Theoretical study of a simple redox system with adsorption of the reactants on a rotating disk electrode

J. Electroanal. Chem., 124 (1981) 19--33 Elsevier Sequoia S.A., Lausanne - - Printed in The Netherlands

19

THEORETICAL S T U D Y OF A SIMPLE R E D O X SYSTEM WITH ADSORPTION OF THE REACTANTS ON A ROTATING DISK ELECTRODE PART I. THE REACTION PATH IN THE CASE OF A LANGMUIRIAN ADSORPTION EQUILIBRIUM

E. LAVIRON

Laboratoire de Synthdse et d'Electrosynthdse Organomdtallique (associd au C.N.R.S., LA 33), Facultd des Sciences, 6 bd Gabriel, 21100 Dijon (France) (Received 6th N o v e m b e r 1980)

ABSTRACT A theoretical t r e a t m e n t is given o f the i--E curves o n a rotating disk electrode for a simple r e d o x reaction O + ne ~ R w h e n b o t h O and R can be adsorbed. It is assumed that a Langmuir isotherm is o b e y e d and that the adsorption rate is n o t a limiting factor. The relative importance of the surface (between adsorbed species) and of the h e t e r o g e n e o u s (between nonadsorbed species) electrochemical reactions depends on the ratio of the rate constants k s and kh for the t w o processes, on the p r o d u c t and on the ratio o f the adsorption coefficients o f O and R, and on the potential. Practical implications of the t h e o r y are discussed on the basis o f a relationship established by Brown and A n s o n b e t w e e n k s and kh. It is shown that, with the above assumptions, electrochemical reactions should take place in m o s t cases via the adsorbed species in aqueous m e d i u m . Conversely, in n o n - a q u e o u s media, reactions can be found for which the influence of adsorption is negligible, although cases can exist for which the participation of adsorbed species has to be considered. Consequences are discussed concerning the verification o f Marcus' t h e o r y (dependence o f the transfer coefficient on the potential, magnitude of the h e t e r o g e n e o u s rate c o n s t a n t ) and the influence of the electrode material on the h e t e r o g e n e o u s rate constant.

INTRODUCTION

Many substances, and in particular the majority of organic compounds, are adsorbed at the solution--electrode interface [ 1 ], so that in principle during a

(ro)

l

0.~

.

0,o, - ¢~

Fig. 1. The reaction scheme (see t e x t for the definition of the symbols). 0 0 2 2 - 0 7 2 8 / 8 1 / 0 0 0 0 - - 0 0 0 0 / $ 0 2 . 5 0 , © 1981, Elsevier Sequoia S.A.

20 simple redox reaction O + n e ~ R either adsorbed species (surface reaction) or non-adsorbed species (heterogeneous reaction) can be involved. There is also a possibility of direct electron exchange between the adsorbed and non-adsorbed systems. A general scheme for this simple case is shown in Fig. 1. The mass transport equations (in general the diffusion equations) have to be solved with adequate initial and boundary conditions. The problem is usually mathematically very complex; in the case of transient methods, analytic solutions can be obtained only in the case of a linear isotherm with an immobile electrode [ 2],' or when both O and R are strongly adsorbed [3,4]. Numerical solutions have been obtained when these conditions are n o t fulfilled; it is usually assumed that the adsorption reactions are always at equilibrium. Most studies concern the case where the electrochemical reactions are also at equilibrium; if not, it has always been assumed that the reaction proceeds only through the surface process [5,6] (for an exception, see ref. 7, where the conditions are very approximate, a complete independence between the surface and volume concentrations being assumed). Recently, Brown and Anson have proposed a relationship between the rate constants ks and hh of the surface and of the heterogeneous processes [8]. Although it is n o t possible to distinguish experimentally between the fractions of the current due respectively to the surface and to the heterogeneous reactions, it would be interesting to assess them theoretically on the basis of their formula. This does n o t seem to be possible for the methods mentioned above, but, as will be shown in this paper, the problem can be solved simply in the case of voltammetry on a rotating disk electrode. For this method, only the particular situation where the electrochemical reaction takes place exclusively on the surface and is totally irreversible has been treated as y e t [9]. We will assume that the adsorption reactions are fast (the adsorption equilibrium is always established) and that a Langmuir isotherm is obeyed. We will n o t consider the direct electron exchange between the adsorbed and non-adsorbed systems. Before studying the general problem, we will reformulate the classical [10] case of a simple heterogeneous reaction in order to obtain expressions lending themselves to a direct comparison with the general solution derived when adsorption is present. THE HETEROGENEOUS REACTION We will consider the case where only O is present in the solution. We have [10] for the fluxes ¢o and Cg of O and R: ¢o =

DS-I(cT -- c~)

CR = D8-1c~

(1)

(2)

Here D and 8 are respectively the diffusion coefficient and the thickness of the diffusion layer, assumed to be equal for O and R, CT the concentration of O in solution and c~ and c~ the concentrations near the surface. For the steady state, we have ¢o = Ca, so that Cb + C~ ----CT

(3)

21

On the other hand, CR = D S - l c ~

(4)

= kh(c~)O - ~ - - c~O 1"-~)

Here kh is the rate constant for the heterogeneous reaction, a the transfer coefficient and 0 = exp [ ( n F / R T ) ( E - - E °) ]

