Surface and volume follow-up chemical reactions in electrochemical mechanisms — Double-layer effects

J. Electroanal. Chem., 112 (1980) 175--188 © Elsevier Sequoia S.A., Lausanne -- Printed in The Netherlands

175

S U R F A C E A N D V O L U M E F O L L O W - U P C H E M I C A L R E A C T I O N S IN E L E C T R O C H E M I C A L M E C H A N I S M S - - D O U B L E - L A Y E R E F F E C T S

J.M. SAVt~ANT

Laboratoire d'Electrochimie, Universit$ Paris VII, 2 place Jussieu, 75221 Paris Cedex 05 (France)

(Received 12th March 1980)

ABSTRACT

The association of follow-up chemical reactions with the initial electron-transfer process is a most common feature of organic and coordination electrochemistry. The resulting kinetics is generally analyzed in terms of a volume follow-up reaction kinetically acting within a diffusion process, while the influence of the double layer of the charge-transfer kinetic is treated in the context of an equilibrium situation. Significant deviations from this model appear as the follow-up reaction becomes faster and faster. Two problems arise in this connec- tion. First, as the reaction layer thickness approaches that of the diffuse part of the double layer the equilibrium treatment of the double-layer effect ceases to be valid. Second, for still faster reactions, corresponding to reaction layer thicknesses on the order of molecular dimensions, the follow-up reaction takes a surface rather than a volume character. These problems are treated in the context of stationary electrochemical techniques for EC, ECE and electro- dimerization mechanisms. The implication of these two effects on the kinetic competition between charge transfer and follow-up chemical reaction is discussed.

INTRODUCTION

The o c c u r r e n c e o f chemica l r e ac t i ons f o l l o w i n g t h e init ial e l e c t r o n - t r a n s f e r s tep is one o f t he m o s t c o m m o n f ea tu re s o f o rgan ic a n d c o o r d i n a t i o n e lec t ro- c h e m i s t r y . When these r eac t i ons are i r revers ible and fas t w i th in t h e t i m e scale o f t h e e l e c t r o c h e m i c a l t e c h n i q u e c o n s i d e r e d t h e y will give rise to an i r revers ible wave c o r r e s p o n d i n g to a o n e - e l e c t r o n e x c h a n g e in t h e case o f a f i r s t -o rder EC m e c h a n i s m or in t he case o f a d i m e r i z a t i o n fo l l owing the e l e c t r o n - t r a n s f e r process;

F i r s t -o rde r EC E l e c t r o d i m e r i z a t i o n (1) A + l e ~ B

(2) B -~ C 2B -~ C (2 ' )

A t w o - e l e c t r o n i r revers ible wave will be obse rved w h e n the i m m e d i a t e p r o d u c t o f t h e chemica l r e a c t i o n is easier to r e d u c e (or to ox id ize ) t h a n t h e s ta r t ing mate r i a l , e.g. in t he case o f an E C E r e a c t i o n m e c h a n i s m w h e r e t h e a b o v e EC r e a c t i o n s e q u e n c e is c o m p l e t e d by a f u r t h e r e l e c t r o d e e l e c t r o n t r ans fe r :

C + l e .~ D (3)

T h e k ine t ics o f t he se processes is classically c o n s i d e r e d in t e r m s o f a v o l u m e

176

reaction associated with the diffusion of reactants. In the context of linear diffusion this leads to the concentration profile of the transient species B being a solution of a partial derivative equation of the type:

3 c s / a t = D a 2 c B / a x 2 - - k c s (or kc )

for a t ime-dependent technique such as cyclic voltammetry, step chronoampero- merry or chronopotent iometry and a differential equation of the type:

0 = d2cB/dX 2 _ kCB (or kc )

for a stationary or quasi-stationary technique such as rotating disc electrode voltammetry (RDEV) or classical polarography [ t: time, x: distance to the elec- torde, D" diffusion coefficient, CB(X, t)" concentration of B, k" rate constant of the chemical reaction ], in the context of the Nernst approximation.

On the other hand, the influence of the double layer on the A/B charge-transfer kinetics is generally treated in the context of a quasi-equilibrium situation as in the case where there is no following chemical reaction, i.e. according to the Frumkin correction procedure [ 1]. In the latter case, note that the validity of this approach is essentially based on the fact that the diffusion-layer thickness is large compared to that of the double layer, including its diffuse part (see ref. 2 and refs. therein).

