Investigation of the influence of residual uncompensated resistance and incomplete charging current correction on the calculation of electrode kinetics when global and convolution analysis methods are used

15

Journal of Electroanalytical Chemistry, 366 (1994) 15-21

Investigation of the influence of residual uncompensated resistance and incomplete charging current correction on the calculation of electrode kinetics when global and convolution analysis methods are used * Alan M. Bond

l

* and Peter J. Mahon

Department of Chemical and Analytical Sciences, Deakin University, Geelong, WC. 3217 (Australia)

Keith B. Oldham Trent University, Peterborough, Ont. K9J 788 (Canada)

Cynthia G. Zoski Department of Chemistry, University of Rhode Island, Kingston, RI 02881-0801 (USA) (Received 14 September 1992; in revised form 12 January 1993)

Abstract The global analysis and convolution potential sweep voltammetry (CPSV) procedures

for the analysis of electrode kinetic parameters are compared under the influence of residual uncompensated resistance and incomplete charging current correction. An initial examination of simulated data containing these known artifacts indicates that both procedures are similarly susceptible to the effects of uncompensated resistance. CPSV may be unaffected by the effects of incomplete charging current correction, whereas the global analysis procedure does experience some inaccuracies and a combination of both procedures is recommended in this circumstance. The effect of unequal diffusion coefficients is also examined.

1. Introduction

A global procedure for studying electrochemical processes has been reported for both planar [l] and spherical [2] electrode geometries. In studies using this method it has been demonstrated [l-3] that global kinetic analysis of a single electrochemical experiment enables the mass transport, thermodynamic and kinetic parameters (i.e. the standard heterogeneous charge transfer rate constant k”, the charge transfer coefficient cr, the reversible half-wave potential ,J+, and

Dedicated to our friend and colleague Professor Jan Sluyters on the occasion of his 65th birthday and retirement from the Chair of Electrochemistry at the University of Utrecht and in recognition of his many significant contributions to electrochemistry. l * Present address: Department of Chemistry, La Trobe University, Bundoora, Vie. 3083, Australia. l

0022-0728/94/$7.00 SSDIOO22-0728(93)02707-O

the diffusion coefficient Do of the reactant) to be evaluated accurately for the simple electrode reaction Osoln+n e- =R

soln

(1)

The analysis method utilizes the existence of a unique surface in (m, i, E) space which is independent of the applied experimental technique. Here E, i and m are the electrode potential, current and the temporal semi-integral of i [2,4] respectively. The acknowledged advantage [1,2] of the global approach is that all the data from a single experiment are used which leads to higher precision. Furthermore, the need for a priori knowledge of a kinetic model of electron transfer is unnecessary so that “global analysis” is a powerful tool in the diagnosis as to whether the well known Butler-Volmer kinetic model is appropriate. The procedure is also ideally suited for programming on a digital computer and the calculation is rapid. 0 1994 - Elsevier Sequoia. All rights reserved

16

A.M. Bond et al. / Factors affecting calculation of electrode kinetics

The related technique of convolution potential sweep voltammetry (CPSV) [S-9] has the same objective as global analysis but uses only the relationship between the potential and the resultant semi-integral of the current for the forward sweep of a cyclic voltammogram. This procedure also provides estimates of k” and (Y.The reversible half-wave potential is calculable but the precision is lower than global analysis because E'1,2is evaluated only from a single potential on the reverse sweep of a -cyclic voltammogram [8]. The successful application of either the global analysis or the CPSV procedure requires the electrochemical experiment to provide an accurate faradaic current, free from any artifact. The influence of uncompensated solution resistance R, can be greatly reduced by the use of potentiostats [lo-171, particularly when coupled with additional software and/or hardware methodology to minimize the problem further [S,ll-191. However, in the more common case of undercompensation of the solution resistance, the effect in cyclic voltammetry is to increase the peak separation [16,19-211 in a manner that could be mistaken for an apparent slow rate of electron transfer in the Nicholson analysis [22]. The effect in CPSV is to decrease the slope of the characteristic logarithmic plot used to calculate the electrode kinetic parameters, and this could also be mistaken for an erroneously slow rate of electron transfer [6]. In cyclic voltammetry, analysis of the electrode kinetics for fast electron transfer processes involves entering the kinetic (quasireversible) regime by decreasing the time scale of the experiment. This is achieved by increasing the potential scan rate, which, combined with the double layer capacitance C,, produces an extraneous current equal to the product of the capacitance and the scan rate. This charging current adds to the faradaic current and therefore to the resistance-induced potential error. Additionally, the product R,C, of the resistance and the capacitance introduces a time constant which dilates the voltammogram [6,20]. In the event of complete potentiostatic compensation of the solution resistance (i.e. R, = 0),the only effect of the charging current will be to displace the forward and the reverse branches of the voltammogram along the current axis. Under this condition the offset may be subtracted from the voltammogram [3,5,14,23-261 using a “blank”, under the assumption that the presence or absence of the electroactive species has no effect on the behaviour of the electrode-solution interface. Alternative techniques such as the application of a nonlinear ramp followed by semi-integral analysis have been used for analytical purposes [27] but not yet in kinetic measurements. Again, as in the case of uncom-

pensated resistance, subtraction of the charging current is unlikely to be perfect. In this paper, an investigation of the effects of residual uncompensated resistance and incomplete charging current correction for simulated data with respect to all the parameters calculable in both the global analysis and CPSV procedures is presented. In addition, the influence of unequal diffusion coefficients for 0 and R is addressed. Our motive in examining these interfering effects so carefully is to prepare the ground for a study of the historically controversial question of the potential dependence of (Y [3,8,23,24,26,28-341. In a companion article [35] we report and analyse experimental data for the well studied [2,23,24,26,36-381 reduction of 2-methyl-2nitropropane to its radical anion in acetonitrile: (CH,),CNO,

+ e- x

(CH,),CNO;

(2)