(5)

in which E ° is the standard potential. Let

(6)

x = c~/cT

From eqns. (3), (4) and (6), we deduce: X = 7 [ ( 1 - - X)O - ~ - - X 0 1 - ' ~ ]

(7)

with (8)

7 = k~ ~ D - 1

The current is given by (9)

i = nFACR = nFAD6-1c~

On the plateau of the wave, c~ -~ CT and i -~ il = n F A c v D 5 -1 • The current can be expressed in the adimensional form: I = i / n F A c ~ D ~ -~ = i/il = X

(10)

Equation (7) can be solved for ×, whatever the value of a. When a is equal to 0.5 it can also be solved for 0, by writing it in the form: x(0l/=) 2 + (xlT)O ~ n - (1 - - X ) = 0

(11)

which yields the solution: 01/= =--(27) -1 + [(27)-=+ (1 -- X)/X] In

(12)

the + sign in front of the last member has to be chosen, since 0 is always posi-

"~. 0.2 0"3f 0.1

logT Fig. 2. Variations of E 1/2 with log ~' for a heterogeneous reaction at 25°C: (1) curve defined by eqn. (13); (2) asymptote defined by eqn. (14).

22

tire. The condition t h a t a = 0.5 is n o t too restrictive, since it is usually not very different from this value [11]. The half-wave potential E , n is obtained when X = 0.5; if we designate by 00.s the corresponding value of 0, we have: 1/2

0.5

.~

__(27)-1 + [(27)-2 + 111/2

(13)

When 7 -~ O, 0 -~ O, and it can easily be deduced from eqn. (7) that 1/2 _+ 7

(14)

o.s

The variations o f E , n - - E ° as a function of log 7 (eqn. 13) are shown in Fig. 2. When 7 -~ o% E -+ E°; when 7 -+ 0, E -+ 2.3 ( 2 R T / n F ) log 7 (eqn. 14). The logarithmic analysis of the wave is given in Fig. 3. These graphs allow the reversibility of the reaction to be defined; it is reversible for 7 larger than about 30 and totally irreversible for 7 smaller than about 0.1. REACTION WITH PARTICIPATION OF ADSORBED REACTANTS

Formulation and resolution of the problem

Equations (1)--(3) are still valid. If a Langmuir isotherm is obeyed, we can write for the rate of the surface reaction [4,5,7] : - - d P o / d t = dFR/dt = ks(PO~?--~ - - FR171-~)

(15)

In this equation, k s is the rate constant (in s-'), Fo and FR the superficial concentrations, fl the transfer coefficient and - - E°')]

77 = e x p [ ( n F / R T ) ( E

(16)

E °' being the surface standard potential, defined by [3] E °' = E ° - - ( R T / n F )

(17)

ln(bo/bR)

The adsorption coefficients bo and bR are assumed to be independent of the

100 ,

2--

~

~

- Io

0.1

0

-0.1 -0.2 -0.3 n(E-E°)/V

-0.4

Fig. 3. Logarithmic analysis for a h e t e r o g e n e o u s r e a c t i o n at 25°C. The value of 7 is shown o n e a c h curve.

23 potential. This hypothesis is justified as a first approximation in the potential range where the wave appears and which does n o t exceed a few tenths of a volt. According to eqns. (5), (16) and (17), we have [3]: (18)

~1 = ( b o / b R ) 0

Relationships between the superficial and volume concentrations are given by the isotherms: Po = Fmbocb(1 + boc~o + bRC~) -1

(19)

FR = FmbRC~(1 + boc~o + b R C ~ ) - '

(20)

in which Fm is the maximal superficial concentration, assumed to be equal for O orR. The rate of the heterogenous reaction is still given by kh (c~)O- ~ - - c~O 1-~); in the steady state the flux given by eqns. (1) or (2) is equal to the sum of the rates of the surface and of the heterogeneous reactions: D 6 - ' c~ = ks(Fo~l --~ -- FRrl 1-~) + kh ( c~oO--~ - - c~O 1-c~)

(21)

The problem is n o w described by eqns. (3), (6) and (18)--(21). The current ia due to the surface reaction is ia = n F A k s ( F o r l -~ - - FR~?1-~)

(22)

The current ih due to the heterogeneous reaction is given by (23)

ih = n F A kh ( cbO ---~ - - c~ 01-'~ )

and the total current is defined by (24)

i = ia + ih = n F A ¢ R = n F A D 6 - ' c ~

We will assume in what follows that a = fl = 0.5. If we introduce in eqn. (22) the values of Fo, FR and ~1 from eqns. (18)--(20), we obtain: ia

=

i. ~ ~ - , l ~t~o~-,n n F A k ~ F m ( b o b R ) l n ( 1 + boc~o + ~R~RJ

_

c~O,n)

(25)

It is convenient to formulate the problem by using adimensional quantities. The adimensional currents Ia, Ih and I are obtained by dividing respectively ia, ih and i by n F A c v D S - 1 . Let us define four dimensionless parameters. The first, 7 (cf. eqn. 8) indicates the reversibility of the reaction. The second, a = ksFm/khc w

(26)

is proportional to the ratio of the rate constant for the surface process to that for the heterogeneous process. The third, b = C T ( b o b R ) 1/2

(27)

expresses the adsorbability of the reactants, and the fourth, r = ( b o / b R ) 112

is the ratio of the adsorbabflity of the t w o species.