These are the two basic assumptions underlying the kinetic analysis of the above mechanisms aiming at a quantitative description of the kinetic control of the overall process, by the charge transfer itself and/or by the follow-up chemical reaction [ 3 ]. This is a legitimate approach for processes which are amenable to a complete thermodynamic and kinetic characterization -- involving the separate determination of the standard potential, the transfer coefficient, the standard rate constant of the electron transfer and the rate constant of the chemical reaction -- by use of the above electrochemical techniques. In such conditions, the chemical reaction is indeed not very fast, thus involving a rather large reaction layer. For a first-order EC mechanism, for example rate constants above 104--10 s s -~ are beyond the present capabilities of the techniques cited above, even when used as the lower edge of their time scale. The thickness of the reaction layer is then on the order of 100--300 nm, i.e. much larger than molecular dimensions and even than the diffuse double layer. This ensures the validity of the treatments based on Fick's second law modified by a chemical term with the Frumkin correction taking care of the effect of the double layer on the charge-transfer kinetics.

However, when faster and faster follow-up reactions are involved, two main problems have to be faced. First the t reatment of the double-layer effect under quasi-equilibrium conditions will become less and less valid. The migrative and diffusive movement of B within the diffuse part of the double layer should then be dealt with explicitly. Second, for very fast follow-up reactions involving reaction-layer thicknesses on the order of molecular dimensions the application of Fick's second law ceases to be valid. In the limit, B no longer diffuses (nor migrates} and the follow-up reaction becomes a surface instead of a volume reaction. An additional problem in this context may be the influence of the electric field on the kinetics of the B -~ C reaction in the double-layer region. Discussion of these problems has drawn much attention to the case of systems involving

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chemical steps that precede the charge transfer. Detailed treatments have been given for the movement of reactants within the diffuse double layer and for the effect of the electric field on the chemical reaction, with particular emphasis on acid dissociation reactions preceding the reduction of proton and dissociation of cyano-complexes (see refs. 4 and 5 and refs. therein). The surface character of the associated chemical reactions have also been considered for similar reaction schemes, as well as for parallel reactions, i.e. catalytic processes (ref. 6 and refs. therein). However, the surface processes dealt with in these studies result from reactant adsorption at the electrode surface leading to a competit ion between surface and volume reactions. We do not consider this type of situation here, being interested rather in the passage from a volume to a surface process as the chemical reaction becomes faster and faster independently of specific adsorption of reactions on the electrode surface. The fact that past work has concerped antecedent rather than follow-up reactions should presumably be related to the possibility of obtaining kinetic data from the current--potential curves in the first case, while quantitative information is difficult to derive directly from a completly irreversible wave involving a fast follow-up reaction. Indeed, observation of the current--potential curves leads in the latter case to (a) determination of the character of kinetic control of the overall process, (b) measurement of the transfer coefficient and the forward electron transfer rate constant when kinetic control is by the charge transfer, and a combination of the standard potential and the chemical rate constant when kinetic control is by the follow-up reaction, with no possibility of obtaining a separate characterization of each step in both cases.

Recently, however, methods have been proposed and tested which allow an indirect determination of the charge transfer and chemical reaction.characteristics for systems involving very last follow-up reactions. One such method is based upon the determination of the catalytic efficiency when carrying out the redox catalysis of the electrochemical process considered [7--10]. Chemical rates up to 109 s -~ can be reached in this way [10] and the standard potential and the standard rate constant of the electron transfer can be determined for even higher values of k [ 9]. Another method derives from the analysis of the competit ion between homogeneous and heterogeneous electron transfer to C and a first-order reaction involving the same species (e.g. H-atom transfer from the solvent, reaction on an added nucleophile in the case of the reductive cleavage of aromatic halides) [11--13]. Very high values of k may again be reached by these methods. A route is thus open to a more detailed description of the reaction mechanisms of processes involving very fast follow-up reactions. This renders worthwhile an investigation of the double-layer effect on such reactions and of the transition between volume and surface processes. It is the purpose of the present paper to discuss these problems, having mainly in mind the question of the kinetic control of the overall reaction, i.e. how the double-layer and surface effects will affect the passage from one kinetic control to the other as compared to that found in the case where these effects are neglected [ 3].

FIRST-ORDER EC MECHANISM WITH A SURFACE FOLLOW-UP CHEMICAL REACTION

Assuming, asdone clasically, that the electron transfer occurs at the outer Helmholtz plane (oHp), its kinetics are given by [1]

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i / F S = ks e x p [ - - ~ ( F / R T ) ( E - - E ° - -¢2)] {(CA)o-- (CB)O e x p [ ( F / R T ) ( E - E ° --¢2)]}

where i is the current flowing through the electrode, E the potential of the electrode, ¢: the potential at the oHp, the potential of the solution beyond the diffuse part of the double layer being taken as the origin of potentials. E ° is the standard potential of the A/B couple, ~ the transfer coefficient and k~ the true standard rate constant of the electron transfer; (CA)o and (CB)O are the volume concentrations of A and B at the oHp. An alternative formulation in terms of surface concentrations, F A and FB is as follows:

i / F S - (ks~R) e x p [ - - ( o ~ F / R T ) ( E - - E ° - - ¢2)] (FA-- FB e x p [ ( F / R T ) ( E - - E ° - - d ) 2 ) ] }

(1)

R being the average size of the A and B molecules, i.e. their radius if they are regarded as spheres. The time variation of FB is then given by:

dFB/dt = ( k ~ / R ) e x p [ - - ~ ( F / R T ) ( Z - - E ° - ¢2)] F A

- - ( ( k s ~ R ) exp[(1 - - o ~ ) ( F / R T ) ( E - - E ° - - ¢ 2 ) ] + k} FB (2)

Since reaction (2) is assumed to be very fast, the stationary-state approximation holds for B: dFB/dt = 0.

