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Product distribution in preparative scale electrolysis
J. Electroanal. Chem., 125 (1981) 1--21 Elsevier Sequoia S.A., Lausanne -- Printed in The Netherlands
PRODUCT
DISTRIBUTION
IN PREPARATIVE
1
SCALE ELECTROLYSIS
P A R T V. E C R E A C T I O N S C H E M E S F O L L O W E D B Y C O M P E T I T I O N BETWEEN DIMERIZATION AND FIRST-ORDER DEACTIVATION OR FURTHER ELECTRON TRANSFER
C. AMATORE and J.M. SAVt~ANT Laboratoire d'Electrochimie de l'Universitd de Paris 7, 2, place Jussieu, 75251 Paris Cedex 05 (France) (Received 24th November 1980)
ABSTRACT The problem of product distribution is analyzed for two reaction schemes involving the dimerization of the product formed upon electron transfer followed by a first-order chemical reaction. The first of these involves a competition between dimerization and first-order deactivation. The product distribution then depends upon a single competition parameter containing the rates of the three chemical reactions and the operational parameters, substrate concentration and diffusion layer thickness for the potentiostatic electrolysis regimes and current density for galvanostatic electrolysis. In the second reaction scheme dimerization competes with further electron transfer. The latter reaction may occur at the electrode and/ or in the solution leading to a three-cornered Dim--ECE--Disp competition. A detailed analysis of the limiting Dim--ECE and Dim--Disp behavior is carried out as a function of a single competition parameter having a different expression for each case. This allows the product distribution to be predicted as a function of the intrinsic (rates) and operational (concentration, diffusion layer thickness, current density) parameters for the two limiting situations. In the general case a zone diagram is given which represents the effect of the parameters on the outcome of the three-cornered competition. The reductive cleavage of aromatic and aliphatic organic molecules is discussed in this context, illustrating the application of the results to experimental systems. INTRODUCTION A s in P a r t s I - - I V o f t h i s s e r i e s [ 1 - - 4 ] w e c o n s i d e r a r e a c t i o n s c h e m e i n v o l v i n g as f i r s t s t e p s t h e u p t a k e o r r e m o v a l o f o n e e l e c t r o n f r o m t h e s t a r t i n g m o l e c u l e followed by a fast and irreversible follow-up reaction: A+ 1 e~
B
(0)
kl B-+ C
(1)
leading to a species C that can undergo a further first-order reaction: C ~ F
(3)
or a further electron transfer at the electrode: C+ le~D 0022-0728]8110000-0000/$02.50 © 1981 Elsevier Sequoia S.A.
(4)
or in the solution: ~d
B+C-* A+D
(5)
In addition, it is now considered that the intermediate C can also undergo a dimerization reaction: k2
2C -+ E
(2)
leading to an electroinactive dimer, E. This problem was originally investigated in the c o n t e x t of a study of the reduction mechanism of aromatic thiocarbonates [ 5,6]:
ArS--C--SAr + 1 e ~ ArS--C--SAr" 11 li S S
(0)
ArS--C--SAr" -+ ArS- + S==C--SAr I} S
(1)
where a conceivable competition was between the dimerization of the resulting neutral radical and its further cleavage: 2 S----C--SAr -+ ArS--C--C--SAr IP II S S
(2)
S----C--SAt -~ CS + ArS"
(3)
Similar types of competition may, however, be involved in a large variety of electrochemical reactions, e.g. cleavage of aromatic or aliphatic carbon-heteroatom bond, reduction of aromatic carbonyl compounds in neutral media, oxidative decarboxylation of carboxylates (Kolbe reaction) or more generally oxidative cleavage reactions. Although the whole reaction mechanism involving reactions (0)--(5) can be thoroughly analyzed, we will divide, for simplicity, the presentation of the results according to two different reaction schemes: Scheme I corresponding to the reaction sequence (0) + (1) + (2) + (3), and Scheme II to the reaction sequence (0) + (I) + (2) + (4) + (5). In this context, the purpose of the present report is to relate the product distribution (E vs. F or E vs. D) to the intrinsic (rates) and operational (concentration, diffusion layer thickness) parameters. As in the other cases treated in Parts I--IV of the series, the various chemical steps are considered as so fast that the thickness of the corresponding reaction layer can be regarded as small compared to that of the diffusion layer. The same three electrolysis regimes will be dealt with: constant concentration potentiostatic electrolysis (CCPE), exhaustive potentiostatic electrolysis (EPE) at the plateau of the reduction wave (the electrode concentration of the substrate is regarded as equal to zero, and constant concentration galvanostatic electrolysis (CCGE). Basic assumptions, symbolism, adimensionalization and resolution procedures are the same as previously described [ I--4].
3 SCHEME I T h e p r o b l e m can be f o r m u l a t e d in d i m e n s i o n l e s s t e r m s as a set o f d i f f e r e n t i a l e q u a t i o n s . F o r 0 < y < p ' , i.e. inside the r e a c t i o n layer: d2a/dy 2 = 0
(6)
d 2 b / d y 2 = ~,1 b
(7)
d 2 c / d y 2 = - - ~ , l b + 2~2c 2 + ~3c
(8)
d 2e / d y 2
(9)
= --)k2 c 2
(10)
d2 f / d Y 2 = --)~3c
f o r # ' ~< y ~ 1, i.e. in t h e r e m a i n d e r o f t h e d i f f u s i o n layer: d2a/dy 2 = 0
(11)
d2 e / d y 2 = 0
(12)
d2f/dy 2 = 0
(13)
b=c=0 t o g e t h e r w i t h the f o l l o w i n g b o u n d a r y c o n d i t i o n s :
Electrolysis regime
EPE
y =0
a=0 (da/dy) + (db/dy) = O,
CCPE
CCGE da/dy = ~0
(dc/dy) = (de/dy) = (df/dy) = 0
y --->p'-
b~0, c-->0 and continuity conditions for a, e, f, and da/dy, de/dy, df]dy
y=l
a=a b
a=l
dab/dr = --( da/dy ) r ab=l for r = 0 e = e b, deb/d~- = --(de/dY)l- , e(r=O b )=0 f = fb, dfb/dT = _ ( d f / d y ) r ' f~r=o) = 0
I t is a s s u m e d t h a t r e a c t i o n s (2) a n d (3) are fast as c o m p a r e d t o r e a c t i o n (1) so t h a t t h e r e is n o a c c u m u l a t i o n o f C at a n y p o i n t o f t h e s o l u t i o n . T h e stat i o n a r y - s t a t e a s s u m p t i o n c a n t h u s be a p p l i e d t o t h e c o n c e n t r a t i o n o f C: d 2 c / d y 2 = ~ x b - - 2~2c 2 - - ~3c = 0
(14)
L i n e a r c o m b i n a t i o n o f eqns. (6), (9), (10) a n d (14) o n o n e h a n d , a n d eqns. (11), (12) a n d (13), o n t h e o t h e r , leads t o d2(a+b+2e+f)/dy
2 =0
4 which then appears valid from y = 0 to y = 1. Integration, taking the boundary conditions for y = 0 into account, then gives: (da/dy)w + 2 ( d e / d y ) w + ( d f / d y ) w
= (da/dy)o
+ (db/dy)o
= 0
i.e. 2(de/dy)l-
+ (df/dy)w
= --(da/dy)w
which expresses the conservation of matter during electrolysis. Since integration of eqn. (6) leads to (da/dy)w
(15)
= a b - - ao
we finally obtain: 2(de/dy)w
+ (df/dy)w
(16)
= ao - - a b
For each electrolysis regime the number of electrons per molecule is one, since the formation of both the dimer E and the deactivation product F requires one electron per molecule of starting material. However, the expressions for the yields in E and F depend upon each particular regime of electrolysis. CCPE
The bulk concentration of A is constant, a b = 1, and the concentration of A at the electrode surface is equal to zero, a0 = 0. It follows that 2(deb/dr)
+
(df/dr) = - - 2 ( d e / d y ) w
-- (df/dy)w
= 1
and thus: 2e b + f b =T i.e. the ( 2 c E + CF) in the bulk of the solution increases linearly with time. With the following change of variables: b* = bh]/2, c* = c?~3?~1/2, y , = y?,I/2 and 3' = 2X2X]/2 X~2, integration of eqn. (7) leads to b* = exp(---y*) and thus from eqn. (14): c* = (27)-1([1 + 43' exp(--Y*)] 1/2 -- 1} Integration of eqn. (10) then gives: (df/dy)l- = --(23,) -1 f ~ ([1 + 43' exp(--y*)] i/2 _ 1} dy* 0
i.e. dfb/dr = _(df/dy)l_
= 3"-1
{(1 + 43') 1/2 - - I n [ 1 + (1 + 43') s/2] -- 1 + In 2}
This equation is independent from r, showing that the bulk concentration of F, like (2CbE+ CbF)varies proportionally with time. Thus, the yields in E and F as defined b y
E RCCeE = 2eb/(2e b + fb) = 1 - - R E CCPE
are given b y R E CCPE = 1 - - R
F = 1 - - 7 -1 ((1 + 47) 1/2 - - I n [ 1 + (1 + 4 7 ) 1 / 2 ] - - 1 + l n 2) CCPE
T h e y d e p e n d u p o n the single p a r a m e t e r : 7 = 2k~ :2 k2 ~.-32 = 2 k ] / 2 k 2 k ~ 2 D l / 2 c ° 5 -1
as r e p r e s e n t e d in Fig. l a . As 7 increases the yield in d i m e r increases u p t o 1 while it goes t o zero as 7 decreases. EPE
T h e c o n c e n t r a t i o n o f A is n o longer c o n s t a n t : dab/dT = - - ( d a / d y ) w = --a b
since a0 = 0
(see eqn. 15). Thus, a b = e x p ( - - r ) . It t h e n follows (eqn. 16) t h a t 2(deb/dr) + (dfb/dr) = --2(de/dy)w
--(df/dy)l-
= exp(--r)
and thus t h a t 2e b + fb = 1 -- e x p ( - - r ) I n t r o d u c i n g the following variables and p a r a m e t e r s : b ° = b* exp(~),
c D = c* e x p ( r ) ,
7 ° = 7 exp(--r)
we obtain: d f b / d r = - - ( d f / d y ) l - = (7°) -1 ((1 + 47D) 1/2 - - l n [ 1
+ (1 + 4 7 ° ) 1/2 ] - - 1 + In 2)
X exp(--r) It thus follows t h a t t h e yields are n o w t i m e d e p e n d e n t t h e y are given b y 7
R~pE = I - - R F EPE = ( 7 -
V°) -1 f
RECPE(~?) dT?
T°
w h e r e t h e d e p e n d e n c e with t i m e c o m e s f r o m 7 D= 7 e x p ( - - r ) . The yields can t h u s be derived f r o m t h e i n t e g r a t i o n o f the f u n c t i o n s o b t a i n e d in t h e CCPE case. In t h e case o f c o m p l e t e e x h a u s t i o n o f t h e starting material t h e i r expression b e c o m e s T
R E p z ( r - * ~ ) = 1 --RFpE (T-~ oo)= 7 -1 f REcPE(~?) dr/ o T h e d e p e n d e n c e o n 7 is t h e n as s h o w n in Fig. l b . CCGE
We n o w have: 2(deb/dr) + (dfb/dr) = ~0,
'LO RE Eor RE CCP CCGE/
-2
0
log 7 or log7 a 4
2
1.0
logz b
2
,
,o
,,
2
,,
f
......
