J. Phys. Chem. 1993,97, 10879-10888
10879
Reactions Occurring during Mixing. Basic Reaction Schemes and Examples in Solvated Electron Reductions Claude P. Andrieux and Jean-Michel Sadant' Laboratoire d'Electrochimie Mol6culaire de l'Uniuersit6 de Paris 7 , Unit6 Associde au CNRS No. 438, 2 place Jussieu, 75251 Paris Cedex 05, France Received: March 22, 1993; In Final Form: July 12, 1993"
A model is proposed that allows the quantitative rationalization of product distributions in reactions occurring during the mixing of two reactants. Illustrating examples have been selected having mostly in mind reactions triggered by pieces of metal delivering solvated electrons in the reactant solution. It may, however, be applied to many other types of reactions occurring during mixing.
Many chemical reactions occur during mixing of the reactants. When two solutions each containing one of the two reactants are put together and these react rapidly with no agitation or moderate agitation, as obtained with conventional magnetic stirrers, typical mixing time can be estimated as follows. In the absence of large convective instabilities' as, for example, when the two solutions to be mixed are separated by a well-defined interface or when the reaction involves solvated electrons coming out of a piece of metal introduced in the reactant solution, a diffusion layer builds up at steady state, as sketched in Figure 1 parts a and b, respectively. The concentration of the two reactants, A and E, is homogenized thanks to natural or forced convection outside of a diffusion layer with a thickness of the order of 10-3-10-2 cm.2 Usually, molecules have a diffusion coefficient of the order of 10-5 cm2 cm-1. It follows that the travel of the A or E molecules across the diffusion layer is typically of the order of seconds or fractions of a second. Many reactions involve much lower lifetimes leading, at steady state, to concentration profiles of the type shown in Figure 1. The concentration profiles are linear or, conversely, equal to zero, outside a thin reaction layer, thin as compared to the diffusion layer, within which the two reactants coexist with concentrations, which, albeit small, both differ from zero. This model of reactions occurring during mixing will be developed in the following sections starting with the simplest of all conceivabiz reaction schemes, A + E B. The concentration profiles inside the reaction layer and the variation of the bulk concentrationof reactants and product with time will becomputed. Then several more complicated reaction schemes involving the competitiveformation of several products will be analyzed having mainly in mind application to reactions that are triggered by an initial step where one of the reactants is the solvated electron diffusing out of a piece of metal introduced in the solution. As discussed in a parallel paper,3 these reactions have several interestingapplicationsin radical chemistry as, for example, S R N ~ aromatic nucleophilic substitutions4J and radical cyclizations.6 The precise understanding of selectivity-structure relationships for these reactions requires the application of the model developed here. Although the illustrating examples are taken in this particular area of chemistry,the model that we develop hereinafter may also be applied with only slight adaptations to any kind of reaction occurring during the mixing of the reactants, at least in the absence of strong hydrodynamic instabilities. The Nernst approximation was made throughout the following applications of the model. It consists of replacing the continuous concentration profiles that develop between the interface and the bulk of the solution by two linear segments that intersect at one boundary of the "diffusion layer" as sketched in Figure 1. The
-
*Abstract published in Advance ACS Absfracts, September 1, 1993.
thickness of the diffusion layer, C, is thus related to the hydrodynamics of the natural or forced convection outside the diffusion layer. Applications of this approximation to electrochemical reactions have shown that it works satisfactorily not only for analyzing the current responses of microelectrodesunder steady-state conditions (rotating disk electrodes)' but also for predicting product distributions in preparative-scaleelectrolyses? As will appear in the following sections, application of the model requires the value of e, or of a parameter containing C, to be known for many reaction schemes. It may then be determined from a reference reaction for which the pertinent rate constants are known. These procedures are illustrated in ref 3.