That analysis is strengthened by knowledge, gained in the present study, of the severity of interference from residual uncompensated resistance, incomplete charging current compensation and unequal diffusion coefficients. 2. Theory The theory of global analysis has been comprehensively presented previously [1,2], so only the equations essential to the analysis will be presented here. In the following summary it is not necessary to differentiate between the two electrode geometries that have previously been considered (i.e. planar [ll and spherical [21) but it should be noted that there are differences between the methods for calculating the semi-integral of the current depending upon the electrode geometry and previously the symbols m and p have denoted the semi-integral for the planar and spherical electrode geometries respectively. The polarizing signal is a cyclic ramp, as this is the simplest method to obtain two (i, m) pairs at each potential. If Tand m’ are the values of the current and its semi-integral at some potential E on the forward sweep with Tand &i as the corresponding values at the same potential during the reverse sweep, then the equations i*=(&i&G-)/(Gi-4)

(3)

and m*=(r%T-rZll)/(Y-tT)

(4)

are used to calculate i *, which is the value that the faradaic current would have attained at potential E in

A.M. Bond et al. / Factors affecting calculationof electrodekinetics

the absence of transport impediments, and m*, which is proportional to what the surface concentration of R would have been in the absence of kinetic impediment. Because the theory shows that there is a linear relationship 1

1

erm( -nFE;,,/RT)

(5)

M

m*=z+

between l/m * and exp(nFE/RT), linear regression permits Do to be found from the intercept: intercept = l/M = l/( nAFfi&)

(6)

where A is the electrode surface area and cb is the bulk concentration of 0. E42 can be calculated from - RTl, E;,z - nF

intercept (

slope

1

The second graph in the global analysis procedure is of ln(i*) vs. (-nF/RTXE - E;,,). Because of the relationship In i*=(-mF/RT)(E-E;,,)+ln

I

(8)

where I = nAF@cO( Do/D,)“‘2

(9)

this graph will be linear if (Y is constant but curved otherwise. The regression analysis of relationship (8) will provide information about (Yand the magnitude of the standard rate constant k”. Further discussion of this topic will be deferred to the companion article L35l. The equivalent treatment of CPSV data is described in refs. 5-9. Under quasireversible conditions the relationship gln&=a(E-E;/,)-$1

n;

(10)

applies, where ~ =

M-ffi[l+ /0,/o, exp(nF/RT)(E-El,,)] 2 1

(11) As in the corresponding global analysis treatment, this equation describes a linear graph of (RT/nF) ln(4) vs. E - Ei,2 if (Y is constant, but a curve if cy is potential dependent. The appropriate data analysis is similar to that for the global method and is described in ref. 35. In CPSV, E;,* is calculated from the 6i value corresponding to the potential at which ris zero [8].

17

3. Simulations

The simulation program was based on the finite difference procedures developed by Feldberg 1391, modified to include a potential-dependent charge transfer coefficient that obeys the equation [24] (Y= cx”+ (da/dE)(

E -E”)

(12)

where (Y’ is the value of the charge transfer coefficient at the standard potential, and da/dE is a constant. To simulate the distortion created by uncompensated resistance, the simulated voltammetric data were processed by adding in the potential error [20] as follows E’=E-iR,

(13)

where E’ is the measured potential with the resistive error added. The sign of the error is dependent on whether undercompensation (positive) or overcompensation (negative) of the potentiostat was simulated. The current data were then interpolated linearly so that a set of matched data pairs for the forward and reverse sweeps was obtained as required with global analysis [l]. The semi-integration was performed using the modified algorithm described in ref. 2. In accordance with the use of dimensionless parameters throughout this paper, the following equation expresses the dimensionless resistance and enables a comparison to be made with previous descriptions of uncompensated resistance in the literature [12]: ( Dov)1’2nFAc~R,

After inserting all of the parametric values, given below, eqn. (14) reduces to p’ = 0.0132 R-i R,. The simulation of incomplete correction of the charging current was achieved by applying a constant current offset to the reverse sweep of the voltammogram equal to some chosen percentage of the forward peak current. A positive or negative offset may be simulated which corresponds to overcorrection and undercorrection respectively. Experimentally both situations may arise. Only the reverse half of the voltammogram is adjusted because of the manner in which the experimental data are acquired 1351. A series of voltammograms was simulated, the parameters being chosen to match the experimental parameters used in studying reaction (2) at a spherical electrode. Thus we generally chose Do = D, = 3.33 X 10m9 m2 s-l, A = 1.9 x 10m6 m2 (corresponding to an electrode radius of 3.9 x 10m4 m), (Y’= 0.425, T = 293 K, E” = - 2.000 V and a potential range of - 1.790 to -2.240 V with a potential increment of 0.001 V. The concentration used in all the simulations was 0.500 mol

A.M. Bond et al. / Factors affecting calculation of electrode kinetics

18

rne3. In those simulations in which (Y was potential dependent, we used values of da/dE = 0.10, 0.20 and 0.30 V-‘, which cover the range reported in the literature [3,26]. The scan rate was 100 V s-l and the standard rate constant was varied over a twentyfold range, so that the dimensionless rate parameter [40]

(15) spanned the range 0.0700 to 1.40. Under these simulation conditions A is well within the range 10m2 5 A I 10 corresponding to quasi-reversibility [40]. At a scan rate of 100 V s-l this range of A corresponds to varying k” from 0.025 cm s-i to 0.51 cm s-l. 4. Results and discussion The voltammograms were subjected to spherical global analysis and to CPSV treatment, using data analysis procedures that matched those used on the experimental data reported in ref. 35. Thus, even though no errors are present in the simulated data, the same weighted regressions were performed which are appropriate to error-containing experimental data. Figures 1 and 2 are typical of cyclic voltammograms produced by the simulation program, before and after convolution. The results of applying spherical global

loo

0

-100 5

3

1

-1

-I

-1.85

-1.95

-2.05

-2.15

4!25

Potential/V Fig. 1. (I) Simulated cyclic voltammogram for A = 1.40, E” = - 2.000 V, U’ = 0.425, da/dE = 0.000 V-‘, Do = D, = 3.33 X lo-’ m* s-l, A = 1.9X 10m6 mz and T = 293 K. (II) Convolved voltammogram.