Fig. 1. C o m p e t i t i o n between dimerization and first-order deactivation (Scheme I). Variations of the dimer yield with the c o m p e t i t i o n parameter 3/= 2k ]/2 k2 k32D 1/2 c ° ~ - 1 , or 7 a = 2k ~/2k2 k32 D -1/2 ( i° / FS), for constant concentration potentiostatic and galvanostatic electrolysis (a), and exhaustive potentiostatic electrolysis (b).
the dimensionless i m p o s e d c u r r e n t density. H e n c e : 2e b + fb = ~ 0 r P e r f o r m i n g a d i f f e r e n t c h a n g e o f variables and d e f i n i t i o n o f a n e w c o m p e t i t i o n parameter: b ~ = b*/~o,
c a = c*/~o,
7,, = . y ~ o
n o w leads t o - - ( d f / d y ) l - = (dfb/dT) = ( ~ ° / 7 ~ ) { (1 + 47~) i n - - l n [ 1 + (1 + 47~) ~/2 ] -- 1 + In 2} which shows t h a t R~CGE(7 ~) -----R E CCPE( 7 : 7 z~) The variations o f t h e yields with the p a r a m e t e r 7 ~ = 2h ~/2k2k~2 D -112(i°/FS) are the same as t h e variations o f t h e yields with 7 = 2 k ~ n h 2 h - 3 2 D U 2 c ° 6 -1
in the CCPE case (Fig. l a ) .
SCHEME II
The p r o b l e m is n o w f o r m u l a t e d as the f o l l o w i n g set o f differential e q u a t i o n s . For 0 < y < p': d2a/dy 2 = --XdbC
(17)
d 2 b / d y 2 = ~ . l b + 3.dbC
(18)
d 2 c / d y 2 = --~.l b + ~.dbC + 2 ~ 2 c 2
(19)
dZd/dy 2 = --~.d b c
(20)
d 2 e / d y 2 = --?~2c 2
(21)
and f o r p ' ~ < y <
1:
d2a/dy 2 = 0
(22)
d2 d / d y z = 0
(23)
d2e/dy 2 = 0
(24)
b=c=O
with the f o l l o w i n g b o u n d a r y c o n d i t i o n s :
Electrolysis regime
EPE
y =0
a =0
CCPE
CCGE (da/dy) + (dc/dy) = ~o
( d a / d y ) + (db/dy) = O,
( d c / d y ) + ( d d / d y ) = O,
(de/dy) = 0
y ~p'-
b -+ 0, c --> 0, and the continuity conditions for a, d, e, and d a / d y , d d / d y , de/dy
y=l
a=~ b da b [dT = --(da/dy )lab =
a=l
1 for ~"= 0
e = e b , d e b / d T = - - ( d e / d Y ) l . , ( e b = 0 for T = 0) f = f o , d f b / d z = _ ( d f / d y ) l _ , (fb = 0 for ~"= 0)
Linear c o m b i n a t i o n o f eqns. ( 1 7 ) - - ( 2 1 ) a n d o f eqns. ( 2 2 ) - - ( 2 4 ) leads t o d2(a + b + c + d + 2 e ) / d y 2 = 0 valid f r o m y = 0 t o y = 1. Taking into a c c o u n t the b o u n d a r y c o n d i t i o n s at y = 0 it t h e n follows t h a t (dd/dy)w
+ 2(de/dy)w
= --(da/dy)l-
= --(a b
--au,)
~ ao--a b
(25)
w h i c h expresses the c o n s e r v a t i o n o f m a t t e r d u r i n g electrolysis. The n u m b e r o f electrons per m o l e c u l e will n o w vary f r o m one t o t w o according t o w h e t h e r d i m e r i z a t i o n or e l e c t r o n transfer (ECE or Disp) is t h e p r e d o m i -
nant pathway. The formation of half a molecule of E from one molecule of A requires one electron, while the formation of one molecule of D from one molecule of A requires two electrons. Thus: nap = (2cE + 2CD)/(2CE
+ CD) =
1 + R D = 2 --R E
The exact expression of the yields as function of the various parameters depends upon each particular electrolysis regime. CCPE
The bulk concentration of A being constant and its electrode concentration equal to zero: ( d d b / d T ) + 2(deb/dr) = - - ( d d / d y ) w - - 2 ( d e / d y ) l -
=1
and thus d b + 2e b = T On the other hand, integrating the sum of eqns. (18) and (20) and taking into account the boundary conditions at y = 0 and y = p' leads to 1
( d d b / d T ) = - - ( d d / d y ) ~ - = - - ( d b / d y ) o + ( d c / d y ) o -- ;k~ f o
b dy
(26)
the right-hand side of which is time independent. Thus: d b = ----T(dd/dy)l-
and therefore the yields are given by RE D CCeE = 1 -- RCCPE = 2eb/(2e b + d b) = 1 + ( d d / d y h -
(27)
Equations (26) and (27) show that the yields can be obtained by simultaneous integration of the two differential equations pertaining to b and c (18) and (19). This is more conveniently carried out after the following change of variables and the introduction of two competition parameters: b* = b;k~n ,
c* = c;k~)2 ,
y* = y;k~ n
O = ~13/4 ~,21/2 = k 13/4k21/2 D -1/4 c 0-1/2 51/2 p = }kd ~k13/4 ~k21/2 : k d k l 3 / 4 k ~ l / 2
D1/4cO 1/2 5-1/2
(28)
(29)
Equations (18) and (19) then become: d 2 b * / d y .2 = b* + ( p / o ) b * c *
(30)
d 2 c * / d y .2 = --b* + ( p / a ) b * c * + a-2(c*) 2
(31)
with, as boundary conditions, y* = 0,
c* = 0
y*~oo,
b*'~-O
and
c*~O
and
1=J 0
b* dy*
(32)
the latter condition deriving from the integration of the sum of eqns. (17) and (18). The yields will then be obtained from the integration of this system as REccPE ---- 1 - - R DCCPE= 2 + ( d b * / d y * ) o - (dc*/dy*)0 They thus depend upon only two competition parameters o and p as defined by eqns. (28) and (29). The parameter p = p / a = h d ~ 3~2, features the competition between homogeneous (Disp) and heterogeneous (ECE) electron transfers as discussed in detail in Part I of this series [ 1]. Thus, if ( p / o ) ~ 0, the heterogeneous electron transfer will predominate over the homogeneous electron transfer and the initial three-cornered Dim--ECE--Disp competition will be reduced to a two-term Dim--ECE competition. Conversely, when ( p / a ) -~ co, a two-term Dim--Disp competition will be observed. For both of these limiting behaviors the competition will depend upon a single parameter. The D i m - - E C E c o m p e t i t i o n . The system of eqns. (30) and (31) becomes: d2b*/dy .2 = b*
(33)
d 2 c * / d y .2 = --b* + a-2(c*) 2
Integration of eqn. (33), according to (32), leads to b* = exp(--y*)
and
(db*/dy*)o = - - 1
It thus remains to integrate: d 2c*/dy *2 = --exp(--y* ) + 0-2 (c*)2 with y =0,
c =0
y ->0%
C ->0
the yields being given by R EccPE = 1 -- R DccPE = 1 -- (dc*/dy*)0 This was carried out, as a function of o, numerically following an iterative finite difference procedure adapted from the Crank--Nicholson m e t h o d as described in the Appendix. The resulting variations of R~CPE with o = k~14k~lnD-l14c°-i/251n are shown in Fig. 2a. The D i m - - D i s p c o m p e t i t i o n corresponds to ( p / o ) -* ¢~. The electron transfer to C occurs now in the solution so that the stationary state assumption can be applied to C: 2~2c 2 + ~ d b C - - ~ l b = 0
and thus: d2b*/dy .2 = b* + ( p b * / 2 ) ( [ ( p b * ) 2 + 45*] 1/2 - - p b * }
10
0.5
a
-2 0 i i i i i i
2
b
logic
c
\ \ co . log e
d
\
x k2w\ \ % kl/ . -2 Disp " / ~ Ece/D,SP /
/ Dir~/Di,p /
'~..--~, \
-2 Dim
I )
Dim/Ece
-~
, I-,~
,
Ece \\
o,
"..
,o~,,
Fig. 2. C o m p e t i t i o n b e t w e e n dimerization and further electron transfer ( S c h e m e II). Constant concentration potentiostatic electrolysis: (a) D i m - - E C E competition: variations of the dimer yield with the parameter a = k l3/4 k~ 1/2 D - 114 c 0 - 1/2 ~ 1/2 ; (b) D i m - - D i s p c o m p e t i t i o n : variations of the dimer yield with the parameter p = k d k 1 3/4 k~ 1/2 D 1/4 c 0 1/2 ~ - 1/2 ; (c) zone diagram for a 5% accuracy on yield determinations (KG: general case).
with oo
f o
b* dy* = 1
and y* ~ ~:
b* ~ 0
the yields being given by R E CCPE
=
1
--R D CCPE = 2 + ( d b * / d y * ) o
The differential equation was solved numerically along the same procedure as above. The resulting variations of R E c ~ with the parameter p = kdk[3/4k-: ~n X D U 4 c ° u25-1n are shown in Fig. 2b. In the g e n e r a l case where ( p / o ) does not take the extreme value, the threecornered Dim--ECE--Disp competition depends upon the t w o parameters o and p. A competition zone diagram (Fig. 2c) has been drawn for a 5% accuracy on yield determinations, on the basis of the above results and of those obtained for the ECE--Disp competition [ 1 ]. It shows the extent of the zone corresponding to the general case (KG) and h o w one passes from one type of limiting behavior to the other.
11 EPE
Since n o w the bulk c o n c e n t r a t i o n of A is n o t held c o n s t a n t and since a0 = 0, eqn. (25) leads to dab /dT = - - ( da / d y ) w = - - a b
and thus to a b = exp(--r)
It follows t h a t (eqn. 25) (ddb/dT) + 2(deb/dr) = --(dd/dy)w
-- 2(de/dy)l-
= exp(---r)
Thus: d b + 2e b = 1 -- exp(--r) On the other hand, introducing a n e w expression of the variables and parameters: b ° = b* exp(T),
C° = c* exp(r),
o ° = o exp(r/2),
p° = p exp(--r/2)
results in the following f o r m u l a t i o n of the problem: d2bD/dy*2 = b D + (pD/o~)bD c~ d~cD/dy*~ = --bO + ( p ° / a ° ) b % °
+ (c~/a°) 2
with, as b o u n d a r y conditions, y* = O,
c °= 0
y*-~o%
b°-~0
and
c°-*0
and ~ b ~ dy* ~ 1 0
( d d b / d T ) is n o w given by
(ddb/dr) = - - ( d d / d y ) l -
= [ ( d b ° / d Y * )o - - ( d c ° / d Y * )o + 1 ] exp(--r)
and thus: T
db(T) = - - f
[1 + ( d b ° / d y * ) o - - ( d c ° / d y * ) o ] ( n ) exp(--T) d~
0 where the d e p e n d e n c e of the first t e r m in the right-hand side on ~ is t h r o u g h a ~ and p~. The yields are t h e n given by R~FE = 1 --RDpE = 2 e b ( r ) / [ d b ( r ) + 2eb(r)] = 1 --db(T)/[1 i.e.
