Since D is about 1O-5 cm s-i. the shortest available life-time will thus be ... why an enormous gain towards short life-times can be obtained. For life-times down.
J Electroanal. Chem., 205 (1986) 43-58 Elsevier Sequoia S.A., Lausanne - Printed
HOMOGENEOUS REACTIONS ELECTRON
CLAUDE
REDOX CATALYSIS
TRANSFERS
P. ANDRIEUX
43 m The Netherlands
FOLLOWED
and JEAN-MICHEL
OF ELECTROCHEMICAL
BY A VERY FAST CHEMICAL
STEP
SAVBANT
Laboratorre d’Electrochrmle de I’UniversitC de Pans 7, lJnlr&AssocrCe au C.N R.S. No. 438, “Electrochrmie Molkdarre”, 75221 Paris Cedex 05 (France) (Received
13th December
1985; in revised form 29th January
1986)
ABSTRACT Homogeneous redox catalysis of electrochemical reactions can be used as a means for characterizing very short-lived intermediates formed upon electron transfer with the electron-transfer reagent. What happens when the reaction following the electron-transfer step is so fast that it takes place within the molecular diffusion layer (solvent cage) is discussed and illustrated with previously reported experimental data. The problem of the transition between a sequential electron transfer-chemical reaction mechanism and a concerted mechanism is also addressed.
INTRODUCTION
Homogeneous catalysis of electrochemical reactions consists of the electrochemical generation of a chemically stable molecular reductant (or oxidant) diffusing in the solution and able to reduce (or oxidize) the substrate in place of the electrode. It is then usually possible to carry out the reduction (or oxidation) with a smaller overpotential than that required by the direct electrochemical process. We use the term “redox catalysis” to designate the cases where the only role of the electrochemically generated mediator is to shuttle electrons (or holes) from the electrode to the substrate [ 11. On the contrary, “chemical catalysis” would designate the cases where the transient formation of an adduct between the substrate and the active form of the mediator is an essential step of the catalytic process [1,2]. In the first case, the origin of the catalysis stems from the three-dimensional dispersion of the electron-transfer agent as opposed to the two-dimensional distribution of the electrons (or holes) supplied by the electrode surface [2]. Redox catalysis thus results from physical rather than chemical factors. Besides its applications in preparative scale electrochemistry [3], redox catalysis has recently been used as a tool for investigating mechanistic and kinetic problems, namely to characterize short-lived intermediates which are out of reach of the conventional electrochemical techniques [2,4-211. 0022-0728/86/$03
50
? 1986 Elsevier Sequoia
S.A.
44 PRINCIPLE
OF HOMOGENEOUS
REDOX
CATALYSIS
Although the actual mechanisms are more complex in most cases, the simple “EC” mechanism suffices for illustrating the principle of the method [4,5,7,8]. The heterogeneous or homogeneous electron-transfer steps, both designated by E. are followed by a first-order irreversible chemical step (C): Direct electrochemical
Mediated process: P+e-PQ hF, (E) Q+ALFB+P ,
process:
A+e-+B B2C
(Cl
where P/Q is the mediator (catalyst) couple, A is the substrate, B is the reduction intermediate and C is the product. E,& and E& will designate the standard potentials of the subscript redox couples. Since one of the main purposes of the present discussion is to compare direct and indirect electrochemical reactions involving the same reaction mechanism, it is appropriate to use the same set of symbolic letters (here “EC” [1,4,5,22-241) in both cases. We can likewise compare “ECE” [4,5,7,8,22,23,25], “CE” [2], etc. direct and indirect electrochemical reactions. This symbolism is different from the one which is sometimes used for catalysis of electrochemical reactions and in which the reaction sequence P + e- P Q, Q + A -j B + P is itself viewed as an “EC catalytic” reaction mechanism [26] by reference to the fact that the second reaction (designated by C) follows the electrode electrontransfer reaction (designated by E). This nomenclature is in fact somewhat confusing since the Q + A + B + P reaction both follows and precedes the electron-transfer step, being actually parallel to it. It is thus a “CE” and an “EC” mechanism at