq 21 or 22 for zX ) (1 or (2. The experimental data should comprise several
Figure 7. Plot of (F/RT)(Epc - E°) vs Pw0.929 (a, top) for zA ) -1, zX ) 1, DA ) DX, and DB/DA ) 0.5 (lowest line), 1 (middle line), and 2 (highest line); (b, middle) for zA ) -8, zX ) 1, DA ) DX, and DB/DA ) 0.5 (lowest line), 1 (middle line), and 2 (highest line); (c, bottom) for zA ) -1000, zX ) 1, DA ) DX, and DB/DA ) 0.5 (lowest line), 1 (middle line), and 2 (highest line). See Table 3.
TABLE 3: Least-Squares Evaluation of bz and m for p ) 0.929, zX ) 1, DA ) DX, and Variable DB (See Eq 19) Figure
zA
DB/DA
bz
m
sd
Figure 7a
-1 -1 -1 -8 -8 -8 -1000 -1000 -1000
2 1 0.5 2 1 0.5 2 1 0.5
0.1853 -0.5824 -1.2997 -0.8116 -1.0205 -1.3125 -0.9834 -1.1053 -1.3074
-1.351 -1.345 -1.335 -1.340 -1.349 -1.351 -1.336 -1.346 -1.350
2.0 × 10-2 8.4 × 10-3 8.5 × 10-3 6.2 × 10-3 7.3 × 10-3 7.9 × 10-3 1.5 × 10-2 1.0 × 10-2 8.4 × 10-3
Figure 7b Figure 7c
CVs obtained for different values of Ipc, (1 - γ), and/or R0ref (the components of Pw, see eq 18). It is not necessary to use all the terms comprising Pwsjust the terms that are varied during the acquisition of the CVs. For example, if the experiment is carried out with fixed xref and γ while varying the scan rate, plot Epc vs |Ipc|0.929. We have not discussed the potentially serious problem of selecting an appropriate reference electrode and the potential problem of liquid junctions. Bento et al.11 have suggested a number of strategies to deal with this vexatious
Reversible Cyclic Voltammetric Responses
J. Phys. Chem. B, Vol. 102, No. 49, 1998 9973
problem. Experiments carried out at different concentrations should give the same values of E° if, and only if, the junction potential is invariant and, if and only if there is no complexation. The junction potential can be minimized by using an internal reference involving a second redox couple with a neutral component whose presence in the system introduces no additional ions (it will require additional simulations to characterize the behavior of the reference system). Hopefully, neither of the redox partners is susceptible to complexation and/or pair formation with any of the ions in the system. Using a counterion that forms a reversible insoluble redox salt/metal (e.g. Cl- with Ag/AgCl or Hg/Hg2Cl2) is also a possibility. An Interesting Theoretical Observation. When the diffusion coefficients for all species are identical, electroeneutrality obtains and when no species cross the outer Helmhotz plane (as would happen with dissolution, plating or adsorption) Oldham10 showed that during steady-state diffusion-migration transport the following condition obtains at any distance from the electrode, for any number of species of any charge:
∑j cj ) ∑j cj,bulk
(24)
We have also observed that the same constraint obtains for a cyclic voltammetric perturbation. Jaworski et al. noticed, but did not comment upon, the same constraint in a chronoamperometric analysis.13,14 The generality of this observation will be examined in a future publication.22 Conclusions The analysis of the cyclic voltammetric behavior of a reversible redox system without added electrolyte can be summarized as follows (for systems where the initial species is the oxidized redox component and ne ) 1 for the redox process): 1. Cyclic voltammetric responses have been simulated with perfect and partial compensation (for IR). ∆IRref (the change in the solution resistance between the working and reference electrodes effected by the CV perturbation) is negligible for zA e -3 and zA g 4. When zA is negative, ∆Rref is negative; this decrease in solution resistance could destabilize the system when the compensation (using a potentiostat with positive feedback) is close to unity. 2. In most cases, useful data will be obtained only with use of a three-electrode system, a potentiostat, and positive feedback to compensate for ohmic drop, IRref, between the working and reference electrodes. 3. While the focus of this paper has been on the analysis of a single (forward) CV scan, the information in the entire CV is important: it can confirm the overall reversibility of the electrochemical process. Furthermore, E° can be roughly approximated by averaging Epc and Epa. 4. E° can be determined by direct matching of simulated and experimental CVs. Furthermore, diffusion coefficient values might be estimated as well. 5. E° can be correlated to measurable CV parameters by a simple linear equation:
Epc ) E° +
RTbz 1.347RT 0.929 Pw F F
(25)
where
Pw ) (F/RT) |Ipc|(1 - γ)R0ref
(26)
The slope of a plot of Epc vs Pw0.929 is independent of the relative
