cyclic voltammetry with asymmetrical potential scan: a ... - Science Direct

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Acta, 1967.Vol. 12.pp. 999to 1030. l’qamon Prea Ltd. Printedin Nortbem Ireland

CYCLIC VOLTAMMETRY WITH ASYMMETRICAL POTENTIAL SCAN: A SIMPLE APPROACH TO MECHANISMS INVOLVING MODERATELY FAST CHEMICAL REACTIONS* J. M. SAWANT Laboratoire de Chimie de 1’Ecole Normale Sup&ieure, Paris 5, France Abstract-Application of trapexoidal and triangular potentia&an methods to the mechanistic analysis of chemical reactions coupled with electron transfer is investigated. For pure diffusion-controlled processes as well as for first-order catalytic reaction schemes, theoretical relations are established in the general case. For various more complex reaction schemes, it is shown that the use of a very asymmetric potential scan leads to very simple analysis criteria for characterixing qualitatively and quantitatively the ratsdetermining chemical step. The method is particularly suited to the study of mechanisms involving chemical reactions following the charge transfer or interposed between two consecutive charge transfers. Second or higher-order reactions are readily characterized. Comparison with the symmetric triangular scan technique is discussed. R&arm&L’application des m&hodes de balayage trap&oidal et triangulaire a l’analyse m&a&i ue des reactions chimiques associ&s a des transferts &ctroniques est etudi6e. En ce qui conceme f, phenomcneS de diiion pure ou les reactions catalytiques du premier ordre, les relations thtoriques sont &ablies dans le cas g&&al. Pour ce qui est des schemas &action&s plus complexca on montre que l’emploi d’un eyage t&s aaym&riqm conduit A des crib d’analyse. t&s aimplee qui permettent de cam&her qualitativement et quantitativemcnt l%tape chimique cinctiquement dhrminante. La m&l-&e convient particulihment h l’hude des m&hanismes comprenant des reactions qui suivent le transfert de charge ou qui sent interpo&s emre deux transferts de charge successifs. Les reactions d’ordre sup&ieur a un sont aisement caract&&es . La comparaison avec la m&rode de balayage triangulaire sym&rique est discutec. zparrmmeafassmts_Man untersuchte die Anwendbarkeit der potentiostatischcn Trapezoid- und Dreiecksspannungsmethoden xur Aufkl&ung der Kinetik von geko pelten chemischen und Durchtrit&eaktionen. Fiir den Fall der r&en Diffusionskontrolle un 8 von katalytischen Reaktionen emter Ordmmg wurden ahguneine Bexiehungen abgeleiit. Es wird +gt,. dass fUr komplexere Reaktionsmodelle die Anwendung von stark asymmetrischen Potentral%ndcnmgen emfachc Ktiterien liefert, welche die qualitative und quantitative Eestimmung des geschwindr ‘tsbestimmenden Reaktionsschrittes erlauben. Die Methode ist spexiell gee&net, um Mec!z ‘smenxu untersuchen, bei denen eine Folgmaktion auftritt oder bei welchen eine chemische Reaktion xwis&en xwei Dumhtrimueaktionen stat&det. Reaktionen haherer Ordnung k&nun leicht bestimmt werden. Man diskutiert den Vergleich mit der symmetrkchen Dreiecksspannungsmethode. possibilities allowed by using stationary electrode-relaxation techniques in the study of electrochemical mechanisms preferably to conventional polarography arise mainly from the following points : (i) Polarization curves can be analysed over a much MAJOR

larger time-scale as it is possible to shorten the electrolysis duration by about two orders of magnitude. Thus faster reaction stages, electron transfer or chemical reaction, can be observed using these techniques rather than classical polarography. Moreover, even for moderately fast stages, which could be detected by polarography, more detailed information is to be expected from relaxation techniques as the diffusion rate can be varied in a decidedly wider range. (ii) The direction of the electron transfer can be reversed in a single microelectrolysis allowing an easier characterization of reaction scheme involving chemical stages consecutive to the electron-transfer process. * Mamrscript received 16 September 1966. 999

