Global kinetic analysis of cyclic voltammograms at a spherical electrode

J. Electroanal. Chem, 291 (1991) 1-17 Elsevier Sequoia S.A., Lausanne

Global kinetic analysis of cyclic voltammograms at a spherical electrode Cynthia Department

G. Zoski of Chemistry,

Keith B. Oldham Department (Received

University of Rhode Island

*, Peter J. Mahon,

of Chemical and Analytical

Kingston, RI 02881-0801 (USA)

Terry L.E. Henderson

and Alan M. Bond **

Sciences, Deakin University, Geelong. Victoria 3217 (Australia)

9 May 1990; in revised form 10 July 1990)

Abstract

The global method for analyzing electrochemical processes controlled by diffusion and by the kinetics of heterogeneous electron transfer is redeveloped for application to electrodes having a spherical geometry. Heretofore, the method has been restricted to experiments in which planar diffusion is the sole mode of transport to the electrode surface. The global method is based on a three-dimensional concept and has the advantage of using all of the data available from a voltammogram to measure the heterogeneous charge-transfer rate constant, the reversible halfwave potential, the diffusion coefficient, and the charge-transfer coefficient. Cyclic voltammetry of 2-methyl-2nitropropane reduction at a mercury sphere is reported. The parameters measured by spherical global analysis are concordant and in excellent agreement with literature values.

INTRODUCTION

Several years ago, a three-dimensional method for studying electrochemical processes controlled jointly by diffusion and kinetics was reported. The so-called “global analysis” [l], based on a concept pioneered by Reinmuth [2], provides a unique surface in (m, i, E) space that is independent of the details of the experimental method selected. Here E and i are the electrode potential and current, while m is the temporal semiintegral of i.

l Permanent Canada. l * Permanent

address: address:

0022-0728/91/$03.50

Department LaTrobe

of Chemistry,

University,

Trent

Chemistry

0 1991 - Elsevier Se&oia

University,

Department, S.A.

Peterborough, Bundoora.

Victoria

Ontario

K9J 7B8

3083, Australia.

2

The global analysis has been demonstrated [1,3] to be an efficient method of determining the standard heterogeneous charge-transfer rate constant, the chargetransfer coefficient, and the reversible halfwave potential for a simple electrode reaction, as well as the electroreactant diffusion coefficient. More recently [4], the global analysis has been extended to accommodate electrode reactions complicated by homogeneous chemical reactions. Three advantages exist for this technique compared with alternative electrochemical methods of evaluating kinetic and thermodynamic parameters. First, the global analysis uses all the data from a single cyclic voltammogram, thus affording high precision. In contrast, the traditional cyclovoltammetric technique [5] relies on the separation of the two peaks of each cyclic voltammogram so that the majority of the experimental data remains unused. Similarly, potentiostatic methods [6-91 access the kinetics at only one or two potentials in a single experiment. A related convolution method [lo] processes only half of the acquired experimental data. Secondly, the global analysis is based on fundamental electrochemical diffusive transport and kinetic principles which are familar to most experimental electrochemists. Moreover, the global procedure can be easily programmed allowing rapid on-line calculation of kinetic and thermodynamic parameters by digital computer. The simplicity and speed of the global procedure are distinct advantages over the more laborious simulation optimization approach [ll-141 or complex nonlinear fitting technique [15-171 which are of a complexity that will appeal to few electrochemists. Similar in objective, though distinct in concept, is the three-dimensional procedure advocated by Parker [18] under the name “normalized potentialsweep voltammetry”. Thirdly, the global method is not based on any particular model of electrode kinetics. It is usual in electrode kinetic studies to assume that the rate of the electrode reaction is related to the electrode potential and the surface concentrations of oxidized and reduced species via the well-known Butler-Volmer equations. This assumption, which underlies semiintegral linear sweep [19,20] and most other methods of determining kinetic parameters, is not necessary in the global approach. Thus, kinetic and thermodynamic parameters can be determined without the prejudice of a preassumed relationship between the rate constant and potential. In fact, the three-dimensional method provides a useful diagnosis of whether Butler-Volmer kinetics is appropriate in a particular electrode reaction. A disadvantage of the global analysis heretofore is that the method is restricted to the evaluation of data acquired at planar electrodes. For reasons of mathematical simplicity, shrouded planar electrodes are assumed in most voltammetric derivations though they are seldom used in practice due to fabrication difficulties. Instead, the static or hanging mercury drop is nowadays a favourite electrode of many experimental voltammetrists. These spherical electrodes cannot always be approximated adequately by a planar model [21-311. The purpose of this article is to redevelop the global analysis for application to spherical electrodes. The technique is then applied to the well-studied [12-14,32-361

3

reduction

of 2-methyl-2nitropropane

&H&NO, to its radical GENERAL

(CH,),~NO;

+ e-+ anion

(1)

in acetonitrile.

