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J. E:ecrroanal. Chem. 181 (1984) 65-81 Elsevier Sequoia S.A.. Lausanne - Printed in The Netherlands 65 THE APPLICATION OF CONVOLUTION POTENTIAL SWEEP VOLTAiMMETRY AT A ROTATTNG DISC ELECTIRODE TO SIMPLE ELECTRON TRANSFER KINETICS J-L. VALDES and H.Y. CHEH Department 01 Chemical Engineering and Applied Chemistry Colum5ia Vniuersily, New York, NY 10027 (U.S.A.) (Received lltb April 1984; in revised form 26th June 1984) ABSTRACT Convolution potential sweep volranunetry at a rotating disc electrode (CPSV-RDE) is shown to be an effective technique in the study of simple electron transfer reac:ions and can be applied to the measurement of kinetic parameters, in particular the standard rate constant and the transfer coefficient. A digital computer data acquisition system is employed to measure experimental current response transients to an imposed linear potential sweep. A convolution transformation, based on a mass transport model of the rotating disc electrode, affords experimental concentrations of a reacting species at the electrode surface. This informalion can be used to characterize a reaction in terms of a kinetic model. Two experimental studies were conducted and demonstrate good agreement between the results of tie present technique and those reported in tbe literature. INTRODUCTION Convolution potential sweep voltammetry (CPSV) : _m non-convective systems has been applied to the study of a wide range of electrochemical reaction mechanisms particularly for organic systems [l-4]. The essential feature of this techrtique stems from the ability to determine the surface concentration of a reacting species directly from a convolution transformation of the current response transient, when the electrode potential is changed with time in a predetermined manner. The convolu- tion procedure results as a mathematical means for eliminating the mass transport polarizat.ion effects that is often inherent in experimental LSV current-potential data. The principal advantage of using a convolution analysis in treating LSV data is an increase in the accuracy of mechanism diagnosis and kinetic parameter de- te_rztination. This is a direct consequence of using’ information along the entire current-potential curve, instead of only the peak current and corresponding poten- tial values. Furthermore, -it is not necessary, in the study of potential-dependent phenomena to assume an CI priori knowledge of a kinetic model in order to compare iL with experimental data [5]. This significant advantage of -CPSV over the more OOZZ-0728/84/%03.00 0 1984 Ekvier Sequoia SA.

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Page 1: The application of convolution potential sweep voltammetry at a rotating disc electrode to simple electron transfer kinetics

J. E:ecrroanal. Chem. 181 (1984) 65-81 Elsevier Sequoia S.A.. Lausanne - Printed in The Netherlands

65

THE APPLICATION OF CONVOLUTION POTENTIAL SWEEP VOLTAiMMETRY AT A ROTATTNG DISC ELECTIRODE TO SIMPLE ELECTRON TRANSFER KINETICS

J-L. VALDES and H.Y. CHEH

Department 01 Chemical Engineering and Applied Chemistry Colum5ia Vniuersily, New York, NY 10027 (U.S.A.)

(Received lltb April 1984; in revised form 26th June 1984)

ABSTRACT

Convolution potential sweep volranunetry at a rotating disc electrode (CPSV-RDE) is shown to be an effective technique in the study of simple electron transfer reac:ions and can be applied to the measurement of kinetic parameters, in particular the standard rate constant and the transfer coefficient. A

digital computer data acquisition system is employed to measure experimental current response transients to an imposed linear potential sweep. A convolution transformation, based on a mass transport model of the rotating disc electrode, affords experimental concentrations of a reacting species at the electrode surface. This informalion can be used to characterize a reaction in terms of a kinetic model. Two experimental studies were conducted and demonstrate good agreement between the results of tie present technique and those reported in tbe literature.

INTRODUCTION

Convolution potential sweep voltammetry (CPSV) : _m non-convective systems has been applied to the study of a wide range of electrochemical reaction mechanisms particularly for organic systems [l-4]. The essential feature of this techrtique stems from the ability to determine the surface concentration of a reacting species directly from a convolution transformation of the current response transient, when the electrode potential is changed with time in a predetermined manner. The convolu- tion procedure results as a mathematical means for eliminating the mass transport polarizat.ion effects that is often inherent in experimental LSV current-potential data.

