e--c-P-e-p: protonation (2) is the rate-determining step; k~ >> k v Z ° and thus. k a p ---- K e k p Z °. This leads to the wave equations and the Ep, E~ expressions ...
Electroanalytical Chemistry and Interracial Electrochemistry, 53 (1974) 165 186
165
( Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
ELECTROCHEMICAL CYCLIZATION I. FORMAL KINETICS FOR VOLTAMMETRIC STUDIES (LINEAR SWEEP, ROTATING DISC, POLAROGRAPHY)
C. P. ANDRIELrX and J. M. SAVI~ANT
Laboratoire d'Electrochimie de I'Universitd de Paris VII, 2 place Jussieu, 75221 Paris Cedex 05 (France) (Received 5th February 1974)
The present study deals with electrochemical cyclization processes occurring to molecules bearing the same functional group twice and where the cyclization step is not preceded by any irreversible chemical or electrochemical step. Typical examples of such processes are the intramolecular pinacolization of dicarbonyl compounds1 : IOH • jC-~
R--CO--(CR/~)n--CO--R + 2e- + 2 H * ~ (CR2)n.......J/(~'.-I C~ R
and the electrohydrocyclization of bis-activated olefins2-4" .....~CH-- CH,a--X X--CH =CH--(CR;~)n--CH=CH--X + 2e-* 2H÷ -Q""(CR~)n I ~ C H --CH2--X
(where X is an activating group such as -COE Et or -CN). On the contrary, in the reductive cyclization of the 1-3 dihalide 5- 11 B r ~ B r
+2e- --~ , ~
the reductive cleavage of one or two C-Br bonds is likely to be irreversible, so that no direct information can be obtained from kinetic measurements on the cyclization step itself. It is assumed that the cyclization process is complete, i.e. corresponds to a two-electron overall exchange along a single wave. The electron transfer from the electrode to such large molecules is probably fast i however, if the cyclization process itself is very fast, the electron transfer may become rate determining. In such conditions nothing could be said about the mechanism of the cyclization. It is thus assumed that the cyclization reaction rate is such that the initial electron transfer remains Nernstian. Having mainly in mind compounds such as aromatic ketones and activated olefins and media of low acidity, e.g. acetonitrile (ACN) and dimethylformamide (DMF) with some water present, it can be assumed that the protonation reactions
166
C . P . A N D R I E U X , J. M. SAVI~ANT
that may interfere starting from a molecule A, do not concern either the anion-radical A T 12-14 which is a relatively weak base nor the open-chain dianion A 2- which behaves approximately as two independent anion-radicals as regards basicity. On the contrary the cyclized dianion C 2- is a strong base. Although the cyclized anionradical C ; is most probably less basic its protonation cannot be completely discarded. In these conditions, the possible reaction schemes for the electrohydrocyclization (EHC) are of two types (Table 1), In the first one the cyclization step involves the open-chain anion-radical A T which is converted into a cyclized anionradical C ~ whereas in the second one the cyclizing species is the open-chain dianion A 2- which gives the cyclized dianion C 2-. In each case the second electron donation step may occur either at the electrode surface or through an electron transfer in solution. These various reaction schemes are represented on Table 1, where the symbols for the thermodynamic and/or kinetic characteristics of each step are given. In the last column of the Table a conventional designation for each reaction scheme TABLE 1 N O M E N C L A T U R E OF T H E R E A C T I O N S C H E M E S
Reaction schemes
Rate det. step
Desiqnation
Electrode electron transfer
Protonation (3)
e-c-e-P-p
C ~ + l e - ~ C 2- (E~)
Cyclization (1)
e-C e-p-p
Initial step ( c o m m o n for all reaction .schemes) A + l e - ~ A "
(E °)
Cyclization of the anion-radical. Protonation of the cyclized dianion k~
(1)A T ~C
;(K,)
k,~ kp
(3) C 2- + T H ---, C H - + T
CH- +TH ~ C H 2 + T -
Solution electron transfer
Protonation (3)
e-¢-d -P-p
k,, (2) C T + A ; ~ C 2- + A ( g d ) ka
Sol. el. trans. (2)
e-c. D - p - p
Cyclization (1)
e - C < i - p -p
Cyclization of the anion-radical. Protonation of the cyclized anion-radical k~ (1) A ~ ~=~ C ~ (K~) k,~
Electrode electron transfer
Protonation (2)
e-c-P--e-p
C H ' + l e- --+ C H - (E,)
Cyclization (1)
e-C.-p e - p
Solution electron transfer
Sol. el. trans. (3)
e--c-p-D-p
Protonation (2)
e--c- P - d - p
Cyclization (1)
e -C-p-d--p
Electrode electron transfer
Protonation (3)
e-e--c-P-p
A ~ + l e - ~ A 2- (E °)
Cyclization (2)
e - e - C -p-p