When A is a charged species, it both diffuses and migrates in the diffuse double layer. As a first approximation, let us assume that the potential profile is linear in the diffuse part of the double layer:

¢ = .---(¢~/p) x + ¢:2

which corresponds to an equivalent thickness of the diffuse double layer equal to p (p is then the Debye--Hiickel length). The distance x is defined starting from the oHp as the origin.

Thus, in the context of a stationary electrochemical technique such as RDEV or polarography:

for 0 < x < p: d2cA/dX 2 + ( z F / R T ) ( d c A / d x ) ( d ¢ / d x ) + ( z F / R T ) CA(d2¢/dx 2) = 0

(3)

i.e. in the context of a linearization of the potential profile as defined above

d2cA/dX 2 - ( z F / R T ) ( ¢ 2 / p ) dcA /dX = 0

for p < x < 6: d 2 c A / d X 2 = 0 (4)

where z is the charge number of A and ~ is the diffusion-layer thickness: = 4.98 D '/3 v 1/6 co - i n for RDEV, 6 being expressed in cm, D and v in cm 2 s-'

and co in rev min- ' [ 14 ].

= (3~rDO/7) 'n for polarography (0: drop time)

The boundary conditions are as follows: CA = C o (bulk concentrat ion of A) for x = 6. Across the external boundary of the diffuse double layer, i.e. for x = p, the flux of A is conserved. For x = 0 the elimination of FB between eqns. (1) and (2), with reintroduction of (CA)O = FA/R, leads to:

i / F S = D[ (dcA/dX)o -- ( z F / R T ) ( ¢ 2 / p ) (cA)o]

179

= ks e x p [ - - ~ ( F / R T ) ( E - - E ° - - ¢2)1 (CA)O/{ 1 + (ks/~R) X exp[(1 -- a ) ( F / R T ) ( E ---E ° -- ~2)]} (5)

Integration of eqn. (3) then leads to

CA = (CA)o + (dcA/dX)o { 1 -- e x p [ ( z F ¢ ~ / R T ) ( x / p ) ] } / [ ( z F / R T ) ( - - C 2 / p ) ]

On the other hand,

( d c A / d X ) p + = [c o - - ( C A ) p ] / ( 8 - - p ) ( 6 )

From the conservation of the flux at x = p"

(dcA/dX )p+ = (dcA/dX)p_ + [ ( z F / R T ) ( - - C : / p ) ] (CA)p _

= (dcA/dx)o + [ ( z F / R T ) - ¢2/P)] (CA)o = i / F S D

Thus:

(CA)p = (CA)o e x p ( z F ¢ 2 / R T ) + (dcA/dx)p × [1--exp(zF¢:/RT)]/[(zF/RT)/(----C2/p)]

As shown earlier [ 2], owing to the fact tha t the diffuse double-layer thickness is small as compared to the diffusion layer (p < < 6), the second term of the right-hand side is negligible for all practical purposes. In other words, the quasi- equilibrium approach of the effect of the double layer is valid as far as A is concerned. It follows that, in eqn. (5), (CA)o may be replaced by (cA)p e x p ( - z ~ p ~ / R T ) . Here, (cA)p can itself be expressed as a funct ion of the current: (CA)p/C ° = (i, - - i ) / i , as follows f rom eqn. (6), where i~ = F S c ° D / 6 is the limiting cu,-~nt of the vol tammetr ic or polarographic wave.

It then follows that the equat ion of the wave is

(i, - - i)]i = e x p [ ( a F / R T ) ( E - -E*) ] + p~ e x p [ ( F / R T ) ( E - - E*)] (7)

with

E *= E ° + ( R T / a F ) ln(k~ pp 6/19) (S)

and

P s = (ksapp) 1/a exp[(z -- 1 ) ( F ¢ 2 / R T ) ] ( k R ) - ' 6 ' / ~ - ' D - ' / ~ + 1

o r

Ps = (ks exp[--z(1 - - a ) ( F ¢ : / R T ) ] } ~/~ (kR) -1 5'~'~-~D-1/'~+1 (9)

where the apparent s tandard rate constant , k~ pp, is related to the true rate constant k~ in the usual way"

k~ pp = ks exp[(a - - z ) ( F ¢ : / R T ) ] (10)

The characteristics of the wave depend upon the parameter Ps which features the compet i t ion of the charge transfer and the surface react ion for the kinetic control of the overall process" when p-~ 0, eqn. (7) becomes

E = E* + ( R T / a F ) ln[(il -- i)/i] (11)

which corresponds to pure kinetic control by the charge transfer. The E~/: = E* has the classical expression (eqn. 8) and the log -plot exhibits a 60/a mV slope.