The principal advantage of using a convolution analysis in treating LSV data is an increase in the accuracy of mechanism diagnosis and kinetic parameter de- te_rztination. This is a direct consequence of using’ information along the entire current-potential curve, instead of only the peak current and corresponding poten- tial values. Furthermore, -it is not necessary, in the study of potential-dependent phenomena to assume an CI priori knowledge of a kinetic model in order to compare iL with experimental data [5]. This significant advantage of -CPSV over the more

OOZZ-0728/84/%03.00 0 1984 Ekvier Sequoia SA.

Page 2: The application of convolution potential sweep voltammetry at a rotating disc electrode to simple electron transfer kinetics

common technique of LSV is attributed to the availability of experimental reactant concentrations at the electrode surface.

The application of convolution analysis to a rotating disk electrode (RDE) enhances the technique of conventional CPSV. This is accomplished by introducing a hydrodynamic regime which is well-characterized and consequently amenable to simple mathematical treatment. Of primary importance is the eradication/provided by a RD% system, of the undesirable effects of natural convection in electrochemical experiments. Complications arising from the effects of natural convection requires experiments to have short duration times so as to avoid the onset of this phenome- non. A RDE system also provides for an increase in the magnitude and stability of the observed mass transfer rates, and hence currents as compared to quiescent solutions.

In conventional CPSV, the primary factor which determines the diffusion rate is the sweep rate. CPSV-RDE extends the capabilities of the basic technique by introducing an additional independent variable, the rotational speed of the disc, which can significantly alter the transport of species to the electrode surface. This facility can be very useful in the study of quasi-reversible and irreversible reactions in which the diffusion process competes with the kinetics of electron transfer. The extent of this competition is determined by the variation in the diffusion rate. However, unlike conventional CPSV, an effective diffusion rate can be determined with CPSV-RDE through a dimensionless sweep rate which is a function of both the potential sweep rate and the rotational speed of the disc. Consequently, CPSV-RDE offers a greater degree of versatility in kinetic analysis by providing the rotational speed of the disc as a complementary variable to the potential.sweep rate.

A diagnostic analysis is often achieved in LSV-RDE through a linear correlation of the reduced peak current with the square root of the dimensionless sweep rate. The value of the corresponding slope distinguishes between a reversible, irreversible and quasi-reversible reactions. Unfortunately, the simple square root correlation is valid only in a limited range of dimensionless sweep rates. There is a transition region in which the reduced peak current is a complicated function of the dimension- less sweep rate. but this is clearly unsuitable for diagnostic purposes. CPSV-RDE does not suffer from this limitation and therefore any range of sweep rates can be investigated and analyzed through the convolution procedure. Furthermore, CPSV- RDE is capable of diagnostic analyses even for small dimensionless sweep rates in which current plateaux instead of current peaks are afforded. The latter current response is necessary for simple IRDE. The application of CPSV-RDE is thus expected to achieve a more accurate and versatile techmique for investigating the kinetics and mechanisms of electrode processes.

The study of electrochemical reactions using relaxation methods or transient techniques has become increasingly popular in the last thirty years. Among the various techniques that can be used include: chronopotentiometry, chronoamperom- etry, cbronocoulometry, linear potential sweep voltammetry and cyclic voltammetry. Extensive treatments of these subjects can be found in the paper by Nichoison and Sham [6] and in the monograph by Macdonald [7]. The linear potential sweep and

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cyclic voltammetric techniques were first introduced by Matheson and Michols [8] in 1938 and were described theoreticaIIy by Randles [9] and Sevick [lo] some time later. These two techniqltes have since become a way of obtaining a rapid “spectrum” of an electrochemical charge transfer system and provides for a detailed examination of reaction mechanisms. Recent developments in the area of CPSV to stagnant systems have greatly extended the usefulness of the basic method and it is now considered as one of the most powerful of electrochemical techniques. The applicaticn of CPSV to a RDE system will be a continuing effort in the search for new methods of analyzing electrochemical reactions.

The mathematical analysis of electrochemical problems in stagnant systems under transient conditions involves solving Fick’s second law of diffusion, wi’h the appropriate boundary conditions. In a RDE system however, the situation is indeed more complicated and a convective diffusion equation is required to describe the mass transport characteristics properly in the vicinity of the electrode surface. In contrast to the previous case of pure diffusion, the convective-diffusion equation is more complex and often causes difficulty in obtaining a solution. An exact analytical solution to this problem is known only for certain simple boundary conditions [ll]. As a result, approximate methods of solution have been employed to deve!op the mathematical formulations, in the form of convolution integrals, which provide the surface concentration of a reacting species from experimental current-potential data.