--exp(--r)]
12
b -2
G
0
Ioga
c V",
T'o
~,oo~ "y,,: /
/
\\
Ece \\\
-2
I.
2
loge
Dim -4
Dim/Ece -2
\ l°2ga
0
Fig. 3. Competition between dimerization and further electron transfer (Scheme II). Exhaustive potentiostatic electrolysis: (a) Dim--ECE competition: variations of the dimer yield with the parameter 0 = k ~ / 4 k - a l / 2 D - 1 / 4 c ° - 1 / 2 6 1 / 2 . (b) Dim--Disp competition: variations of the dimer yield with the parameter p = k d k ~ 3 / 4 k ~ l / 2 D 1 / 4 c ° 1/25-1/2 ; (c) zone diagram for a 5% accuracy on yield determinations (KG: general case).
1
R~eE(T) = 1 --REDpE(T) = 211 - - e x p ( - - T ) ] - ' f
REcPE(O/r/, pr/)'r/dr/
exp(--'r/2)
and can thus be obtained as a function of time by integration of the yields in the CCPE case. For complete exhaustion of the solution, the yields are more simply expressed as 1 REpE:I
--
RDE : 2 :
E RccPE (o/vl, p~)~? d~
(34)
0
Figure 3 represents the yield in E for such conditions for the Dim--ECE competition as a function of the parameter a (Fig. 3a), for the Dim--Disp competition as a function of the parameter p (Fig. 3b) and the zone diagram showing the various types of competition (Fig. 3c), these results being based on those obtained in the CCPE case through eqn. (34). CCGE
Appropriate change of variables and definition of parameters are now: b ~ = b*/~I,0,
c a = c*/T o
OA = 0 ~ 0 -1:2 = h~/4 h ~l/2 D1/4 (iO /FS)-i/2
(35)
13
pa = p q2o 1/2 = kd k ~3Z4k-21nD-11, ( iO/FS)ln
(36)
Equations (30) and (31) keep the same form with b*, c*, a and p being replaced by b a, c a, o a and p~ respectively. The boundary conditions are now:
c a= 0
y* = 0: and
(dca/dy *) -- (dba/dy*) = 1 y* -+ oo:
b a -> O,
(37)
c a -* 0
eqn. (37) n o w replacing eqn. (32). Here, b a and c a are n o t dependent upon r and thus are ( d d / d y ) l - and (de/dy)l- since the integration of a linear combination of eqns. (19)--(21) leads to
(dd/dy)w + 2(de/dy)~- = _xpo ;
b a dy*
o
and integration of the sum of eqns. (18) and (20) gives
(dd/dy)l- = kO°[(dbA/dy*)o--(dca/dY*)o + ;
ba dy*] = ¢2°(--1+ ;
o
o
ba dy *)
The yields are thus also time independent, although the individual bulk concentrations of E and D are proportional to time: oo
R CEC G E = 1
--R D CCGE = 2 --
1/f
b a dy*
(38)
0
The Dim--ECE competition with negligible interference of the homogeneous electron transfer will be observed for (pa/o ~) -~ O. The system then depends upon the single parameter o a as defined by eqn. (36). It is entirely defined by d2ca/dy* 2 = [(dca/dy,)o -- 1 ] exp(--y*) + (ca/o~) 2 with, as boundary conditions, y * =0: y
*
-->oo:
CZX
C
a
=0 -->~
the yields being given by R CCGE E =
1 -- R D CC6E =
2-
[1 --(dc~/dy*)o] -1
The differential equation was solved numerically by the same finite difference method as in the preceding cases (see Appendix). The resulting variations of the yield in E with the intrinsic and operational parameters as contained in o a are shown in Fig. 4a. The Dim--Disp competition will be observed when (pa/o ~) -> oo. The system then depends upon the single parameter pa (eqn. 37). The stationary-state assumption can indeed be applied to C leading to
14
log da
log e
a
/
/
• k~",'y D*Sl / \ ,,
)irn/D
EcelDisp /
/
\ \
)
/
Ece
N\\\ Dim -4
Dim/Ece 2
0
x log a'' 2
Fig. 4. Competition between dimerization and further electron transfer (Scheme II). Constant concentration galvanostatic electrolysis: (a) Dim--ECE competition: variations of the dimer yield with the parameter 0 ~ = k~/4k~l/2D1/4 ( i ° / F S ) - l l 2 ; (b) Dim--Disp competition: variations of the dimer yield with the parameter pa = k d k T 3 / 4 k ~ l / 2 D - l / 4 ( i o / F S ) l / 2 ; (c) zone diagram for a 5% accuracy on yield determinations (KG: general case).
b~= (p~/o~)b~c ~ + (e~/o~) 2 a n d thus to d 2ba/dy*2 = b ~ + ( p ~ b A / 2 ) { [(pab~)2 + 4b ~]u2 _ p ~ b a} with for y* = O,
dbA/dy * = -1
and for y* -~ ~o,
b ~ -* 0
the yields still being given by eqn. (38). The numerical resolution along the same finite difference m e t h o d as previously (see Appendix) leads to the variations of the yield in E with the parameter p~ represented in Fig. 4b. Figure 4c shows the zone diagram summing up the variations of the yields with the two competition parameters o ~ and p~. DISCUSSION Scheme I
The yield in dimer is an increasing function of the parameter 7 = k ~/2 k 2 k ] 2 DU2 c o ~-I
15 for the potentiostatic regimes and of "/~ = k ~/2 k 2k-32D -1:2 ( i 0 / F S )
for the galvanostatic regime. The increase of this yield upon increasing c o or decreasing 5 in the first case and upon increasing the current density in the second, reflect the second-order character of the dimerization as opposed to the first-order character of the deactivation reaction: the higher the reduction flux of A the higher the formation of B and thus of C and the more efficient the dimerization process. From a practical viewpoint, the working curves of Fig. 1 will help in selecting the appropriate experimental conditions for optimizing the yields in either the dimer or the first-order deactivation product. As expected, the yield in dimer increases with k2 and decreases with k3. It is noted that the latter rate interferes through its square. The fact that the dimer formation increases with k, was less straightforwardly predictable. It again reflects the second-order character of the dimerization process as opposed to the first-order character of the deactivation reaction: for a given value of the reduction flux of A, the higher k,, the larger the production of C from B and thus the more efficient the dimerization process. The working curves of Fig. 1 also provide a mean for controlling the correctness of the predicted mechanism through the variations of the yields with the operational parameters. At the same time they allow the determination of the rate factor k ~:2 k2k-32 from which a given rate constant can be derived once the other two are known from independent sources. SCHEMEII
When C can undergo a further electron transfer Concurrently with the dimerization process, the above analysis has shown the practical importance of the nature of the electron transfer, homogeneous or heterogeneous. The role of the operational as well as of the intrinsic factors is indeed entirely different in each case. The passage from competition with an ECE electron transfer and a Disp electron transfer is favored by an increase in initial concentration and a decrease in the diffusion layer thickness, but the most important effect is that of k, (see Figs. 2c, 3c, 4c): the faster the B -> C decomposition the closer to the electrode is C formed and the more efficient is reduction by the electrode rather than by B. In the case of a Dim--ECE competition, the yield of dimer is a decreasing function of the parameter o = k~/4 k ~ ' : 2 D -':4 c ° - ' / 2 5 '/2 for the two potentiostatic regimes, and of o " = k ~ 4 k ~ ' n D ' / 4 ( i ° / F S ) -':2 for the galvanostatic regime. The effect of c o and 5 in the first case and of the current density in the second case reflects the second-order character of the dimerization as opposed to the first-order character of the ECE process for the same reasons as for scheme I. The effect of k2 is straightforward, while t h a t of kl can be rationalized as follows. The higher is k , , the closer to the electrode is C formed and the more efficient its further reduction at the electrode at the expense of dimerization. For the Dim--Disp competition the yield of dimer is a decreasing function of the parameter p = kdk-13:4k-2 i n D ' : 4 c ° ,/2 5-,:2 for the two potentiostatic regimes and of p:' = k d k ~ 3 / 4 k - 2 1 n D - ' / 4 ( i ° / F S ) ' / 2 for the galvanostatic. The effect of the
16 operational parameters is now the opposite to what it was in the Dim--ECE case: the higher the reduction flux of A ( c ° t , 5~, (i°/S) ~ ) the more efficient the reduction at the expense of dimerization. These effects are not easy to predict on an intuitive basis since we deal with two concurrent second-order reactions differently located in the reaction sequence. While the effect of kd and k2 are straightforward, that of k ~ derives from the fact that the larger is k 1, the higher the conversion of B into C, which favors the coupling of 2C at the expense of the reaction of C with B. The zone diagrams (Figs. 2c, 3c, 4c) provide an overall view of the effect of the intrinsic and operational parameters on the three-cornered Dim--ECE--Disp competition. In the two limiting Dim--ECE and Dim--Disp situations the working curves of Figs. 2--4 can be used for checking the correctness of the reaction mechanism, optimizing the yield in the desired product and determining rate constants along the same lines as decribed for scheme I. As mentioned in the introduction, the present analysis was used to investigate the reduction mechanism of aromatic trithiocarbonates [ 5]. Let us now use these results to discuss some aspects of reductive cleavage reactions: RX+le~
RX:
kl
RX: -~ R" + X-
(0) (1)
(where R is an aromatic or aliphatic group and X a leaving group attached to R by a carbon--heteroatom bond). More specifically, let us discuss the possible interference of radical--radical dimerization in the reaction sequence following the initial cleavage step: k2
2 R" -~ RR
(2)
The following reactions may compete with this dimerization process: (1) H-atom transfer from the solvent: k3 R + SH -~ RH + S"
(3)
(2) Electron transfer to R ' , at the electrode: R" + 1 e ~ R-
(4)
or in the solution: kd R" + RX" -~ R- + RX
(5)
the latter two reactions being complete by proton abstraction from the reaction medium. The outcome of this four-cornered competition depends upon the structure of the starting RX. If R is an aromatic group, for most of the X groups (e.g. in the case of halides, quaternary ammoniums and phosphoniums, ternary sulfoniums, sulfides, s u l f o n e s . . . ) the standard potential of the R " / R - couple is very positive to the reduction potential of RX. For the Dim--ECE competition this corresponds exactly to the conditions we have assumed in the above treatment, i.e. the concentration of R" (C) is zero at the electrode surface. For the Dim--Disp competi_tion this means that the homogeneous electron transfer between R" and RX" (5) is close to the diffusion limit (k d -~ k d i f ) .