(28)

24 By taking into a c c o u n t eqns. (3), (6) and (18}-{20), eqn. (21) becomes:

X = 7{I + o[b-' + r(l --X) + r-Ix] -I } {(1 --×)0 -'/2 --X01/2)}

(29)

The adimensional expressions of the currents take the form: X= X

(30)

Ih = ~,((1 -- X)0-'/2 _ X0,n)

(31)

I a = "),o(b-' + r ( l - - X) + r - l x ) - ' ( ( l

- - X)0-'/2 _ X0,~2)

(32)

The coverages 0 o = Fo/Fm and OR = FR/Fm are obtained directly from eqns. (19) and (20): 0o = r(l -- x)(b-' + r(l --X) + r-'x)-'

(33)

OR = r - ' x ( b - ' + r(l -- X) + r-'x)-'

(34)

Let M = 1 + o[b-' + r(1--X) + r-'x]-'

(35)

Equation (29) can be written in the form: X(0an)2 + (x/TM)O ' n - - { I --×) = 0

(36)

The solution of this equation is 0 '/2 = --(27M)-' + [(2~/M)-2 + (I -- X)/X]'n

(37)

The plus sign in front of the last term results from the fact that 0 is always positive. Equation (37) gives the equation of the wave. Examples of variations of I, 0o and 0R are given in Figs. 5 and 6. DISCUSSION S h a p e o f the wave In the general case, w hen r ¢ 1, the shape o f the wave is di fferent from t h a t o f the wave f o r a heterogeneous process (cf. eqn. 12), because the term M in eqn. (37) is a f u n c t i o n of X. If r = 1 (O and R are equally adsorbed), M = 1 + o ( b - ' + 1 ) - ' , which is i n d e p e n d e n t of X; in t hat case the wave has the same shape as f o r a heterogeneous process, with a rate c o n s t a n t 7M instead o f 7. The distortions i nt r oduc e d by adsorption are best appreciated by carrying o u t the logarithmic analysis o f the wave. A few examples are given in Fig. 4. The half-wave p o t e n t i a l This is obtained when X = 0.5. L e t us designate by m the corresponding value of M; we have: m = 1 + o ( b - ' + r/2 + 1 / 2 r ) - '

(38)

and 0 0.5 , n = - - ( 2 7 m ) - ' + ((27m) -2 + 1) ' n

(39)

25

2-

o

1

IT- o 8'

1

q-

--2 O.1

I O

I -O.1

I -0.2 n(E-E°)/V

I

-0.3

I

-0.4

Fig. 4. E x a m p l e o f l o g a r i t h m i c analysis for a process w i t h a d s o r p t i o n at 25°C; 9' = 10-3, o = 1.2 x 106, r = 10 -4. (1) b = 1 0 - s ; (2) b = 1 0 - 4 ; (3) b = 10 -3. (a) Process w i t h a d s o r p t i o n ; (h) l o g a r i t h m i c analysis for a h e t e r o g e n e o u s process for w h i c h t h e half-wave p o t e n t i a l is t h e same.

Comparison with eqn. (13) shows that the half-wave potential has the same expression as for a heterogeneous process, with a rate constant 7m instead of 7. Since we always have m > 1, this means that adsorption always increases the apparent reversibility o f the reaction. This can be explained physically by the fact that adsorption offers a supplementary path for the global process. It is worth noting that E,/2 c a n n o t become more positive than E ° when the reaction becomes reversible (when 7m -* 0% 0 -~ 1, E -~ E°). This seems to be in contradiction to accustomed ideas about the effects of adsorption, according to which prewaves appear when R is more strongly adsorbed than O when the electrochemical reaction is reversible, i.e. the reaction can take place at potentials more positive than E °. This apparent paradox is a consequence of the steady state. As all the molecules of O which undergo the reduction through the adsorbed species desorb, no gain in energy results from adsorption; the final energetic balance is the same whatever the path of the reaction, so that there is no effect on E , n . Reduction at a potential more positive than E ° is possible only with transient methods, where the reduction of adsorbed O to adsorbed R results in a gain in energy. THE REACTION PATH

This can be studied by calculating the ratio: (40)

p = ia/i = Ia/I

which represents the percentage of the reaction passing via the surface path. In view of eqns. (29)--(32), we have: p =

o [ e + b - ' + r ( 1 - - X) +

r-Ix]-'

(41)

26

E f f e c t o f the p o t e n t i a l on the reaction p a t h

If r = 1, the reaction path is independent of the potential (cf. Fig. 5), since p = a [ a + b -1 + 1]-1

(42)

If r ¢ 1, p depends on E, and tends towards different limits p÷ and p_ when the potential tends respectively towards +oo (foot of the wave) and --~ (plateau of the wave) (cf. Fig. 6). When E -~ +oo, X -* 0, and we have:

(43)

p+ = a ( o + b -1 + r ) -1

When E -* - ~ , X -* 1, so that p_ = o(o + b -1 + r-l) -l

(44)