The first approximate solution of the transient convective-diffusion equation was obtained by Levich [12], through a successive approximation method. In the analy- sis, JAvich assumes as a first order approximation that the convection is insignificant during the initial stages of electrolysis. This simplification is tantamount to a pure diffusion model and can easily be solved analytically. In the next approximation, a linearized convection term is introduced through the utilization of the concentration profile determined by the pure diffusion model. The solution of this expanded equation permits the determination of characteristic times when the rs.tes of mass transport by diffusion only become comparable to that by convection and diffusion.

Filinovskii and Kiryanov [13] also developed an approximate analytical method for solving the non-stationary convective-diffusion equation. A transformation of variables produces a term proportional to the fourth power of the distance to the electrode. Elimination of this term reduces the problem to a canonical equation for the Airy flunctions. Krylov and Babak [ll] obtained an exact analytical solution in the form of a power series of parabolic cylindrical functions_ A comparison of the accurate solutions obtained by Krylov and Babak with that of Filinovskii, under conditions of constant current or mass flux, yields a maximum difference of 7%.

A Ner-nst diffusion layer model, originally proposed by Siver [14,15] and subse- quently used by other investigators, has proven to be a relatively accurate and simple method in describing the mass transport to a RDE for certain conditions. Numerical calculations by Hale [16], concerning the problem of convective-diffusion to a RDE under galvanostatic conditions reveals that Siver’s approximation is only about 4% too high in the predicted concentration profile, over most of the range of times considered.

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The surface concentration obtained from a convolution analysis of the-current response transient depends upon which mass transport model is employed. It is thus not surprising that tlic analysis of a reaction scheme can be significantly affected by this choice. Consequently, it is necessary to compare the results of the various mass transport models in order to insure maximum accuracy in the convolution analysis.

The purpose of ile present paper is to extend the theory of CPSV to a RDE system and investigate the feasibility of this new method in studying the kinetics of simple electrode processes. The theoretical results provide a basis for the analysis of two separate redox reactions, the reduction of ferricyanide to ferrocyanide and the reduction of ferric to ferrous ion.

THEORETICAL: MASS TRANSPORT

Consider the following simple reaction scheme:

O+ne-tiR (I)

which may exhibit reversible, quasi-reversible, or irreversible kinetic characteristics. The transport of reacting species 0 to the surface of a RDE is governed by the following convective-diffusion equation,

at/at + LLac/a2 =m%/a2 (2)

where c(z, t) represents the concentration of a reacting species, u,(z) the component of the electrolyte velocity which is normal to the disc, z the distance to the electrode surface, and D the diffusion coefficient of the species in question. Introducing the following dimensionless variables,

C(<, T> = 1 - 45,4/c” (3)

5 = z/6, (4)

7 = oz/s; (5)

and for large Schmidt number, where the diffusion boundary layer is much greater than the momentum boundary layer [12], the velocity U,(E) near the disc surface is given by the following expression,

0, = - 0.5102( 03/vyz' (6) where gN is the thickness of the Nemst diffusion layer, o is the rotational speed of the disc, Y the kinematic viscosity of the electrolyte and cb the bulk concentration of the reacting species. Equation (2) is transformed into the following,

ac/.ar=a'c/a5"+Qg2~c/ag (7)

here a is. a numerical coxtant equal to 2.136, and is derived from using an approximate velocity profile for o, in the vicinity of the disc.

In a LSV experiment, the electrode is potentiostatieally swept linearly with time at a rate u. The electrode potential at any moment in time is given by the following

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69

expression,

E(t)=Ei-or (8)

where Ei is the initial potential of the working electrode. The current response to such a transient reveals information concerning the kinetics of reactions occurring at or near the electrode surface. In the theory of LSV, the solution of a particular problem can only be achieved if the reaction mechanism is assumed a priori, such that a set of appropriate boundary conditions complements eqn. (7). In contrast, we find that with CPSV-RDE and simple electron transfer reactioiis, the boundar) conditions to eqn. (7) can be generalized in the following manner,

c=o at 7=0 and 520 (9)

c-+0 at 7>0 and [+oo (10) X/&$= -i(T)/i,= -q(7) at ok-0 and <=O (11)

where i, is the mass transport limiting current density and is defined by:

i! = nFDrb/6,, (12)

Note that the function *(T) is experimentally given by performing LSV on a RDE, and it is fundamental in carrying out a CPSV analysis.