1000

J. M. SAVE&NT

The forced convection techniques, eg the rotating disk method, have the same potentialities. In particular, the use of a ring-disk electrode is then to be paralleled with current (or potential) reversal under non-stationary conditions. Among the relaxation methods, linear sweep chronoamperometry is increasingly used in mechanistic analysis of electron-transfer processes coupled with homogeneous chemical reactions,14 although derivation of theoretical relations allowing a quantitative use of this technique is rather backward as compared to potentiostatic and galvanostatic techniques. Indeed mathematical difficulties are encountered in solving diffusion problems under linear potential-sweep conditions, since a non-linear equation expressing either equilibrium, or the kinetics of the electron transfer, has to be introduced in the boundary conditions. In the case of the simple linear scan method, a rather complete theoretical analysis is available at the present time for the basic reaction schemes involving first-order chemical reactions, antecedent,@ consecutive,loJ1 or paralle14~u~1sto the electron transfer or interposed between two consecutive electron transfers.ls For higher order reactions a general theory of chemical polarization has been recently developedl” and applied to practically important reaction schemes (dimeriza%on and monomer&&ion). In order to reverse the net electron-transfer process a symmetrical triangular potential scan is generally applied to the working electrode. Theoretical analysis of polarization phenomena in the second part of the potential scan is rather involved and an essentially numerical approach to the problem has been described so far in the case of first-order chemical reactions only. ll*U For simple linear scan conditions it has been shown,s*l* in the case of a first order reaction preceding or following the electron transfer, that simpliEed treatments can be considered when the two parameters K (equilibrium constant) and A=RT-.-

nF

ki+kb V

(5, k, are the rate constants of the chemical reaction, Yis the sweep rate) reach limiting values. Areas of experimental validity of these simplified treatments (within a given approximation on the measured quantity) can then be deftned in a log K/log iz diagram. Such a diagram corresponding to a preceding reaction and a 5 per cent error on peak-height measurement is recalled in Fig. 1. In the case of a parallel reaction the diagram becomes unidimensional, as the equilibrium constant is assumed to be very low. For higher order reactions, the same type of analysis is still possible by introducing the parameters x and 1, which contain, besides equilibrium and rate constants, initial concentration terms. It should be emphasized that a fundamentally analogous approach ought to be valid in the case of a double potential scan. When no special relation between scanning rates or electrolysis durations of each part of the scan has been set, it is likely to occur that the first stage of the scan refers to one region of the diagram while the second stage refers to the adjacent one. In the particular case of a symmetrical triangular scan, the point representing the system remains in the same region during both parts of the scan. In the present state of theoretical analysis,ll it is to be noted that for follow-up reactions simple determinations of the rate constanPl concerns only zone KO, ie moderately fast reactions. Moreover the polarographic half-wave potential of the redox system needs to be independently known and thus the chemical reaction has to be slow enough for the representative

Asymmetrical potential

scan.

Mechanisms involving moderately fast reactions

1001

point to be shifted from zone KO to zone DO when raising the sweep-rate. For such a system, information provided by the simple linear scan analysis is theoretically sufIicient for the rate constant to be determined.lO Higher precision is however to be expected from measurements performed with a symmetrical triangular scan, since the

Fro. 1. Zone diagram de6ning the kinetic character of the currentonthebasisofaSper cent accumcy for peak or top-of-wave current. Working

electrode

potential

FIG. 2. Definition of the input-signal form and characteristics.

observed quantity, the ratio of anodic and cathodic peak currents, is much more sensitive to changes of rate constant and sweep rate than is peak potential in single linear scan analysis. The use of an asymmetrical scan preferably to a symmetrical one allows a simpler approach to mechanistic analysis of chemical reactions coupled with a charge transfer provided that the reaction is moderately fast. Indeed, the input signal can be selected in order to secure the following conditions of polarization: in the first part of the scan