THEORY

Consider a stationary spherical electrode of radius r, which, for times t < 0, is held at a potential such that no reaction occurs. Subsequently to t = 0, potentials are encountered such that the n-electron reduction O(soln)

+ n e-*

R(soln)

(2)

occurs. In this section we will not prescribe any particular polarizing signal, though later we will consider a triangular potential ramp. The electroreactant 0 is initially present at a bulk concentration cb but the product R is initially absent, so that ci = 0. Both species are soluble only in the electrolyte solution which is unstirred and contains excess supporting electrolyte to ensure that diffusion is the sole transport mode. The symbols Do and D, represent the diffusion coefficients of species 0 and R respectively. Reaction (2) is assumed to occur bidirectionally with the forward rate proportional to c& the concentration of species 0 at the electrode surface, and the backward rate proportional to ck, the concentration of species R at the electrode surface. The net rate of the electron-transfer reaction, measured by the current i according to Faraday’s Law, can be equated to the difference between these two first order reaction rates i/nAF

= rate = k,c& - k,c”,

(3)

where A is the area of the spherical electrode, F is Faraday’s constant, and k, and k, are the two rate constants. These two constants are expected to depend on potential in the sense that k, will become larger, but k, smaller, as the potential is made more negative. If eqn. (3) is to apply at equilibrium, then the two rate constants must be in the ratio k,/k,=exp((nF/RT)[E-E”])

(4

where E” is the standard (more strictly the formal, or conditional) potential, in order that the Nemst equation be obeyed when the rate is zero. Being soluble in the electrolyte solution, 0 and R diffuse in a transport field bounded by a convex spherical surface, the electrode. Under such circumstances, it has been demonstrated [37] that the faradaic current is related to the concentrations of 0 and R at the electrode surface by i=nAF&-$&(cb-c&)+F(cL-c&)=nAF&$ci+Fc&

4

where d’/*/d t’/* represents the semidifferentiation operator with respect to time t. Each term involving r0 is a correction for the curvature of the electrode. Laplace transformation [38] of eqn. (5) leads to

[

1

We now make the approximation difference between Do and D,,

wi>

= nAFgP’{

h + G/r0

1 1

&+T JORP{c;}

&+,- JDO9{&-c&}=nAFJD,

Y{i}=nAFJDO

that, in the correction terms only, we can ignore the so that

& - c;}

= nAFJD,P{

(7)

c;}

where D represents some mean diffusion coefficient. Laplace inversion that multiplies U{ i} in the left-hand side of eqn. (7) gives [29] -$ where the f function f(x)

= l/&G

Hence,

exp($)

erfc(%)=$f($)

by (9)

use of the convolution

After combination

theorem

with eqn. (7), Laplace

c&) = nAF&cS,

of the factor

(8)

- exp( x) erfc(J;;)

making

nAFK($,-

is defined

(6)

[39], we are led to

inversion

= $$(T)

produces

f( D’ti

“)

d7 = p

(11)

The integral in eqns. (10) and (11) is technically known as a convolution integral; the procedure which uses it has been called “spherical convolution” [29,40]. The symbol p is used to represent the spherical convolution integral in eqn. (11). This equation relates the concentrations of 0 and R at the surface of a spherical electrode to the faradaic current. By combining eqns. (3) (4) and (11) to eliminate cb and cS,, one arrives at the universal equation

i=

nAF&

- p

exp((nF/RT)[E-E”])

(12)

which provides a unique relationship between i, ~1, and E, irrespective of the experimental technique employed. Special cases of eqn. (12) arise when ~1or i is equal to zero. When p = 0, i adopts the special value i * = nAFc:k,

(13)