17
It is noted that op = o ~ p ~ = k d / k 2
Thus, in the present case: op = oAp ~ = kdi~/k2
Since the dimerization rate is at maximum at the diffusion limit it follows that op
= o~p ~ ~> 1
(39)
The outcome of the competition between dimerization and heterogeneous and/ or homogeneous electron transfer in these conditions can be derived from the zone diagrams as shown on Figs. 2c, 3c and 4c: eqn. (39) implies that the points representing the competition are located above the dashed line. The corresponding region mostly involves the Disp, ECE and Disp/ECE zones, i.e. corresponds to the predominance of electron transfer over dimerization. The maximal yield in dimer would be obtained for o ~ = 1 and p~ = 1 in the CCGE case. This would correspond to a yield equal to 20% (Fig. 4a). The obtention of such a yield would imply that the dimerization reaction is diffusion controlled and that the values of k l, D and i° are such that k l = ~2/3 D - 1 / 3 ( i° / F S ) 2/3 "dif
In most cases the dimer yield is expected to be much lower. This holds for cases in which the H-atom transfer from the solvent can be neglected, as for example in the case of liquid ammonia [7,8]. In organic solvents, H-atom transfer interferes in competition with heterogeneous and homogeneous electron transfer [9]. This will further reduce the yield of dimer, the exact o u t c o m e of the competition between dimerization and H-atom transfer being given by the above treatment of Scheme I. We thus n o w understand why the yield of dimer found in the reductive cleavage of aromatic c o m p o u n d s is n o t generally more than a few percent at maximum. It was, for example, found to be of the order of 5% in the reduction of tetraphenylphosphonium in DMF [ 10,11 ] and a few percent dimer was detected in the reduction of halopyrenes [12] and 9-haloanthracenes [13]. A completely different picture may be observed in the reductive cleavage of aliphatic compounds. An interesting example inthis connection is the reduction of the diphenylmethyl and triphenylmethyl-p-nitrophenylsulfide in DMF [14,15] where high yields of dimers are obtained. In this case, the presence of the nitro group in the para position considerably facilitates the uptake of the first electron. The reduction potential is then slightly negative (diphenylmethyl) or even positive (triphenylmethyl) to the R ' / R - standard potential. Dimerization can thus c o m p e t e successfully with both heterogeneous and homogeneous electron transfer to R ' . On the other hand, H-atom abstraction by this t y p e of R ° radicals is certainly much less efficient than for aryl radicals which thus allows the dimerization process also to compete successfully with H-atom transfer.
18 ACKNOWLEDGEMENT This work was supported in part by the C.N.R.S. (A.T.P. "Energie et Mati6res Premi6res" 1979. Electrochimie S61ective). APPENDIX (1) The first differential equation to be numerically solved (Dim--ECE for CCPE) can be expressed as (y* and c* are replaced by y and c for simplicity) d 2 c / d y 2 =--exp(--y) + 0 -2 c 2
(A1)
with y = 0:
c=0
(A2)
y -. ~o:
c -~ 0
(A3)
the yield to be calculated being R = 1 -- ( d c / d y ) o . The space interval [0, p"] is divided into N elementary intervals of length Ay. The value of C at y = j Ay is noted as cj. The differential equation is then expressed under finite difference form [16] as (Cj-- i --
2 c i + c i + i ) / ( A y ) 2 = --exp(--j Ay) + o - 2 c }
Its resolution is carried out by iteration. Let c~'-1 (0 ~< j ~< N) be the values obtained at the (n -- 1)th iteration. For the nth iteration the above equation is linearized according to (c'~-1 - - 2c'~ + c ~ + l ) / ( A y ) 2 = --exp(--jAy) + (o-: c~ -' )c~'
(A4)
with c~ = 0
(A5)
and the condition corresponding to y -+ p", i.e. to j = N. The latter is obtained as follows. When y >> 1 the first term in the right-hand side of eqn. (A1) can be neglected, leading to: d2c/dy 2 ~ o-2c 2
the integration of which, taking eqn. (A3) into account leads to dc/dy
-~ --(2/3) In o -1 c 3n
The value of p" is selected for this approximation to be valid. We t o o k tt" = 20 which correspond to exp(--p") ~ 2 X 10 -9 . The linearized finite difference expression of the above equation is then: (c~v -- c~v-1)/Ay = --[(2/3) '/: o - ' (c~v- 1 ) ' n ] cTv
(A6)
The (N + 1)c~ (0 ~
19
w h e n b o t h the following c o n d i t i o n s are fulfilled: [C~/ - - c~--ll •
1 0 -6 C~/
[ ( d c / d y ) ~ - - ( d c / d y ) ' ~ - l [ <~ 1 0 -6 (dc/dy)'~
T h e value o f (dc/dy)'~ used to calculate the yield: R = 1 - - (dc/dy)'~
is o b t a i n e d f r o m the following six-term e x p a n s i o n which is m o r e a c c u r a t e t h a n a single t w o - t e r m e x p a n s i o n : (dc/dy)3 = [ 3 0 0 ( c ~ - - c ~ ) + 200c~ - - 7 5 c ~ + 1 2 c ~ ] / 6 0 A y T h e whole calculation is started for the highest value o f o, 104 in the present case, f o r which the values used t o start the iterations are t a k e n as cj = 1 -- e x p ( - - j A y ) ,
j = 0, N
which c o r r e s p o n d s t o a p u r e ECE situation. T h e f u r t h e r calculations are t h e n carried o u t for decreasing values o f o, the cj o b t a i n e d for t h e last value o f o serving as starting values for the n e x t value o f o. This p r o c e d u r e , used with N = 500, i.e. Ay = 4 X 10 .2 leads to a rapid c o n v e r g e n c e . (2) T h e differential e q u a t i o n c o r r e s p o n d i n g to the D i m - - E C E c o m p e t i t i o n in the CCGE case: d2 c / d y 2 = [ ( d c / d y ) o - - 1] e x p ( - - y ) + o-2 c 2
(A7)
with y=O:
c=O
y --> ~ :
c~ 0
was c o m p u t e d according t o the same p r o c e d u r e , the linearized finite d i f f e r e n c e expression o f eqn. (A7) at the n t h iteration being (c~'-1 -- 2c~ + C~+l)/(Ay) 2 = [(dc/dy)~ -1 -- 1] X exp(--j Ay) + ( 0 - 2 c ' ; - 1 ) c ' ~ (3) T h e Dim--Disp c o m p e t i t i o n leads t o the same differential e q u a t i o n for b o t h the CCPE and CCGE c o n d i t i o n s : d 2 b / d y 2 = b + ( p b / 2 ) { [ ( p b ) 2 + 4b1'/2 _ p b }
(A8)
with y - + oo:
b -~ 0
(A9)
and b dy = l,
in the CCPE case
(A10)
o
or y=O:
( d b / d y ) = --1,
in t h e CCGE case
(All)
2O The same procedure as above is followed. The differential equation is expressed under finite difference form and linearized as n (b?-1 -- 25? + 5j+l)/(Ay )2 = 57 [1
+ (p/2){ [(pb?-l) 2 + 4b? -1] 1/2 _
pb?-l} ]
for 1 < j ~< N -- 1, eqn. (A9) becomes 5~v = 0 and eqn. (A10) N--1
b~=(2/Ay)--2
~
5? -1
j=l
while eq. (A11) is expressed as b~ = [60Ay + 300(b? -- b~) + 200b~ -- 75b~ + 12b~]/137 A set of (N + 1) linear equations involving the (N + 1)b? (0 < j ~< N) is then obtained. It is solved as in the preceding case. The whole calculation is carried out for a series of increasing values of p. For the lowest p value (10 -4 ), the iteration process is initiated, taking bi = exp(--jAy) The bj values obtained for the last value of p are used to start the calculation for the next value of p. The iteration process is stopped when the following conditions are fulfilled: Ib~ - - b ~ - l l < 1 0 -6 b~
and the yield in dimer is finally obtained as
R = 2 + (db/dy)~ in the CCPE case, and
R=2--1
bdy
in the CCGE case, where (db/dy)~ and
(db/dy)7) = [--137b~
[f~b
dy] n are determined according to:
+ 300(5? --b~) + 200b~ -- 75b~ + 12b~]/(60Ay)
and
[;
b dyln = (AY)I (b'~/2) + N~lJ=lb?l
respectively. REFERENCES 1 C. A m a t o r e and J . M . Sav~ant, J. E l e c t r o a n a l . C h e m . , 1 2 3 ( 1 9 8 1 ) 1 8 9 . 2 C. A m a t o r e and J . M . S a v 6 a n t , J. E l e c t r o a n a l . C h e m . , 1 2 3 ( 1 9 8 1 ) 2 0 3 . 3 C. A m a t o r e , F. M ' H a l l a a n d J . M . S a v 6 a n t , J. E l e c t r o a n a l . C h e m . , 1 2 3 ( 1 9 8 1 ) 2 1 9 .
21
4 5 6 7 8 9 10 11 12 13 14 15 16
C. A m a t o r e , J. Pinson, J.M. Savdant a n d A. Thi~bault, J. Electoanal. Chem., 1 2 3 ( 1 9 8 1 ) 2 3 1 . M. Falsig, H. Lurid, L. N a d j o a n d J.M. S a y , a n t , A c t a Chim. S c a n d , Set. 34B ( 1 9 8 0 ) 6 8 5 . M. Falsig, H. L u n d , L. N a d j o a n d J.M. S a y , a n t , Nouv. J. Chim., 4 ( 1 9 8 0 ) 445. J.M. S a y , a n t a n d A. T h i d b a u l t , J. E l e c t r o a n a l . C h e m . , 89 ( 1 9 7 8 ) 335. C. Amatore~ J. C h a u s s a r d , J. Pinson, J.M. Sav~ant a n d A. Thi~bault, J. A m . C h e m . Soc., 101 ( 1 9 7 9 ) 6012. F. M'Halla, J. P i n s o n a n d J.M. S a y , a n t , J. A m . C h e m . Soc., 1 0 2 ( 1 9 8 0 ) 4 1 2 0 . J.M. S a y , a n t a n d Su K h a c Binh, E l e c t r o c h i m . A c t a , 20 ( 1 9 7 5 ) 21. J.M. S a y , a n t a n d Su K h a c Binh, J. Org. C h e m . , 42 ( 1 9 7 7 ) 1 2 4 2 . J. G r i m s h a w a n d J. T r o c h a - G r i m s h a w , J. Chem. Soc. Perkin Trans. II, ( 1 9 7 5 ) 215. O. H a m m e r i c h , p e r s o n a l c o m m u n i c a t i o n , 1 9 7 9 . G. F a r n i a , M.G. Severin, G. C a p o b i a n c o a n d E. VianeUo, J. C h e m . Soc. Perkin Trans. II, ( 1 9 7 8 ) 1. G. F a r n i a , M.G. Severin, G. C a p o b i a n c o a n d E. Vianello, 31st Meeting o f the ISE, Venice, S e p t e m b e r 1 9 8 0 , A b s t r a c t No. 28E. J. C r a n k , M a t h e m a t i c s of D i f f u s i o n , C l a r e n d o n Press~ L o n d o n , 1 9 6 4 .