values of the diffusion coefficients; the intercept (RTbz/F) is dependent upon the relative values of the diffusion coefficients. See eqs 18 and 23 and associated discussion. Throughout this analysis we have used a one-electron reversible reduction as the model system. The analyses for a one-electron oxidation are clearly analogous: the signs of bz and m ()-1.347) will be reversed; subscripts for anodic and cathodic peaks will be exchanged; signs of zA and zX must be changed and excluded values of zA will be -1 and 0, the initial species will be B and its counterion; X, the initial (normalized) potential would be -0.3(F/RT), and the initial sign of ν will be positive. The present work has focused on extraction of E° values from cyclic voltammetric data. We have not discussed just how diffusion coefficients of individual species A, B, and X, might be determinedssteady-state limiting current measurements at microelectrodes may provide the most direct determination, at least of the initial species. Finally, simulations or computations which consider uncompensated resistance but do not consider the effects of migration on transport or the effects of changes in depletion layer may still provide adequate approximations for larger values of |zA| (see Figures 2 and 5). Acknowledgment. The authors thank Prof. Keith Oldham for numerous comments, questions, and suggestions during the evolution of this work and for making available some unpublished analyses. S.W.F. gratefully acknowledges the support of the U.S. Department of Energy, Contract No. DE-AC0298CH10886. S.W.F. also thanks the Department of Chemistry, Monash University, Clayton, Australia, for their hospitality April-May 1997. A.M.B. thanks the Australian Research Council for support provided via a Special Investigator Award (1997-1999). Glossary area (cm)2 area of the working electrode cA, cB, cX (mol/cm3) analytical bulk concentration of species A, B, and X; since we focus on reductions (see eq 6) we assume that cB ) 0; electroneutrality (eq 8) effects cAzA + cXzX ) 0 time- and distance-dependent concentration of cj (mol/cm3) jth species (see, e.g., eq 7) Dj (cm2/s) diffusion coefficient of jth species (see, e.g., eq 7) DA, DB, DX (cm2/s) diffusion coefficients of species A, B, and X Dall (cm2/s) diffusion coefficient value when DA ) DB ) DX Dmax (cm2/s) maximum diffusion coefficient in system E° (V) formal potential of operative redox couple ECV (V) potential required by CV protocol ECV,rev (V) potential at which ν changes sign starting potential for CV (and often the final ECV,start (V) potential) Eeff (V) effective potential of working electrode generic peak potential, cathodic peak potential, Ep, Epc, Epa (V) and anodic peak potential, respectively ∆Ep,rev (V) separation of peak potentials for a reversible electron transfer potential step used in simulation ∆Estep (V) F (C/equiv) Faraday’s constant (96497 C/equiv) I, Ipc, Ipa (A) current, cathodic peak current, and anodic peak current, respectively i, ipc, ipa (A/cm2) current density, cathodic peak current density, and anodic peak current density, respectively
9974 J. Phys. Chem. B, Vol. 102, No. 49, 1998 m ne nhalf p Pu Pw R (J mol-1 T-1) Rref (ohms) R0ref (ohms) ∆Rref (ohms)
Ru (ohms) T (K) ν (V/s) xref (cm) zA, zB, zX
zj γ δ (cm) E (V/cm) σbulk (ohm cm) τ (s) φ2 (V) χp, χpc, χpa
slope of working curve (see eq 20) number of electrons in a redox reaction (ne ) 1 for examples considered in this paper) number of half-cycles in CV (nhalf ) 2 in this work17) exponent in working curve equation (see eq 19) uncompensated resistance parameter (see eq 17) working curve parameter (see eq 18) gas constant (8.3144 J mol-1 T-1) resistance between working and reference electrodes during the CV perturbation resistance between working and reference electrodes prior to CV perturbation change in resistance between working and reference electrodes during the CV perturbation (effected by changes in the composition of the depletion layer) uncompensated resistance at start of CV (see eq 16) temperature scan rate of CV distance between working and reference electrodes charges associated with species A, B, and X; since we limit discussion to one-electron transfers, zB ) zA - 1 charge of jth species (see, e.g., eq 7) fraction of resistance R0ref that is compensated approximate thickness of depletion region (see eq 14) electric field (see eq 7) resistivity of bulk solution (see eq 10) duration of CV17 potential drop across the diffuse double layer peak current function, cathodic peak current function and anodic peak current function, respectively (see eq 11)
References and Notes (1) Potentiometry is also a viable approach when both components of a redox couple are stable and easily preparedsnot always the case. (2) Cyclic voltammetric protocol changes the potential of the working electrode between specified potential limits with dV/dt ) (ν. (3) Throughout this paper we assume the IUPAC sign convention: reduction currents are negative, oxidation currents are positive.