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The variation of Eu2 with the rotation speed in RDEV is then a cathodic shift of 30/a mV for a tenfold increase of co. The double-layer effect then corresponds to the Frumkin correction. When p ~ ~, eqn. (7) takes the form:

E = E ° + (1 --z)¢2 + (RT/F) ln(kR~/D) + (RT/F) l n [ ( i l - i)/i] (12)

which features the pure kinetic control by the follow-up surface reaction. The slope of the log -plot is now 60 mV and

E1/2 = E ° + (1 - - z ) ~2 + (RT/F) ln(kRS/D) (13)

The second term in the right-hand side of the latter equation represents the effect of the double layer which results both from a static ~-effect on the charge transfer and from a dynamic ~-effect on A. There is no dynamic ~-effect on B since it does not migrate in the diffuse double layer, being chemically destroyed in the very place where it has been formed. The E ~/2 now shifts cathodicaUy by 30 mV for a tenfold increase of co.

An image of the transition between these two limiting controls as Ps varies is provided by Fig. 1 in the case where a - 0.5"

exp[(O.5F/RT)(E~n - -E*)] + Ps e x p [ ( F / R T ) ( E l n - - E * ) ] = 1 (14)

A kinetic zone diagram based on the variations of E1/2 can be drawn showing three zones" KP for pure kinetic control by the follow-up reaction; IR for the kinetic control by the charge transfer; and KI for mixed kinetic control (Fig. 2). The boundary lines between these zones correspond to a 2 mV uncertainty on the measurement of the half-wave potential: when, in the intermediate zone KI, E ~/2 reaches a value 2 mV below that corresponding to either IR or KP, the system is considered to enter the corresponding zone. It is then found than the

r~ - 3

-1

- 2

÷ -4

v

-5

- 6

log p Fig. 1. First-order EC mechanism. Half-wave potential as a function of the parameter p featuring the kinetic compet i t ion between charge transfer and follow-up reaction.

181

KP

4

- r 1/3

E 2

,.0 1 e~

o ~ 0

__O _'

R

g g ~ g 9 ~o ~ ~S-

log(klS -~)

Fig. 2. F i r s t -o rde r EC mechan i sm. Kinet ic z o n e d iagram for the c o m p e t i t i o n b e t w e e n charge t ransfer and a sur face or a vo lume fo l low-up chemica l react ion. Po t en t i a l d i f fe rence be tween the oHp and the so lu t ion : ¢2 = - - 0 . 1 2 V; th ickness o f the diffuse doub le layer: p = 0.7 nm, uncha rged s ta r t ing mate r i a l (z --.. 0) , d i f fus ion coef f i c ien t : D = 9 x 10 -6 c m 2 s - 1 , o~ - 0.5.

passage f rom KI to IR corresponds to log p = - -1 .372 and f rom KI to KP to log p = 2.194. The diagram has been drawn for a typical value of R, 0.3 nm, for z = 0 (uncharged starting material) and ¢ 2 = --0.12 V having in mind processes such as the reduct ion of aromatic halides [ 9--13]. It is seen tha t the double- layer effect is such tha t the chemical react ion does not compete with a charge transfer that would be characterized by k~ pp, but by a larger value of the s tandard rate constant . This will be true as soon as z < 1. For z = 1 the compe- t i t ion will be with k~ pp which is > k~. For z > 1, compet i t ion will involve a s tandard rate constant still larger than ks, a l though smaller than k~ pp"

F I R S T - O R D E R EC M E C H A N I S M W I ~ H A V O L U M E F O L L O W - U P R E A C T I O N - E F F E C T OF D O U B L E L A Y E R ON T H E K I N E T I C S

The differential equat ions pertaining to A remain the same as in the preceding t r ea tmen t (eqns. 3 and 4) as well as the boundary condi t ion for x = (CA = C°) and the conservation of the flux for x = p. The main differences regard

B which is now considered to undergo a volume follow-up react ion which occurs concomi tan t ly with diffusion and also with migrat ion in the diffuse par t of the double layer:

for 0 < x < p" d : c B / d X - - ( z - - 1 ) ( F / R T ) ( ¢ 2 / p ) ( d c B / d x ) - - k C B = 0 (15)

for p < x < 5" d 2 c ~ / d x - - kcB = 0 (16)

with the following boundary condit ions:

x = 5" cB = 0; x = p" conservation of the flux

1 8 2

X = 0 •

i / F S = D[ (dc A/dX )o "- ( z F / R T) (¢2 /p )(cA )o ]

= - - D [ ( d c B / d x ) o - - (z - - 1 ) ( F / R T ) ( ¢ 2 / p )(Cs)o ]