It is worth noting that these values are independent of the reversibility if o is kept constant. If r > 1, p. < p_, the reaction proceeds more and more through the surface reaction from the f o o t to the plateau of the wave (cf. Fig. 6). The situation is reversed if r < 1. This effect can be explained as follows. When the potential is increased negatively, the direct effect of the potential (cf. eqn. 21, with ~ = /3) is the same for the surface and for the heterogeneous reactions, since ~-1/2 increases proportionally to 0-1/2 and ~ 2 decreases proportionally to 0 i n (cf. eqn. 18). The relative rates of the surface and of the heterogeneous reactions are then governed by the relative variations of Fo and c~ and of FR and c~. If, for example, r = 1 it can easily be deduced from eqn. (33) that 0o = (1 -- X)/ (1 + b-l), so that the relative a m o u n t of Fo and c~ remains constant, since 1 -- X represents the adimensional concentration of c~. In the same way Oa =

2

/ /

0.5

0.5

1

I 0.1

0

- 0.1

n (E - E°)/V

I

-0.2

0.1

0 -0.1 n(E-E°)/V

-0.2

Fig. 5. Example of variations of I, p, 0 o and OR: (1) I; ( 2 ) p ; (3,4) 0 o and OR; 7 = 10 -1 , O = 1.2X 106 , b = 1 0 - s , r = l a t 2 5 ° C . Fig. 6. Example of variations of I, p, 0 o and OR: (1) I; ( 2 ) p ; (3) 0o; (4) OR; 7 = 10-1, o = 1.2× 106 , b = 1 0 - s , ~ = 1 0 s a t 2 5 ° C .

27

X/(1 + b -1) remains proportional to c~. This explains why there is no change in the reaction path. When r > 1, it can be easily deduced from eqns. (33) and (34) that 0o(Fo) decreases more slowly than 1 -- X(Cb) and that 0R(FR) increases more slowly than X(c~) ( for example, this can be done by studying the value of X for which 0o has decreased, or OR increased, by half). The surface reaction path is thus favoured when the potential becomes more negative. The reverse becomes true when r < 1. E f f e c t o f the adsorbability parameter b The parameters p_ and p+ are equal to zero when b = 0, and they increase when b increases, but they reach a limit which we will designate by p ~ and p+.¢ when b -+ oo. p ~ . = p(p + r-l) -1

(45)

p+~ = p(p + r) -1

(46)

E f f e c t o f the adsorbability ratio r When r increases, i.e. when O becomes more adsorbed relatively to R, p . decreases, i.e. less current flows along the adsorbed path. This paradoxical result can be rationalized as follows. At constant potential at the f o o t of the wave, an increase in r produces two effects: (a) it increases the ratio of Fo to c~, which favours the adsorbed reaction; (b) it also increases r / f o r a given value of E (eqn. 18), which tends to decrease the rate of the surface reaction (cf. eqns. 15 or 22). If for example we consider an irreversible process at the f o o t of the wave, we have for the surface reaction d F R / d t = ksFOr/-1/2 = (bobR)l/2(1 + bocSo)O -1/2

if (bobR) in, c~ and 0-1/2 are constant, the value of this expression decreases when bo/bR increases; the effect on the potential is greater than the effect on Fo.

When r increases p_ increases, which can be explained by considering the influence of the potential (see above), which is more marked when O is strongly adsorbed. Effects o f the concentration c T When CT --->0, p -* O/(O + b -i) and m -~ 1 + ob. When CT increases, p and m decrease monotonically, and when CT -* ~o, p -. 0 and m -~ 1; the heterogeneous path thus becomes favoured when CT increases. CONSIDERATIONS ON THE REACTION PATH IN THE CASE OF ORGANIC COMPOUNDS

A quantitative evaluation of the reaction path requires the introduction of reasonable values of the parameters o, b and r.

28 P a r a m e t e r s b and r

These depend on the value of bo and bR. In aqueous medium, organic compounds are usually markedly adsorbed at potentials n o t too distant from the potential of maximal adsorption. Let us assume t h a t a Langmuir isotherm is obeyed (F/Fro = B c / ( 1 + B c ) for a single substance). The adsorption coefficient represents the inverse of the concentration which is necessary for the coverage F/Fro to be equal to 0.5. A value of 10 4 cm 3 mo1-1 for B at the potential of maximal adsorption corresponds thus to a weakly adsorbed substance (c = 10 -1 mol 1-1 at half coverage), whereas a value of 101° cm 3 mol -~ corresponds to a strong adsorption. Adsorption in non-aqueous solvents, although weaker than in water, can still be appreciable [ 12]. A Frumkin isotherm is usually followed, but if B is estimated from the value of c at half coverage, values of the order of 10 3 to 10 s cm 3 mo1-1 are f o u n d [12]. Parameter a

Values of Fm and of k s/kh are required in order to evaluate this parameter. For organic molecules, F m is of the order of 10 -l° to 3 × 10 -I° mol cm-2; we shall take in what follows F m = 2 × 10 -l° mol cm -2. Brown and Anson [8] have calculated recently a value of the ratio ks/k h by assuming t h a t the rate constants have the form: ks = As e x p ( - - X J 4 R T )

(47)

and kh = A n e x p ( - - X h / 4 R T )

(48)

In these equations As and Ah are preexponential factors and hs/4 and Xh/4 reorganization energies. They assumed that Xs = Xh, and used for As the value k T / h = 6 X 1012 s -I at 25°C; asAh can be taken as 104 cm s -1 [11] they obtained: ks = 6 × 108 kh s-'

(49)