Bond and Feldberg (4) Bard, A. J.; Faulkner, L. R. Electrochemical Methods: Fundamentals and Applications; J. Wiley and Sons: New York, 1980; pp 571573. (5) For simplicity of presentation we limit our discussions to reversible one-electron transfers. Generalization of the Nernst equation by replacing (F/RT) by (neF/RT) is valid only when peak separations are 2.218/nesrarely met when ne ) 2 and seldom when ne > 2. Consideration of quasi-reversible electron transfer, already complicated by introduction of heterogeneous rate parameters, is further complicated by diffuse double layer effects. (6) Way, D. M.; Bond, A. M.; Wedd, A. G. Inorg. Chem. 1997, 36, 2826. (7) There are always going to be some ions present associated with impurities and/or with the solvent itself (e.g., from autoionization such as H2O a H+ + OH-). The implicit presumption in these analyses is that the concentration of adventitious ions will be a small fraction of the analyte concentration. (8) Norton, J. D.; White, H. S.; Feldberg, S. W. J. Phys. Chem. 1990, 90, 6772. (9) Smith, C. P.; White, H. S. Anal. Chem.1993, 65, 3343. (10) Oldham, K. B. J. Electroanal. Chem. 1992, 337, 91. (11) Bento, M. F.; Thouin, L.; Amatore, C.; Montenegro, I. J. Electroanal. Chem. 1998, 443, 137. (12) Bento, M. F.; Thouin, L.; Amatore, J. Electroanal. Chem. 1998, 446, 91. (13) Jaworski, A.; Donten, M.; Stojek, Z. J. Electroanal. Chem. 1996, 407, 75. (14) Jaworski, A.; Donten, M.; Stojek, Z. J. Electroanal. Chem. 1997, 420, 307 (a correction to ref 13). (15) Nicholson, R. S. Anal. Chem. 1965, 37, 667, 1351. (16) The validity of the electroneutrality assumption hangs on the presumption that the thickness of the diffuse double layer is very small compared to the dimensions of the depletion layers likely to be involved in the cyclic voltammetric perturbation. A rule of thumb relationship (assuming a 1:1 electrolyte, D ) 10-5 cm2/s and T ) 300 K) is δdouble layer/δdepletion layer ≈ 10-4[ν(V/s)/c1:1(M)]1/2 (17) A conservative constraint is: xref g 6(Dmaxτ)1/2; or equivalently xref g 6[nhalfDmax|{ECV,rev - ECV,start}/ν|]1/2 where Dmax is the largest operative diffusion coefficient in the system, τ is the duration of the voltammogram, and nhalf is the number of half-cycles (nhalf ) 2 for examples in this paper). (18) Nicholson, R. S.; Shain, I. Anal. Chem. 1964, 36, 706. The general current function, χ, is defined as: χ ) i/FcA{(F/RT)DA|ν|}1/2. (19) Relative diffusion coefficients are defined by the ratios Dj/DA, for all species. (20) Rudolph, M In Physical Electrochemistry, Principles, Methods, and Applications; Rubinstein, I., Ed.; Marcel Dekker: New York, 1995. (21) In a chronoamperometric experiment with perfect compensation the value of I∆Rref will be absolutely constant; Eeff will not equal ECA (the constant potential specified by the chronoamperometric protocol) but it will be also be constant, even if limiting current conditions do not obtain! Note that the analysis of Jaworski et al.13,14 is for uncompensated chronoamperometry. (22) Oldham, K. B.; Feldberg, S. W. J. Phys. Chem., submitted for publication.