Results and Discussion
-
kE 1. Concentration Profiles for the A + E B Reaction Scheme. In the application to reactions where E represents solvated electrons diffusing out of a piece of metal, the latter takes the place of the portion of space designated by "E solution" in Figure 1. The concentration of E at the boundary between this region of space and the diffusion layer is a constant that we designate by The concentration of A in the region designated by "A solution" is and may vary with time in a way that will be described in the next section. The product, B, may diffuse out of the reaction layer where it is generated and penetrate the region "A solution" but not the region "E~olution".~ If we assume that the diffusion coefficients of E, A, and B are the same (common value a),the concentration profilesof A and E obey the following set of differential equations and boundary conditions.
e.
x = o : C,=c",,
C,=O
(3) (4)
where the Cvalues are the space-dependent concentrations of the subscript species and x and C the distances that are defined in Figure 1. We consider the situation that is met when the rate constant k~ is so large as to render the reaction layer thickness small (mathematically infinitely small) as compared to the diffusion layer thickness.10 As sketched in Figure 1, the diffusion layer
0022-3654/93/2097-10879$04.00/0 0 1993 American Chemical Society
Andrieux and Savhnt
10880 The Journal of Physical Chemistry, Vol. 97,No. 41, 1993
b
a I I
Q
x
: d
: i
reaction layer diffusion layer.
E solution
0
: :
:
x
: d :
reaction layer diffusion layer
Esolution :
A solution
A solution
Figure 1. Reaction occurring during mixing of reactants A and E yielding product B. Concentration profiles a t the onset of the steady state. (a) A, E, and B are soluble in both initial solutions, (b) A, E, and B are soluble in the initial A solution; A and B are insoluble in the initial E solution (e.g., a piece of metal from which the electrons diffuse out in the A solution).
may then be subdivided into three zones. The central zone is the reaction layer where E and A coexist although their concentrations are small. In the zone to the left of the reaction layer, the concentration of A is zero while the concentration profile of E is linear. Conversely, in the zone to the right of the reaction layer, the concentration of E is zero and the concentration profile of A is linear. The prolongations of the linear concentration profile of E on the left-hand side and of the linear concentration profile of A on the right-hand side intersect the x-axis a t the same point ( x = d) which defines the center of the reaction layer. This geometry implies that
e=-
e,
a=-
c
and multiply it by
c*
AB^/^
(7)
e,
where
eq 5 becomes 1 1 -+--=
>> 1
z = -x - d
e,,Le., introducing
CE and
Le., since y
a-e=y-1 (12) We now perform a secondchangeof thespaceandconcentration variables aiming a t converting the space and concentration variables that have infinitely small values within the reaction layer into variables that are commensurate with 1. We thus define a space variable, z, which has as origin the center of the reaction layer,
Normalizing the distance by C, i.e., introducing
and the concentrations by
analysis under these particular conditions noting that for other problems the treatment may be adapted by removing thecondition expressed by eq 10 and replacing it by eq 8. With the simple reaction scheme discussed in this section subtraction of eq 1 from eq 2 and integration taking account of the boundary conditions leads, in dimensionless terms, to
1 Similarly the concentration variables a and e are changed into
where y is the excess factor defined as
e,
y=-
(9)
e,
As seen below (daldy)! = -(de/dy)o for the simple reaction schemeconsidered in this section. For theother reaction schemes dealt with in the following sections they are of the same order of magnitude. In the case of solvated electrons diffusing out from a piece of alkali metal is very large (of the order of 10 MI'), very much larger than the current values of Thus y and therefore eq 8 reduces to
-
c.
($)o
These changes of variables12 allow the reformulation of eqs 1-4 as
=- 1
implying that the center of the reaction layer is close to the righthand boundary of the diffusion layer. We will continue the
d2e* - a*e* = 0 dz*2 d2a* - a*e* = 0 dz*2 z* = - w :
de* dz* - - 1,
a*
=O
t 19)
The Journal of Physical Chemistry, Vol. 97, No. 41, 1993 10881
Reactions Occurring during Mixing
where the "reaction time"
t,
is defined as
SBCO,
t, = -
vt C A
After time t,, the reactant E would spill over the diffusion layer and penetrate the A solution. The concentration of the product B inside the diffusion layer obeys the following differential equations.
-2
-1
1
0 Z'
2
Figure 2. A and E dimensionless concentration profiles in the reaction layer. a*(z*) and e * ( z * ) represent functions for the simple reaction scheme A
-.