The analytical solution to eqn. (7) has been determined by Filinovskii and Kiryanov [13] using an approximate method for the case of a given surface concentration and the case of constant current density. In an analogous manner, an expression for the surface concentration can be derived subject to the generalized boundary conditions given by eqns. (9)-(11). The Laplace transfcr.m method of solution can be applied to eqns. (7) and (9)-1:ll). An expression for the concentra- tion of a reacting species at the surface of a PDE is found to be,

C(0, s) = -~(s)Ai(s/a”3)/~“3Ai’(S/a2’3) 03)

where Ai and Ai’ represent tbe Airy function and its first logarithmic derivative respectively, and s is the Laplace integration parameter. In order to facilitate the Laplace inversion process, an interpo!ation formula for the first logarithmic deriva- tive of the Airy function [13] has been used,

Ai(s)/Ai’(s) z - (1-t s)/&G (14)

where for /3 = 1.877, eqn. (14) is accurate to 0.1 5% over &he entire range of real s values. Substitution of eqn. (14) into eqn. (13) and then applying the Convolution Theorem [17] along with the first translation property of the inverse Lapiace transform, the following expression for the surface concentration is obtained:

c(o, 7) =p(x) i exp[ -pa”/3( 7 - X)]

G-X + Q’/~(B - l)“? exp[ -a1/5( 7 - X)]

x erf [ c7”3 (p - l)“‘m] dX 05)

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70

Introducing a dimensionless sweep rate u, and electrode potential l, -.

0 = nFvt3;/RTD (16)

~=07-11 (17)

and:

N = (nF/RT)( Ei - E”) (18)

where E” is the standard electrode potential. Defining a new variable of integration

P by:

p=x+u 09)

eqn. (16) becomes,

1 dp

where the identity of the error function,

(21)

has been used in this derivation for ease in the numerical computation of convolu- tion integrals. The singularity encountered at the upper limit in the first integral for the surface concentration in eqn. (20) can be removed through an integration by parts. The resuhant Rieman-Stieltjes integral can be solved numerically and the procedure is discussed in a later section.

In a ‘RDE system, the Nemst-Siver approximation is often employed. This model assumes that mass transport occurs by diffusion only and within a layer immediately adjacent to the electrode surface of thickness 6,. The Nernst diffusion layer thickness for large Schmidt numbers, typically the case for aqueous electrolytic solutions, was derived by Levich [?2] for the problem of steady state mass transfer to a RDE and is given by,

6, = 1 612~‘/3,1/6,-‘/z (22)

Retaining the same dimensionless variables given in eqns. (3-5) and applying the assumptions of the Nemst model, eqn. (7) and the subsequent generalized boundary

Page 7: The application of convolution potential sweep voltammetry at a rotating disc electrode to simple electron transfer kinetics

conditions take on

ac,jar = a’c/ag’

c-=0

C+O

acjat = -~(~)/i,

71

the following form,

at r=O and 520

at T),O and {=l

F--*(T) at T>O and ,$=O

(23)

(24

(25)

(26)

The analytical solution to eqn. (23) has been examined by Andricacos and co-workers [18-211 under various forms of kinetic behavior for LSV to a RDE. A similar approach can be used by applying the Lapiace Transform method to eqns. (23-26), whereby an expression for the surface concentration in terms of the experimental diml:nsionless current function, ‘k, is obtained:

C(0,~)=~TV(X)~&[0,~i(7-X)]dX (27)

where X is a dummy variable of integration and ez is one of the theta functions [22] defined as,

e,[o,Ti(T--A)] = /+__ (1 +2e(-l)iexp[+j VT7 i=l

1 (28)

Introducing the same dimensionless variables as before, eqn. (27) becomes,

where,

\k*(p)=*(p)x *+2 g (-l)‘exp 2 1 j=l ( 11

(29)

The singularity encountered at the upper limit in eqn. (29), is now eliminated by an integration by parts and this technique equally applies in the case of eqn. (20) as discussed previously. furthermore, since the convolution integral of the experimental function q must be computed ncmzrically it is convenient to discretize the integral in eqn. (29) into n segments of iength 6 by letting,

p = 6E (31) n= ({+24)/a (32)