Solution electron transfer
Protonation (3)
e -d-c-P-p
k,, (I) 2A ~ ~=~ A 2 - + A ( K d ) ka
Cyclization (2)
e-d-C.-p-p
Sol. el. trans. (I)
e-D-c-p-p
ko
(2) C" + TH ~=~ C H ' + T k/, k
C H - + T H ~=~ C H 2 + T k,
C H ' + A T --. C H - + A kd
Cyclization of the dianion. Protonation of the dianion k~ (2) A 2- ~ C 2- (K,) k,; ku
(3) C 2- + T H --* C H - + T C H - + T H ~=~ C H 2 + T -
167
ELECTROCYCLIZATION. F O R M A L KINETICS. I
is given. It follows the same lines as in the study of electrohydrodimerization 15, the capital letter representing the rate-determining step. It has been shown recently 16 on the example of dinitro compounds of the type :
in ACN and D M F , that, as soon as n = 3, the difference in the standard potentials of the two successive reductions, AE ° = E~ - E~, becomes practically equal to (RT/F) In 4 i.e. the statistical factor arising from the difference in the symmetry numbers of A, A ~, A 2-. Even when n = 1 or 2, AE ° is not far from this value (for example 70.5 mV at 20°C for n = 1 in ACN16). It follows that in the solution electron-transfer reaction : 2 A ~ ~=e A + A 2K a = e x p { -(F/RT)AE °} is not very small (the value for n = 1 is of the order of 0.1). This is the reason why e - d - c - p - p - t y p e mechanisms must be considered as being a priori as probable as the other ones listed in Table 1 although the repulsion between the two negative charges of A 2- might be larger in diketones and in bis-activated olefins than in the corresponding di-p-nitrophenyl compounds leading to a larger AE ° and then a smaller Ka. For the same reason, the e - e - c - p - p - t y p e mechanisms are likely to give rise to a single two-electron wave. As regards the reducibility of the cyclized anion-radical C ~ let us compare the structures of the bifunctional species to those of the monofunctional ones :
A
~Z M
~- ....
~-Z O
~.
-M"r
)
e~.. Z
CT
> C - - Z H Or > C H - - Z " N~H"
It is seen that C ~ is of the same type as the protonated radical M H ' . Since this last species is definitely more reducible than the starting material M (see ref. 17 and refs. therein) it follows that C ~ is most probably markedly more reducible than A itself. Thus the e--c - e - p - p and e - c ~ l - p - p - t y p e mechanisms will lead to a single twoelectron wave when the cyclization reaction interferes effectively. The protonated cyclized radical C H ' , being more reducible than C ~, is a ]brtiori more reducible than A ~ . It follows that when CH" is formed the current arising from the reduction of C ~ is negligible as compared to that corresponding to the reduction of CH" itself in the e-C-p--e-p-type mechanisms. For the same reason the solution electron transfer C H ' + A ~ ---+C H - + A is much more likely to occur than: C ~ + A ~ ~ C 2- + A in the e - C - p - d - p - t y p e mechanisms.
168
C. P. A N D R I E U X , J. M. SAVI~ANT
The basicity of the cyclized dianion C 2 - is so large that the first protonation reaction leading to C H - can be considered as irreversible. It follows that the second protonation step (CH- + TH ~ CH 2 + T - ) has no kinetic influence on the whole process. The medium is considered as being either buffered or unbuffered. In this last case the proton donor content, Z °, is assumed to be large enough in comparison to the initial concentration so as to remain practically unaffected by the course of the electrolysis. The diffusion coefficients of the open-chain species A, A ~ and A = will be assumed to be the same (D). The diffusion coefficient of the cyclized species C ~ and C 2- are also considered as equal (De). Particular attention will be devoted to settling the diagnostic criteria pertaining to the variations of the peak potential (Ep)in linear sweep voltammetry (LSV) with the sweep rate (v), initial concentration (c °) and concentration (Z °) of the proton donor TH. However, the logarithmic analysis of the waves and the variations of the half-wave potential (E½) in rotating disc electrode voltammetry (RDEV) and classical polarography (CP) with the rotation speed (co) or the drop time (0), the initial concentration and the proton donor content are also dealt with. The Nernst diffusion layer approximation is used for the calculation of the wave equations in both RDEV and CP (see ref. 18 and refs. therein), the diffusion layer thickness being : 6 = 1.61D~v~to -½ for RDEV TABLE 2 f = F/R T Current
i: current; i,: limiting current in R D E V and CI~ M a s s transfer operators
LSV
RDEV.