= ks e x p [ - - a ( F / R T ) ( E -- E ° - - ¢2)] {(CA)o -- (CB)o e x p [ ( F / R T ) ( E - - E ° --¢2)] }

(17)

Integrat ion of (16) leads to:

(cB)p + (D ike2 ) ~n (dcB/dx)p = 0

and

CB = (CB)p e x p [ - - ( k / D ) ~n (x - - p ) ] for p < x < ti

if it is assumed that at x = ti, no t only cB = 0 but also dcB/dx = O. This amounts to assuming tha t the follow-up reaction is sufficiently fast for the reaction layer thickness ( D / k ) ~n to be small as compared to the diffusion-layer thickness 6. In other words, k is large enought for "pure kinet ic" condit ions (see e.g. ref. 3) to be achieved.

The integration of eqn. (15), taking the conservation of the flux at x = p into account , leads to (see Appendix)

[(CB)o[C°](k~2[D) 1/:= (ilia) h ( k 6 : / D ) (18)

where

h = 1 + 2t3(exp[(1 + fl~)ln el - - exp[- - (1 + t 32) el} / ( [1 + t3 + (1 + ~2)i/21

exp[(1 +/32) ~/2 e] - - [1 + i3 -- (1 +/32) ~/2] exp[- - (1 + 132) ~/z e]} (19)

t3 = [(z -- 1 ) / x ] ( F / R T ) - - ¢ 2 ) D ~/2 p-~ k -~/2 and e = k ~/2 p D - i n

On the other hand, the same t rea tment for A as before leads to

(CA)o/C °= [(i~ -- i)/i~] exp[ - - - zF¢2 /RT] (20)

Elimination of (CA)o and (CB)o f rom eqns. (17), (18) and (20) then gives the equat ion of the vol tammogram:

(il - - i)/i = e x p [ ( a F / R T ) ( E -- E*)] + Pv e x p [ ( F / R T ) ( E - -E*) ]

with

E *= E ° + (RT /aF) ln ( I e~PP~/D)

and

Pv = g (k6 2/D)(kaPp)l/°~k-1/2~ 1/a-~ D-~/a+ 1/2 (21)

where

g(k8 2/D) = h(le8 2/D)~/2 exp[(z -- 1)(F¢2/RT)] (22)

The vol tammogram equat ion is thus formally the same as in the case of a surface reaction. What differs f rom one case to the other is the expression for the parameter p. Thus:

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(1) Under pure kinetic control by the charge transfer (p -~ 0) the equation of the wave is exactly the same as before (eqn. 11).

(2) Under kinetic control by the follow-up chemical reaction (p ~ ~), the equation of the wave becomes

E = E ° + (RT/2F) ln(k~ 2/D) -- (RT[F) ln[g(k52[D)] + (RT/F) l n [ ( i ~ - i)[i]

i.e. the log-plot will have a 60 mV slope, as for the surface reaction, and the half-wave potential will be given by

E m = E ° + (RT/2F) ln(k52/D) -- (RT/F) In g(k62/D)

Two limiting situations are found for extreme values of the dimensionless chemical rate parameter k62/D:

(1) If k is small, but still being large enough for the pure kinetic conditions to be achieved, g(k6 ~/D) -~ 1. The classical expression for a wave controlled by the follow-up chemical reaction without influence of the double layer is found"

E,n = E ° + (RT/2F) ln(k62/D)

This corresponds to the reaction layer thickness, (D/k)~n, being much larger than the thickness of the diffuse double layer p.

(2) Conversely, for very large values of kd 2/D, i.e. for reaction layers becom- ing much thinner that the diffuse part of the double layer: g(k5 ~/D) exp[(z -- 1)(F¢2/RT) and thus

Ein = E ° + (1 - - z ) ¢2 + (RT /2F) In(k62/D)

The situation then closely resembles that found for a rate-controlling surface reaction (eqn. 13), the radius R of the molecule being replaced by the reaction layer thickness (D/k)ll2.

The transition between pure kinetic control by the charge transfer and pure kinetic control by the follow-up chemical reaction can again be represented as far as half-wave potentials are concerned, by the diagram of Fig. 1, corresponding to a --- 0.5, where p now represents Pv. A kinetic zone diagram can then be constructed along the same procedure as for the surface reaction implying, in the present case, the numerical computat ion of the g function (eqns. 19--21). The transition between the IR dnd the KI zones will thus correspond to log p~ = --1.372, and that between KP and KI to log Pv = 2.194. A typical example is given in Fig. 2 for z = 0, ¢2 = --0.12 V, D = 9 × 10 -6 cm 2 s-~ and p = 0.7 nm. The latter figure corresponds to a 0.1 M concentration of a 1-1 supporting electro- lyre in solvent such as acetonitrile or dimethylformamide [15].