-I -1.75

-1.85

-1.95

-2.05

-2.15

-2.25

Potential/V Fig. 2. (I) Simulated cyclic voltammogram for A = 0.0700 with other simulation parameters as listed in Fig. 1. (II) Convolved voltammogram.

analysis and the CPSV treatment to many simulated cyclic voltammograms have been assembled in Tables 1-7. In these tables the headings “Global” and “Window” both relate to global analysis but utilize two different weighting schemes, as described in ref. 35. The parameters Do and Ef,2 are listed under the “Linear” analysis section in these tables but it should be noted that the calculation of these parameters, as given by eqns. (5)-(71, is independent of the calculation of the kinetic parameters and therefore the parameters D, and E;,2 are common to both “Linear” and “Quadratic” analyses. By inspection of the data contained in Table 1, it can be seen that a linear analysis in the presence of a potential-dependent LYwill generate a value of (Y that is smaller than cy” and that becomes increasingly inaccurate as A decreases or as da/dE increases. However, estimation of (Y’ and da/dE by quadratic regression analysis is quite accurate. Furthermore, with increasing kinetic control (i.e. as A decreases), linear analysis produces a high A value, whereas the quadratic analysis yields a value of A close to the true value. Therefore, failure to include a potential-dependent (Y will result in the overestimation of k” when A is small. Of course, these simulated data do not contain iR, drop, charging current or noise, so that the conclusions are only relevant to the analysis of ideal data. Normally, in the analysis of cyclovoltammetric data, the values of D, and D, are assumed to be equal.

ffo (da/d

A

E)/V-’

E;,z /V

a0

A

(da/dE)/V-’

a0

A

Ei,z /V

a0

A

Ei,z /V ‘4 a0 (da/dE)/V-’

A a0

Calculated parameter

0.0746 0.367 - 1.999 0.0693 0.426 0.301

0.0729 0.386 - 1.999 0.0692 0.425 0.201

0.0710 0.406 - 1.999 0.0692 0.425 0.101

0.0754 0.366 -2.000 0.0699 0.425 0.301

0.0733 0.387 -2.000 0.0698 0.425 0.202

0.0715 0.406 - 2.000 0.0698 0.425 0.102

0.140 0.381 -2.000 0.136 0.426 0.303

0.138 0.396 -2.000 0.136 0.426 0.203

0.137 0.411 - 2.000 0.136 0.426 0.104

0.135 0.396 -2.000 0.136 0.426 0.304

0.135 0.405 - 2.000 0.136 0.426 0.205

0.135 0.415 -2.fMO 0.136 0.426 0.105

Window

0.142 0.383 - 2.000 0.138 0.424 0.296

0.139 0.398 -2.000 0.137 0.426 0.204

0.139 0.412 -2.000 0.138 0.425 0.100

CPSV

0.671 0.414 - 2.000 0.682 0.428 0.294

0.674 0.419 - 2.000 0.682 0.428 0.195

0.678 0.424 - 2.000 0.681 0.428 0.096

Global

A = 0.700

a0 = 0.425, E” = -2.000 V, T = 293 K and electrode area 1.9~ 10d6 m’.

0.0702 0.386 -2.000 0.0692 0.426 0.302

0.0701 0.398 -2sKm 0.0692 0.426 0.202

0.0697 0.411 -2.ooo 0.0692 0.426 0.103

Global

CPSV

Global

Window

A = 0.140

A = 0.0700

a Simulation parameters: D = 3.33 X 10m9 m* s-l,

(Quadratic)

da/dE=0.3V-’ (Linear)

(Quadratic)

da/dE=0.2V-’ (Linear)

(Quadratic)

da/dE = 0.1 V-’ (Linear)

Potentialdependent a (calculation)

a a

0.670 0.420 -2.000 0.685 0.427 0.286

0.675 0.423 -2.000 0.685 0.428 0.189

0.680 0.425 - 2.000 0.685 0.428 0.091

Window

TABLE 1. Determination of electrode kinetic parameters by analysis of digitally simulated data containing a potential-dependent

0.697 0.413 - 2.000 0.705 0.422 0.287

0.697 0.418 -2.000 0.703 0.424 0.196

0.697 0.422 -2.000 0.701 0.425 0.106

CPSV

1.29 0.423 -2.000 1.31 0.429 0.270

1.30 0.425 -2.000 1.33 0.429 0.172

1.31 0.427 -2.000 1.31 0.429 0.074

Global

A = 1.40

1.29 0.426 -2.000 1.32 0.429 0.258

1.30 0.427 -2.000 1.32 0.429 0.162

1.31 0.428 -2.000 1.32 0.429 0.065

Window

1.35 0.425 -2.000 1.37 0.427 0.333

1.38 0.418 -2.OcKl 1.39 0.419 0.164

1.38 0.420 -2.000 1.39 0.420 0.070

CPSV

[ E:

% !k R a %

3

1

2

Q

F \ ti R 6 a

$

%

20

A.M. Bond et al. / Factors affecting calculation of electrode kinetics

However, in practice this assumption may not be valid and consequential errors are expected to be especially troublesome, at spherical electrodes where the assumption is made that the diffusion coefficients of 0 and R are equal when applying the small spherical correction term [2]. The influence of unequal diffusion coefficients for 0 and R was therefore examined in order to determine whether neglect of this factor would have an effect on the calculation of /co and (Y. With unequal diffusion coefficients, the kinetic parameter may be redefined [41] as a/2 + (

R

I

n \/;;

(16)

Voltammetric curves with two values of F, each with three differing degrees of electrode sphericity uSsphere defined as [42] %sphere=

(l/r,)dT

(17)

TABLE 2. Determination of electrode kinetic parameters Do = 2.500 x 10e9 mz s-* and D, = 1.250 x 10m9 mz s-t Sphericity factor a (calculation) Ussphere = 0.0010 (Linear)

Parameter

Ei,a/V

cy ffc (da/dE)/V-’

5 (da/dE)/V-’