PRODUCT
DISTRIBUTION
IN PREPARATIVE
1
SCALE ELECTROLYSIS
P A R T V. E C R E A C T I O N S C H E M E S F O L L O W E D B Y C O M P E T I T I O N BETWEEN DIMERIZATION AND FIRST-ORDER DEACTIVATION OR FURTHER ELECTRON TRANSFER
C. AMATORE and J.M. SAVt~ANT Laboratoire d'Electrochimie de l'Universitd de Paris 7, 2, place Jussieu, 75251 Paris Cedex 05 (France) (Received 24th November 1980)
ABSTRACT The problem of product distribution is analyzed for two reaction schemes involving the dimerization of the product formed upon electron transfer followed by a first-order chemical reaction. The first of these involves a competition between dimerization and first-order deactivation. The product distribution then depends upon a single competition parameter containing the rates of the three chemical reactions and the operational parameters, substrate concentration and diffusion layer thickness for the potentiostatic electrolysis regimes and current density for galvanostatic electrolysis. In the second reaction scheme dimerization competes with further electron transfer. The latter reaction may occur at the electrode and/ or in the solution leading to a three-cornered Dim--ECE--Disp competition. A detailed analysis of the limiting Dim--ECE and Dim--Disp behavior is carried out as a function of a single competition parameter having a different expression for each case. This allows the product distribution to be predicted as a function of the intrinsic (rates) and operational (concentration, diffusion layer thickness, current density) parameters for the two limiting situations. In the general case a zone diagram is given which represents the effect of the parameters on the outcome of the three-cornered competition. The reductive cleavage of aromatic and aliphatic organic molecules is discussed in this context, illustrating the application of the results to experimental systems. INTRODUCTION A s in P a r t s I - - I V o f t h i s s e r i e s [ 1 - - 4 ] w e c o n s i d e r a r e a c t i o n s c h e m e i n v o l v i n g as f i r s t s t e p s t h e u p t a k e o r r e m o v a l o f o n e e l e c t r o n f r o m t h e s t a r t i n g m o l e c u l e followed by a fast and irreversible follow-up reaction: A+ 1 e~
B
(0)
kl B-+ C
(1)
leading to a species C that can undergo a further first-order reaction: C ~ F
(3)
or a further electron transfer at the electrode: C+ le~D 0022-0728]8110000-0000/$02.50 © 1981 Elsevier Sequoia S.A.
(4)
or in the solution: ~d
B+C-* A+D
(5)
In addition, it is now considered that the intermediate C can also undergo a dimerization reaction: k2
2C -+ E
(2)
leading to an electroinactive dimer, E. This problem was originally investigated in the c o n t e x t of a study of the reduction mechanism of aromatic thiocarbonates [ 5,6]:
ArS--C--SAr + 1 e ~ ArS--C--SAr" 11 li S S
(0)
ArS--C--SAr" -+ ArS- + S==C--SAr I} S
(1)
where a conceivable competition was between the dimerization of the resulting neutral radical and its further cleavage: 2 S----C--SAr -+ ArS--C--C--SAr IP II S S
(2)
S----C--SAt -~ CS + ArS"
(3)
Similar types of competition may, however, be involved in a large variety of electrochemical reactions, e.g. cleavage of aromatic or aliphatic carbon-heteroatom bond, reduction of aromatic carbonyl compounds in neutral media, oxidative decarboxylation of carboxylates (Kolbe reaction) or more generally oxidative cleavage reactions. Although the whole reaction mechanism involving reactions (0)--(5) can be thoroughly analyzed, we will divide, for simplicity, the presentation of the results according to two different reaction schemes: Scheme I corresponding to the reaction sequence (0) + (1) + (2) + (3), and Scheme II to the reaction sequence (0) + (I) + (2) + (4) + (5). In this context, the purpose of the present report is to relate the product distribution (E vs. F or E vs. D) to the intrinsic (rates) and operational (concentration, diffusion layer thickness) parameters. As in the other cases treated in Parts I--IV of the series, the various chemical steps are considered as so fast that the thickness of the corresponding reaction layer can be regarded as small compared to that of the diffusion layer. The same three electrolysis regimes will be dealt with: constant concentration potentiostatic electrolysis (CCPE), exhaustive potentiostatic electrolysis (EPE) at the plateau of the reduction wave (the electrode concentration of the substrate is regarded as equal to zero, and constant concentration galvanostatic electrolysis (CCGE). Basic assumptions, symbolism, adimensionalization and resolution procedures are the same as previously described [ I--4].
3 SCHEME I T h e p r o b l e m can be f o r m u l a t e d in d i m e n s i o n l e s s t e r m s as a set o f d i f f e r e n t i a l e q u a t i o n s . F o r 0 < y < p ' , i.e. inside the r e a c t i o n layer: d2a/dy 2 = 0
(6)
d 2 b / d y 2 = ~,1 b
(7)
d 2 c / d y 2 = - - ~ , l b + 2~2c 2 + ~3c
(8)
d 2e / d y 2
(9)
= --)k2 c 2
(10)
d2 f / d Y 2 = --)~3c
f o r # ' ~< y ~ 1, i.e. in t h e r e m a i n d e r o f t h e d i f f u s i o n layer: d2a/dy 2 = 0
(11)
d2 e / d y 2 = 0
(12)
d2f/dy 2 = 0
(13)
b=c=0 t o g e t h e r w i t h the f o l l o w i n g b o u n d a r y c o n d i t i o n s :
Electrolysis regime
EPE
y =0
a=0 (da/dy) + (db/dy) = O,
CCPE
CCGE da/dy = ~0
(dc/dy) = (de/dy) = (df/dy) = 0
y --->p'-
b~0, c-->0 and continuity conditions for a, e, f, and da/dy, de/dy, df]dy
y=l
a=a b
a=l
dab/dr = --( da/dy ) r ab=l for r = 0 e = e b, deb/d~- = --(de/dY)l- , e(r=O b )=0 f = fb, dfb/dT = _ ( d f / d y ) r ' f~r=o) = 0
I t is a s s u m e d t h a t r e a c t i o n s (2) a n d (3) are fast as c o m p a r e d t o r e a c t i o n (1) so t h a t t h e r e is n o a c c u m u l a t i o n o f C at a n y p o i n t o f t h e s o l u t i o n . T h e stat i o n a r y - s t a t e a s s u m p t i o n c a n t h u s be a p p l i e d t o t h e c o n c e n t r a t i o n o f C: d 2 c / d y 2 = ~ x b - - 2~2c 2 - - ~3c = 0
(14)
L i n e a r c o m b i n a t i o n o f eqns. (6), (9), (10) a n d (14) o n o n e h a n d , a n d eqns. (11), (12) a n d (13), o n t h e o t h e r , leads t o d2(a+b+2e+f)/dy
2 =0
4 which then appears valid from y = 0 to y = 1. Integration, taking the boundary conditions for y = 0 into account, then gives: (da/dy)w + 2 ( d e / d y ) w + ( d f / d y ) w
= (da/dy)o
+ (db/dy)o
= 0
i.e. 2(de/dy)l-
+ (df/dy)w
= --(da/dy)w
which expresses the conservation of matter during electrolysis. Since integration of eqn. (6) leads to (da/dy)w
(15)
= a b - - ao
we finally obtain: 2(de/dy)w
+ (df/dy)w
(16)
= ao - - a b
For each electrolysis regime the number of electrons per molecule is one, since the formation of both the dimer E and the deactivation product F requires one electron per molecule of starting material. However, the expressions for the yields in E and F depend upon each particular regime of electrolysis. CCPE
The bulk concentration of A is constant, a b = 1, and the concentration of A at the electrode surface is equal to zero, a0 = 0. It follows that 2(deb/dr)
+
(df/dr) = - - 2 ( d e / d y ) w
-- (df/dy)w
= 1
and thus: 2e b + f b =T i.e. the ( 2 c E + CF) in the bulk of the solution increases linearly with time. With the following change of variables: b* = bh]/2, c* = c?~3?~1/2, y , = y?,I/2 and 3' = 2X2X]/2 X~2, integration of eqn. (7) leads to b* = exp(---y*) and thus from eqn. (14): c* = (27)-1([1 + 43' exp(--Y*)] 1/2 -- 1} Integration of eqn. (10) then gives: (df/dy)l- = --(23,) -1 f ~ ([1 + 43' exp(--y*)] i/2 _ 1} dy* 0
i.e. dfb/dr = _(df/dy)l_
= 3"-1
{(1 + 43') 1/2 - - I n [ 1 + (1 + 43') s/2] -- 1 + In 2}
This equation is independent from r, showing that the bulk concentration of F, like (2CbE+ CbF)varies proportionally with time. Thus, the yields in E and F as defined b y
E RCCeE = 2eb/(2e b + fb) = 1 - - R E CCPE
are given b y R E CCPE = 1 - - R
F = 1 - - 7 -1 ((1 + 47) 1/2 - - I n [ 1 + (1 + 4 7 ) 1 / 2 ] - - 1 + l n 2) CCPE
T h e y d e p e n d u p o n the single p a r a m e t e r : 7 = 2k~ :2 k2 ~.-32 = 2 k ] / 2 k 2 k ~ 2 D l / 2 c ° 5 -1
as r e p r e s e n t e d in Fig. l a . As 7 increases the yield in d i m e r increases u p t o 1 while it goes t o zero as 7 decreases. EPE
T h e c o n c e n t r a t i o n o f A is n o longer c o n s t a n t : dab/dT = - - ( d a / d y ) w = --a b
since a0 = 0
(see eqn. 15). Thus, a b = e x p ( - - r ) . It t h e n follows (eqn. 16) t h a t 2(deb/dr) + (dfb/dr) = --2(de/dy)w
--(df/dy)l-
= exp(--r)
and thus t h a t 2e b + fb = 1 -- e x p ( - - r ) I n t r o d u c i n g the following variables and p a r a m e t e r s : b ° = b* exp(~),
c D = c* e x p ( r ) ,
7 ° = 7 exp(--r)
we obtain: d f b / d r = - - ( d f / d y ) l - = (7°) -1 ((1 + 47D) 1/2 - - l n [ 1
+ (1 + 4 7 ° ) 1/2 ] - - 1 + In 2)
X exp(--r) It thus follows t h a t t h e yields are n o w t i m e d e p e n d e n t t h e y are given b y 7
R~pE = I - - R F EPE = ( 7 -
V°) -1 f
RECPE(~?) dT?
T°
w h e r e t h e d e p e n d e n c e with t i m e c o m e s f r o m 7 D= 7 e x p ( - - r ) . The yields can t h u s be derived f r o m t h e i n t e g r a t i o n o f the f u n c t i o n s o b t a i n e d in t h e CCPE case. In t h e case o f c o m p l e t e e x h a u s t i o n o f t h e starting material t h e i r expression b e c o m e s T
R E p z ( r - * ~ ) = 1 --RFpE (T-~ oo)= 7 -1 f REcPE(~?) dr/ o T h e d e p e n d e n c e o n 7 is t h e n as s h o w n in Fig. l b . CCGE
We n o w have: 2(deb/dr) + (dfb/dr) = ~0,
'LO RE Eor RE CCP CCGE/
-2
0
log 7 or log7 a 4
2
1.0
logz b
2
,
,o
,,
2
,,
f
......