the same time. Because of this, and also taking into account that the electrochemical response is very different in this case as compared to the CE and EC direct electrochemical reactions, we prefer to use the term catalytic or mediated and, within this framework, to reserve terms like EC, CE, etc. to describe the reaction sequence involved in the mediated process. The catalytic enhancement of the P/Q wave upon addition of the substrate to the solution is a function of the kinetics of the two coupled reactions (E) and (C). Theories have been established that allow one to relate the magnitude of the catalytic increase of the current to the rate constants of reactions (E) and (C) [1.22-251. There are two limiting regimes for the kinetics of the (E)-(C) reaction sequence: (1) When k, -=ck_,c”, (c”, is the bulk concentration of the catalyst), the rate-determining step is (C) while (E) acts as a pre-equilibrium. The overall rate constant which can be derived from the catalytic current is then kc(kr/k-n) (2) When,
= kc exp{(F/RT)(Ga conversely,
k, a k_,
- Go)} c”p, the rate-determining
step is the forward
45
reaction (E) and thus the rate constant that can be derived from the catalytic current is then k,. Mixed kinetic control can also be observed. Under these conditions, variation of the catalyst concentration is a means for shifting the system from one control to the other, since the backward reaction (E) is second order whereas reaction (C) is first order (decreasing c”p shifts the system towards control by reaction (E) and vice versa). Using the redox catalysis data along these lines together with the direct electrochemical data has allowed the determination of values of the rate constant kc which are far above those obtainable from the usual electrochemical techniques [7-12,16,18-211. While the latter are restricted to intermediates (B) with life-times longer than about 1O-4 s. the redox catalytic method is able to tackle the nanosecond time range. Why this huge gain in performance is possible is worth examining in some detail. In direct electrochemistry, the ability to detect a short-lived intermediate is related to the minimal thickness of the diffusion layer. The reaction layer, i.e. the portion of the space adjacent to the electrode where B decays, has a thickness of about (D/kc)‘/‘. Detection of B will then be possible until the reaction layer is about one third of the diffusion layer thickness [26]. It is possible to decrease the diffusion layer thickness by going to short times with transient techniques or to high rotation rates in rotating electrode voltammetry. It is not, however, possible to go much below 10e4 cm in the context of common electrochemical techniques. Since D is about 1O-5 cm s-i. the shortest available life-time will thus be about 10m4 s-‘. The homogeneous reduction process can also be viewed as a kind of electrolysis, where the Q molecule plays the role of the electrode. An essential difference. however, is that the corresponding diffusion layer thickness is of the order of magnitude of molecular sizes, i.e. a few tenths of a nm. This is the essential reason why an enormous gain towards short life-times can be obtained. For life-times down to the nanosecond range, the reaction layer is in fact larger than the “molecular diffusion layer”. The “molecular electrolysis” thus takes place outside the diffusion layer similarly to what occurs in coulometry with slow follow-up chemical reactions
v71. Under these conditions. as long as k_, c”pcan be raised up to a value close to k, within the available range of catalyst concentrations, ci, it is possible to derive h-c from the redox catalysis data. In the case where k_, is close to the diffusion limit. k,, which corresponds to large values of E& - E,&, it follows that the largest attainable value of k, is about k, (cOp),,,, i.e. taking (cz),,, = 5 X lo-’ M and k, = 10” AC’ SC’ (in dimethyl formamide [S]), 5 x 10’ s-l. Larger k,s can no longer be derived from the redox catalysis data since the kinetic control is now by the forward reaction (E). However it is still possible to obtain from them the standard potential of the A/B couple under conditions where the (C) reaction occurs outside the “molecular diffusion layer”. Then [4,7] breaking down the electron-transfer process into three steps: (E):