Their derivation is, however, valid for a static model, in which the molecules are already attached to the electrode surface when the reaction takes place. The situation in our case is different, since the molecules are initially in the solution, adsorb, react and then desorb; under such conditions a limitation can be due to the rate of collision of the molecules with the electrode [ 13]. This reaction scheme has been treated by Mohilner [ 14] on the basis of the theory of Marcus [ 11]. The expression which he obtained for the adsorption current can be written: i a = n F A ~ p Z h e t a s e x p ( - - X / 4 R T ) exp[--(AG~ + A G ~ ) / 2 R T ] ×

× exp[--(AG~ + A G ~ ) / 2 R T ] e x p [ - - n 2 F 2 ( E -- E ° -- A A G q ) 2 / 4 ~ r t T ] ×

× (C~)0-1/2 - - C~01/2}

(50)

In this expression Kp is a transmission coefficient which can be taken as equal

29 t o u n i t y (adiabatic r e a c t i o n ) , Zhe t is the t h e r m a l velocity o f t h e molecules, as the activity o f the a d s o r p t i o n sites, ~ the r e o r g a n i z a t i o n energy, AG~ and AGrq (i = O o r R) the c h a r g e - i n d e p e n d e n t and the c h a r g e - d e p e n d e n t parts o f the s t a n d a r d e l e c t r o c h e m i c a l free energies o f a d s o r p t i o n A G ° ( A G ° = A G ~ + AG~) and AAG q = (AG~) -- A G ~ ) / n F . We have t h e relationships [ 14]: exp(--AV~/R

T) = bo/~±

(51 )

e x p ( - - A G ° / R T ) = bR/~[±

(52)

in which ~/± is the m e a n ionic activity c o e f f i c i e n t . In t h e case o f a Langrnuir i s o t h e r m [ 14] : a s = 1 -- ( F o + FR)/Fm

(53)

In view o f eqns. (19), (20) and ( 5 1 ) - - ( 5 3 ) , t a k i n g into a c c o u n t t h a t the f o u r t h e x p o n e n t i a l in eqn. (50) is little d i f f e r e n t f r o m u n i t y , we o b t a i n f o r the adsorption current: ia = nFAt~pZhe t e x p ( - - ~ / 4 R T ) ~ / 7 ~ l ( b o b a ) i n ~, . s ~ - I i ~~0~' s ~-ln × (1 + bocSo + ~'R'~RI

__ c~01/2)

(54)

If we c o m p a r e this e x p r e s s i o n with eqn. (25), we see t h a t t h e y are e q u i v a l e n t if

ksl'~m = KPZhet~ 1 exp(--~/4RT)

(55)

L e t us assume t h a t k s has the f o r m given b y eqn. (47); t h e r e o r g a n i z a t i o n energies h and Xs s h o u l d be the same, so t h a t we obtain:

As = KPZhet(Fm~/,_+ )-1

(56)

If we take gp = 1, Zhet = 104 c m s -1 [ 1 1 ] , 3'-+ = 1 cm 3 mo1-1 (dilute solutions) and Fm = 2 × 10 -1° m o l c m - : , we o b t a i n As = 5 × 1013 s -~. This value is larger t h a n t h e value 6 × 1012 s -~ t a k e n b y B r o w n and A n s o n , which shows t h a t the rate o f collision o f the m o l e c u l e s with the e l e c t r o d e surface will n o t limit the process; we shall t h e r e f o r e use eqn. (49); we have t h e n : o = k s F m / k h C T = 6 × 10 s × 2 × 1 0 -1° CT1 = 0.12CT 1

(57)

It m u s t be r e m e m b e r e d t h a t the values o f b and (1 are n o t i n d e p e n d e n t since t h e y b o t h c o n t a i n the c o n c e n t r a t i o n c T. The reaction path

In a q u e o u s m e d i u m , bo will o f t e n be n o t very d i f f e r e n t f r o m ba (r ~- 1), at least at p o t e n t i a l s n o t t o o far f r o m the p o i n t o f zero charge or f o r a r o m a t i c systems, because m o l e c u l e s t h e n t e n d t o r e m a i n a d s o r b e d regardless o f t h e i r charge owing t o i n t e r a c t i o n s o f t h e 7r e l e c t r o n s w i t h the surface [ 1 5 ] . F o r a w e a k a d s o r p t i o n , bo = bR ~ 104 c m 3 mol-~; if c T is b e t w e e n 10 -6 and 10 -s m o l cm -3, b = i 0 -2 to 10 -4, so t h a t b -1 ~ r or r -1, and we have (cf. eqns. 43 and 44): p+ = p _ = p o ~ (~(u + b - l ) -1 = o b ( 1 + o b ) -1

(58)

30

As ab = 0.12CT 1 × CT(bobR) u2 = 1.2 × 103, we have P0 --~ 1 ; the reaction takes place practically entirely through the adsorbed species. The shape of the wave will, however, n o t be modified practically by the adsorption, so that it will appear as an ordinary "heterogeneous" wave with an apparent rate constant "),m. We can discuss the problem in more detail by considering semiquantitatively the influence of the potential. Frumkin has proposed (see refs. 16, 17, p. 61 and l b , p. 76) the following expression to describe the variations of the adsorption coefficient B: B M exp[--g(E -- EM) 2]

B =

(59)