5

P*

I+

L!!c M

1

Eg

E

E:,

Fig. 1. Two-dimensional

relationship

between

i*

Fig. 2. Two-dimensional

relationship

between

p* and E when the current

and E when the convolute

E'‘4

p equals zero. equals zero.

where i * represents the faradaic current that would have flowed across the spherical working electrode at potential E if there were no transport impediments whatsoever. Because k, depends on potential, there is a two-dimensional relationship between the current and potential at p = 0, as illustrated in Fig. 1, which contains purely kinetic information. Similarly, when i = 0, p adopts the special value exp((nF/RT)[E-

,u* = nAFck

E”])

-’

(14)

I where p*/nAT& expresses the concentrations that R would have attained at the spherical electrode surface had there been no kinetic impediments to the electrode reaction. Figure 2, containing purely transport and thermodynamic’ information, illustrates the two-dimensional relationship between p* and E at i = 0. Substitution of expressions (13) and (14) into eqn. (12) leads to the very simple relationship i/i*+p/p*=l

(15)

at any constant potential E. This predicted linear interdependency of i and p is shown graphically as Fig. 3. Thus values of i* and CL*may be determined as the intercepts of a plot of i versus the p value at the same potential. Because two noncoincident points determine a straight line uniquely, any experiment that generates two (i, p) pairs at a common potential can be used to determine the intercepts i * and p*. Graphs similar to Figs. 1 and 2 can then be constructed by determining i * and p* values at several potentials. Though the potential E may be

iA i*-

Fig. 3. Linear

relationship

between

i and fi at any constant

potential.

6

varied in many ways, a later section will consider one of the simplest means of scanning the useful potential range. One should appreciate tha? no particular dependence of the electron-transfer rate upon potential was presupposed in deriving eqns. (12)-(15) although first-order kinetics were specified in eqn. (3). A test of this first-order kinetic assumption is provided by rearranging eqn. (14) to 1

1

+ exp( -nFEO/RT)

*EL = nAF&~

nAFc;JD,

exp( nFE/RT)

(16)

which predicts that a plot of l/p* versus exp(nFE/RT) will be linear. Turning to eqn. (13), k, can be calculated from values of i* at several potentials as i */nAFc& thus permitting its potential dependence to be determined. Expression (4) then provides a route for investigating the potential dependence of k,. The dependence of k, and k, on potential is exponential only if Butler-Volmer kinetics are obeyed with a constant transfer coefficient. In such a case, a plot of In i* versus E will be linear. In the next section, we recast expressions (12), (13) and (16) in terms of the Butler-Volmer model. BUTLER-VOLMER

MODEL

The Butler-Volmer k, = k”

equations

exp(( -anF/RT)[E

- E”])

(17)

and k,=k”

exp({(l-a)nF/RT}[E-E’])

(18)

are almost universally believed to express the dependence of the reaction rate constants on potential, though there is not unanimity on whether the charge-transfer coefficient (Y should be regarded as constant. In eqns. (17) and (18), k” is the heterogeneous charge-transfer rate constant at the standard potential E O. Substituting eqn. (17) for k, in the general equation (12) leads to

+ exp ($[E-E;/,l)=I-~[l+exp(~[E-E;,I])]

(19)

where we define I = nAFc:k

o (Do/D,)

a’2

(20)

M = nAFc&&

and the reversible

(21)

halfwave

E1;2 = E o + (RT/2nF)

potential

ln( DdD,)

The significance of these three new constants in the relationships potential are shown in Figs. 1 and 2. TJsually only approximate

(22)

of i * and CL* to values for M and

Fig. 4. A plot of l/p* determined.

versus exp(nFE/RT)

Fig. 5. A plot of ln i * versus

- nFE/RT

from the slope and intercept

from the slope and intercept

of which

E{,2 and

M are

of which a and 2 are determined.