Global

Window

CPSV

0.118 0.501

1.19 0.498 2.503

1.19 0.498 2.510

1.17 0.507

2.009 0.118 0.500 - 0.001

- 2.009 0.118 0.500 0.009

- 2.009 0.118 0.501 - 0.005

- 2.009 1.19 0.498 - 0.035

- 2.009 1.19 0.498 - 0.028

-2.009 1.18 0.505 0.164

0.118 0.500 2.501

0.114 0.499 2.505

0.120 0.499

1.24 0.476 2.513

1.26 0.475 2.553

1.21 0.496

2.009 0.120 0.497 - 0.019

- 2.009 0.114 0.498 -0.011

- 2.009 0.120 0.498 - 0.009

- 2.010 1.22 0.475 - 0.419

- 2.010 1.23 0.474 - 0.360

- 2.009 1.21 0.495 0.066

0.133 0.490 2.507

0.132 0.492 2.507

0.131 0.492

1.66 0.374 2.543

2.14 0.227 2.695

1.60 0.397

2.013 0.133 0.473 - 0.170

- 2.013 0.132 0.485 - 0.086

- 2.013 0.132 0.486 - 0.069

- 2.013 1.59 0.320 - 2.035

- 2.014 3.58 0.211 - 1.843

-

f?l 109Do/m2 s-l (Quadratic)

CPSV

0.118 0.500 2.501

;3L&e;;r; 0.0100

Ei,z/V

ty= 118 . 1Aw-l/*=0.995

= 0.0995

Window

0.118 0.500 2.501

,“o 109Do/m2 s-l

(Quadratic)

by analysis of digitally simulated data having unequal diffusion coefficients

ry = 0 .118, n?r-“2 Global

where r. is the spherical electrode radius, were simulated and then analysed. Results with D, = (l/2)0, (i.e. @o/Da) a/2 = 1.190 where (Y is constant and equal to 0.5) and with D, = (3/2)0, (i.e. (oo/&)“‘2 = 0.904 where cy iS COnStaM and equal to 0.5) are presented in Tables 2 and 3 respectively. The values of AQT-‘I2 for the two cases are close to 0.1 and 1. The standard value used for E” in the simulations was - 2.000 V and the theoretical E;,2 value expected from analyses of the data will be offset from E” by a constant equal to (RT/2nF) ln(D,/D,). Therefore, the expected E;,2 values are respectively equal to - 2.009 V and - 1.995 V in Tables 2 and 3. The data in Tables 2 and 3 for the lower value of ?P reveal that as the degree of sphericity increases, the estimation of ?P becomes slightly inaccurate whereas the evaluation of cy is almost unaffected with only a minimal apparent da/dE being introduced. At the larger value of !P, where the process approaches reversibility, the errors become much more significant due to the dominance

-

b

b

b

b

%phere = O.looo

(Linear)

ry (Yo 109Do/m2 s-t G/,/V

(Quadratic)

5 (da/d El/V-’

-

a Other simulation parameters: cr” = 0.500, da/dE = 0.000 V-l, b CPSV method unable to calculate a value for D, directly.

b

E” = -2.009 V and T= 293 K.

b

- 2.012 1.50 0.363 - 1.706

21

A.M. Bond et al. / Factors affectingcalculationof electrodekinetics

of diffusion control over kinetic control for the voltammetric response. Errors introduced from this factor will become apparent when there are large differences in the diffusion coefficients at spherical electrodes of very small radii and the kinetics approach reversibility. So far, only ideal data have been used in the analysis. The influence of uncompensated resistance on the analysis of data in both global analysis and CPSV is summarized in Tables 4 and 5 where there is no potential dependence of (Y present in the initial simulated voltammogram. Only one weighting scheme is reported for global analysis since the two approaches yielded almost indistinguishable results. For an electrode of spherical geometry, the uncompensated resistance between the working and reference electrodes is described in eqn. (18) [42-451

TABLE 3. Determination of electrode kinetic parameters D, = 2.500 X 10v9 m2 s-l and D, = 3.750 x 10e9 m2 s-l Spheric&y factor a (calculation) asphere

=

Parameter

by analysis of digitally simulated

ly = 0.0899, A.s-- “’ = 0.0995 Global

Window

0.0899 0.500 2.500 - 1.995 0.0897 0.500 0.002

0.0899 0.500 2.500 - 1.995 0.0899 0.500 0.001

ty

2

109Do/m2 s-r G/z/V

(Quadratic)

?P

a0 (da/dE)/V-’

!?’= 0 .899, Air- “’ = 0.995 CPSV

Global

Window

-

0.896 0.503 2.502 - 1.995 0.899 0.503 0.066

0.896 0.502 2.504 - 1.995 0.898 0.502 0.031

-

0.871 0.517 2.495 - 1.994 0.883 0.517 0.292

0.866 0.513 2.477 - 1.994 0.877 0.514 0.174

0.734 0.582 2.453 - 1.992 0.769 0.581 0.953

0.707 0.565 2.342 - 1.991 0.746 0.579 0.715

0.0900

0.500

b

1.995 0.0900 0.500 - 0.001

cy

ffo 109D,/m2

b

s-l

G/z/V

(Quadratic)

=

CPSV

0.888 0.508 b -

1.995 0.895 0.507 0.193

0.0100

(Linear)

%ph,

data having unequal diffusion coefficients

0.0010

(Linear)

ospsphere =

where d is the distance between the working and the reference electrodes. In acetonitrile with 0.1 M (C,H,),NClO,, the specific resistivity p is 128 0 cm [14]. Thus, for an electrode of radius 3.9 X 10m4 m, the maximum value of I?, would be 261 R in this electrolyte. An error of f 10 R would correspond to f 3.8% error in applying resistance compensation. Typically, an error of f5% is thought to be of an acceptable magnitude. Consequently, the values of R, in Tables 4 and 5 which encompass the range +40 0 are experimentally realistic. Residual uncompensated resistance (i.e. resistances that have a positive sign in Tables 4 and 51, as expected, gives rise to apparent kinetics that are slower than they should be with either method of data analysis. With the global analysis method in the presence of uncompensated resistance, (Y’ is underestimated for lower values of A in relation to the expected value of 0.425 but becomes overestimated with larger values of A when a linear analysis is

1.995 0.0897 0.500 0.001

0.883 0.510 b -

1.995 0.889 0.509 0.203

O.looo

(Linear)

4

(10 109Do/m2 s-* G/z/V

(Quadratic)

P

a0 (da/dE)/V-’

0.0837 0.504 2.490 - 1.992 0.0834 0.509 0.035

0.0837 0.504 2.483 - 1.992 0.0837 0.507 0.024

0.0844 0.503 b -

1.992 0.0843 0.507 0.031

a Other simulation parameters: CZ’= 0.500, da/dE = 0.000 V-r, E” = - 2.000 V and T = 293 K b CPSV method unable to calculate a value for DO-directly.