+ E k,
B. For the definition of symbols, see text.
At the onset of steady state, the concentration profile of B is as sketched in Figure 1. From then on, the concentration of B in the bulk, cog, will increase at the expense of that of A, according to
I-db*\
which implies that a*
- e* = z*
(21) No known analytical functions are solutions of the above set of equations. Numerical computation (see Appendix) leads to the functions a*(z*) and e*(z*) that are represented in Figure 2. Using the equations that define z* (13-15), a*, and e* (7,16) from the actual experimental parameters, one may derive what are the concentrations of A and E inside the reaction layer and what are the dimensions of the reaction layer e*, = a*, = 0.545
(22) and thus the thickness of the reaction layer is of the order of9
and the concentrations of A and E at the center of the reaction layer,
2. Rates of Decay of the Reactant and Appearance of the kE Product in the Bulk Solution for the A + E B Reaction Scheme. The disappearance of the reactant A in the bulk of the solution is governed by eq 25
where b* is a dimensionless variable defined as
where ( C B )is~ the constant value of CBin the left hand side part of the diffusion (Figure 1). With the considered simple reaction scheme both (db*/dz*), = 1 and (da*/dz*), = 1, but this ought not to be the case for more complicated reaction schemes leading to several products as illustrated in the following sections. An experimental procedure frequently used with reactions involving solvated electrons is to add successively little bits of the metal each containing a number of moles smaller than the number of A moles contained in the solution. Each bit of metal added thus disappears until all the reactant A has been consumed in successive steps. If we assume then each piece of metal is a sphere, the disappearance of the piece of metal and of A is given by the following differential equation.
-
where n M is the number of moles of electrons in the metal and n; its initial value; M Eand d~ are the molar mass and desity of the metal, respectively. It follows that the decay of the piece of metal is obtained from eq 34.
with, for t = 0, (34) = CA (26) where t is the time, Vthe volume of the A solution, S the surface area covered by the reaction layer, and CAthe initial concentration of A in the bulk solution. Equation 25 may be rewritten as
Where the characteristic decay time
is defined by
The decay of reactant A thus occurs in a stepwise manner according to the differential equation
If the source of reactant E were infinitely large, S would not vary with time and then the decay of the A concentration would be
e=o
for t 2 t ,
where nA is the number of moles of A contained in the solution and i L its initial value. Thus during the dissolution of the first piece of metal, the number of moles of A decreases from n i to
Andrieux and Savbnt
10882 The Journal of Physical Chemistry, Vol. 97, No. 41, 1993
n i - [nk/(da*/dz*),] according to
3 AIL
The concentration of A remaining after the dissolution of the first piece of metal is thus
I
A
t
. .
CA nf4 During the dissolution of the second pice of metal, the decay curve is the same, making the concentration of A pass from
..
I & 0
I
I
I
I
1
2
3
4
t I
and so forth upon addition of successive pieces of metal until all A has been consumed. An example of the decay of the reactant A is given in Figure 3 in the case where ni/nk = 3 and for (da*/dz*), = 1 as is the case for the simple mechanism under discussion in this section. The appearance of the product, B, follows a converse law. During the dissolution of the first piece of metal
as illustrated in Figure 3 for the simple mechanism under discussion where (db*/dz*), = 1. 3. Two-E Mechanisms (ECE and DISP). We now consider the following reaction scheme (Scheme I) in which the product of the reduction of E by A is converted through a first-order reaction into a second intermediate that may fuither react with the reactant E and/or with the first intermediate. This is
5
tE
2nd
1st
'.
3rd
piece of metal
Figure 3. Decay of the reactant A and appearance of the product B (A + E k, B) for the successive addition of three pieces of metal in the case where the number of moles contained in each piece is one third of the initial number of moles of A in the solution. m, decay of the reactant, A, appearance of the product, nB/nk.