E’1,2 are available, a priori. This is generally due to the wide variation in diffusion coefficient and standard potential values reported in the literature. In spherical global analysis, no value of Ei,2 is required, and only a crude diffusion coefficient. In terms of Butler-Volmer kinetics, eqns. (13) and (16) can be rewritten in forms that emphasize the linear relationships In i * = In I + anFEf,,/RT between In i* and -nFE/RT, 1 *= II

- anFE/RT and for

exp( - nFE;,,/RT) &+

M

(23)

exp

(

nFE RT

between l/p* and exp(nFE/RT). These 5. Hence a plot (or a least-squares exp(nFE/RT) will give a straight exp( -nFEi,,/RT)/M. The M and Ef,2 M = l/intercept

1

(24)

relationships are illustrated in Figs. 4 and regression calculation) of l/p* versus line of intercept l/M and gradient values are then found from (25)

and

(26) as shown in Fig. 4. Similarly, a plot (or a corresponding calculation) of ln i * versus - nFE/RT will give a straight line (if cw is a potential-independent constant) of slope (Y and intercept In I. The kinetically significant result is the I/M ratio, equal to I/M

= I/nAFc;fi

= k”D~-‘~f’2D,“/2

Usually Do = D, so that the I/nAF& standard rate constant k O.

(27) ratio will be within experimental error of the

8 THE POLARIZING

SIGNAL

The simplest means of scarming all useful potentials and obtaining two (i, CL) pairs at each potential is via cyclic voltammetry, as illustrated in Fig. 6. The applied signal consists of a negative-going ramp with slope --u and starting at an initial potential E, where the current is negligible, so that E = E,, - vt

O
(28)

where 0 is the duration of the experiment. The ramp continues until time 8/2 by which time E has reached a value about 100/n mV more negative than E O. The ramp direction is then reversed, so that (29)

8/2
E=E,-v(&t)

Useful data will cease to be collected once the potential has passed a value 100/n mV more positive than E “, though the potential ramp could conveniently return to E,,. If the ramp is simulated by a staircase, voltage increments should be as small and n?merous as possible. If i and F are the values of i and p at some potential on the forward sweep and i and F are the corresponding values at the same potential during the reverse sweep, then the equations i* = (k;-r;i’)/(k-c)

(30)

and

may be used to determine i* and p* from two data pairs. The terms i * and p* are, in fact, the quantities appearing in eqns. (23) and (24) and are defined in Fig. 3.

ib

(b>

u r

cc>

r’ ____---2 --____

-

Fig. 6. (a) Potential versus time (b) curre_nt_versus potential and (c) convolved current versus potential graphs for cyclic voltammetfy. (- - -) I, i, $, and /i at a specific potential.

IMPLEMENTATION

OF THE ANALYSIS

Modern voltammetric instrumentation provides digital current data at a set of equally spaced time intervals. We let A represent the time interval between data and the odd number 2N + 1 be the total number of current data zo, I~,.. ., zj ,..., i,,_,, i,,. Hence 2NA = 8. The initial potential is selected so that the initial current i,, is zero. The potential programme provides E, data at times corresponding to each i,. Thus, for a cyclovoltammetric experiment, the results take the form of Table 1. Hence i, and iZN_j are currents measured at the same potential El: they correspond to an (z 8 pair in the notation of eqns. (30) and (31) as shown in Fig. 6. The fourth column of Table 1 illustrates values of p which result from convolving the current data of the third column while the seventh column results from the conjoined third and sixth columns. The entries pi and CLZN_jconstitute a ($, iL!Lpair in eqns. (30) and (31) and as illustrated in Fig. 6. The convolution integral given in eqn. (11) is computed using a published algorithm [29] which utilizes the equations

J=l,

2, 3,...2N

(32)

where pJ is a member of the ,ii family when J < N and a member of the F family when J > N. The h coefficients are given by - exp( jb - b) erfc,/Jm

t4

+2exp(jb)erfc@

- exp( jb + b) erfc,/‘J’

TABLE

(33)

1

Format of the data for the spherical global analysis. The symbol ~represents during the forward sweep; E the time at each potential during the backward