Q + Az(QA);$;(PB)FB P
+ I’, I>
(C): BzC c
and writing the steady-state assumption for the pairs of reactants, QA and PB, inside the solvent cage, one obtains [28] for the forward global electron-transfer rate constant, k,: 1 -=&+$+
1
kE
t
D
exP{(F/RT)(%
k,
(1)
- -%o)}
where kFt and kaC&are the activation-controlled electron-transfer rate constants. Note in addition that the backward global electron-transfer rate constant, k_,, is given by
limit rate constant, is given by k, = 4nN,DR (where NA is k,, the diffusion Avogadro’s constant, R is the distance of closest approach between the centres of the two reactants and D is twice the average diffusion coefficient of the two reactants), applying the Fick law of spherical diffusion to the mutual movement of the reactant molecules [28]. On the basis of the indications given in refs. 29a and 29b and in footnote 20 in ref. 8, the derivation of eqn. (1) is as follows. The steady-state assumption for QA and PB can be written as (k,
+ kE’)[QA]
- k”‘;[PB]
+ (k,+
-k;‘[QA]
= k,[Q][A]
k”“;)[PB]
= k,[P][B]
Thus.
[QAI=
k,+k”‘f,
+k~~t+k~~[QIIAl+pD+~~+k~~[P][B]
k
D
E
and
WI --=--
d[Ql dbl -=-= dt ==
d[Bl
dt
dt
dt
k,k;’ k, + kg’ + k”‘:, -k,[Ql[A]
+
[Ql[Al+ k,
k,k”c:, + k;,, + k”‘;
PI PI
k-,[PI[Bl
According to Marcus’s theory (301, In kg’ and In k”ck are parabolic functions of it is the driving force, F(E& - E,&). For the purpose of the present discussion, sufficient to approximate the variation of k?k by 1 z=kz
1 exp{[(l
-a)F/RT](EFo-E,“,)}
which means that the activation becomes infinite. The expression 1 -= k;”
1
1 kg exp{(aF/RT)(E&-E&)}
+z
energy tends towards for kE’ ensues:
zero when the driving 1
+Z,,,I exp{(F/RT)(E.&-&)}
force
(21
41
(where kz is the standard rate constant and Z,,, is the bimolecular collision frequency) with a constant value of (Y close to 0.5. This amounts to neglect of the variation of (Y(which is indeed small for organic aromatic compounds [31]). The standard rate constant, kg, i.e. the value of kgt for a zero driving force can be expressed as kg = Zso, exp( -AG k_, c”,) and not too large so that it occurs outside the molecular diffusion layer. What happens when k, is so large that the reaction takes place within the solvent cage is the object of the following discussion. Similar to what has previously been done for an electrode process [33], we will discuss both the case where B diffuses away from P while being converted into C and the case where the reaction is so fast that B has no time to diffuse away from the electron-transfer reactant (which is analogous to a surface reaction [33] in an electrochemical process). We will also address the question of the transition between a two-step EC process and a concerted EC process which would be obtained in the case of an extremely fast follow-up reaction. Redox catalysis data obtained previously [6,7] for the reduction of aromatic halides in non-aqueous solvents will serve to illustrate the discussion.
k(E) VS EFQ RELATIONSHIP WHEN REACTION (C) IS SO FAST AS TO OCCUR MOLECULAR DIFFUSION LAYER OF THE ELECTRON-TRANSFER REAGENT
WITHIN
THE
We follow the same approach as Debye [28d]. omitting. for simplicity, the interaction energy between the two reactants (electrostatic. for example. if both are charged). Reactant A diffuses. in the context of spherical diffusion, towards the molecule Q with a diffusion coefficient, D, equal to the sum of the two individual diffusion coefficients. As represented in Fig. 2, they exchange one electron when they are in contact (the distance of the centres is then R), giving rise to P and B. When kc is not too large. B diffuses away from P and converts into C outside the diffusion layer. For larger k,s, B is converted into C within the diffusion layer (lower pathway in Fig. 2). For even larger values of kc, B is converted into C before diffusing away from P (upper pathway in Fig. 2). In all cases. at steady state (r is the distance from the centre of the electrontransfer reagent molecule) the following equations apply in terms of molecular concentrations around each mediator molecule:
with (cA)= = cq, cA designating Integration leads to
the bulk concentration
(4)
R where the subscript
R means
of A.
r = R.