J. M. SAVEANT

1002

,the system is represented by a point in the KO, KP, KE or KG regions; in the second part this point is shifted to the DO region. As the sweep rate cannot be raised over cu 500 V/s, it clearly appears that only moderately fast reactions can be character&d by this method. We now study the theoretical possibilities of applying the asymmetrical scan method to some reaction schemes that are frequently met in electrochemical mechanism analysis. Electronic devices comprising a fast response potemiostat and a function generator (or equivalent operational-amplifier circuitry) are available for trapezoidal and for triangular scanning; the theory for both is derived. Figure 2 features the two forms of potential scanning considered here. The case of a nernstian charge transfer under pure diffusion control will be treated first as a reference model for more complicated reaction schemes. No particular assumptions concerning u, (or 0) and u, are required in this case for the polarization problem to be solved. The same holds for the fist-order parallel reaction case. For all the other reaction schemes dealt with here, the simplifying conditions stated above will be introduced. The analysis is for a reduction; transposition to oxidation is obvious. PURE DIFFUSION The

CONTROL-NERNSTIAN

CHARGE

TRANSFER

reaction scheme is 0 + R.

Diffusion coefficients are assumed to be equal. We use D, diffusion coefhcients; E,,,normal redox potential of the O/R couple; Co, total initial concentration; E(i), working electrode potential. The current density j(r) will be expressed as

(0 Resolution of the classical set of partial derivative equations and boundary conditions (see for exampleU) leads to

(2) b(x, r) being the ratio between the concentration of substance R and the total initial concentration Co. Charge transfer being nernstian, the general expression of b(0, r) is b(0, t) = b(0, -0)

,(

+ Y(t)

1

(3)

1 + expFT nF B(t) -

&II

where Y(r) is the Heaviside function and b(0, - 0) the value of b(O, t) when t --f 0 from the negative side. Thus 1 Q(0 = (*,)1’2

1 1 + exp nF E

P(0) -

- b(0, -0)

&J

I dz 0 - #la

(4)

Asymmetrical potential scan. Mechanisms involving modnately fast reactions

1003

Altemative initial conditions can be chosen that determine the b(0, -0) value. (i) Electrochemical equilibrium is achieved for t = 0 and then 1

b(0, -0) = b(0, _tO) = l+expS

(5)

RT [E(O) -

EnI

l

Current is consequently given by dr (t - +”

(6) l

(ii) The working electrode potential is suddenly brought up to the E(0) value so that MO, -0) # b(O, +O).

(7)

The most important practical case corresponds to b(0, -0) = 0.

(3)

Then (4) becomes

4(r) =

l

A-

(nty’a

l+e*pK RT MO) -

&I dr * (9) (t - +”

1

Simple linear scan The potential/time

dependence is defmed as E(t) = Ei - ut,

w-0

with Ei = E(O), initial potential, V, sweep rate. Introducing the adimensional forms of potential and current,

5= - g

[E(t) - E,],

u = nF RT [Ei -

J&I,

j

(11) 02)

polarization curves are described with the Y?(l) function by

1

(i): ucosh8

0 (t -drl$I” ’ 2

(13)

2

(ii):

W)

=

5 1 1 dtl l +’ @(E + z#‘* 1 + exp (u) 47P s -ucoshx q (t - 7j)l18’ 0z

(14)

J. M. SAVEANT

1004

(13) corresponds exactly to the current expression given by Matsuda and Ayabeu and (14) to that derived by Nicholson and Shain” Although both equations tend towards the same expression when u + co, Y(E) =

J4#”

s E

drl

1

(15)