0.774 0.544 b -

1.993 0.787 0.547 0.420

(da/dE)/V-’

E)/V-’

/V

/V

El/V-’ -2.000

0.0702 0.441

-2.000

-2.000 0.0699 0.438

0.0696 0.435

-2.000

0.0693 0.433

- 1.999

0.0692 0.425

- 2.000

0.0688 0.428

-2.000

0.0685 0.425

-2.000

0.0683 0.423

- 2.000

0.0714 0.427 - 0.077

0.0711 0.425 - 0.072

0.0707 0.423 - 0.068

0.0704 0.421 - 0.065

0.0692 0.425 0.002

0.0698 0.417 - 0.058

0.0694 0.415 - 0.054

0.0691 0.413 - 0.051

0.0688 0.411 - 0.048

0.146 0.430 - 0.043

0.144 0.428 - 0.047

0.143 0.426 -0.051

0.141 0.424 - 0.055

0.138 0.426 0.005

0.138 0.419 - 0.062

0.137 0.417 - 0.066

0.136 0.415 - 0.070

0.133 0.411 - 0.078

Quadratic

ms-’

-2.000

0.800 0.418

-2.000

- 2.000 0.774 0.421

0.750 0.423

-2.000

0.727 0.425

- 2.000

0.682 0.428

-2.000

0.684 0.429

-2.000

0.664 0.431

-2.000

0.645 0.432

-2.000

0.627 0.433

Linear

0.808 0.425 0.183

0.779 0.425 0.105

0.751 0.424 0.030

0.725 0.424 - 0.041

0.681 0.428 - 0.003

0.677 0.422 - 0.174

0.655 0.422 -0.237

0.635 0.421 - 0.296

0.615 0.420 - 0.352

Quadratic

A = 0.700, k” = 2.52 x 10e3 m SC*

E" = -2.000 V, T = 293 K and electrode area 1.9~ low6 m2.

-2.000

0.145 0.436

-2.000

-2.000 0.144 0.434

0.142 0.433

-2.000

0.141 0.431

- 2.000

0.138 0.425

-2.000

0.136 0.428

- 2.000

0.136 0.426

-2.000

0.135 0.425

-2.000

0.134 0.423

Linear

A = 0.140, k”=5.04x10-4

= 0.000 V-l,

Quadratic

Linear 0.0680 0.421

ms-’

A = 0.0700, k”=2.52x10-4

D = 3.33 X 10e9 m2 s-l, CY’= 0.425, da/dE

Ei,z /V

(da/d

A ffo

Ei,z /V

(da/dE)/V-’

A a0

Ei,z /V

(da/dE)/V-’

A a0

ET,2

A 00 (da/dE)/V-’

Ei,z /V

A as (da /dE)/V-’

E;,2

A as (da/dE)/V-’

Ei,z /V

(da/d

A (10

Ei,a /V

A aa (dcr/dE)/V-’

Ei,z /V

a Simulation parameters:

p’= -0.53 R,=-40fl

p’= -0.40 R,=-30R

p’= -0.26 R,=-20fI

p) = -0.13 R,=-10R

p’= +o.oo R,=+Of2

R,=+lOR

pr = +0.13

p’ = + 0.26 R,= +20 R

p’= +0.40 R,=+300

A

0’ = + 0.53 k,=+40n

a0

Value

Resistance error

-2.000

1.79 0.404

-2.000

- 2.000 1.68 0.410

1.58 0.416

-2.000

1.50 0.421

-2.000

1.34 0.429

-2.000

1.34 0.430

-2.000

1.28 0.434

-2.000

1.22 0.437

-2.000

1.16 0.440

Linear

A = 1.40, /~~=5.04xlO-~

TABLE 4. Determination of electrode kinetic parameters by applying the global analysis method to digitally simulated data a containing uncompensated resistance

1.85 0.417 0.643

1.72 0.419 0.406

0.420 0.205

1.60

1.50 0.421 0.002

1.34 0.429 - 0.025

1.32 0.423 - 0.369

1.25 0.424 - 0.536

1.18 0.424 - 0.675

1.12 0.424 - 0.834

Quadratic

ms-’

w

23

A.M. Bond et al. / Factors affecting calculation of electrode kinetics

TABLE 5. Determination of electrode kinetic parameters by applying the CPSV method to digitally simulated data a containing uncompensated calculated from ref. 8 resistance with I$ Resistance error

Value

+ 40 D

A

Linear 0.0680 0.422

a0 (da/dE)/V-’

Quadratic 0.0697 0.414 - 0.040

0.135 0.423

0.136 0.424

0.0682 0.423

A CYO

Linear

(do/dE)/V-’ - 2.000

%/r/V +20 n

0.0695 0.424

A CT0 (da/d

E)/V-’

+10n

0.0697 0.425

A (Yo (da/dE)/V-’

00

0.0698 0.425

A cr 0

(da/d

El/V-’

EI,,/V -100

A

0

CY

0.0709 0.427

%2/V

A CT0 (da/dE)/V-’ -%2/V

-3on

A a0

0.0721 0.429

-40

n

A cl0 (da/dE)/V-’ %2/V

0.0709 0.426 0.000

0.137 0.420 - 0.038

0.141 0.424

0.141 0.427

0.0708 0.430 0.010

0.144 0.427

0.140 0.420 - 0.034

0.145 0.428

0.141 0.424 - 0.002

-2.000

a Simulation parameters: D = 3.33 X 10e9 m2 s-l, m.2

0.148 0.428

0.141 0.427 0.004

0.425, du/dE

applied. In CPSV the estimation of LYsuffers in a similar manner to global analysis although to a slightly lesser degree. With the global analysis method an apparently negative da/dE value is calculated for all resistance errors except when severe overcompensation occurs under near-reversible conditions. The calculation of the apparent dcu/dE in the presence of resistive errors in CPSV produces results different from the global analysis with a negative dependence only being