Fig. 1. C o m p e t i t i o n between dimerization and first-order deactivation (Scheme I). Variations of the dimer yield with the c o m p e t i t i o n parameter 3/= 2k ]/2 k2 k32D 1/2 c ° ~ - 1 , or 7 a = 2k ~/2k2 k32 D -1/2 ( i° / FS), for constant concentration potentiostatic and galvanostatic electrolysis (a), and exhaustive potentiostatic electrolysis (b).
the dimensionless i m p o s e d c u r r e n t density. H e n c e : 2e b + fb = ~ 0 r P e r f o r m i n g a d i f f e r e n t c h a n g e o f variables and d e f i n i t i o n o f a n e w c o m p e t i t i o n parameter: b ~ = b*/~o,
c a = c*/~o,
7,, = . y ~ o
n o w leads t o - - ( d f / d y ) l - = (dfb/dT) = ( ~ ° / 7 ~ ) { (1 + 47~) i n - - l n [ 1 + (1 + 47~) ~/2 ] -- 1 + In 2} which shows t h a t R~CGE(7 ~) -----R E CCPE( 7 : 7 z~) The variations o f t h e yields with the p a r a m e t e r 7 ~ = 2h ~/2k2k~2 D -112(i°/FS) are the same as t h e variations o f t h e yields with 7 = 2 k ~ n h 2 h - 3 2 D U 2 c ° 6 -1
in the CCPE case (Fig. l a ) .
SCHEME II
The p r o b l e m is n o w f o r m u l a t e d as the f o l l o w i n g set o f differential e q u a t i o n s . For 0 < y < p': d2a/dy 2 = --XdbC
(17)
d 2 b / d y 2 = ~ . l b + 3.dbC
(18)
d 2 c / d y 2 = --~.l b + ~.dbC + 2 ~ 2 c 2
(19)
dZd/dy 2 = --~.d b c
(20)
d 2 e / d y 2 = --?~2c 2
(21)
and f o r p ' ~ < y <
1:
d2a/dy 2 = 0
(22)
d2 d / d y z = 0
(23)
d2e/dy 2 = 0
(24)
b=c=O
with the f o l l o w i n g b o u n d a r y c o n d i t i o n s :
Electrolysis regime
EPE
y =0
a =0
CCPE
CCGE (da/dy) + (dc/dy) = ~o
( d a / d y ) + (db/dy) = O,
( d c / d y ) + ( d d / d y ) = O,
(de/dy) = 0
y ~p'-
b -+ 0, c --> 0, and the continuity conditions for a, d, e, and d a / d y , d d / d y , de/dy
y=l
a=~ b da b [dT = --(da/dy )lab =
a=l
1 for ~"= 0
e = e b , d e b / d T = - - ( d e / d Y ) l . , ( e b = 0 for T = 0) f = f o , d f b / d z = _ ( d f / d y ) l _ , (fb = 0 for ~"= 0)
Linear c o m b i n a t i o n o f eqns. ( 1 7 ) - - ( 2 1 ) a n d o f eqns. ( 2 2 ) - - ( 2 4 ) leads t o d2(a + b + c + d + 2 e ) / d y 2 = 0 valid f r o m y = 0 t o y = 1. Taking into a c c o u n t the b o u n d a r y c o n d i t i o n s at y = 0 it t h e n follows t h a t (dd/dy)w
+ 2(de/dy)w
= --(da/dy)l-
= --(a b
--au,)
~ ao--a b
(25)
w h i c h expresses the c o n s e r v a t i o n o f m a t t e r d u r i n g electrolysis. The n u m b e r o f electrons per m o l e c u l e will n o w vary f r o m one t o t w o according t o w h e t h e r d i m e r i z a t i o n or e l e c t r o n transfer (ECE or Disp) is t h e p r e d o m i -
nant pathway. The formation of half a molecule of E from one molecule of A requires one electron, while the formation of one molecule of D from one molecule of A requires two electrons. Thus: nap = (2cE + 2CD)/(2CE
+ CD) =
1 + R D = 2 --R E
The exact expression of the yields as function of the various parameters depends upon each particular electrolysis regime. CCPE
The bulk concentration of A being constant and its electrode concentration equal to zero: ( d d b / d T ) + 2(deb/dr) = - - ( d d / d y ) w - - 2 ( d e / d y ) l -
=1
and thus d b + 2e b = T On the other hand, integrating the sum of eqns. (18) and (20) and taking into account the boundary conditions at y = 0 and y = p' leads to 1
( d d b / d T ) = - - ( d d / d y ) ~ - = - - ( d b / d y ) o + ( d c / d y ) o -- ;k~ f o
b dy
(26)
the right-hand side of which is time independent. Thus: d b = ----T(dd/dy)l-
and therefore the yields are given by RE D CCeE = 1 -- RCCPE = 2eb/(2e b + d b) = 1 + ( d d / d y h -
(27)
Equations (26) and (27) show that the yields can be obtained by simultaneous integration of the two differential equations pertaining to b and c (18) and (19). This is more conveniently carried out after the following change of variables and the introduction of two competition parameters: b* = b;k~n ,
c* = c;k~)2 ,
y* = y;k~ n
O = ~13/4 ~,21/2 = k 13/4k21/2 D -1/4 c 0-1/2 51/2 p = }kd ~k13/4 ~k21/2 : k d k l 3 / 4 k ~ l / 2
D1/4cO 1/2 5-1/2
(28)
(29)
Equations (18) and (19) then become: d 2 b * / d y .2 = b* + ( p / o ) b * c *
(30)
d 2 c * / d y .2 = --b* + ( p / a ) b * c * + a-2(c*) 2
(31)
with, as boundary conditions, y* = 0,
c* = 0
y*~oo,
b*'~-O
and
c*~O
and
1=J 0
b* dy*
(32)
the latter condition deriving from the integration of the sum of eqns. (17) and (18). The yields will then be obtained from the integration of this system as REccPE ---- 1 - - R DCCPE= 2 + ( d b * / d y * ) o - (dc*/dy*)0 They thus depend upon only two competition parameters o and p as defined by eqns. (28) and (29). The parameter p = p / a = h d ~ 3~2, features the competition between homogeneous (Disp) and heterogeneous (ECE) electron transfers as discussed in detail in Part I of this series [ 1]. Thus, if ( p / o ) ~ 0, the heterogeneous electron transfer will predominate over the homogeneous electron transfer and the initial three-cornered Dim--ECE--Disp competition will be reduced to a two-term Dim--ECE competition. Conversely, when ( p / a ) -~ co, a two-term Dim--Disp competition will be observed. For both of these limiting behaviors the competition will depend upon a single parameter. The D i m - - E C E c o m p e t i t i o n . The system of eqns. (30) and (31) becomes: d2b*/dy .2 = b*
(33)
d 2 c * / d y .2 = --b* + a-2(c*) 2
Integration of eqn. (33), according to (32), leads to b* = exp(--y*)
and
(db*/dy*)o = - - 1
It thus remains to integrate: d 2c*/dy *2 = --exp(--y* ) + 0-2 (c*)2 with y =0,
c =0
y ->0%
C ->0
the yields being given by R EccPE = 1 -- R DccPE = 1 -- (dc*/dy*)0 This was carried out, as a function of o, numerically following an iterative finite difference procedure adapted from the Crank--Nicholson m e t h o d as described in the Appendix. The resulting variations of R~CPE with o = k~14k~lnD-l14c°-i/251n are shown in Fig. 2a. The D i m - - D i s p c o m p e t i t i o n corresponds to ( p / o ) -* ¢~. The electron transfer to C occurs now in the solution so that the stationary state assumption can be applied to C: 2~2c 2 + ~ d b C - - ~ l b = 0
and thus: d2b*/dy .2 = b* + ( p b * / 2 ) ( [ ( p b * ) 2 + 45*] 1/2 - - p b * }
10
0.5
a
-2 0 i i i i i i
2
b
logic
c
\ \ co . log e
d
\
x k2w\ \ % kl/ . -2 Disp " / ~ Ece/D,SP /
/ Dir~/Di,p /
'~..--~, \
-2 Dim
I )
Dim/Ece
-~
, I-,~
,
Ece \\
o,
"..
,o~,,
Fig. 2. C o m p e t i t i o n b e t w e e n dimerization and further electron transfer ( S c h e m e II). Constant concentration potentiostatic electrolysis: (a) D i m - - E C E competition: variations of the dimer yield with the parameter a = k l3/4 k~ 1/2 D - 114 c 0 - 1/2 ~ 1/2 ; (b) D i m - - D i s p c o m p e t i t i o n : variations of the dimer yield with the parameter p = k d k 1 3/4 k~ 1/2 D 1/4 c 0 1/2 ~ - 1/2 ; (c) zone diagram for a 5% accuracy on yield determinations (KG: general case).
with oo
f o
b* dy* = 1
and y* ~ ~:
b* ~ 0
the yields being given by R E CCPE
=
1
--R D CCPE = 2 + ( d b * / d y * ) o
The differential equation was solved numerically along the same procedure as above. The resulting variations of R E c ~ with the parameter p = kdk[3/4k-: ~n X D U 4 c ° u25-1n are shown in Fig. 2b. In the g e n e r a l case where ( p / o ) does not take the extreme value, the threecornered Dim--ECE--Disp competition depends upon the t w o parameters o and p. A competition zone diagram (Fig. 2c) has been drawn for a 5% accuracy on yield determinations, on the basis of the above results and of those obtained for the ECE--Disp competition [ 1 ]. It shows the extent of the zone corresponding to the general case (KG) and h o w one passes from one type of limiting behavior to the other.
11 EPE
Since n o w the bulk c o n c e n t r a t i o n of A is n o t held c o n s t a n t and since a0 = 0, eqn. (25) leads to dab /dT = - - ( da / d y ) w = - - a b
and thus to a b = exp(--r)
It follows t h a t (eqn. 25) (ddb/dT) + 2(deb/dr) = --(dd/dy)w
-- 2(de/dy)l-
= exp(---r)
Thus: d b + 2e b = 1 -- exp(--r) On the other hand, introducing a n e w expression of the variables and parameters: b ° = b* exp(T),
C° = c* exp(r),
o ° = o exp(r/2),
p° = p exp(--r/2)
results in the following f o r m u l a t i o n of the problem: d2bD/dy*2 = b D + (pD/o~)bD c~ d~cD/dy*~ = --bO + ( p ° / a ° ) b % °
+ (c~/a°) 2
with, as b o u n d a r y conditions, y* = O,
c °= 0
y*-~o%
b°-~0
and
c°-*0
and ~ b ~ dy* ~ 1 0
( d d b / d T ) is n o w given by
(ddb/dr) = - - ( d d / d y ) l -
= [ ( d b ° / d Y * )o - - ( d c ° / d Y * )o + 1 ] exp(--r)
and thus: T
db(T) = - - f
[1 + ( d b ° / d y * ) o - - ( d c ° / d y * ) o ] ( n ) exp(--T) d~
0 where the d e p e n d e n c e of the first t e r m in the right-hand side on ~ is t h r o u g h a ~ and p~. The yields are t h e n given by R~FE = 1 --RDpE = 2 e b ( r ) / [ d b ( r ) + 2eb(r)] = 1 --db(T)/[1 i.e.