An integration by parts of eqn. (29) yields,

where the integral in eqn. (33) is of the Rieman-Stieltjes type and can be approxi-

(33)

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72

mated by its corresponding sum [6],

(34)

Substituting eqn. (34) into eqn. (33) and after elimination of the special points i = 0 and i = tr from the summation, the convolution integral for the surface concentration becomes,

C,W) = 2( $)“‘(**(1)fi+ ydi-T[~*(i + 1) -B’(i)]) , where,

q*(i) = \k(i)

f \ i=l

m

1+2 C (-1)‘exp j=O

(35)

(36)

The form of t1 le convolution integral given in eqn. (35) makes it amenable for computation from a discretized form of the current response function q(r).

KINETICS

Reversible reacticts

In a reversible electrode process, the surface concentrations for the oxidized and reduced species are related by the Nemst equation,

E = E” + (RT/tzF) in( cp/cp) (37)

In eqn. (37) the concentration of reacting species at the electrode surface is used instead of the bulk concentration. For a linearly polarized electrode the Nemst

equation in terms of dimensionless variables becomes,

C”(O, T) = 1 - exp( -0 +CR(O, T) exp( -<) (38)

A material balance equation relating the concentrations of oxidized and reduced species can be written as,

cO(0, T) = 1 - CR@, T) (39)

Substitution of eqns. (33) and (39) into eqn. (37) results in the final form of an equation describing a reversible electron transfer reaction,

E= E” +(RT/nF) h-r{(l -CS(9))/CS(+)} (40)

where C,(q) is the convolution integral of the experimentally determined current response function \k, and represents the instantaneous dimensionless surface con- centration of the oxidized species. Note that the superscript for the convolution integral of the oxidized species is excluded for the sake of clarity. A diagnostic criteria in determining the reversibility of a reaction is provided by a convolution

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73

transformation of + and an application of the logarithmic function given in eqn. (40). A plot of the applied potential to a working electrode in a LSV experiment vs. t'he decimal logarithm of the function [l - C,(‘kjlJC,(\k), results in a straight line of slope (RT/nF) or 59.1/n mV at 25 OC. As pointed out by Sav&nt [l], such a test is more efficient than in cyclic voltammetry (CV), where the problem of diagncsing reversibility involves reconstruction of the anodic curve from the extension of the cathodic one beyond the inversion point.

Irreversible reacGons

In the previous section we considered reverzible reactions in which rhe rate of electron transfer was assumed to be infinitely fast. The arguments are now extended to a more complicated case in which the kinetics of the electron exchange process at the electrode surface exerts an influence upon the reaction. Consider the following reaction scheme,

k

Otne-AR (41)

where the observed current density is first order in the surface concentration of

electroactive species and is given by,

i(t) = nFQO(O, t) (42)

A proper non-dimensionalization of eqn. (42), using the convolution integral repre-

sentation for the surface concentration yields,

k, = @‘/&.~)*k/[l- C,@‘k)] (43)

The rate-potential relationship can be easily obtained once &he diffusion coefficient and the Nemst diffusion layer thickness is known.

If a Tafel type potential-dependent behavior for the heterogeneous reaction rate constant, k,, is assumed in the following form,

k, = k: exp{ -aanF(E- E”)/RT} (44)

where k” is the potential independent or “standard” reaction rate constant, and a is the cathodic transfer coefficient, then the followizg logarithmic equation is obtained,

E=E”+(RT/(mF)InA+(RT/cynF)In{[l-.C,(\k)]/~} (45)

where,

A = k”&./D (46)

An inspection of eqn. (45) reveals that it is possible to determine key kinetic parameters from a convolution analysis of a first order irreversible kinetic model. The transfer coefficient, LT, can be derived from the siopz and the standard reaction rate constant, k”, from the intercept,, when the applied ‘potential is piotted against

the logarithm of the function [l - C,(*)J/(\k). Note that unlike I.he reversible

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74

reaction, the logarithmic equation for an irreversible reaction is a function of the un-convoluted current response.

Quasi-reversible readions

Electrochemical reactions often exhibit intermediate range kinetics. These so called quasi-reversible reactions cannot be adequately described by a model which employs either a completely reversible or irreversible kinetics. The kinetics of copper deposition from a CuSO, solution, for example, has been found using LSV on a RDE to lie within the quasi-reversible range 1181. In light of this we will conclude our discussion on the kinetics of simple electrode processes by examining the application of CPSV to a quasi-reversible reaction system occurring on a RDE.