DISCUSSION

Figure 2 provides an overall view of the kinetic competi t ion between charge transfer and follow-up reaction in the case of the reduction of an uncharged starting material in an aprotic solvent such as acetonitrile or DMF for ¢2 = --0.12 V. A typical experimental example of such processes would be the reductive cleavage of aromatic halides [9--11]. The left-hand part of the diagram shows the increasing influence of the double layer on the conditions in which the volume reaction interferes as it becomes faster and faster. The right-hand

184

side shows what happens when the follow-up reaction has a surface character. The dashed lines representing the transition between the volume and the surface situation have a somewhat arbitrary character" deviations from Fick's second law have been assumed to interfere significantly when the reaction layer thickness is twice the molecular radius. Several points appear worth notice upon inspection of the diagram"

(1) The interference of the double-layer effect and then of the surface character of the follow-up reaction, as it becomes faster and faster, results in a situa- t ion that appears more favorable to the chemical reaction as far as kinetic control of the overall process is concerned that if these two effects were neglected. The actual IR/KI and KP/KI boundary lines are indeed located largely below the lines that would have been obtained by extrapolation of the behavior corresponding to slow follow-up reactions.

(2) The double-layer effect becomes significant for follow-up reactions that are not extremely fast. It is seen that as soon as the rate constant is > 106 s-i it can no longer be neglected.

(3) For k comprised between 106 and 3 × 10 ~° s -~, kinetic control of the overall reaction is only slightly dependent upon the value of k. For 8 = 10 -3 cm (co = 2300 rev/min), it is seen that kinetic control is by the follow-up reaction as soon as k~ pp >i 3 cm s-~. Still having in mind organic compounds, the latter figure falls into the upper range of values for the apparent standard rate constants of electron transfer to large aromatic molecules [ 16]. It follows that for chemical rate constants > 106 s -~, the observation of a pure kinetic control by the follow-up reaction appears unlikely.

(4) For the typical value of 5 = 10 -3 cm and for k > 106 s-' the kinetic control will still be entirely by the charge transfer as soon as k app < 3 X 10 -2 cm s -1. In the range 106 < k < 3 X 101° s -i, mixed kinetic control will be observed for k app i> 4 × 10 -2 cm s-~ This is not an uncommon value for aromatic molecules

S •

and may help to clarify previous observations in the haloaromatic series [ 9--10]. For example, it was observed [9] that while kinetic control is entirely by the charge transfer for the reaction of bromobenzene in DMF, it appears to possess a mixed character for iodobenzene in the same conditions. Although it appears likely that the charge transfer is faster for Phi than for PhBr, the above observa- t ion is not easy to rationalize in the context of a t reatment neglecting the effect of the double layer since the follow-up chemical reaction, i.e. the cleavage of the anion radical, will itself be significantly faster in the first case than in the second. A very large increase of k app would then be required when passing from PhBr to Phi in order to overcome the concomitant increase of k. This is no longer necessary with the present t reatment involving the effect of the double layer since, in the range of k values considered, kinetic control is nearly inde- pendent of k. The increase of ks pp implied by the experimental behavior thus looks much more reasonable in the context of the present t rea tment than it is when neglecting the effect of the double layer.

In the case of reductions, still occurring at potentials negative to the point of zero charge (~2 < 0), but involving a charged starting material, the effect noted as point (1) would be even more dramatic for negatively charged species. It would disappear for a +1 charged starting material and be inversed for higher positive charges. It is noted that the above t rea tment can be easily extended to oxidation processes.

185

Other double-layer effects than those treated above will occur if the follow- up reaction is pseudo-first order rather than first order and involves charged electroinactive reactants such as protons or negatively charged coordinating agents. Such effects could be treated in a similar way as for the antecedent reaction scheme [4--5]. Also, for fast follow-up reactions the strong electric field existing in the double-layer region may affect the kinetics of the chemical reaction [ 5]. This effect is likely to be much more significant for reactions involving reactants of opposed charges (see the discussion of the typical case of an acid-- base reaction with a negatively charged base in ref. 5 and refs. cited therein) than for a reaction involving the formation of a charged and an uncharged species as, for example, in the case of the reductive cleavage of aromatic halides.

OTHER MECHANISMS

While EC mechanisms are likely to be found in coordination electrochemistry, where the passage from one oxidation state to the other is often accompanied by a ligand exchange reaction, they are seldom encountered in organic electrochemistry. The product of the chemical follow-up reaction, C, is then indeed quite often easier to reduce, or to oxidize, than the starting material resulting in an overall two-electron process. As far as volume processes are concerned, the further reduction (or oxidation) of C may occur at the electrode or in the solution (from B) according mainly to the fastness of the B -* C reaction, leading to an ECE or to a DISP mechanism [17,18 and refs. therein] if the second electron transfer occurs at the electrode (reaction 3) or in the solution accord-

kD ing to B + C --, A + D respectively. For pure kinetic conditions, the competition between two pathways depends upon the parameter:

c°Dlt2/k 3t2 5 :> 1 -* DISP kD ~ 1 -~ ECE

In a typical REDV experiment with c o = 10 -3 M, ¢~ = 10 4 rev/min, taking for kD its maximal value, i.e. the diffusion limit (~ 10 ~° M -~ s -~), it is seen that the ECE pathway will predominate as soon as k 2> 2 × l0 s s -~. It follows that not only surface problems but also significant double-layer effects could only interfere in the context of an ECE mechanism and not with a DISP mechanism. A similar conclusion holds for typical polarographic or preparative scale conditions.