-
Besides the first reaction, all other reactions taking place in the reaction layer may be assumed to obey the steady-state approximation, Le.,
It follows, with the same normalizations as in section 1, that
SCHEME I
(45) kE
A+E-+B
d2a* -a e (l - X ) = O dz*2
k
B+C with
k'e
x=++--;-[ 1 1
C+E-D
Pa
kb
C
+ B -.A + D'
(DISP)
reminiscent of the ECE-DISP mechanism in molecular electroc h e m i ~ t r y 2 where ~ J ~ the first and second electron transfers involve the electrode surface instead of the reactant E in the present case. For example, an aromatic halide, RX(A), may be reduced by solvated electrons in liquid ammonia into the corresponding anion radical, RX'-(B). The latter cleaves off the halogen ion, X-, yielding the aryl radical R'(C) that may then be reduced by a solvated electron and/or by a RX*-(B) molecule. The products of the two reductions D and D'are here the same, R-(that converts into R H by protonation by the solvent or residual water). The differential equations governing the concentration profiles of E and A are as follows:
(
3+L2-8]L/3 pa*)
(47)
in which the parameter
governs the competition between the reduction of C by B and by E, respectively. Addition of eqs 44 and 45 and integration leads to 2a* - e * 3: z* (49) and thus to
as a reflection of the fact that, whatever the competition between the two pathways, the overall reaction consumes 2 molecules of E per molecule of A. When p 0 (ECE), the E concentration profile is the same as in the simple mechanism discussed in section 1 whereas the A concentration profile is half the symmetrical of the E profile around the center of the reaction layer (Figure 4). (DISP) the E concentration profile slightly When p changes: it is obtained from the preceding by multiplication by
-
with eqs 3 and 4 as boundary conditions.
**
-
The Journal of Physical Chemistry, Vol. 97,, No. 41, 1993 10883
Reactions Occurring during Mixing
-4
-3
-1
-2
0
1
2
h P Figure 5. Yields of D and D' for reaction Scheme I as a function of the competition parameter p (eq 48).
with (53)
Normalization of the D and D' profiles may be carried out as follows:
e e,
and being the concentrations of D and D', respectively, in the left hand side section of the diffusion layer (Figure 1). Thus, when taken into the steady-state approximation eqs 43 and 44 account, eqs 51 and 52 become
-
Fipre4. DimensionlessE and A concentrationprofiles (e*(z*),a*(r*)) for reaction Scheme I in the two limiting cases where p 0 (ECE) and p (DISP).
-
2113 (Figure 4). The A profile is still half the symmetrical of the E profile around the center of the reaction layer. In between these two limiting behaviors, Le., for intermediate values of p, both the A and E profiles vary. The variations are, however, small as are the differences between the two limiting sets of profiles. The competition between the two pathways is better observed upon comparing the productions of D (result of the ECE pathway) and D' (result of the DISP pathway) even though D and D' will be the same species in most practical applications.
d2cDt 2 ) -
dx2
+ k',CBCc
=0
Knowing the a*(z*) and e*(z*) profiles previously computed for any value ofp, one may then integrate numerically eqs 55 and 56 so as to obtain (dd*/dz*), and (ddm/dz*), and thus the yield of D and D' for each mole of A consumed:
P(D) =
(dd*/dz*),
, P(D') =
(da * /dz*) ,
(dd"/dz*), (da*/dz*),
(57)
as a function of the competition parameter p (Figure 5 ) . As expected, the production of D increases with k'E and conversely decreases when k $ increases (eq 47). Less obvious is the role Of kE, the rate constant of the initial step, the production of D decreases when kE increases. The physical reason behind this result is that as kE increases, the amount of E in the reaction layer decreases thus minimizing the C E reaction. The effect of k is also worth noting: the efficiency of the ECE pathway increases with k and vice versa for the DISP pathway. As k increases, the steady-state concentration of B decreases, thus decreasing the efficiency of the DISP pathway at the expense of the ECE pathway. 4. Competition between 1E per A and 2E per A Reaction Pathways. A simple reaction scheme illustrating such a competition is that depicted in Scheme I1 where the intermediate B can either be transformed into C with no further consumption of E or react with E to yield D.
+
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10884 The Journal of Physical Chemistry, Vol. 97, No. 41, 1993 2.5
2
1.5
1
0.5 -2
-3
0
1
0
-1
2
logo Figure 7. Yields of C and D for reaction Scheme I1 as a function of the competition parameter u (eq 60).