--exp(--r)]
12
b -2
G
0
Ioga
c V",
T'o
~,oo~ "y,,: /
/
\\
Ece \\\
-2
I.
2
loge
Dim -4
Dim/Ece -2
\ l°2ga
0
Fig. 3. Competition between dimerization and further electron transfer (Scheme II). Exhaustive potentiostatic electrolysis: (a) Dim--ECE competition: variations of the dimer yield with the parameter 0 = k ~ / 4 k - a l / 2 D - 1 / 4 c ° - 1 / 2 6 1 / 2 . (b) Dim--Disp competition: variations of the dimer yield with the parameter p = k d k ~ 3 / 4 k ~ l / 2 D 1 / 4 c ° 1/25-1/2 ; (c) zone diagram for a 5% accuracy on yield determinations (KG: general case).
1
R~eE(T) = 1 --REDpE(T) = 211 - - e x p ( - - T ) ] - ' f
REcPE(O/r/, pr/)'r/dr/
exp(--'r/2)
and can thus be obtained as a function of time by integration of the yields in the CCPE case. For complete exhaustion of the solution, the yields are more simply expressed as 1 REpE:I
--
RDE : 2 :
E RccPE (o/vl, p~)~? d~
(34)
0
Figure 3 represents the yield in E for such conditions for the Dim--ECE competition as a function of the parameter a (Fig. 3a), for the Dim--Disp competition as a function of the parameter p (Fig. 3b) and the zone diagram showing the various types of competition (Fig. 3c), these results being based on those obtained in the CCPE case through eqn. (34). CCGE
Appropriate change of variables and definition of parameters are now: b ~ = b*/~I,0,
c a = c*/T o
OA = 0 ~ 0 -1:2 = h~/4 h ~l/2 D1/4 (iO /FS)-i/2
(35)
13
pa = p q2o 1/2 = kd k ~3Z4k-21nD-11, ( iO/FS)ln
(36)
Equations (30) and (31) keep the same form with b*, c*, a and p being replaced by b a, c a, o a and p~ respectively. The boundary conditions are now:
c a= 0
y* = 0: and
(dca/dy *) -- (dba/dy*) = 1 y* -+ oo:
b a -> O,
(37)
c a -* 0
eqn. (37) n o w replacing eqn. (32). Here, b a and c a are n o t dependent upon r and thus are ( d d / d y ) l - and (de/dy)l- since the integration of a linear combination of eqns. (19)--(21) leads to
(dd/dy)w + 2(de/dy)~- = _xpo ;
b a dy*
o
and integration of the sum of eqns. (18) and (20) gives
(dd/dy)l- = kO°[(dbA/dy*)o--(dca/dY*)o + ;
ba dy*] = ¢2°(--1+ ;
o
o
ba dy *)
The yields are thus also time independent, although the individual bulk concentrations of E and D are proportional to time: oo
R CEC G E = 1
--R D CCGE = 2 --
1/f
b a dy*
(38)
0
The Dim--ECE competition with negligible interference of the homogeneous electron transfer will be observed for (pa/o ~) -~ O. The system then depends upon the single parameter o a as defined by eqn. (36). It is entirely defined by d2ca/dy* 2 = [(dca/dy,)o -- 1 ] exp(--y*) + (ca/o~) 2 with, as boundary conditions, y * =0: y
*
-->oo:
CZX
C
a
=0 -->~
the yields being given by R CCGE E =
1 -- R D CC6E =
2-
[1 --(dc~/dy*)o] -1
The differential equation was solved numerically by the same finite difference method as in the preceding cases (see Appendix). The resulting variations of the yield in E with the intrinsic and operational parameters as contained in o a are shown in Fig. 4a. The Dim--Disp competition will be observed when (pa/o ~) -> oo. The system then depends upon the single parameter pa (eqn. 37). The stationary-state assumption can indeed be applied to C leading to
14
log da
log e
a
/
/
• k~",'y D*Sl / \ ,,
)irn/D
EcelDisp /
/
\ \
)
/
Ece
N\\\ Dim -4
Dim/Ece 2
0
x log a'' 2
Fig. 4. Competition between dimerization and further electron transfer (Scheme II). Constant concentration galvanostatic electrolysis: (a) Dim--ECE competition: variations of the dimer yield with the parameter 0 ~ = k~/4k~l/2D1/4 ( i ° / F S ) - l l 2 ; (b) Dim--Disp competition: variations of the dimer yield with the parameter pa = k d k T 3 / 4 k ~ l / 2 D - l / 4 ( i o / F S ) l / 2 ; (c) zone diagram for a 5% accuracy on yield determinations (KG: general case).
b~= (p~/o~)b~c ~ + (e~/o~) 2 a n d thus to d 2ba/dy*2 = b ~ + ( p ~ b A / 2 ) { [(pab~)2 + 4b ~]u2 _ p ~ b a} with for y* = O,
dbA/dy * = -1
and for y* -~ ~o,
b ~ -* 0
the yields still being given by eqn. (38). The numerical resolution along the same finite difference m e t h o d as previously (see Appendix) leads to the variations of the yield in E with the parameter p~ represented in Fig. 4b. Figure 4c shows the zone diagram summing up the variations of the yields with the two competition parameters o ~ and p~. DISCUSSION Scheme I
The yield in dimer is an increasing function of the parameter 7 = k ~/2 k 2 k ] 2 DU2 c o ~-I
15 for the potentiostatic regimes and of "/~ = k ~/2 k 2k-32D -1:2 ( i 0 / F S )
for the galvanostatic regime. The increase of this yield upon increasing c o or decreasing 5 in the first case and upon increasing the current density in the second, reflect the second-order character of the dimerization as opposed to the first-order character of the deactivation reaction: the higher the reduction flux of A the higher the formation of B and thus of C and the more efficient the dimerization process. From a practical viewpoint, the working curves of Fig. 1 will help in selecting the appropriate experimental conditions for optimizing the yields in either the dimer or the first-order deactivation product. As expected, the yield in dimer increases with k2 and decreases with k3. It is noted that the latter rate interferes through its square. The fact that the dimer formation increases with k, was less straightforwardly predictable. It again reflects the second-order character of the dimerization process as opposed to the first-order character of the deactivation reaction: for a given value of the reduction flux of A, the higher k,, the larger the production of C from B and thus the more efficient the dimerization process. The working curves of Fig. 1 also provide a mean for controlling the correctness of the predicted mechanism through the variations of the yields with the operational parameters. At the same time they allow the determination of the rate factor k ~:2 k2k-32 from which a given rate constant can be derived once the other two are known from independent sources. SCHEMEII
When C can undergo a further electron transfer Concurrently with the dimerization process, the above analysis has shown the practical importance of the nature of the electron transfer, homogeneous or heterogeneous. The role of the operational as well as of the intrinsic factors is indeed entirely different in each case. The passage from competition with an ECE electron transfer and a Disp electron transfer is favored by an increase in initial concentration and a decrease in the diffusion layer thickness, but the most important effect is that of k, (see Figs. 2c, 3c, 4c): the faster the B -> C decomposition the closer to the electrode is C formed and the more efficient is reduction by the electrode rather than by B. In the case of a Dim--ECE competition, the yield of dimer is a decreasing function of the parameter o = k~/4 k ~ ' : 2 D -':4 c ° - ' / 2 5 '/2 for the two potentiostatic regimes, and of o " = k ~ 4 k ~ ' n D ' / 4 ( i ° / F S ) -':2 for the galvanostatic regime. The effect of c o and 5 in the first case and of the current density in the second case reflects the second-order character of the dimerization as opposed to the first-order character of the ECE process for the same reasons as for scheme I. The effect of k2 is straightforward, while t h a t of kl can be rationalized as follows. The higher is k , , the closer to the electrode is C formed and the more efficient its further reduction at the electrode at the expense of dimerization. For the Dim--Disp competition the yield of dimer is a decreasing function of the parameter p = kdk-13:4k-2 i n D ' : 4 c ° ,/2 5-,:2 for the two potentiostatic regimes and of p:' = k d k ~ 3 / 4 k - 2 1 n D - ' / 4 ( i ° / F S ) ' / 2 for the galvanostatic. The effect of the
16 operational parameters is now the opposite to what it was in the Dim--ECE case: the higher the reduction flux of A ( c ° t , 5~, (i°/S) ~ ) the more efficient the reduction at the expense of dimerization. These effects are not easy to predict on an intuitive basis since we deal with two concurrent second-order reactions differently located in the reaction sequence. While the effect of kd and k2 are straightforward, that of k ~ derives from the fact that the larger is k 1, the higher the conversion of B into C, which favors the coupling of 2C at the expense of the reaction of C with B. The zone diagrams (Figs. 2c, 3c, 4c) provide an overall view of the effect of the intrinsic and operational parameters on the three-cornered Dim--ECE--Disp competition. In the two limiting Dim--ECE and Dim--Disp situations the working curves of Figs. 2--4 can be used for checking the correctness of the reaction mechanism, optimizing the yield in the desired product and determining rate constants along the same lines as decribed for scheme I. As mentioned in the introduction, the present analysis was used to investigate the reduction mechanism of aromatic trithiocarbonates [ 5]. Let us now use these results to discuss some aspects of reductive cleavage reactions: RX+le~
RX:
kl
RX: -~ R" + X-
(0) (1)
(where R is an aromatic or aliphatic group and X a leaving group attached to R by a carbon--heteroatom bond). More specifically, let us discuss the possible interference of radical--radical dimerization in the reaction sequence following the initial cleavage step: k2
2 R" -~ RR
(2)
The following reactions may compete with this dimerization process: (1) H-atom transfer from the solvent: k3 R + SH -~ RH + S"
(3)
(2) Electron transfer to R ' , at the electrode: R" + 1 e ~ R-
(4)
or in the solution: kd R" + RX" -~ R- + RX
(5)
the latter two reactions being complete by proton abstraction from the reaction medium. The outcome of this four-cornered competition depends upon the structure of the starting RX. If R is an aromatic group, for most of the X groups (e.g. in the case of halides, quaternary ammoniums and phosphoniums, ternary sulfoniums, sulfides, s u l f o n e s . . . ) the standard potential of the R " / R - couple is very positive to the reduction potential of RX. For the Dim--ECE competition this corresponds exactly to the conditions we have assumed in the above treatment, i.e. the concentration of R" (C) is zero at the electrode surface. For the Dim--Disp competi_tion this means that the homogeneous electron transfer between R" and RX" (5) is close to the diffusion limit (k d -~ k d i f ) .