0.667 0.427

0.683 0.427

0.698 0.426

0.729 0.423

0.144 0.428 0.011

0.747 0.423

0.662 0.421 -0.195

0.782 0.418

0.680 0.424 - 0.107

0.801 0.419 -2.000

= 0.000 V-l,

1.21 0.433

1.18 0.428 - 0.531

1.26 0.432

1.24 0.430 - 0.359

1.35 0.424

1.34 0.423 - 0.270

-2.000 0.698 0.426 0.017

1.41 0.422

1.41 0.422 - 0.015

- 2.000 0.730 0.423 0.024

1.49 0.425

1.49 0.426 0.152

-2.000 0.751 0.427 0.125

1.60 0.417

1.62 0.417 0.294

-2.000 0.787 0.422 0.167

1.69 0.418

1.73 0.418 0.540

-2.000

-2.000 0.147 0.423 0.038

1.13 0.426 - 0.694

-2.000

- 2.000 0.144 0.432 0.031

1.16 0.433

Quadratic

-2.000

-2.000

- 2.001 (YO =

0.644 0.418 - 0.279

-2.000

-2.000 0.0717 0.435 - 0.027

0.651 0.427

Linear

- 2.000

-2.000

-2.001 0.0718 0.431 0.015

0.616 0.423 - 0.293

-2.000

- 2.000

- 2.001 0.0721 0.430

0.140 0.425

0.623 0.432

Quadratic

-2.000

-2.000

- 2.000

(da/dE)/V-’ -%2/V

0.0698 0.425 0.002

- 2.000 0.0709 0.428

0.137 0.417 - 0.054

-2.000

-2.000

(da/dE)/V-’

-200

0.700 0.421 - 0.018

-2.000

-%2/V

0.137 0.425

Linear

- 2.000

-2.000

- 2.000

G/r/V

0.136 0.414 - 0.070

- 2.000 0.0699 0.418 - 0.028

A = 1.40

A = 0.700 Quadratic

-2.000

-2.000

&r/V +30 n

A = 0.140

A = 0.0700

0.811 0.425 0.275

1.84 0.410

1.89 0.408 0.721

-2.000

E” = - 2.000 V, T = 293 K and electrode area 1.9 X 10m6

observed under conditions of residual uncompensated resistance. Overall, the presence of uncompensated resistance will cause an underestimation of da/dE and if overcompensation does occur, only CPSV will produce a positive da/dE when A is maintained at the lower end of the range investigated here. The estimation of E;,, is generally unaffected by the presence of any resistive error. The consequence of incorrect charging current com-

A.M. Bond et al. / Factors affecting calculation of electrode kinetics

24

pensation is reported in Tables 6 and 7 where the simulations were performed without the addition of a potential-dependent Q. Global analysis consistently predicts a more accurate value of Et2 than does CPSV, as expected from the earlier discussion. From eqn. (101, which relates to CPSV, it can be seen that accurate evaluation of E;,2 is a necessity if the “kinetic correction” to m is to be assessed correctly. This then influences the calculation of k”, (Y’ and da/dE. In

contrast, in global analysis the value of Ei,2 is basically only necessary for an accurate determination of k” and does not affect either (Y’ or da/dE. If E42 is known accurately, then in the presence of a charging current correction error, CPSV will of course be more accurate than the global analysis method when the arbitrary zero current procedure, as discussed in ref. 35, is applied to the experimental data. In this sense CPSV could be envisaged to be superior to global analysis.

TABLE 6. Determination of electrode kinetic parameters by applying the global analysis method to digitally simulated data a containing incomplete charging current correction Current offset -4%

Value

A = 0.0700 Linear 0.0620 0.453

A

a0 (da/d

E)/V-’

-3%

0.0731 0.447

A

a0 (da/dE)/V-’

A aa (da/dE)/V-’

0.0704 0.432

A a0 (da/dE)/V-’

0.0692 0.425

A a0 (da/d

E)/V-’

+l%

0.0681 0.418

A a0 (do/d

E)/V-’ 0.0657 0.419

A a0 (da/d

El/V-’

G/2/V

+3%

A a0 (da/dE)/V-’ EiJV

+4%

A

0.0648 0.397

0.0692 0.425 0.002

0.148 0.405 - 0.375

0.143 0.443

0.140 0.435

0.138 0.425

0.0668 0.432 0.073

0.137 0.415

0.145 0.411 - 0.253

0.136 0.405

0.141 0.418 -0.126

0.135 0.395

0.138 0.426 0.005

0.134 0.386

0.135 0.434 0.128

a0 = 0.425, du/dE

0.779 0.437

0.729 0.433

0.682 0.428

0.642 0.412

0.132 0.441 0.241

0.611 0.390

0.738 0.405 - 1.129

0.586 0.369

0.707 0.414 - 0.587

0.564 0.351 - 1.998

= 0.000 V-l,

E” = -2.000

1.86 0.436

1.70 0.383 - 3.728

1.67 0.434

1.55 0.397 - 2.372

1.50 0.431

1.43 0.410 - 1.189

-2.000 0.681 0.428 - 0.003

1.34 0.429

1.34 0.429 - 0.025

- 2.000 0.660 0.442 0.561

1.19 0.414

1.26 0.451 1.046

- 1.999 0.638 0.451 0.953

1.08 0.382

1.18 0.461 1.597

- 1.999 0.615 0.457 1.231

- 1.999 0.126 0.454 0.433

1.87 0.369 - 5.349

- 2.001

- 1.999 0.129 0.448 0.342

2.09 0.438

Quadratic

-2.001

- 1.999

- 1.997 m2 s-l,

0.772 0.398 - 1.687

- 2.000

- 1.998 0.0597 0.452 0.261

0.831 0.440

Linear

- 2.001

-2.000

- 1.998 0.0620 0.445 0.202

0.810 0.391 - 2.282

- 2.001

- 1.999 0.0644 0.439 0.139

0.887 0.443

A = 1.40 Quadratic

- 2.001

-2.000

- 1.994

a Simulation parameters: D = 3.33 X lo-’ m2.