--ucosh8 2 (l - $l” ’ 02

(13) gives perhaps a better representation of actual conditions than (14) since in present experimental practice potential is maintained at its initial value for quite a long time compared with the scan duration. Numerical computation of (13) leads to the peak values Ypp and &, as functions of u (Table 1). These data provide the exact dependence of polarization curves on initial potential. Previously reported values of u above which the influence of the initial potential is inconsiderabler1*16 are not precisely defined, as neither the observed quantity, nor the measurement precision, are indicated. Mean error in experimental practice can be reasonably estimated as 5 per cent on peak height and a 2 mV on peak or half-peak position. Under these conditions the corresponding limiting values of the initial potential are Er=E,+*mVat25”C(u

n

= 2*5), concerning peak current,

(16)

and El = E,, + 42 mV at 25°C (u = l-6), concerning peak potential,

(17)

Ei = E,, + 74 mV at 25’C (u = 2*9), concerning half-peak potential,

(18)

for a one-electron process. Turning back to the general case, ie, whatever the time-dependence of the working electrode potential, (5) can be considered as giving the best representation of actual conditions as in the simple linear sweep case. Let us assume that the potential input signal in composed of two successive time-functions, E(t) = El(t)

for

0 < 1< 8

The current/potential

(19)

t>e I ’

E(t) = E2(t) for curves are then given by

dr

for 0 < t < 89

(20)

(t - #” 1

+ exp 2

d7

[E&) - E,]

(t - #”

for t > 8. (21) ac is the current function during the first part of the scan, Qs is the current function corresponding to the second part of the scan defmed to the extension of the @, curve which would have been obtained if the El(t) scanning were maintained beyond the

Asymmetrical potential scan. Mechanisms involving moderately fast reactions

switching time 0. Current/potential curves corresponding to trapezoidal triangular scanning (Fig. 2) are readily obtained applying (20) and (21). Trapezoidal

1005

and to

scan EI(~) = Ei + W)(E,

-

E&

Es(t) = E, + v,(t -

0)

(0 < t < e),

(22)

(t > e>.

(23)

,1

In the cathodic stage

@c(t) =

1

-L

(,ty

[

1

-

I + exp $$-(E,-EE,)

1 +expg(E,-EJ

(24)

ie, the classical expression of diffusion current under potentiostatic control. During the anodic stage current is conveniently expressed through the function ‘I?&), deIined in the same manner as Y!(l) (12), 1E

Y&2) =

z2

1

dg s U8cosh2 9 (& - $1’2 ’ _

02

with t2 = g

242 =

[E,(t) -

zT[E, -

E,] = g

u,(t - e) - up,

(26)

&I,

(27)

(28)

witija the anodic current density as measured from the extension of the prolonged cathodic curve. Anodic polarization curves are thus identical to the cathodic curves obtained with the same sweep rate, initial potential Ei being replaced by the switching TAFILE 1 I4

0

‘r,

0.265

&I

1.57

&,,p f0.23

1 0.361 1.27 -0.49

1.3

1.5

0.381

0.392

1.22 -0.64

1.20 -0.72

2 0.413 1.16 -0.88

3 0.434 1.12 -1.03

4

5

0442

0445

1.11 -_1.70

1.11 -1.09

6

I

20

0446

0446

1.11 1.11 -1.09 -1.09

1.11 -1.09

0446

potential E,. Consequently the anodic curve dependence on switching potential is given by Table 1. Independence of peak height from switching potential is thus reached within 5 per cent when E, 6 E, - 64 mV at 25°C. n

(29)

J. M. SAVEANT

1006

Similarly peak potential becomes independent from switching potential within 2 mV at 25% for a one-electron process when E, < E,, - 41 mV,

(30)

E,
a

(33)

With

Ei = E, + vJI.

(34) The cathodic current is of course the same as that obtained in simple linear scan experiments, ie, when initial potential is positive enough,

l 51 =

El(?) - E,,

Yc =

dg

jc

(35)

(36)

withj,, cathodic current density. During the anodic stage, current is expressed by the function Ya as measured from the extension of the prolonged cathodic curve.