17
It is noted that op = o ~ p ~ = k d / k 2
Thus, in the present case: op = oAp ~ = kdi~/k2
Since the dimerization rate is at maximum at the diffusion limit it follows that op
= o~p ~ ~> 1
(39)
The outcome of the competition between dimerization and heterogeneous and/ or homogeneous electron transfer in these conditions can be derived from the zone diagrams as shown on Figs. 2c, 3c and 4c: eqn. (39) implies that the points representing the competition are located above the dashed line. The corresponding region mostly involves the Disp, ECE and Disp/ECE zones, i.e. corresponds to the predominance of electron transfer over dimerization. The maximal yield in dimer would be obtained for o ~ = 1 and p~ = 1 in the CCGE case. This would correspond to a yield equal to 20% (Fig. 4a). The obtention of such a yield would imply that the dimerization reaction is diffusion controlled and that the values of k l, D and i° are such that k l = ~2/3 D - 1 / 3 ( i° / F S ) 2/3 "dif
In most cases the dimer yield is expected to be much lower. This holds for cases in which the H-atom transfer from the solvent can be neglected, as for example in the case of liquid ammonia [7,8]. In organic solvents, H-atom transfer interferes in competition with heterogeneous and homogeneous electron transfer [9]. This will further reduce the yield of dimer, the exact o u t c o m e of the competition between dimerization and H-atom transfer being given by the above treatment of Scheme I. We thus n o w understand why the yield of dimer found in the reductive cleavage of aromatic c o m p o u n d s is n o t generally more than a few percent at maximum. It was, for example, found to be of the order of 5% in the reduction of tetraphenylphosphonium in DMF [ 10,11 ] and a few percent dimer was detected in the reduction of halopyrenes [12] and 9-haloanthracenes [13]. A completely different picture may be observed in the reductive cleavage of aliphatic compounds. An interesting example inthis connection is the reduction of the diphenylmethyl and triphenylmethyl-p-nitrophenylsulfide in DMF [14,15] where high yields of dimers are obtained. In this case, the presence of the nitro group in the para position considerably facilitates the uptake of the first electron. The reduction potential is then slightly negative (diphenylmethyl) or even positive (triphenylmethyl) to the R ' / R - standard potential. Dimerization can thus c o m p e t e successfully with both heterogeneous and homogeneous electron transfer to R ' . On the other hand, H-atom abstraction by this t y p e of R ° radicals is certainly much less efficient than for aryl radicals which thus allows the dimerization process also to compete successfully with H-atom transfer.
18 ACKNOWLEDGEMENT This work was supported in part by the C.N.R.S. (A.T.P. "Energie et Mati6res Premi6res" 1979. Electrochimie S61ective). APPENDIX (1) The first differential equation to be numerically solved (Dim--ECE for CCPE) can be expressed as (y* and c* are replaced by y and c for simplicity) d 2 c / d y 2 =--exp(--y) + 0 -2 c 2
(A1)
with y = 0:
c=0
(A2)
y -. ~o:
c -~ 0
(A3)
the yield to be calculated being R = 1 -- ( d c / d y ) o . The space interval [0, p"] is divided into N elementary intervals of length Ay. The value of C at y = j Ay is noted as cj. The differential equation is then expressed under finite difference form [16] as (Cj-- i --
2 c i + c i + i ) / ( A y ) 2 = --exp(--j Ay) + o - 2 c }
Its resolution is carried out by iteration. Let c~'-1 (0 ~< j ~< N) be the values obtained at the (n -- 1)th iteration. For the nth iteration the above equation is linearized according to (c'~-1 - - 2c'~ + c ~ + l ) / ( A y ) 2 = --exp(--jAy) + (o-: c~ -' )c~'
(A4)
with c~ = 0
(A5)
and the condition corresponding to y -+ p", i.e. to j = N. The latter is obtained as follows. When y >> 1 the first term in the right-hand side of eqn. (A1) can be neglected, leading to: d2c/dy 2 ~ o-2c 2
the integration of which, taking eqn. (A3) into account leads to dc/dy
-~ --(2/3) In o -1 c 3n
The value of p" is selected for this approximation to be valid. We t o o k tt" = 20 which correspond to exp(--p") ~ 2 X 10 -9 . The linearized finite difference expression of the above equation is then: (c~v -- c~v-1)/Ay = --[(2/3) '/: o - ' (c~v- 1 ) ' n ] cTv
(A6)
The (N + 1)c~ (0 ~
19
w h e n b o t h the following c o n d i t i o n s are fulfilled: [C~/ - - c~--ll •
1 0 -6 C~/
[ ( d c / d y ) ~ - - ( d c / d y ) ' ~ - l [ <~ 1 0 -6 (dc/dy)'~
T h e value o f (dc/dy)'~ used to calculate the yield: R = 1 - - (dc/dy)'~
is o b t a i n e d f r o m the following six-term e x p a n s i o n which is m o r e a c c u r a t e t h a n a single t w o - t e r m e x p a n s i o n : (dc/dy)3 = [ 3 0 0 ( c ~ - - c ~ ) + 200c~ - - 7 5 c ~ + 1 2 c ~ ] / 6 0 A y T h e whole calculation is started for the highest value o f o, 104 in the present case, f o r which the values used t o start the iterations are t a k e n as cj = 1 -- e x p ( - - j A y ) ,
j = 0, N
which c o r r e s p o n d s t o a p u r e ECE situation. T h e f u r t h e r calculations are t h e n carried o u t for decreasing values o f o, the cj o b t a i n e d for t h e last value o f o serving as starting values for the n e x t value o f o. This p r o c e d u r e , used with N = 500, i.e. Ay = 4 X 10 .2 leads to a rapid c o n v e r g e n c e . (2) T h e differential e q u a t i o n c o r r e s p o n d i n g to the D i m - - E C E c o m p e t i t i o n in the CCGE case: d2 c / d y 2 = [ ( d c / d y ) o - - 1] e x p ( - - y ) + o-2 c 2
(A7)
with y=O:
c=O
y --> ~ :
c~ 0
was c o m p u t e d according t o the same p r o c e d u r e , the linearized finite d i f f e r e n c e expression o f eqn. (A7) at the n t h iteration being (c~'-1 -- 2c~ + C~+l)/(Ay) 2 = [(dc/dy)~ -1 -- 1] X exp(--j Ay) + ( 0 - 2 c ' ; - 1 ) c ' ~ (3) T h e Dim--Disp c o m p e t i t i o n leads t o the same differential e q u a t i o n for b o t h the CCPE and CCGE c o n d i t i o n s : d 2 b / d y 2 = b + ( p b / 2 ) { [ ( p b ) 2 + 4b1'/2 _ p b }
(A8)
with y - + oo:
b -~ 0
(A9)
and b dy = l,
in the CCPE case
(A10)
o
or y=O:
( d b / d y ) = --1,
in t h e CCGE case
(All)
2O The same procedure as above is followed. The differential equation is expressed under finite difference form and linearized as n (b?-1 -- 25? + 5j+l)/(Ay )2 = 57 [1
+ (p/2){ [(pb?-l) 2 + 4b? -1] 1/2 _
pb?-l} ]
for 1 < j ~< N -- 1, eqn. (A9) becomes 5~v = 0 and eqn. (A10) N--1
b~=(2/Ay)--2
~
5? -1
j=l
while eq. (A11) is expressed as b~ = [60Ay + 300(b? -- b~) + 200b~ -- 75b~ + 12b~]/137 A set of (N + 1) linear equations involving the (N + 1)b? (0 < j ~< N) is then obtained. It is solved as in the preceding case. The whole calculation is carried out for a series of increasing values of p. For the lowest p value (10 -4 ), the iteration process is initiated, taking bi = exp(--jAy) The bj values obtained for the last value of p are used to start the calculation for the next value of p. The iteration process is stopped when the following conditions are fulfilled: Ib~ - - b ~ - l l < 1 0 -6 b~
and the yield in dimer is finally obtained as
R = 2 + (db/dy)~ in the CCPE case, and
R=2--1
bdy
in the CCGE case, where (db/dy)~ and
(db/dy)7) = [--137b~
[f~b
dy] n are determined according to:
+ 300(5? --b~) + 200b~ -- 75b~ + 12b~]/(60Ay)
and
[;
b dyln = (AY)I (b'~/2) + N~lJ=lb?l
respectively. REFERENCES 1 C. A m a t o r e and J . M . Sav~ant, J. E l e c t r o a n a l . C h e m . , 1 2 3 ( 1 9 8 1 ) 1 8 9 . 2 C. A m a t o r e and J . M . S a v 6 a n t , J. E l e c t r o a n a l . C h e m . , 1 2 3 ( 1 9 8 1 ) 2 0 3 . 3 C. A m a t o r e , F. M ' H a l l a a n d J . M . S a v 6 a n t , J. E l e c t r o a n a l . C h e m . , 1 2 3 ( 1 9 8 1 ) 2 1 9 .
21
4 5 6 7 8 9 10 11 12 13 14 15 16
C. A m a t o r e , J. Pinson, J.M. Savdant a n d A. Thi~bault, J. Electoanal. Chem., 1 2 3 ( 1 9 8 1 ) 2 3 1 . M. Falsig, H. Lurid, L. N a d j o a n d J.M. S a y , a n t , A c t a Chim. S c a n d , Set. 34B ( 1 9 8 0 ) 6 8 5 . M. Falsig, H. L u n d , L. N a d j o a n d J.M. S a y , a n t , Nouv. J. Chim., 4 ( 1 9 8 0 ) 445. J.M. S a y , a n t a n d A. T h i d b a u l t , J. E l e c t r o a n a l . C h e m . , 89 ( 1 9 7 8 ) 335. C. Amatore~ J. C h a u s s a r d , J. Pinson, J.M. Sav~ant a n d A. Thi~bault, J. A m . C h e m . Soc., 101 ( 1 9 7 9 ) 6012. F. M'Halla, J. P i n s o n a n d J.M. S a y , a n t , J. A m . C h e m . Soc., 1 0 2 ( 1 9 8 0 ) 4 1 2 0 . J.M. S a y , a n t a n d Su K h a c Binh, E l e c t r o c h i m . A c t a , 20 ( 1 9 7 5 ) 21. J.M. S a y , a n t a n d Su K h a c Binh, J. Org. C h e m . , 42 ( 1 9 7 7 ) 1 2 4 2 . J. G r i m s h a w a n d J. T r o c h a - G r i m s h a w , J. Chem. Soc. Perkin Trans. II, ( 1 9 7 5 ) 215. O. H a m m e r i c h , p e r s o n a l c o m m u n i c a t i o n , 1 9 7 9 . G. F a r n i a , M.G. Severin, G. C a p o b i a n c o a n d E. VianeUo, J. C h e m . Soc. Perkin Trans. II, ( 1 9 7 8 ) 1. G. F a r n i a , M.G. Severin, G. C a p o b i a n c o a n d E. Vianello, 31st Meeting o f the ISE, Venice, S e p t e m b e r 1 9 8 0 , A b s t r a c t No. 28E. J. C r a n k , M a t h e m a t i c s of D i f f u s i o n , C l a r e n d o n Press~ L o n d o n , 1 9 6 4 .