0.146 0.451

Linear

- 2.001

- 2.000

- 1.996

:d:,dE),V-’ Ei&V

0.0716 0.418 - 0.074

- 1.997 0.0659 0.417

0.152 0.399 - 0.494

- 2.001

- 1.998

Ei&V +2%

0.0741 0.412 -0.151

- 1.999

Ei,r/V

0.149 0.457

Quadratic

-2.001

-2.001

Ei&V 0%

0.0765 0.406 - 0.228

- 2.002

G/,/V

-1%

0.0717 0.440

Linear

- 2.002

- 2.003

EI/,/V -2%

0.0790 0.400 - 0.305

- 2.004

E:,s/V

A = 0.700

A = 0.140 Quadratic

0.999 0.355

1.10 0.468 1.944

- 1.998 0.593 0.461 1.442

0.933 0.334

1.04 0.472 2.196

- 1.998 V, T= 293 K and electrode area 1.9 X 10m6

A.M. Bond et al. / Factors affecting calculationof electrodekinetics

25

TABLE 7. Determination of electrode kinetic parameters by applying the CPSV method to digitally simulated data a containing incomplete charging current correction with Ei,r calculated from ref. 8 Current offset

Value

-4%

A

A = 0.0700 Linear 0.0963 0.412

2 (da/dE)/V-’

-3%

A 120

0.0879 0.417

(da/dE)/V-’

A a0

0.0818 0.420

(da/dE)/V-’

A

0.0749 0.423

cr 0

(da/dE)/V-’

0%

A 00 (da/d

0.0698 0.425 E)/V-’

Ei,r/V +1%

A CYO

0.0652 0.427

E;,s/V

A (Yo (da/dE)/V-’ E,T,,/V

+3%

A ffo

0.0557 0.430

A

ff 0 (da/dE)/V-’ Ei,,/V

0.0753 0.418 - 0.026

0.188 0.360 - 0.346

0.173 0.407

0.161 0.413

0.150 0.419

0.0698 0.425 0.002

0.141 0.424

0.175 0.380 -0.231

0.132 0.429

0.162 0.396 -0.142

0.124 0.433

0.151 0.411 - 0.064

0.116 0.437

0.141 0.424 - 0.002

- 1.986

a Simulation parameters: D = 3.33 X 10e9 m2 s-l, m’. b Errors too great to perform complete analysis.

0.109 0.440

1.10 0.129 - 2.913

0.934 0.356

0.813 0.381

0.698 0.426

0.609 0.461

0.123 0.445 0.085

0.537 0.487

0.935 0.252 - 1.699

0.477 0.508

0.794 0.361 - 0.589

0.427 0.525 - 1.991

a0 = 0.425, da/d E = 0.000 V-r,

b b

b

b

b

b

b b

b

b

b

b b

1.92 0.360

1.77 0.253 - 2.213

- 2.003 0.698 0.426 0.017

1.41 0.422

1.41 0.422 - 0.015

- 2.000 0.617 0.470 0.323

1.11 0.500

1.14 0.504 0.636

- 1.998 0.548 0.505 0.511

0.908 0.548

0.940 0.558 0.901

- 1.995 0.488 0.533 0.638

- 1.993 0.108 0.461 0.146

b

Quadratic

b

- 1.995 0.115 0.453 0.118

b

b

- 1.998

- 1.989

Equations (3) and (4) show that the global analysis procedure relies on the difference between the forward and reverse sweep values of i and m to calculate the values of i * and m* at each potential. However, if a current offset is present on the reverse potential scan then this will result in the erroneous estimation of i* and m*. The results of this effect can be observed in Table 6 for global analysis where, even for low values of A, an apparently significant dcu/dE is introduced.

1.15 0.459

Linear

b

- 2.000 0.131 0.435 0.046

- 1.992 0.0520 0.446 0.072

1.33 0.010 - 3.862

- 2.002

- 1.994 0.0559 0.442 0.059

Quadratic

- 2.005

- 1.997 0.0601 0.437 0.043

1.32 0.639

- 2.007

- 2.000 0.0648 0.432 0.025

Linear

A = 1.40

-2.009

- 2.003

- 1.989 0.0532 0.431

Quadratic

- 2.005

- 1.993

(da/dE)/V-’ G/,/V +4%

0.0828 0.407 - 0.071

- 1.996 0.0609 0.428

0.186 0.400

- 2.008

- 2.000

(da/dE)/V-’

-I-2%

0.0897 0.397 -0.115

- 2.003

EI,z/V

Linear

-2.011

- 2.008

Ei,,/V -1%

0.0989 0.383 -0.181

-2.011

Ei,,/V -2%

Quadratic

- 2.015

Ei,,/V

A = 0.700

A = 0.140

0.758 0.581

0.788 0.597 1.037

- 1.992 0.437 0.557 0.730

0.643 0.605

0.669 0.629 1.115

- 1.990

E” = - 2.000 V, T = 293 K and electrode area 1.9 X 10e6

However, the apparent value of da/d E is only positive if the voltammogram is overcorrected where this corresponds to an upward displacement on the current scale of the reverse sweep. Interestingly, as the current of the reverse sweep is displaced upwards (i.e. overcorrection), the apparent value of (Y’ decreases for the linear analysis but increases for the quadratic analysis. However, as expected, both methods give the same value when no charging current error is present. The value

26

A.M. Bond et al. / Factors affecting calculation of electrode kinetics

estimated for A is linked directly to the calculation of I& and is therefore underestimated when the calculated potential is less than -2.000 V and overestimated when II;/2 is more negative than -2.000 V which occurs as a result of undercorrection. As usual, the results become less accurate as A approaches the reversible limit. Examination of Table 7 indicates that the errors in the determination of cr” are different for CPSV compared with global analysis. Generally, the apparent (Y’ value tends to increase as the reverse scan becomes further displaced negatively when using either linear or quadratic analysis, in contrast to global analysis, with the quadratic form of analysis being much more sensitive to such an effect. The estimation of A from CPSV, using either linear or quadratic analysis, suffers similarly to global analysis but with greater errors and this is because the calculation of E;,* is less accurate using the one point analysis from ref. 8 in comparison with the value calculated from global analysis. The apparent potential dependence of (Yfollows a similar trend to global analysis and although the relative magnitude of the error is less at lower values of A, this is reversed for higher values of A. In summary, global analysis is more sensitive to the influence of a charging current correction error but is likely to produce a more accurate estimation of E;,2 than CPSV. Consequently, a comparison of results obtained for CPSV analysis using the values of Ei,2 obtained from CPSV and global analysis will give a good indication as to the occurrence of a charging current correction error. 5. Conclusion An examination of the influence of the errors that may be present in a series of experiments to determine the electrode kinetics of a reaction demonstrates that reported results are likely to be dependent upon the method of calculation when experimental artifacts are present. The CPSV method is less affected by charging current subtraction errors if the more accurate EI,* value calculated from global analysis is used, whereas the calculation of kinetic parameters, particularly da/d E, from global analysis in this circumstance could contain significant error. Consequently, a comparison of results obtained by both the global analysis and CPSV methods is extremely useful to ascertain whether any artifact is affecting the outcome of either or both methods. Acknowledgments