%oshs

lg 2 0

6, u, and Y* being defined by (26X28). us4

When cc

(38) the anodic current function Y* tends towards the cathodic current function Y, if the same &scale is used. The ratio of anodic peak current (determined with the prolonged cathodic curve as a base line) to cathodic peak current is

(39) and the differences in peak potentials and in half-peak potentials are A&, = 2.22 g nF

W) AEpla = 2.18 g nF

Asymmetrical potential scan. Mechaniws involving moderately fast reactions

1007

AEp is independent of both Scanning rates. (39) and (40) appear as reversibility criteria of an electrochemical system when studied by the triangular scan method. Of course, for a symmetrical triangular scan, the peak-current ratio is equal to unity as previously shown. U~17 (38) is practically firlGlled for a finite value of parameter z+, which depends on the sweep-rate ratio. The entire polarization curve depends thus simultaneously on u, and I& as featured by (37). In Table 2 are reported the values of the peak-current ratio expressed by the Y function ratio

(41) of the peak-potential

separation expressed by (42)

and of the half-peak potential separation expressed by Iw4P/e

-

meI

=

gT

AJ%IB,

as functions of the distance between switching potential and normal potential expressed through the parameter u, and of the sweep-rate ratio vr/v2. Retaining the same TABLE2

vl,vx / \ O*OOl

le3

0.852 2.33 1.73 --0.860 2.34 1 a73

1

le5 @879 2.31 1.82

2

3

O-926

2.27 I.97

4

5

/6

o-991 2.22 2.16

0.988 2.22 2.18

1

I

0.886 2.31 1.82

0.930

0.975

2.23 2.12

0993 2.22 2.16

0998 2.22 2.18

0.1

0.915 2.38 1.74

0693.5 2.34 1.83

0.966 2.29 199

0991 2.24 2-13

0998 2.23 2.17

1wO 2-22 2.18

1

1.083 2.12 1.99

1.065 2.12 2.10

I.036 2.16 2.25

1.009 2.21 2.23

1.002 222 2.20

1GO2 2.22 2.19

10

1.034 2.14 2.34

1.025 2.17 2.53

1.013 2.20 2.41

1.004 2.22 220

1*002 2.22 2.19

1.002 2.22 2.19

0.01

2.27 1.97

--

a3

IQ00 222 2.18

l-002

(The three numbers in each square represent successively(vl/v.#‘*(j&/(i&,, nFIRT A& nF/RT

AEP”.

precision criteria as previously, an estimate can be made of u, as a function of t)llt)g, above which either (Y&?l?,Jp, or (~F/RT)AEP or (nF/RT)AEpla are practically independent of switching potential. The results of this estimation are reported in Fig. 3. Points situated above line (C) correspond to practical independence of peak-current ratio on switching potential. Points above line (P) show the conditions for independence of peak-potential separation on switching potential. The same situation with respect to half-peak potential is reached above line (S).

1008

J. M. SAWANT

It can be concluded, from the preceding considerations, that triangular and trapezoidal scan methods have essentially the same potentialities in checking the reversibility of an electrochemical system. However with triangular scan procedure, anodic and cathodic linear sweep curves are obtained in the same experiment allowing the reversibility test to be achieved more conveniently. In both cases current/potential curves can be expressed in closed forms, (25) and (37), which are easier to handle than the corresponding integral equations (see for example the numerical approach of these

FIG. 3. Diagram defining the wnes of independence of the polarization curves from switching potential in the case of a triangular scan for pure diffusion conditions. (c):Limiting curves for a 5 pez cent accuracy on peak current. (p) : Limiting curve for a 2 mV error on peak potential. (5): Limiting curve for a 2 mV error on half-peak potential.

equations by Nicholson and Sham in the particular case of a symmetric triangular scan.n FIRST-ORDER The reaction

REACTION

scheme considered (4 0 zy=

PARALLEL

TO ELECTRON

TRANSFER

now is

R (nernstian)

R+Z’E“O.