Turning thus to the ECE reaction scheme, the current--potential curves can be derived from those corresponding to the EC scheme with the same kinetic characteristics by simply doubling the current at each potential, provided k is sufficiently large for pure kinetic conditions to be achieved. In the case of a volume reaction, this results from the fact that the sum of the B and C fluxes is constant and very close to zero throughout the diffusion layer since the concentrations of B and C are both very small when k~ /D is large, i.e. when the reaction layer, (D/k)~2, is small compared to the diffusion layer t~. The contri- bution of C to the total current thus corresponds to the opposite of the B flux at the electrode surface, i.e. to the flux of A. The contributions of C and A to the total current are therefore equal, resulting in a current intensity which is

186

twice that of the EC mechanism. For a surface reaction"

d F c / d t = k F B = i A / F S

showing again tha t the cont r ibut ion of C is equal to tha t of A and thus the cur rent is twice tha t of the EC scheme.

The above analysis of kinetic control of the overall react ion in the con tex t of an EC mechanism can therefore be t ransposed wi thout changes to an ECE mechanism. This legitimates the preceding discussion of the reduct ion of ben- zene halides which actually follows an ECE ra ther than an EC mechanism.

For e l e c t r o d i m e r i z a t i o n reactions involving the coupling of two molecules of the first reduct ion (or oxidat ion) product , B, a similar, a l though more compli- cated, t r ea tment as for the first-order EC scheme could be carried out. However, this does not appear necessary for the following reasons. For a volume dimeriza- t ion, and ignoring for the m o m e n t the double-layer effect, integration of the derivative equat ion involving B"

D ( d 2 c s / d x 2) - - k c ~ = 0

taking into account the boundary condi t ion (cB)8 = 0, leads for a fast reaction ("pure kinetic condi t ions") to

(CB)o/C ° = ( 3 D / 2 k c ° 52) 1/3 ( i / i l ) 2/3

A reaction layer can then be defined from the slope at x = 0 of the B profile"

= ~ [ ( C s ) o / C ° ) ] / ( i / i , )

i.e.:

p = ( 3 D ~ 1 2 k c O ) ' / 3 ( i , / i ) 1/3

An est imation of the lower l imit of p is obta ined as follows: k is taken as equal to the diffusion limit, i.e. k ~< 101° M -~ s, i l / i = 1, i.e. its minimal value, and c o < 0.1 M and 5 = 10 -2 cm would correspond to current condit ions in preparative scale electrolysis. In RDEV 5/c o is also at least equal to 0.1 cm M-~ corresponding to smaller values of both c o and 5. Taking D = 9 X 10 -6 cm 2 s-~ it follows tha t tt i> 30 nm, i.e. much larger than the molecular dimensions. It can thus be concluded tha t the dimerizat ion react ion will no t take a surface character. It is also no ted tha t the above est imation of the lower limit of the reaction layer leads to a value which would correspond to a first-order rate constant equal to 106 s-~ in the contex t of a first-order EC mechanism for which the double-layer effect was shown to be small. It follows that significant double-layer effects are no t expected in the case of an electrodimerizat ion process. It must , however, be emphasized again tha t these conclusions, as well as the preceding ones, hold in the case where the reactants are not adsorbed specifically at the electrode surface.

ACKNOWLEDGEMENT

This work was suppor ted in par t by the C.N.R.S. (Equipe de Recherche Associ6e 309 "Elec t rochimie Mol6culaire"). Drs. C.P. Andr ieux and C. Amatore are thanked for helpful discussions on this paper.

187

APPENDIX

The purpose of this A p p e n d i x is to es tabl ish eqns. (18- -19) . i t is conven ien t to give a d imens ionless f o r m u l a t i o n to the p r o b l e m t h r o u g h the fo l lowing changes of variables"

y = x / ~ , p' = p / ~ , 7 = - - ( F ¢ 2 / R T ) 5 / p , k = k ~ 2 / D

b = CB/C ° , ~ = i 5 / F S c ° D = i/il

For 0 < y < p "

d 2 b / d y 2 + (z ~ l ) 7 ( d b / d y ) ~ k b = 0

with

= ( d b / d y ) o + (z -- 1 )Tbo

In the Laplace plane (b -~ b, y -~ p) , the d i f ferent ia l e q u a t i o n becomes"

l) = ( p b o - - ~ ) / [ p : + (z - - 1 ) T p -- hi

i.e."