Consider the following reaction scherde,

O+ne-2R kb

(47)

where a quasi-reversible kinetic model is used to describe the current-potential behavior of reaction (47) in the following form,

i(t) = nFk { ~~(0, t) - ~~(0, t) exp[ nF( E - EO)/RT] } (48)

Using the same procedure of non-dimensionalization as for the case of a reversible and irreversible reaction, eqn. (48) becomes,

(49)

If a Butler-Volmer model is used to describe the kinetics of the quasi-reversible reaction, eqn. (48) can be written in the form,

i(r) = nFk”{ cO(O, f) exp[ -anF( E - E=‘)/RT] - ~~(0, t)

x exp[(l - a)nF( E - P)/RT] } (50)

and upon introducing the convolution quantities, eqn. (50) becomes,

E=E”+EF lnh+EF In I- C,(q) -exp(--5)C,(*k)

\k > (51)

Kinetic parameters are determined from the slope and intercept, much in the same manner 2s for an irreversible reaction. However, unlike the case of an irreversible reaction, the logarithmic equation for a quasi-reversible reaction is a function of the dimensionless electrode potential S. It is evidenr that a reaction which can be characterized by a quasi-reversible kinetic model cannot in general be described by another kinetic model. Thus the distinct logarithmic functions provided by the convolution analysis provides a clear diagnosis of the kinetics of a simple reaction. However, conditions of experimental accuracy may not provide conclusive evidence necessary for the discrimination between kinetics models. In this case, it is essential

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75

that in addition to having experimental da!a fit a particular kinetic model, it must not agree with any other convolution analysis within experimental error.

EXPERIMENTAL

An experimental study was conducted on a RDE to test the applicability of CPSV to mechanism diagnosis and kinetic parameter determination of simple electron transfer reactions. The reduction of 0.01 M ferricyanide [Fe(CN),]‘- in 1.0 M KC1 and the reduction of ferric ion Fe3+ to ferrous ion Fe’+ from a 0.075 solution of FeCl, were chosen for this purpose.

Pretreatment of the Pt disc working electrode consists in polishing with an aqueous solution of 0.3 pm powdered alumina followed by a thorough rinse with distilled water in an ultrasonic bath. Electrochemical cleaning of the working electrode was performed by alternating cathodic and anodic polarizations between oxygen and hydrogen evolution from a 0.5 _M solution of sulfuric acid.

A Pine Instrument analytical rotator attached to Pt disc electrode with a surface area of 0.461 cm’ was used as the RDE. The counter electrode was made from a Pt coil and a Fisher Scientific saturated calomel electrode served as the reference electrode. Experiments were conducted at room temperature, and in each experiment nitrogen gas was bubbled through the electrolyte for appro.ximately 15 rnin and it was kept above the solution during measurements. The kinematic viscosity of the ferricyanide and ferric ion solutions were measured with an Ostwald viicometer, and were found to be 0.85 and 0.90 cSt (1 cSt = ‘LO-’ m’ s-‘), respectively. The diffusion coefficients for the reacting species were obtained from the literature and are 7.4 x 10M6 cm’/8 for the ferricyanide [13] and 4.6 X 10d6 cm’/s for the ferric ion.

An EC0 potentiostat, modulated by a Tacussel GSTP2 function generator for linear sweep voltammetry, was used to perform these experiments. The conversion of the analogue current response transient signal into a digital form was achieved with a Plessy Peripherals (RT-11) microcomputl:r, in conjunction with an ADAC data acquisition package. A sufficient number >f A/D conversions insured that each experimental curve could be digitized int3 at least 1000 points. The computer initiated the start of each experiment by tri, ooering a linear sweep on the function generator. In this way we were able to determine, with sufficient accuracy, the duration of each experiment.

Current response transients obtained experimentally at a rotational speed of 103 r.p.m. for the reduction of ferricyanide and the reduction of ferric ion are shown in Figs. 1 and 2, respectively. This rotational speed for the RDE was chosen because the magnitude of the observed currents offered the greatest flexibility in the A/D conversion process. However, sweep rates rangin g from 250 V/min to 1 V/mm were used to investigate the aforementioned reaction systems.