Here, B M is the value of the adsorption coefficient at the potential E M of the m a x i m u m adsorption and g a coefficient. In principle, the influence of the potential should thus cause a deformation of the current which we have calculated above, but we will still assume that the adsorption coefficient is practically constant in the potential interval corresponding to the front of the wave, and study the influence of the location of t h e wave in the potential range. From eqn. (58) we can calculate the value of b corresponding to a given value of p0; if we designate by B the c o m m o n value of b o and bR, b = CTB, and we obtain: B = (aCT)-'po/(1 --P0) = 8.33p0/(1 --P0)

(60)

since OCT = ksFm/kh = 0.12. We have represented in Fig. 7 the variations of B (eqn. (59) and the limits given by eqn. (60). This graph shows that if O and R are still equally adsorbed, the reaction takes place mostly in the adsorbed state whatever the half-wave potential, i f g ~ 2 or 3, which is usually the case [17, p. 61, and 18].

2 = _ 9~/. o

_ ~

O 10°/o

_ XXX\\

-6

I

I

2. 0.5

+-1

\ ~

10,\', +_1.5

E-EM/V Fig. 7. V a r i a t i o n s o f log B a c c o r d i n g t o eqn. ( 5 9 ) ; B M = 104. T h e h o r i z o n t a l lines give t h e value o f P0 ( p e r c e n t a g e o f r e a c t i o n via t h e a d s o r b e d species). T h e n u m b e r o n each curve is t h e value of g.

31

Many organic c o m p o u n d s are m or e strongly adsorbed than assumed here, and the path through the adsorbed species should be a fortiori m ore c o m m o n . When B increases, the curves of Fig. 7 are shifted upwards, whereas the limits indicated by the horizontal lines remain at the same value. In water the reaction is usually complex, since it consists o f a series of electronic exchange and of protonations, but the different intermediates are usually adsorbed, so that the conclusion that adsorption should play a central role should still hold. If O is much less strongly adsorbed than R, or vice versa, and if adsorption is weak, the heterogeneous path can becom e favoured (cf. nonaqueous media). In non-aqueous medium, only one electron is usually exchanged. R e d u c t i o n of uncharged species occurs often at rather negative potentials, t o give a radical ion which should be m uc h less adsorbed than the reactant. If we assume for example that bo = 103 and bR = 10 -7, we can still apply eqn. (58), and we find p+ ~- p_ = 1.2 X 10 -3, i.e. the reaction will take place mainly through the nonadsorbed species. At positive potentials, or even at negative potentials if one starts from a positively charged form, it will be possible to find a situation where the adsorption o f bot h c o m p o n e n t s can be similar, so t hat the adsorption could play a role, as in water. CONCLUSION

Our results, which show the necessity of eventually considering the influence o f adsorption, even in cases where it could be t h o u g h t at first sight n o t to operate, have been obtained for a r o t a t i n g disk electrode. T h e y should, however, hold in the main for ot her electrochemical methods. The adsorption rates have been considered as infinitely fast in our t reat m ent , so t hat the results should be considered as an u p p e r limit; physical adsorption is indeed a fast process, which can n o t be measured by present m e t h o d s [19], so that our conclusion might reasonably apply t o t hat case. We will consider the influence of the adsorption rate in a subsequent paper. The relationship of Brown and Anson should give a reasonable order of magnitude for the ratio of the surface and h o m o g e n e o u s rate constants, even if one c a n n o t exclude t hat the reorganization energies should be somewhat different in bot h cases and t h a t that, relative to the surface process, should depend on the electrode material. It can seem surprising at first sight t hat the reaction has a t e n d e n c y to pass through the adsorbed species rather than through the non-adsorbed species, even at low adsorbabilities. This must be seen as a consequence of a c o n c e n t r a t i o n effect due to adsorption, which has already been shown to operate in the case of surface kinetic currents [17]. As shown above, the rate of the heterogeneous and of the surface reactions are of the same order of magnitude (cf. the values of the " a p p a r e n t " pre-exponential factors in b o t h cases); the reaction with the larger c o n c e n t r a t i o n will thus be favoured. L e t us consider for example a 10 -7 mol cm -3 solution, and let us calculate the c o n c e n t r a t i o n c s (equivalent to a superficial concentration) in a one-molecule-thick layer near the electrode. If we take 10 -7 cm f or the diameter of a molecule, we find cs = 10 -7 × 10 -7 = 10 -14 tool cm -2. On the o t h e r hand, if we consider a weakly adsorbed substance (B = 104 cm 3 mol-1), we have for a 10 -7 mol cm -3 solution F = F m B c / ( 1 + B c )