2
where the competition parameter u is defined as
1.5
kkE1I3 4213 (J=-
k&
1
eE2/3a1/3
Examples of the E and A profiles are given in Figure 6 for three typical values of the competition parameter u. The production of C and D is then given by
OS
0
P(C) = 2 2
+ (de*/dz*)-, (da*/dz*),
P(C) 1.s
P(C)
1
-
, P(D) = -
-
0 and P(D)
-
1 and P(D)
-
(de*/dz*)-, -1 (da*/dz*),
1,2E/A when u 0, E/A when u
-
-
(61)
0
Figure6. Dimensionless E and A concentrationprofiles (e*@*), a*(z* for reaction Scheme I1 as a function of the competition parameter
The variations of P(C) and P(D) with the competition parameter u are shown in Figure 7. The role of the rate constant of the initial reaction, kE, is again worth emphasizing: because the amount of E in the reaction layer decreases as kE increases, the reduction of the intermediate B becomes decreasingly efficient. The following reaction scheme (Scheme 111) is another example, of more practical importance, where the number of E molecules consumed per A molecule varies from one to two according to the competition between various steps.
SCHEME I1
SCHEME 111
0.5
0
-2
-1
1
0
2
z*
ke
ke
A+E-B
A+E-B
k
k
B-+C
B-C
kh
k'e
B+E-D
B+E-D
When the same strategy as in sections 1 and 3 is followed, it appears that the dimensionless profiles of A and E, defined exactly in the same way, obey the following differential equations:
ko
d2e* 2e* + ua*e* = --dz*2
e*
d2a* ---*e* dz *
2C-G
+u
=
0
k "E
C+E-F
(59)
The system is governed by the following equations describing the diffusion of A and E and the steady-state approximation for B and C.
Reactions Occurring during Mixing
The Journal of Physical Chemistry, Vol. 97, No. 41, 1993 10885 are known from the numerical resolution of eqs 66 and 67, yields the value of (dd*/dz*), and thus
P(D) =
(dd* /dz*)
(11
(da*/dz*),
Thus, the yields of the three products may be obtained from eqs 7 4 and 76.
The same transformations as already made in sections 1 and 3 lead to the following dimensionless formulation of the differential equations that govern the profiles of E and A. --
e*
ThevariationofP(D),P(F),and P(G) with the twocompetition parameters is discussed in detail in ref 3 together with an experimental example of reaction Scheme I11 involving the cyclization and reduction of aliphatic and aromatic radicals generated from reductive cleavage of the parent halides by solvated electrons in liquid ammonia. 5. E-Catalyzed Reactions. We now discuss reactions in which the reactant E is used as a catalyst to trigger a transformation of A and in which E may also be involved in side reactions where it may react with intermediates deriving from A as in the following reaction scheme.
+u
with
SCHEME IV
5
1 pe* --[
= 2 a* (1 +
+)ll2
-13
(68)
The two dimensionless parameters are given by
ku
A+E+B k
B-C
and
k’e
B+E-D kb
B+C-D+F and govern the competition between the transformation of B into C and the reaction of B with E and the competition between the reduction of C and its dimerization. Since
There are two products, D and F. The second and third reactions constitute the propagation loop of a chain process that consumes a negligibly small amount of E which thus serves in this respect as an initiator (catalyst) in the first step. The last two reactions may thus be viewed as termination steps of this chain process. Product F is the result of the chain process and product D that of the termination steps. The modified Fick’s law equations governing the E and A profiles are
xi-
d2CG dx2
+ k D c= 0
(73)
it follows that, in terms of product yields,
P(D+F) = -
(de* /dz* )-, - 1, (da* /dz *) ,
and the equations resulting from the steady-state approximation on B and C are
Equation 71 may be transformed in the same way as in the preceding sections into d2d* dz*2
* u
+ e*
=0
(75)
the computation of which, once the n * ( z * ) and e*(z*) functions
With the introduction of the same dimensionless variables as in sections 1 and 3, one obtains the following equations that allow
Andrieux and S a v h n t
10886 The Journal of Physical Chemistry, Vol. 97, No. 41, 1993 1
2.5
0.9
0.8
2
0.7 0.6
1.5
0.5 0.4
1
0.3 0.2
0.5
0. I 0 -3
0
0
-I
-2
1
2
logo Figure 9. Yields of D and F for reaction Scheme IV as a function of the competition parameter u (eq 86) for a = fl = 1.