RESULTS AND DISCUSSION

The current response transients for the reduction of ferricyanide and ferric ion at several different sweep rates as shown in Figs. 1 and 2, respectively, are charactetis-

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76

tic of LSV. In kcordance with the theory of LSV at a R3E system presented by several authors [19-211, peak currents are observed for sufficiently iarge valu& of the dimensionless sweep rate, G. On the other hand, low enough polarization rates afford current plateaux instead of current peaks and the former are asymptotic to a limiting current density given by eqn. (12). These phenomena reveal the importance

2.5 u=42

._ a

0.5 -

0.6 0.5 0.4 0.3 Potential E/V VS. NHE

Fig. 1. Dimensionless current transients for reduction of fenicyanide at various dimensionless sweep rates.

Potential E/V vs. NHE

Fig. 2. Dimensionless current transients for reduction of ferric ion at various dimensionless sweep rates.

Page 13: The application of convolution potential sweep voltammetry at a rotating disc electrode to simple electron transfer kinetics

of mass transport in.supplying an electroactive species to the electrode at a rate which is commensurate with its depletion by a surface reaction.

The surface concentration of the reacting species is obtained from a convolution transformation of the experimental current response function \k. At this point, it is appropriate that a comparison analysis of the various transport models be made in order to ascertain their effectiveness in describing the mass transport characteristics to a RDE. For this purpose, we have chosen the case of ferricyanide reduction at a = 8.33. The Nernst diffusion layer model and the Filinovskii mass transport model have been employed through- the use of eqn. (35) and eqn. (20), respectively. In addition, a numerical technique (Appendix I), has been used to solve the complete convective-diffusion equation, eqn. (7), with boundary conditions, eqns. (9)-( 1 I). In each case the dimensionless surface concentration is obtained throughout the course of the experiment and these results are shown in Fig. 3.

It is apparent from Fig. 3, that for short times all three curves coincide within experimental uncertainty and this suggests that the simpler diffusion model is sufficiently adequate to describe the mass transport characteristics. However, as time goes on, the Nemst model underpredicts the surface concentration by as much as 5% from the numerical solution, in agreement with the results obtained by Hale under galvanostatic conditions. An explanation for this discrepancy may be that as time increases during an experimental run’ the concentration gradient develops to a di.stance from the electrode surface where both convection and diffusion play a commensurably important role in mass transport. As a result it is reasonable to expect that under these circumstances a model which does not take into account

3 1.0 L? c .o 2 ; 0.8 -

8

E E .g 05- L

2

2 2%

E 0.3 - -z

5 E ._ D

0.0 1.0 3.0 4. 2

0

Fig. 3. Dimensionless surface concentration for reduction of fenicyanide at o = 8.33 using a Numerical solution of eqn. (7), (curve 1). Filinovskii mass transport model (curve 2). and Nemst diffusion model (curve 3).

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78

convection will tend to predict a lower surface concentraticxl than the actual. It is for this reason that the Filinovskii mass transport model, which deviates less than 1% from the numerical solution, is preferred in performing subsequent kinetic analyses. The numerical solution, although used here for the purposes of comparison, is not an efficient method for determining the surface concentration and thus carrying out a convolution analysis. The reason for this is primarily based on the fact that the numerical method involves solving large sets of ordinary differential equations which neecesitate much more computer time than for the convolution integrals.

Once the surface concentration is determined, a kinetic model can be applied to characterize a reaction system. A convolution analysis on the reduction of ferric ion was performed for a reversible kinetic model using eqn. (40). The logarithmic analysis for this reaction is shown in Fig. 4, for several dimensionless sweep rates. An average slope of 0.062 mV compares favorably to the theoretical value of 0.059 mV for a completely reversible system. Ng [23] has found this system to behave in a reversible manner using the technique of LSV at a RDE.

A similar convolution analysis performed on the reduction of ferricyanide for a reversible kinetic model yields an average slope of 0.083 mV, in substantial disagree- ment with the theoretical value of 0.059 mV. In fact this result is not surprising since several investigators [24,25] have reported that a reversible model does not ade- quately describe the kinetics of this system. However, when an irreversible kinetic model is applied to the reduction of ferricyanide, eqn. (45), a linear regression analysis affords a correlation coefficient of 0.999. ,The corresponding logarithmic convolution analysis curve is shown in Fig. 5. Furthermore, the application of eqn. (45) for a logarithmic analysis of an irreversible reaction also yields values for the cathodic transfer coefficient (r and the dimensionless kinetic parameter A, from

2.t

l!