49
8 C
P
+
0
0
C
Q
Fig. 2 SpherIcal
diffusion
The same decomposition
representation
of the E-C
reactlon
diffusion equation applies occurs outside the diffusion
sequence.
likewise layer:
for
B in the
case
where
its
(5) with (c,), = cn, cn designating Integration then leads to
the bulk concentration
of B.
(6) On the other hand, the rate of consumption mediator molecule can be written as hR=D
i
2
i
of A and the appearance
= kF’( CA)R - k$(
CB)R
of B at each
(7)
R
R
kF’ and k?f, being the molecular rate constants for the reduction of A and the oxidation of B at the surface of the mediator. respectively. It follows from eqns. (6) and (7) that
R
R
(4aDR) kg’ = (4aDR)
(4mDR)kY;
+ kg’ + k”‘:, cA - (4rDR ) + kg’ + k”‘;_ ”
(8)
The global rate of consumption of A molecules and production of B molecules at each mediator molecule thus appears as the difference of two terms featuring the
50
reduction of A and the oxidation of B, respectively. The former arises when the mediator is in the form of Q and the latter when it is in the form of P. Upon extension to the overall set of P and Q molecules present in the solution, the first term is multiplied by co and the second by cr. Expressing the rates in terms of moles, one then finally obtains eqn. (1) from eqn. (8). In the opposite case, where k, is so large that B collapses at the surface of P without diffusing away, eqn. (4) is still valid but not eqns. (5) and (6). Equation (7) still applies for the flux of A but not for that of B which is then equal to zero. We can thus write the steady-state assumption for B, which leads to
where I/ is the maximal volume occupied by the B molecules when they are in contact with P. Combination of eqns. (4) (7) (first part) and (9) then leads, in terms of molar concentrations, to
which replaces
eqn. (1). i.e. taking eqn. (2) into account,
which replaces eqn. (3). This is equivalent to eqn. (2) in ref. 34, making k,, = k,, = k, on the basis of the spherical diffusion approximation. For smaller values of k,, B diffuses within the molecular diffusion layer while being converted into C, eqn. (5) is then replaced by ~~~~~~~~~~~ ____ \ dr2 r with. as boundary
d(CB)r
___
dr
conditions,
eqn. (7) and (c,),
= 0. Integration
then leads to
and thus to 1 -l+l+ G-k”,”
1 k,
k,(l
+ Rk;“/D”*)
eXp{
(F/RT)(
E;B
-
E&o>}
(12)
51
in place of eqns. (1) and (10). Taking
eqn. (2) into account:
1
1
1
exp{(F/RT)(E&-Ego))
k,(l+Rk[y”2/D”2)+%
which replaces eqns. (3) and (11). Figure 3 gives a representation of the variations of k, with Ego which combines eqns. (3). (11) and (13) according to the value of k,. In all cases, the diagram breaks down into three asymptotes with 0. l/120 and l/60 mV’ slopes. (The fourth asymptote shown on the right-hand portion of the diagram arises for reasons described in the next section.) The first two are the same whatever the value of X-c, whereas the location of the third is a function of k,. The latter is represented in Fig. 4 under the form of the variation of ( EFo - E&,) (E& is the abscissa of the intersection of the l/60 mV-’ asymptote with log k,) with k,. Section a of the curve (low values of k,) represents eqn. (13) and section b (high values of kc). eqn.