32 ~- 2 X 10 -1° X 104 X 10 -7 = 2 X 10 -13 mol cm-2; even at low adsorbability, the concentration in the adsorbed layer is larger than that in the layer adjacent to the electrode, and the difference increases when B increases. When CT increases, as mentioned above (paragraph on the influence of CT), the heterogeneous path becomes relatively more favoured because cs increases more rapidly than F, which eventually reaches a limit. This effect is, however, small when the adsorbability is low, and should become significant for concentrations which cannot be used in practice; it can also play a role when the adsorbability is high, so that full coverage is reached for small values of CT. Among the implications of our results, we will mention more particularly the verifications of the theories of electron transfer at electrodes [ 11,20], and the influence of the electrode material on the rate constants. The theories mentioned above predict that the transfer coefficient for a simple redox reaction should vary linearly with the potential. Experimental conditions must then be such that the adsorption path be negligible (a similar behaviour should be observed for the surface reaction [ 14], but the corrections for the influence of the potential at the outer Helmholtz plane should be replaced by a factor containing the electrical dependent part of the adsorption energies, in any case only one type of reaction should be treated at a time). When aromatic compounds are used, their reduction potential should be negative relative to the point of zero charge and n o t too near it. In the different studies previously carried out with organic compounds, this condition is fulfilled [21]. The same theories also yield a relationship between the rate constant for a homogeneous electron exchange in solution and that for a heterogeneous electron exchange at the electrode surface [11,20]. As shown above, adsorption causes an increase in the apparent heterogeneous rate constant, so that this possibility should be completely excluded; this seems to be the case for the majority of the examples studied [22]. It has been shown by Parsons t h a t the heterogeneous rate constant should be independent of the electrode material (for metallic electrodes) if there is no adsorption, whereas it might depend on it if the species are adsorbed [23]. As shown by eqn. (38), the apparent heterogeneous rate constant depends on the adsorption parameters b and r through the multiplying factor m. Attempts have been made to verify t h a t the rate constant is independent of the electrode material by using quinones in non-aqueous media [23b,24] on the grounds that there should be no adsorption under these conditions. The half-wave potentials of these compounds lie between about --0.4 and --0.6 V vs. a saturated calomel electrode, i.e. in the region of the point of zero charge. Under these conditions it c a n n o t be excluded that both the quinone and the radical resulting from its reduction are adsorbed, even though weakly (hydroquinone is adsorbed in nonaqueous media [12a]). The effects of adsorption can be best appreciated by considering the apparent heterogeneous rate constant ~/m (eqn. 38). In the case of a weak adsorption, b -1 > > r if r is n o t too different from unity, and the expression of m reduces to (cf. eqns. 27 and 57) m = 1 + ab = 1 + O.12(bobR)

In

(61)

If for example we assume that bo = bR = 10 {very weak adsorption), we find that m = 2.2; for bo = bR = 102, m = 13. From one case to the other, the rate