E

F

7 I

r;

z

Eo -4

0

0

0

4

Pl

E2

A 2A

i2

P2

2NA (2N -l)A (2N -2)A

Ej

jA

‘J

PJ

EN-2

EN-l EN

jN-2)A (N-l)A NA

i2N- j)A

the time at each potential sweep

t I

F

P*

i*

j2N

p2N

0

0

i2N-l

PZN-I

6

i,*

i2N-2

PZN-2

a;

i2*

i2N_j

PZN-j

Pf

j;

i,_,

PN-2

jN+2)A

I,,,

pN+2

pN+2

‘*

;;+,

iN-I

PN-1

cN+ljA

iN+l

pN+l

P%+,

iZ+l

iN

PN

10

forlgj
\i

i

+ b - exp( b) erfcfi

(34)

with b = DA/r,‘. The referenced algorithm, and the computer programme based on it, were adapted for the present work with the only change being replacement by of the command “Set a = A/(289455nVb2)” found in the “Set a = Am/r,b2” original (Appendix B of reference [29]). Data processing initially makes use of eqns. (30) and (31) in the form .* _ 1.i -

P2iv-jij-

Pji2N-j

J=

1, 2, 3,...

N

J=

1, 2, 3,...

N

p2N-j-pj

and pf=

p2N-j'j-

pj'2N-1

ij - i2N_j

which are the algebraic equivalents of the graphical extrapolations shown in Fig. 3. Next, from a plot of l/p; versus exp(nP’E,/RT) similar to Fig. 4, one finds values for M and Et,= according to eqns. (25) and (26). Similarly, OLand I are easily found as in Fig. 5. Finally, the standard rate from a plot of In if versus -nFE,/RT constant k o is found from the I/M ratio given by expression (27). The many points used in constructing Figs. 4 and 5 have a wide range of inherent reliabilities and require some form of weighting in order to fit the “best” straight line through all the available data. A weighted-least-squares procedure [1,39,41] was used in this study. TESTING OF THE ANALYSIS

To verify the spherical global analysis concept, we first employed simulated cyclic voltammograms which were generated using an explicit finite difference technique [42]. Both the simulation and the analysis assumed Butler-Volmer kinetics and semiinfinite spherical diffusion. Table 2 summarizes the results of subjecting the simulated current data to the spherical global analysis. The table considers three values of k” for each value of the sphericity factor [30] defined as u = (l/r,)dv

(37)

Thus, u values of 0.0010, 0.0100, and 0.1000 correspond to a spherical radius of 8.014 x lop4 m, 8.014 X low5 m, and 8.014 X lop6 m respectively under the conditions assumed in Table 2. Inspection of this table illustrates the success of the spherical global analysis in determining kinetic and thermodynamic parameters of an electrode reaction over a range of electrode sizes and rate constant values. The need to take sphericity into account in the global analysis is established in Table 3. This table lists the results when simulated spherical data are processed using the global analysis based on a planar geometry [l]. As the sphericity factor IJ decreases (corresponding to a decrease in electrode curvature), the variables D, El,,, a, and

11 TABLE

2

Results of subjecting simulated cyclic voltammetric data to the spherical global analysis. The simulations assumed values of (I = 0.5000, Do = D, = 2.500 X 10m9 m* SC’, Err,* = - 0.05000 V, II = 100.000 V s-l, and T = 298.15 K. Three sphericities e and three rate constants k o are considered Values recovered

Values assumed 0

102ko/m

0.0010

0.0550 0.5500 5.5000

0.0100

0.1000

s-l

109D/m2

s-l

G/2/V

a

lO*k “/m

2.500 2.502 2.503

- 0.05000 - 0.05005 - 0.05006

0.4999 0.5010 0.5110

0.05500 0.5506 5.528

0.0550 0.5500 5.5000

2.500 2.501 2.502

- 0.05000 - 0.05005 - 0.05006

0.5000 0.5009 0.5086

0.05501 0.5510 5.558

0.0550 0.5500 5.500

2.495 2.498 2.499

- 0.05002 - 0.05007 - 0.05008

0.5006 0.4999 0.4968

0.05502 0.5537 5.814

s-t

k” take values which are numerically closer to the input values used in the simulations. Of experimental interest is the question of how well the spherical global analysis is able to cope with current data that contain extraneous contributions, such ‘as spurious “ noise”. To investigate such a situation, data sets were created by adding to each ij in Table 1 a gaussian noise component distributed about a mean of zero with a root-mean-square of 2% of the forward peak current of each cyclic voltammogram. The gaussian noise was generated pseudorandomly using a standard method [8,39]. The effect of subjecting the resulting noisy data to the spherical global analysis is tabulated in Table 4. Evidently, the analysis is able to extract kinetic parameters containing only modest errors, even when the faradaic data are contaminated by noise. Smoothing the current data using a nine-point weighted moving-average [43], as applied to experimental data, resulted in retrieved kinetic