lo&/M-‘r-‘)
9
logk;
________
8
7
6
5
4
3
2 I
I b
tiu mv
Fig. 3. Variations of k, wth E& as a function of k, (the number accordmg to eqns. (l), (9) and (10). for typical values of k, (10” R (0.3 nm). Temp. 302 K.
on each hne IS the value of kc m s - ’ ) M-’ SC’). D (2x lo-’ cm’ SC]) and
52
kc.610'*s-'
iog(k,/s-‘) I
13
Fig. 4. Vanation
of (E&
- E&),
wth
kc.Same numerical
values as m Fig. 3. (a) Equation
(12); (b) eqn.
(10).
(11). Figure 4 thus shows the error on the determination of EiB as a function of k, when the conversion of B into C within the solvent cage is ignored. The diagrams in Figs. 3 and 4 were drawn for the following numerical values: k, = 10” M-’ s-l, D=2X10p5 cm* s-i, ZsO, = 3 x 10” M-i s-‘, which are typical of an organic aprotic solvent (DMF) and of aromatic substrates and mediators, and we considered the case of a large catalyst (0.3 nm radius) and of a smaller substrate (0.15 nm radius). Thus NAV = 0.22 1 and R = 0.45 nm. The value of kz in Fig. 3 is arbitrary and does not affect the generality of the analysis. From left to right, the first two portions of the working curve represent two limiting behaviours. The actual curve would involve a smooth transition between them. This simply reflects the fact that the actual curve corresponds to a diffusion process that occurs over too short distances, as compared to molecular size, for the Fick law to be obeyed strictly [34]. Taking for the maximal value of k,, 6 X 10” s-l (kT/h or frequency of collision with the solvent), we find the error on E& cannot exceed 83 mV (at 302 K). As shown in the next two sections, the maximum error is significantly less than this figure in most practical cases.
TRANSITION
BETWEEN
SUCCESSIVE
AND CONCERTED
E-C
STEPS
When the rate of the C step is so large that B collapses immediately after electron transfer has occurred without having time to diffuse away. the question arises of whether the electron transfer and the follow-up reaction are actually successive steps or rather occur along a concerted mechanism. If B collapses within the encounter complex PB before PB has time to collide with a solvent molecule, the energy required to convert B into C has to be gained during the initial collision, meaning that the E and C steps are concerted. Thus 6 X lo’* s-i (kT/h at 302 K, or alternatively the collision frequency with a solvent molecule) can be taken as the
53
0
EAC
reaction
coordinate
Fig. 5. Sequential and concerted EC mechanisms. Parabohc representation surfaces as a function of E&.
of the potential energy
of k, over which the EC process can be considered as concerted. Under these conditions, the 60 mV slope diffusion line in the log k,-E& diagram disappears (Fig. 3) the equation replacing eqn. (1). (10) or (12) being
value
l/k,
= l/k;‘+
l/k,
k:’ now being the activation-controlled forward rate constant of the concerted EC process. A new l/60 mV_’ diffusion line (not represented in Fig. 3) may reappear but at more positive values of E&, corresponding to the replacement of E,& by E&. (i.e. shifted by an amount corresponding to the driving force of the B --, C reaction):
1 -l+l+ G-kg’
1 k,
k,
exp{(~/RTN%-E&))
For slower B --$ C reactions, the potential energy surfaces are as represented schematically in Fig. 5. The reaction coordinate is a composite of the reaction coordinate of reaction B + C and of external (solvation) and internal factors controlling the electron-transfer step. Upon increasing E&, the system may pass from a situation where B is an actual intermediate (upper curve) to a situation where
Fig. 6. Sequential and concerted as a function of E&.
EC mechamsms.