33

constant is multiplied by 5.9. Experimentally determined values of the heterogeneous rate constant obtained on the same metal show discrepancies [22,23b,24,25] which could be due to differences in adsorption, owing, for example, to differences in the purity of the solutions. On the other hand, in one study [24], the material of the electrode was found to have a significant effect (the rate constants vary by about one order of magnitude), which could be explained as above, since the adsorption coefficients may vary from one material to the other. In any case, quinones, which are possibly adsorbed, do not seem to be the best suited to verify that the heterogeneous reaction rate should be independent of the electrode material. REFERENCES 1 (a) M. G o u y , A n n . Claim. P h y s . , 8 ( 1 9 0 6 ) 2 9 1 ; R . P a y n e , J. E l e c t r o a n a l . Claem., 4 1 ( 1 9 7 3 ) 2 7 7 ; (b) B.B. D a m a s k i n , O . A . P e t r i a n d V . V . B a t r a k o v , A d s o r p t i o n o f O r g a n i c C o m p o u n d s a t E l e c t r o d e s , P l e n u m Press, N e w Y o r k , 1 9 7 1 . 2 R. G u i d e l l i , J. E l e c t r o a n a l , Claem., 1 8 ( 1 9 6 8 ) 5. 3 E. L a v i r o n , Bull. Soc. Claim. F t . ( 1 9 6 7 ) , 3 7 1 7 ; ( 1 9 6 8 ) 2 2 5 6 ; ( 1 9 6 9 ) 1 7 9 8 ; J. E l e c t r o a n a l . Claem., 5 2 (1974) 355 and 395; 63 (1975) 245; 105 (1979) 25 and 35. 4 E. L a v i r o n , J. E l e c t r o a n a l . Claem., 9 7 ( 1 9 7 9 ) 1 3 5 ; 1 0 1 ( 1 9 7 9 ) 1 9 . 5 R . G u i d e l l i , J. Plays. C h e m . , 7 4 ( 1 9 7 0 ) 9 5 ; H. M a t s u d a a n d P. D e l a r t a y , C o l l e c t . Czecla. Claem. C o m mun., 25 (1960) 2977. 6 R . G u i d e l l i a n d G. P e z z a t i n i , J . E l e c t r o a n a l . C h e m . , 8 4 ( 1 9 7 7 ) 2 1 1 ; V . G , Levicla, B.I. K a l k i n a n d E . D . B e l o k o l o s , E l e k t r o k h i m i y a , 1 ( 1 9 6 5 ) 1 2 7 3 ; K. H o l u b , C o l l e c t . C z e c h . Claem. C o m m u n . , 3 1 ( 1 9 6 6 ) 1 4 6 1 ; J. E l e c t r o a n a l . Claem., 1 6 ( 1 9 6 8 ) 4 3 3 ; E.M. P o d g a e t s k i i , E l e k t r o k l a i m i y a , 1 0 ( 1 9 7 4 ) 2 0 9 ; K . K. H o l u b a n d J. K o r y t a , C o l l e c t . C z e c h . C h e m , C o m m u n . , 3 0 ( 1 9 6 5 ) 3 7 8 5 ; M. S e n d a a n d P. D e l a h a y , J. Plays. C h e m . , 6 5 ( 1 9 6 1 ) 1 5 8 0 . 7 H . A . L a i t i n e n a n d J . E . B . R a n d l e s , T r a n s . F a r a d a y S o c . , 51 ( 1 9 5 5 ) 5 4 . 8 A.P. B r o w n a n d F.C. A n s o n , J. E l e c t r o a n a l . Claem., 9 2 ( 1 9 7 8 ) 1 3 3 . 9 E.M. P o d g a e t s k i i a n d V . Y u . F i l i n o v s k i i , E l e k t r o k h l m i y a , 6 ( 1 9 7 0 ) 1 1 7 8 , 1 0 P. D e l a h a y , N e w I n s t r u m e n t a l M e t h o d s in E l e c t r o c h e m i s t r y , I n t e r s c i e n c e , N e w Y o r k , 1 9 6 5 , p. 2 1 7 . 11 R . A . M a r c u s , J. Claem. Plays., 4 3 ( 1 9 6 5 ) 6 7 9 . 1 2 (a) U. G a u n i t z a n d W. L o r e n z , C o l l e c t . C z e c h . Claem. C o m m u n . , 3 6 ( 1 9 7 1 ) 7 9 6 ; (b) V . D . B e z u g l y i , L . A . , K o r s h i k o v a n d V.B. T i t o v a , E l e k t r o k l a i m i y a , 6 ( 1 9 7 0 ) 1 1 5 0 ; U. P a l m a n d A . A l u m a a , J. E l e c t r o a n a l . C h e m . , 9 0 ( 1 9 7 8 ) 2 1 9 ; R . I . K a g a n o v i c h , B.B. D a m a s k i n a n d M . K . Kaislleva, E l e k t r o k l a i m i y a , 6 ( 1 9 7 0 ) 1 3 5 9 ; R . V . I v a n o v a , L . N . K u z n e t s o v a a n d B.B. D a m a s k i n , E l e k t r o k l a i m i y a , 1 3 ( 1 9 7 7 ) 1 8 8 1 . 1 3 K . B . O l d h a m , J. E l e c t r o a n a l . C h e m . , 1 6 ( 1 9 6 8 ) 1 2 5 . 1 4 D.M. Molailner, J. Plays. C h e m . , 7 3 ( 1 9 6 9 ) 2 6 5 2 . 1 5 A. F r u m k i n a n d B. D a m a s k i n , P u r e A p p l . C h e m . , 1 5 ( 1 9 6 7 ) 2 6 3 ; B.E. C o n w a y a n d R . G . B a r r a d a s , Electroclaim. Acta, 5 (1961) 319. 1 6 A . N . F r u m k i n , Z. P h y s . C h e m . , 3 5 ( 1 9 2 6 ) 7 9 2 . 1 7 S.G. M a i r a n o v s k i i , C a t a l y t i c a n d K i n e t i c w a v e s in P o l a r o g r a p l a y , P l e n u m Press, N e w Y o r k , 1 9 6 8 . 1 8 J . A . V . B u t l e r , P r o c . R. S o c , , 1 2 2 A ( 1 9 2 9 ) 3 9 9 . 19 P. D e l a h a y , D o u b l e L a y e r a n d E l e c t r o d e K i n e t i c s , I n t e r s c i e n c e , N e w Y o r k , 1 9 6 5 , p. 1 1 7 . 2 0 N.S. H u s h , J. Claem. Plays., 2 8 ( 1 9 5 8 ) 9 6 2 ; V . G . L e v i c h , in P. Delalaay a n d D.W. T o b i a s ( E d s . ) , A d v a n c e s i n E ~ c t r o c l a e m i s t r y a n d Claemical E n g i n e e r i n g , V o l . 4 , I n t e r s c i e n c e , N e w Y o r k , 1 9 5 6 , p. 2 4 9 ; R . R . D o g o n a d z e , in N.S. H u s h ( E d . ) , R e a c t i o n s o f M o l e c u l e a t E l e c t r o d e s , W i l e y = I n t e r s c i e n c e , N e w Y o r k , 1 9 7 1 , p. 1 3 5 . 21 J . M . S a v d a n t a n d D. Tessier, J. E l e c t r o a n a l . Claem., 6 5 ( 1 9 7 5 ) 5 7 ; J. Plays Claem., 81 ( 1 9 7 7 ) 2 1 9 2 ; 8 2 ( 1 9 7 8 ) 1 7 2 3 ; J. G a r r e a u , J . M . S a v 6 a n t a n d D. Tessier, J. Plays. Claem., 8 3 ( 1 9 7 9 ) 3 0 0 3 . 2 2 H. K o j i m a a n d A . J . B a r d , J. A m . Claem. S o c . , 9 7 ( 1 9 7 5 ) 6 3 1 7 . 2 3 (a) R . P a r s o n s , S u r f . Sci., 2 ( 1 9 6 4 ) 4 1 8 ; (b) A. C a p o n a n d R . P a r s o n s , J. E l e c t r o a n a l . Claem., 4 6 (1973) 215. 2 4 T.W. R o s a n s k e a n d D . H . E v a n s , J. E l e c t r o a n a l . Claem., 7 2 ( 1 9 7 6 ) 2 7 7 . 2 5 R . S a m u e l s s o n a n d M. S h a r p , E l e c t r o c h i m . A c t a , 2 3 ( 1 9 7 8 ) 3 1 5 .