The authors wish to express their appreciation to the Australian Research Council, the Natural Sciences

and Engineering Research Council of Canada and the Donors of the Petroleum Research Fund, administered by the American Chemical Society for their financial contribution to this project. An Australian Postgraduate Research Award (PJM) is also gratefully acknowledged. References 1 A.M. Bond, T.L.E. Henderson and K.B. Oldham, J. Electroanal. Chem., 191 (1985) 75. 2 C.G. Zoski, K.B. Oldham, P.J. Mahon, T.L.E. Henderson and A.M. Bond, J. Electroanal. Chem., 297 (1991) 1. 3 M.R. Anderson and D.H. Evans, J. Electroanal. Chem., 230 (1987) 273. 4 K.B. Oldham and J. Spanier, J. Electroanal. Chem., 26 (1970) 331. 5 J.C. Imbeaux and J.M. Saveant, J. Electroanal. Chem., 44 (1973) 169. 6 L. Nadio, J.M. Saveant and D. Tessier, J. Electroanal. Chem., 52 (1974) 403. 7 J.M. Saveant and D. Tessier, J. Electroanal. Chem., 61 (1975) 251. 8 J.M. Saveant and D. Tessier, J. Electroanal. Chem., 65 (1975) 57. 9 J.M. Saveant and D. Tessier, J. Electroanal. Chem., 77 (1977) 225. 10 E.R. Brown, D.E. Smith and G.L. Booman, Anal. Chem., 40 (1968) 1411. 11 E.R. Brown, H.L. Hung, T.G. McCord, D.E. Smith and G.L. Booman, Anal. Chem., 40 (1968) 1424. 12 J.C. Imbeaux and J.M. Saveant, J. Electroanal. Chem., 28 (1970) 325. 13 D. Garreau and J.M. Saveant, J. Electroanal. Chem., 35 (1972) 309. 14 P.E. Whitson, H.W. VandenBorn and D.H. Evans, Anal. Chem., 45 (1973) 1299. 15 D. Garreau and J.M. Saveant, J. Electroanal. Chem., 50 (1974) 1. 16 P. He and L.R. Faulkner, Anal. Chem., 58 (1986) 517. 17 D. Garreau, P. Hapiot and J.M. Saveant, J. Electroanal. Chem., 281 (1990) 73. 18 D. Brim, J. Electroanal. Chem., 88 (1978) 309. 19 E. AIhberg and V.D. Parker, J. Electroanal. Chem., 107 (1980) 197. 20 D.F. Milner and M.J. Weaver, Anal. Chim. Acta, 198 (1987) 245. 21 L.K. Safford and M.J. Weaver, J. Electroanal. Chem., 261 (1989) 241. 22 R.S. Nicholson, Anal. Chem., 37 (19651 1351. 23 J.M. Saveant and D. Tessier, J. Phys. Chem., 81 (1977) 2192. 24 D.A. Corrigan and D.H. Evans, J. Electroanal. Chem., 106 (1980) 287. 25 J.O. Howell, W.G. Kuhr, R.E. Ensman and R.M. Wightman, J. Electroanal. Chem., 207 (1986) 77. 26 D.A. Corrigan and D.H. Evans, J. Electroanal. Chem., 233 (1987) 161. 27 K.B. Oldham and C.G. Zoski, J. Electroanal. Chem., 145 (1983) 265. 28 R.A. Marcus, J. Chem. Phys., 43 (1965) 679. 29 R. Bonnateree and G. Cauquis, J. Electroanal. Chem., 35 (1972) 287. 30 M.J. Weaver and F.C. Anson, J. Phys. Chem., 80 (1976) 1861. 31 J.M. Saveant and D. Tessier, Faraday Discuss. Chem. Sot., 74 (1982) 57.

A.M. Bond et al. / Factors affecting calculation of electrode kinetics 32 R.A. Petersen and D.H. Evans, J. Electroanal. Chem., 222 (1987)

33 34 35 36 37 38

129. H.H. Bauer, J. Electroanal. Chem., 16 (1968) 419. Z. Samec and J. Weber, J. Electroanal. Chem., 77 (1977) 163. A.M. Bond, P.J. Mahon, E.A. Maxwell, K.B. Oldham and C.G. Zoski, in preparation. C. Amatore, J.M. Saveant and D. Tessier, J. Electroanal. Chem., 146 (1983) 37. AK Hoffman, W.G. Hodgson, D.L. Maricle and W.H. Jura, J. Am. Chem. Sot., 86 (1964) 631. M.E. Peover and J.S. Powell, J. Electroanal. Chem., 20 (1969) 427.

21

39 S.W. Feldberg, in A.J. Bard (Ed.), Electroanalytical Chemistry, Vol. 3, Marcel Dekker, New York, 1969, pp. 199-296. 40 H. Matsuda and Y. Ayabe, Z. Elektrochem., 59 (1955) 494. 41 A.J. Bard and L.R. Faulkner, Electrochemical Methods, Wiley, New York, 1980, Chapter 6. 42 J.C. Myland, K.B. Oldham and C.G. Zoski, J. Electroanal. Chem., 206 (1986) 1. 43 R.M. Wightman and D.O. Wipf, in A.J. Bard (Ed.), Electroanalytical Chemistry, Vol. 15, Marcel Dekker, New York, 1989, pp. 267-353. 44 C. Kasper, Trans. Electrochem. Sot., 77 (1940) 353. 45 L. Nemec, J. Electroanal. Chem., 8 (1964) 166.