3

0" 3- I

2 -I 0.

-1.8

2-

3-

O-

& 1 -4.0 -2.0 -0.0 2.0 I:

10

Fig. 4. Convolution analysis using logarithmic eqn. (40) for a reversible ekxrode rraciion.

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79

-7 1.0

3

-’ t 0” 3

‘I .- 8 .

8 A

8

.

8

L

a

El

0 A

o- 11.7 19.5

2.9

0.5 1.5 2.5 3.5 4.5 ?

Fig. 5. Convolution analysis using logarithmic eqn. (45> for an irreversible Tafel-type e!ectrods reaction.

which the standard rate constant k” is easi!y obtained. The values of these kinetic p;uameters are Listed in Table 1, along with those obtained by Jahn and Vielstich [26] using extrapolation methods and by Randles and Sornerton [9] using an ac method. The present results using the method of CPSV at a RDE compare favorably with those reported in the literature.

TABLE 1

Kinetic parameters for reduction of ferricyanide obtained by wrious methods

Method Cathodic transfer Standard coefficient rate constant k”/cm s-’

CPSV-RDE Jahn-Vielstich

E” Randles-Somerton ac

[91

0.67 5.2x lo-’ 0.61 5.0x10-’

9.0x10-’

CONCLUSIONS

The application of CPSV to a RDE system has been developed and applied to the study of simple electron transfer reaction. Theoretical expressions for the surface concentrations in the form of convolution integrals were derived for a Nernst diffusion model and a Filinovskii mass transport model. Logarithmic convolution

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80

analysis equations were formulated for a reversible, irreversible, and a quasi-reversi- ble reaction and these kinetic models have been tested on the reduction of fer- ricyanidc and the reduction of ferric to ferrous ion. Comparison of the present results with those found in the literature are in good agreement and the method also demonstrates viability in determining kinetic parameters. Current work is now focussed on extending CPSV-RDE to include more complex mechanisms where both chemical and electrochemical reactions can occur.

APPENDIX. NUMERICAL SOLUTION OF CONVECTIVE-DIFFUSION EQUATION

The complete convective-diffusion equation and the corresponding boundary conditions are,

ac/ar=a2c/ag'ic52ac/ag (Al) c(t. O)=O (AZ) C(u3,7)=0 (A3)

$0, T) = -q(T) (A41

Replacing the spatial partial derivatives in eqn. (Al) by a central finite difference approximation results in the following,

a”c/ag= (c;,, - 2Ci -f C,_,)/A~2 (A5)

aC/aeE (Ci+, - Ci_,)/245 (‘46)

eqn. (Al) becomes an ordinary differential equation and is given by,

dCi/dr = 7 A;_ { c. r+l -2C,+ Ci_t} ++ai’A&{Ci+t-C-r) t.47)

where the value of A.$ is determined by stability and accuracy considerations [27]. Using boundary condition eqn. (A3), the concentration of reacting species in the bulk of solution is given by,

dC,/dr = (l/A.$‘){C,_, - 2C,} -an’AEC,-, , ( A8 ) The current response function \k(?) specifies boundary condition eqn. (A4) for every value of T. This makes it possible to Lrtegrate the differential equation which predicts the surface concentration. The equation for the surface concentration becomes:

dC,/dT= (2/At2){CI - Ca-A&e(~)j (A%

where C, is the concentration of reacting species a distance A,$ away from :he electrode surface. Finally, for the region located between the surface of the electrode and the bulk of the solution,

dC,/dT = (l/A<‘){ C,,, -2Ci+Ci_,}+$ai2A<{Ci+,-C,_,} (AlO)

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81

Although this region is of less concern than the surface for electron transfer reactions, eqns. (AlO) are coupled to the surface and bulk equations and must be included to complete the integration. A finite region of interest in the vicinity of the electrode is dii\ided into 50 submtcrJSs and the equivalent number of ordinary differential equaticns are integrated using an explicit form of the 4th order Runge-Kutta method. The results are shown in Fig. 3, where curve number 1 represents the surface concentration obtained form the numerical solution.

ACKNOWLEDGEMENTS

The authors wish to thank the support of the E?oion Corporation Educational Foundation. One of us (J.L.V.) was the recipient of an Exxon Fellowship during the course of this work.

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