TABLE

3

Kinetic and thermodynamic values calculated when simulated cyclic voltammetric data for a spherical electrode are analyzed using the plunar version of the global analysis. Simulation parameters are identical to those listed in Table 2. A rate constant of 0.5500 X lo-* m s-’ was used in these simulations Values assumed d

Values recovered

0.0010 0.0100 0.1000

2.680 3.116 4.968

109D/m2

s-r

472

/v

- 0.04922 - 0.04777 - 0.04481

a

102ko/m

0.5120 0.4733 0.3883

0.4439 0.3277 0.2174

s-t

12 TABLE 4 Results of

subjecting noisy simulated cyclic voltammetric data to the spherical global analysis. Simulation parameters were identical to those listed in Table 2. A sphericity of CJ= 0.1000 was used

Values assumed 102k “/m s-l

Values recovered 109D/m2 SC’

452

0.0550 0.5500 5.5000

2.363 2.423 2.447

- 0.03712 - 0.04890 - 0.04931

and thermodynamic Table 4.

values

which

were

/V

slightly

a

102k O/m s-’

0.4887 0.5039 0.5126

0.4459 0.5563 5.724

better

than

those

reported

in

EXPERIMENTAL

Solutions of 5.00 mM (CH,),CNO, in 0.100 M (C,H,),NClO, were prepared in acetonitrile (Mallinckrodt, ChromAR-HPLC grade) freshly distilled daily from calcium hydride. Tetraethylammonium perchlorate (Southwestern Analytical Chemicals, Inc.) was recrystallized from methanol, dried in vacua, and stored over P205. Bubbling with nitrogen prior to each experiment was used to deoxygenate the solutions. The water-jacketed cell was thermostatted at 20” C throughout and all experiments were performed in a Faraday cage. A Metrohm E410 hanging mercury drop electrode served as the working electrode. The radius of the spherical drop, determined from the averaged mass of ten drops, was z 3.9 x 10e4 m in these experiments. A platinum ring served as the counter electrode and the reference electrode consisted of a silver strip in a CH,CN solution containing 0.010 M AgNO, and 0.100 M (C,H,),NClO,. A PAR 175 waveform generator produced the triangular voltage which was input to an Amel 551 potentiostat. Positive feedback was applied in all experiments. The degree of compensation was adjusted until the cyclic voltammogram for the reversible reduction of p-nitrotoluene under identical experimental conditions exhibited a reversible shape. This method of iR compensation has been suggested by Whitson et al. [44]. Data were acquired at time intervals corresponding to 1 mV increments using a Gould Model 4035 digital storage oscilloscope and transferred to an Olivetti Ml4 personal computer. Experiments performed on blank solutions of supporting electrolyte only, or on solutions containing both supporting electrolyte and (CH,),CNO,, were replicated three times and the averaged currents were stored serially. Current data were background corrected, interpolated, and smoothed using a nine-point weighted moving-average. Interpolation of i data was necessary because the forward and reverse scan rates were not exactly equal and because reversal did not always occur at a sampled point. The smoothed current data were spherically convolved off-line to generate the required (< ji, i, and ~3 quartets for the determination of sets of i * and p* values.

13

RESULTS AND DISCUSSION

A typical experimental cyclic voltammogram for the 2-methyl-2-nitropropane reduction and the result of spherically convolving this curve are shown in Figs. 7 and 8 respectively. The two-dimensional relationships which exist between i* and

60 )

1.60

1

1.90

2.10

2.00

2.20

-E /V Fig. 7. () An experimental cyclic voltammogram for the reduction of 5.00 mM (CH,),CNO, 0.1000 M (&H,),NClO, +CH,CN at a sweep rate of 20 V s-l. (- - -) Calculated i* data.

6

in

6 /---

5 . 4

3

--I

f

YI

~

1.60

1.90

2.00

2.10

2.20 ’

-E IV Fig. 8. ‘Ihe convolved current-versus-potential (- - -) Calculated p* data.

graph ( -)

of the

cyclicvoltammogram

of Fig. 1.