b = [ ( rbo -- ~ ) exp ( ry ) -- ( s b o - - ~ ) e x p ( s y ) ] / ( r - - s)

where

r = ( - - (z -- 1 ) 7 + [(z -- 1)272 + 4;~1 ~/2 }/2

and

s = ( - - (z -- 1 ) ~ / - [(z -- 1)27: + 4k] '/2 }/2

The f lux of B is thus for 0 < y < p "

f s = ( d b / d y ) + ( z - 1 ) 7 b

= { ( r b o - ~ ) [ r + (z -- 1 )7 ] e x p ( r y ) - - ( s b o - - ~ ) [ s + ( z - 1 ) 7 ] e x p ( s y ) } / ( r - - s )

(fb)p: = (fb)p+ = (db/dy)p,+ = - - k ~2 bp,

the l a t t e r e q u a t i o n deriving f r o m the r e so lu t ion of the d i f fus ion e q u a t i o n valid f o r p ' < y < l .

I t fol lows t h a t bok ~z2 = ~ h ( k ) wi th

h (k) = [r + (z - - 1 ) 7 + k 1/2] e x p ( r p ' ) - - [s + (z - - 1 )7 + k '/2] exp( sp ' ) k,/2 r[r + ( z - - 1 ) 7 + k lj2] e x p ( r p ' ) - s[s + ( z - 1 ) 7 + k 1/2] exp( sp ' )

since

r[r + ( z - 1 ) 7 + k '/2 ] = ~,1/2(r + } kl/2)

and

s[s + ( z --- 1 ) 7 + k 1/2] = kl/2(s + k 1/2)

h(k) = [r + (z - - 1 ) 7 + k 1/2] e x p ( r p ' ) - - [s + (z -- 1 ) 7 + k 1/2] exp( sp ' )

i.e. •

(r + k 1/2) e x p ( r p ' ) - - (s + k 112) exp( sp ' )

1 8 8

h(;~) = 1 + ( z - 1)~/ e x p ( r p ' ) - - exp(sp ' )

(r + k 1/2) exp( rp ' ) - - (s + k ~/2) exp(sp ' )

which leads to eqn. (19) in t roduc ing ~ = (z -- 1)V/2k 1/2 and e = kl/2p '.

REFERENCES

1 P. Delahay, Double Layer and Elec t rode Kinetics, Intersmence, New York, 1965, p. 154. 2 P. Delahay, Double Layer and Elec t rode Kinetics, Interscience, New York, 1965, p. 208. 3 L. Nadjo and J.M. Say,ant , J. Electroanal , Chem., 48 (1973) 113. 4 P. Delahay, Double Layer and Elec t rode Kinetics, Interscience, New York, 1965, pp. 207, 227. 5 H.W. Ntirnberg and G. Wolf, J. Electroanal . Chem., 21 (1969) 99. 6 S.G. Mairanovskii , Catalyt ic and Kinet ic Waves in Polarography, Plenum Press, New York, 1968. 7 C.P. Andrieux, j .M. Dumas-Bouchia t and J.M. Say ,an t , J. Electroanal . Chem., 87 (1978) 39. 8 C.P. A n d n e u x , J.M. Dumas-Bouchia t and J.M. Say ,an t , J. Electroanal . Chem., 87 (1978) 55. 9 C.P. Andrieu~, C. Blocman, J.M. Dumas-Bouchia t and J.M. Say,ant , J. Am. Chem. Soc., 101 (1979)

3431. 10 C.P. Andr ieux, C. Blocman, J.M. Dumas-Bouchia t , F. M'Halla and J.M. Say ,an t , J. Am. Chem. Soc.,

102 (1980) 3806. 11 F. M'HaUa, J. Pinson and J.M. Say ,an t , J. Am. Chem. Soc., 102 (1980) 4120. 12 C. Amatore , J. Chaussard, J. Pinson, J.M. Say ,an t and A. Thi~bault , J. Am. Chem. Soc., 101 (1979)

6012. 13 C. Amatore , J.M. Say ,an t and A. Thi~bault , J. Electroanal . Chem., 103 (1979) 303. 14 J. Albery, E lec t rode Kinetics, Clarendon Press, Oxford, 1975, p. 53. 15 P. Delahay, Double Layer and Elec t rode Kinetms, Interscience, New York, 1965, p. 43. 16 H. Koj ima and A.J. Bard, J. Am. Chem. Soc., 97 (1975) 6317. 17 C. Amatore and J.M. Say ,an t , J. Electroanal . Chem., 85 (1977) 27. 18 C. Amatore and J.M. Say ,an t , J. Electroanal . Chem., 86 (1978) 227.

Documents

Surface and volume follow-up chemical reactions in electrochemical mechanisms — Double-layer effects