The oxidizing agent Z is in strong excess so that the chemical reaction can be considered as a first order one, k being the pseudo&&order rate constant. All the other notations are the same as above. The general expression of the current function @ is now

s

texp [-k(t

0

- T)l y (t - 7y2

+ k[b(O, T) -

b(0, -0)]

d7. I

(49

Asymmetrical potential scald Mechanisms involving moderately fast reactions

1009

Assuming as before that electrochemical equilibrium is achieved at time zero, it follows that

1

+k

1

-

1 + exp @&E(T) - E,]

exp E-W

41 dT

-

(t,-,Ty

1 + exp FT [E(O) -

. (45)

When the potential scanning is composed of two Merent potential/time G(t) and E,(t) (19), the polarixation curves are decked by

functions,

1 1 + exp FT [Eh) 1

+k

-

1

-

exP F-W-

7)ld7

(t - Ty

1 + exp FT [E,(O)-

1 + exp ‘~TI&(~) - &I

t I((

EnI

(46)

for 0 < t -c 8, and -d

@ n--- ;,s

B d7

1

l+expE RT MT) 1

+k

1 + exp $$(T)

- &I

1 + exp $E&)

- J&l I

1

- &I

1

-

1 + exp f$4&)

exp k-k(r - T)l dT (t

- 41

-

T)ln

II (47) for t > 8. Simple linear scan

exp [-‘(t -v>l drl ‘I

Performing the adimensionalixation of current and potential functions as defined by (11) and (12), the polarization curves are

+A

1 1 1 + exp (-7) - 1 + exp u

(E - .r1Y2

(48)

1

=RTk --.

nF v

When u +

(49)

00, Y becomes the expression previously derived for catalytic currents corresponding to a first order reaction and a nernstian charge transfer.ls The u value 7

1010

J. M. SA~EANT

above which polarization curves are independent of initial potential is now a function of parameter f. When iz is very small pure diffusion situation is attained and dependence of peak-height, peak-potential and half-peak potential on u has been described above. On the other hand when ;Zincreases pure kinetic conditions tend to be achieved’ and then: 1 1 Y I-00 = pn > ’ ( l+exp(-E)-l+expu Thus when u > 3 (ie Ei > E, + (0.077/n) V at 2X!)

the limiting current value is independent of initial potential to within 5 per cent. Similarly when u > 4 (ie E, > E,, + O-102 V at 25°C for a monoelectronic process), half-wave potential is independent of initial potential to within 2 mV. Thus if I( is made greater than 3, the practical independence of limiting current is achieved for the entire set of polarization curves. On the other hand when u is greater than 4 half-peak potentials are no longer dependent on initial potential whatever the value of iz. Trapezoidal scan The cathodic and anodic current functions are readily derived from general formulae (46) and (47) taking relations (22) and (23) into account, 0c=

expt--k0

1

WP2

1 +exprT nF [EV--&J

1

I+

exp gT(Ei - E,)

p”

+

1

erfW)‘lBl

1 + exp

FT[E, -

+.

(51)

EnI

The oc function tends toward the expression of the current/time function previously derived16 under potentiostatic control corresponding to the top of the polarographic wave (E,+ -co) provided the initial potential Ei is made positive enough for the second term in brackets to be neglected aa = P

1 1 i( d 1 + iz,[ 1 1 + exp u -U* 4cosha 0‘7 1 + exp (-7) z jl x

exP

[--w*((2 - p

q)l

(S2)

dq

’

with &. and us as in (26), (27), \r, as in (28), and 2, = (RT/nF)(k/u&. As in the pure diffusion case, anodic polarization curves, as measured to the extension of the prolonged cathodic curves, are identical to those obtained with a singlelinear cathodic scan with the same sweep rate, the initial potential Ei being replaced by the switching potential E,. Conditions of independence of limiting current and half-wave potential from switching potential are thus the same as those stated above for simple linear scan experiments.