0

>O

1000

exp(nF(E-E'dRT)

Fig. 9. Graph of l/jr* versus exp(nF(E - E*)/RT) for the data from Fig. 8. (. . .) Selection of the ) Weighted regression line. The potential E * is an arbitrary potential used experimental points. (to keep the values of the exponentials within computational limits.

E, and between p* and E are also shown as the dashed lines in these figures. These data were used in generating the relationships between l/p* and exp(nFE/RT) shown as Fig. 9. Though the first of these diagrams displays considerable scatter, the predicted linear relationships are generally obeyed. The information contained in the slopes and intercepts of graphs such as these is summarized in Figs. 4 and 5. Table 5 lists the results from Figs. 9 and 10 and similar graphs from current data recorded at several sweep rates. These sweep rates were chosen to avoid the slow decomposition of the 2-methyl-2-nitropropane radical anion in acetonitrile [12,32]. Not only are the tabulated results remarkably consistent, but the small discrepancies that occur show no trend with sweep rate. The experimental values measured in this work display excellent agreement with values of D = (2.5-3.0) x lop9 m2 s-l, E’,,2 = -2.00 V, cu= 0.45, and k” = (3.2-3.7) x 10e4 m s-l reported in the litera-

TABLE 5 A summary of experimental results for the reduction of 2-methyl-2-~tropropane values represent standard deviations

in acetonitrile. The f

v/v s-1

109D/mz s-’

E&r/V

a

104k O/m s-r

10 20 50 100 200

4.06 3.71 3.90 3.93 3.76

-

0.421 0.440 0.430 0.426 0.430

2.96 3.00 3.18 2.88 3.45

averages

3.9*0.1

- 1.9~*0.~5

0.429 f 0.007

3.1+0.2

1.989 1.994 1.998 1.990 1.999

15

6-

a

-a -nF(E-E* )/RT

Fig. 10. Graph of In i* versus - nF( E - E *)/RT for the data in Fig. 7. The dots represent a selection of experimental points and the solid line is a weighted regression line. The potential E* is described in Fig. 9.

ture [10,12,13] for experiments under the same conditions. Neither our data, nor those in the literature, employed double layer corrections. All workers have also assumed that the background charging current is the same in the presence and absence of the electroactive 2-methyl-2nitropropane. Because the three-dimensional global analysis samples the entire potential spectrum and uses all available data, it enables a rigorous test of electron transfer models to be made. The linearity shown in Figs. 9 and 10 provides strong evidence that the Butler-Volmer model of electrode kinetics is obeyed for the 2-methyl-2nitropropane system. Moreover, the linear relationship between In i * and -nFE/RT apparent in Fig. 10 suggests that (Yis not potential dependent. This result contradicts previous findings [10,14] in which cy was found to vary with potential. The reasons for this discrepancy are unknown. However, it can be noted that the reported dependence on a is a small effect and that the following differences apply: (a) the previously used method [lo] of data analysis processed only the data from the forward half of the cyclic voltarnmogram whereas our method processes all of the data; (b) sphericity is built into our method whereas a different method of correction is used in the earlier study [lo] or experimental attempts were made to avoid its presence [14]; (c) uncompensated resistance effects may not be the same in the different methods. While the question of the dependence of (Yon potential is an intriguing one, it is not the main theme of this paper. Further studies specifically aimed at addressing this discrepancy are needed before the question of the dependence of (Yon potential (or otherwise) can be satisfactorily resolved.

16 SUMMARY

We have redeveloped global analysis to accommodate the curvature inherent in spherical electrodes. Spherical global analysis retains all of the advantages of its planar counterpart, which include: (1) All the data from a single experiment are used which improves the accuracy and minimizes noise effects; (2) All parameters are calculated from a single experiment so that irreproducibilities between runs are avoided; (3) The analysis is objective and, unlike simulation-curve-fitting methods, requires no recursive operations or subjective judgement; (4) The analysis provides an internal check on the validity of Butler-Volmer kinetics. The global analysis shares with other electrochemical methods of determining transport and kinetic parameters the requirement of accurate, albeit non-trivial, compensation for capacitance and solution resistance, as well as the absence of homogeneous chemical reactions. Studies are in progress in our laboratories to examine the effects of uncompensated resistance and charging current, as well as following chemical reactions, on the global analysis. ACKNOWLEDGEMENTS

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