2
with
I .5
P(D) = 1
(dd*/dz*) (da*/dz*),
and P(F) = 1 -P(D)
(84)
where 0.5
the competition parameters being defined by the following equations:
0
10
8 6
4
2 0
-2
2
1
0
-1
z* Figure 8. E (e*@*) and A ( a * ( z * ) ) dimensionlessconcentration profiles for reaction Scheme IV for three typical values of the competition parameter u corresponding respectively to P(D) - 0.9, P(F) = 0.1; P(D) = 0.5, P(F) = 0.5; P(D) - 0.1, P(F) = 0.9 in the case where a = fl = 1.
Typical examples of E and A profiles are shown in Figure 8, whereas the variations of P(D) and P(F) with the competition parameter u are depicted in Figure 9. We again note particularly the role of kE: an increase of k E favors the formation of the chain product F a t the expense of the termination product E for the same reasons as discussed in the preceding sections. A particularly interesting application of reaction scheme IV is reactions triggered by solvated electrons in liquid ammonia in the case of fast cleaving substrate anion radicals as discussed in ref 3. In this type of reaction, additional termination steps are sometimes observed as depicted in Scheme V.
SCHEME V ke
A+E-B k
the computation of the a* and e* profiles and the of the product yields.
--u*e*( d2e* dz*2
1+
kD
A+C-B+F
X 1+-
=0
a : )
~
B-C
1+-+-
k'e
B+E+D kb
B+C-+D+F k'k
C+E-G k'b
d2d* - a*e*(1 +:)(l+;) dz*'
u
2C+D+G
x
1+2+?
There is now an additional termination product, G. The governing equations are the following:
The Journal of Physical Chemistry, Vol. 97, No. 41, 1993 10887
Appendix The numerical resolution of each of set dimensionless differential equations and boundary conditions corresponding to each reaction scheme was carried out according to the following procedures inspired from those previously used14 in the analysis of catalysis at redox polymer coatings on electrode surfaces.k The space variable, z*, was varied between -5 and +5 where the boundary conditions for z* = --co and I* = -co were respectively applied. This interval is divided in equal subintervals,Az* (usually 100and in few cases 200). Each differential equation is converted into a finite differenceequation using the following approximation of the second derivative (z* = jAz*). 1
+-Xa The first derivative is approximated by:
+
d2d* -- a*e* dz*2
(1 + $ ) ( l + f )
X
u
B
e*
1+-+-
-
The computation of the e*(z*) and a*(z*) profiles in thesimple mechanism, A E B, was carried out as follows. From eqs 17 and 21 we obtain
=O
X 1 +a
This equation may be numerically solved by one or the other of the following procedures. We try a value of e*(z*= -5 Az*) and calculate the value of e*(z*=5) through the set of finite difference equations:
+
+
Where X is the positive solution of the following third-degree equation:
and the competition parameters are defined as follows:
The variation of the D, F, and G yields derived with the above competition parameters is discussed in details in ref 3.
The starting value, e*(z*= -5 Az*), is changed until the boundary condition, e*(5) = 0 is satisfied. An alternative procedure is to start from a guessed value of e*(z*=O) and, using the fact that a*(z*=O) = e*(z*=O) and a*(&*) = e*(-&*), and calculate through the set of eq 101 the valueofe*(z*=5). Thestartingvalueofe*(z*=O) is then iterated until e*(z*=5) = 0. For more complicated reaction schemes, the following procedure was employed, taking as an illustrating example reaction Scheme IV, Le., eqs 81-85. We start as if the e*(z*) profile was the same as in the preceding simple reaction scheme. We thus obtain a starting a*(z*) profile by application of eq 81. Using thevalueof a*(z*=O) thusestimatedeq82 is iterated asa function of the value of a*(-Az*) until a*(z*=5) = 0. The a*(z*) profile is thus reintroduced into eq 8 1 so as to obtain a new e*(z*) profile. The procedure is then repeated until stable e* and a* profiles obeying the boundary conditions are reached which usually required 4 or 5 iterations. These e*(z*) and a*(z*) profiles are then used to compute the d*(z*) profiles by numerical integration of eq 83 so as to obtain (dd*/dz*), and, finally, the yields of D and F (eq 4).
Conclusions
References and Notes
The above discussion has illustrated the applicationof the model of a reaction occurring during mixing by several examples that were mostly selected having in mind reactions that are triggered by piecesof metal able to inject electronsinto the reactant solution. Reference 3 provides a series of practical applicationsof the model to radical reactions triggered by the introduction of pieces of an alkali metal in liquid ammonia. The model was shown there to successfully unravel and rationalize quantitatively the leaving group effects observed in these reactions. It may be used as well to rationalizeproduct distributionsin many other types of reactions occurring during the mixing of the two reactants.