Linear
representation
of the potential
energy
surfaces
the concerted EC mechanism (lower curve) is more advantageous energetically than the sequential EC mechanism [35]. It is thus expected that the l/60 mV-’ slope diffusion line be replaced by a 1,420 mV-’ line corresponding to the rate of the EC process under activation control (Fig. 3). Still another l/60 mV-’ slope diffusion line (not represented in Fig. 3) may appear upon increasing E& but this would correspond, as discussed above, to much more positive values of E& and thus to very small values of k,. An estimate of the position where the new activation control line replaces the l/60 mV-’ slope diffusion line can be obtained as shown in Fig. 6. The potential energy curves are approximated by straight lines having the same slope for all three stages, A, B and C. The equation AG:c
= AG,f, + AG,‘,
(14)
can thus be used to estimate the value of E& where the system passes from a sequential A-B-C mechanism to a concerted A-C mechanism. We do not intend to provide here a detailed description of the transition between the sequential A-B-C and concerted A-C mechanisms. This would require us to
55
know whether the A-B step passes through a Marcus “inverted region” before the A and C potential free energy curves intersect. The existence of an “inverted electron transfers involving region” has not yet been detected in intermolecular aromatic reactants [35], which has been interpreted in terms of quantum corrections reducing the inverted effects significantly [30]. Note, however, that evidence for inverted region behaviour has been provided recently for intramolecular electron transfers in organic molecules [36]. The only point we wish to emphasize here is that a decrease in driving force should result in the passage from a sequential to a concerted electron transfer-bond breaking mechanism. In this context, eqn. (14) provides a correct estimation of the location of the additional l/120 mV line. even though it corresponds to a discontinuity in the activation Gibbs energy of the overall reaction. The straight lines represented in Fig. 3 ensue. They show how one passes, upon AB reaction) to a increasing E&, from a l/120 mV_’ slope (activation-controlled l/60 mV_’ slope (mixed-controlled by the diffusion of B and its simultaneous conversion into C) and again to a l/120 mV_’ slope (activation-controlled AC reaction) as a function of the standard potential of the catalyst couple. The location of the right-hand l/120 mV-’ slope asymptotes is derived from the activation Gibbs energy AG& , this being zero for k, = 6 X 1012 SC’. REDOX CATALYSIS ORGANIC APROTIC
OF THE SOLVENTS
REDUCTIVE
CLEAVAGE
OF
AROMATIC
HALIDES
IN
We now discuss, as an example illustrating the preceding analysis, the results obtained previously in the redox catalysis of the reduction of aromatic halides [7,8]. In the original treatment [7,8], the possible effect of kc on the determination of the standard potential as sketched in Figs. 1 and 3 was ignored. It is thus worth re-examining these data in order to estimate the magnitude of the possible ensuing on that of the homogeerrors on the determination of EiB and, as a consequence, neous and heterogeneous standard rate constants for electron transfer. In all the cases where it was possible to determine both E,& and k, [8], kc was smaller than 5 x 10’ s-t. Thus the error in E,&, as determined by the method sketched in Fig. 1, was less than 3 mV (as immediately derived from the working curve in Fig. 3) in these cases. The problem really arises for faster cleavage reactions, i.e. in the chloro- and bromobenzene and pyridine series [7], where the exact value of k, is not known (note that an early redox catalysis investigation of the reduction of chlorobenzene in DMF concluded erroneously, due to insufficient accuracy in the numerical treatment, that k, was of the order of 10’ s-i [5]). Comparison of the theoretical log k,-E& diagram (Fig. 3) to the experimental diagrams obtained previously for the bromo- and chlorobenzenes and pyridines [7] provides an estimate of the upper limit of kc in each case (Table 1): the lowest point in the experimental k, vs. E& plots is taken as the maximal value of k, below which the concerted electron transfer-bond breaking process could occur.
56 TABLE
1
Homogeneous and heterogeneous pyridines m DMF
electron-transfer
Compound
kc/s-’
E&JV
PhBr PhCl 2-PyBr 3-PyBr 2-PyCI 3-PyCl