Asymmetrical potential scan. Mechanhs

1011

involving moderately fast reactions

In the first part of the scan current is, of course, the same as which was obtained in linear sweep conditions for the same sweep-rate. The anodic current function is now given by

This YS function becomes identical to the expression of the cathodic current function that would have been obtained in a simple linear scan experiment using the sweep rate as, provided U, --+ co, ie, the switching potential E, is made decidedly more negative that the normal potential E,,. In these conditions, ratio of anodic to cathodic peakcurrents depends on the &values corresponding to u1and a,. The ratio between anodic peak-current and cathodic limiting current is simpler to handle as it is equal to Y(E)/ G/e, which has been previously calculated! In the particular case of a symmetrical input signal, anodic and cathodic functions are identical, as shown previously by numerical analy~is.~ (53) provides the current/potentical curves under a closed form which is easier to handle that the corresponding integral equation when, for instance, independence conditions from switching potential are looked for. For pure catalytic conditions (53) becomes

(

‘rE&) = A;‘*

1

l+exp(-M-l+expfl? u2 [,:

(1 + Z)%,]

-

(54)

The anodic curve is then wave-shaped, the limiting ‘3!* value for &-w co being A?jt2, independent of the switching potential. The iniIuence of this latter factor is then restricted to the fact that the whole anodic curve has to be measured to the extension of the cathodic one as considered beyond the switching time 8. Half-wave potentials depend on the switching potential more markedly. If a 2 mV error (at 25°C for n = 1) is assumed, conditions of experimental independence of half-wave potential on switching potential are tl > 3.3 + 0~08(uJu3 (55) 2 1 +

(Q.hJ

E, < E, - 25.6 “‘1 ~~u~~

mV.

Characteristic features of a first-order parallel reaction mechanism as studied by trapezoidal or triangular scan methods can be smmmuized as follows: 1. The anodic current/potential curve, as measured to the extension of the cathodic curve that would be obtained beyond the switching time, is exactly superimposable on the cathodic polarization curve provided the switching potential is made sutEciently

1012

J. M. SAW

more negative than the normal redox potential. The same behaviour has been met in pure diffusion conditions, but in the present case both cathodic and anodic curves are featured by a non-zero limiting current whose value is independent of sweep rates and proportional to the square root of the strong oxidixing agent concentration. Waveshaped curves are observed in the anodic range as welI as in the cathodic one, if the chemical reaction is fast enough as compared to the sweep. Qualitative diagnosis of the reaction scheme thus appears somewhat easier when cyclic scan procedures are used instead of simple Iinear scan. 2. From a quantitative point of view, ie, when rate-constant measurements are concerned, no fundamentaIly new information is gained using trapezoidal or triangular scan instead of a single Iinear sweep. However, a rapid and convenient estimate of rate constant is provided by measuring the ratio of the cathodic Iimiting current to the anodic peak current corresponding to pure diffusion conditions. FIRST-ORDER The

REACTION

CONSECUTIVE

TO ELECTRON

TRANSFER

reaction scheme is OeR (M

R+I

(K = M,).

Whatever the potential/time function is, concentrations of 0 and R are given, through their adimensional expressions a and b, by a(0, t) = 1 - J-

’

dr,

@(T)

7P f 0 (t - p

(57) (58)

Hence, the current function Q(t) is the solution of the integral equation exp [-k(t

-

T)]

@&??(t) -

E,]) + 1) (t %) _ +a

l+K

dT = l* (5%

If the potential scan is composed of two different potential/time &(t) (see (19)), the polarization curves are defined by + em [--k(t -

41

~T&W - &I) + 1) (I(D,(::,s -

l+K atO 8,

functions G(t) and

-

d7 = 1

d7

41

l+K - JU) + 1) (t :$a

d7 (61)

Asymmetrical potential scan. Mc&anisms involving moderately fast reactions

1013

Qe and 0,

being defined as previously. Attention will bc focused on the current function