(1) Miller, D. G.; Vitagliano, V. J . Phys. Chem. 1986, 90, 1706 and references cited therin. (2) (a) Andrieux, C. P.; Savbnt, J.-M. Electrochemical Reactions. In Investigation ofRates andMechanism of Reactionr, Techniquesof Chemistry; Bernasconi, C. F., Ed.; Wiley: New York, 1986; Vol. VI/4E, Part 2, pp 305-390. (b) Andrieux, C. P.; Hapiot, P.; Savbnt, I.-M. J . Eicctroanai. Chem., in press. (c) Andrieux, C. P.; SavCant. J.-M. Catalysis at Redox Polymer Electrodes. In Molecular Design ofElectrodeSurfaces, Techniques of Chemistry; Murray, R. W., Ed.; Wiley: New York, 1992; Vol. XXII, Chapter V, pp 207-270. (3) Andrieux, C. P.; Savtant, J.-M. J . Am. Chem. Soc., in press. (4) (a) Bunnett, J. F.Acc. Chem. Res. 1978, Zl, 413. (b) Rossi, R. A.; Rossi, R. H. AromaticSubstitution by theSml Mechanism; ACS Monograph 178; The American Chemical Society: Washington, DC, 1983.
10888 The Journal of Physical Chemistry, Vol. 97, No. 41, 1993 ( 5 ) (a) Bard, R. R.; Bunnett, J. F.; Creary, X.;Tremelling, M. J. J . Am. Chem. SOC.1980, 102, 2852. (b) Tremelling, M. J.; Bunnett, J. F. J . Am.
Chem. SOC.1980, 102,1315. (6) Meijs, G. F., Bunnett, J. F.; Beckwith, A. L. J. J. Am. Chem SOC. 1986, 108,4899. (7) Albery, W. J. Electrode Kinetics; Claredon Press: Oxford, 1975. (8) (a) Amatore, C.; Savtant, J.-M. J . Electroanal. Chem. 1981, 123, 189. (b) Amatore, C.; Savtant, J.-M. J . Electroanal. Chem. 1981,123,203. (c) Amatore, C.; M’Halla, F.; Savtant, J.-M. J . Electroanal. Chem. 1981, 123, 219. (d) Amatore, C.; Pinson, ,J.; Savbant, J.-M.; Thiebault, A. J. Electroanal. Chem. 1981, 123, 231. (e) Amatore, C.; Savtant, J.-M. J. Electroanal. Chem. 1981, 125, 23. (f) Amatore, C.; Savbant, J.-M. J. Electroanal. Chem. 1981, 226, 1. (g) SavCnt, J.-M. J . Electroanal. Chem. 1987,236,31. (h) Vincent, M. L.; Peters, D. G. J . Electroanal. Chem. 1993, 344, 25. (9) This boundary condition would have to be modified to allow free penetration in cases where the reaction involves the mixing of two solutions, in which both A and E are soluble.
Andrieux and S a v h t (10) Similar to that corresponding to “pure kinetic conditions” in electrochemical reactions” or in catalytic reactions within redox polymer coatings on electrode surfaces.” (1 1) (a) Lepoutre, G.; Debacker, M.; Demortier, A. J . Chim.Phys. 1974, 71,113. (b)Sienko, M. J.InSolurionsMttal-Ammonlac.PropritttsPhysicoChimiques; Lepoutre, G., Sienko, M. J., Eds.; Benjamin: New York, 1964; pp 24-40. (12) The appearance of a I / , power in these expressions instead of */z electrochemical systems under ‘pure kinetic conditions”l0 derives from the fact that the A + E reaction is a bimoleclar reaction whereas the abovementioned electrochemical systems consist of a first-order homogeneous reaction following a fast electron transfer at the electrode surface. (13) Amatore, C.; Gareil, M.; Savtant, J.-M. J. Electroanal. Chcm. 1984, 176, 311. (14) Andrieux, C. P.; Dumas-Bouchiat, J.-M.; Savtant, J.-M. J . Electroanal. Chem. 1982, 131, 1.