(3 02 Msubscrip~ = -- 0t + Dsubscrlp I 0X~
Msubscri~ = D~bscript ~ 2
d2 --
V(x)
d
t: time, x: space V(x): c o m p o n e n t of the fluid velocity perpendicular to the electrode surface Equilibrium constants
Ksubscriot =
ksubscript/k~ubscrlpI, K o = exp f ( E ~ - E~s)
Dimensionless variables and parameters
Potential Eo) , u = 2f(E, - Eo) (LSV) E: electrode potential; E,: initial potential Eo: characteristic potential depending on the reaction scheme (see Table 3) ¢ = - 2f(E-
Current i ~u -
FSc°D½~f i v ½ (LSV)
Convolution integrals
f ~supcrscript t/~up 1
pup . . . . ip* l t/Jsupcrscripl2 =
7[ - •
....
-- Usuper~ctipl 1
lot
2 (~su~,~,ipt I _ q)- t dr/
ELECTROCYCLIZATION. FORMAL KINETICS. I
169
(3 in cm, D in cm 2 s- 1, v: kinematic viscosity in cm 2 s- l, 09: angular rotation speed in tad s - I ) and 6 =
DO
for CP
The notations and definitions are given in Table 2. CYCLIZATION OF THE ANION-RADICAL: PROTONATION OF THE CYCLIZED DIANION Electrode electron transfer
The derivative equation system to be solved is: M(CA)
= 0
M(CA ) = k ~ c A - k ' c c c-
M~(cc--) =--k~CA +k'~ccMc (Cc~- ) = kp Z ° Cc2-
with the initial and boundary conditions: t=0, x~>0and/or*x=o%t/>0:
CA=C o, C A - = C c - = C c 2 - = 0
X = 0 , t />0:
CA~CA = e x p f ( E - - E
°)
c c-/Cc2- = exp f ( E - E °) (~cA/~x) + (~cA /~x) = 0
(~Cc I~x) + (&c~-I~x)
= 0
The current is: i = F S {O (t3CA/t~X)o + O c (0Cc /dX)o }
(the subscript 0 means that the gradients are taken for x = 0 ) e - c - e - P - p : protonation (3) is the rate-determining step. The above system becomes: M (CA) = 0 M~ (cc2- ) = kp Z ° Cc~-
with the same initial and boundary conditions for x = oo and for." x = O, t ~ 0 : D (OCA/e?X)+ D r (OCc~-/Ox) = 0
CA/Cc2-=ex p 2 f { E - -
t E~12 E ° +
1
since both electron transfers and the coupling step are considered to be at equilibrium : i = 2 F S D (OCA/?~X)o = -- 2 F S D c (0Cc2-/OX)o
The polarization problem can thus be defined as corresponding to a 2e* and: LSV,or: RDEV. Conditions concerning time are absent for RDEV and also for CP in the Quasi-stationary approximation.
Equation of the L S V wave ~v, ~v' ~v-~p/2 Definition of E o = f - i ~ + E E~=
e~-d-p-p
e-c-D-p-p
e-c--d--P-p
e-C-e-p-p
e-c--e-P-p
~cK2~
I In 1
Tf
k~Z ° D~ ; D
e o + ~ln
1
1 k¢ ~/;
2 -1 ~ e x p ( - { ) = 1 - 2 - 1 1 ~ u 0.992. 0.78, 1.85
;
1 Kckec °
~' + ~Inys
1
2 -~ ~ e x p ( - ~ ) = l - 2 - 1 1 ~ 1.054, 0.902, 1.51
1 1 kpZ ° E° + - - I n - - K c K e 3f 3f v
2 -1 ~ U e x p ( - 3 ~ / 2 ) = l _ 2 - 1 1 ~ , 1.215, 0.655, 1.23
Llnlk¢ E~t + 2 f f v
2 -1 ~ e x p ( _ ~ ) = 1 _ 2 - 1 l~U 0.992,0.78, 1.85
E~I + E ° ~2---+
2-1 ~u e x p ( _ 2 ¢ ) = 1 _ 2 - 1 l~U 1.403, 0.563, 0.925 ij - i
I
it - i
" m"
1 1 k¢6 z Et° + ~-fln~ D
l
o 62
i~i~
2~(i,-i)
E ° + ~-fln ~Kckac
1
E=E, + ~In
I
i 1 K~KdkoZ ° 62 + ~-fln ~ D
2
E=E½ + ~ . l n
-
i
il - i
k~62
E° + ~ - f l n ~
1
E=E~ + ~ l n
E o1 +E~o 1 De fi2 2 + ~ln K2kpZ °-D D
...j
E=E~ + ~t.ln
1
Log. eqn. of the R D E V - C P wave
Cyclization of the anion-radical. Protonation of the cyclized dianion
Reaction scheme
TABLE 3
E° + ( 2 f ) -1 ln(31t/la)kcO
E ° - 1.473 f - ' + ( 2 f ) - ' In f - i kc v " ' E ° + ( 2 f ) -1 In 1 . 2 9 6 D - t v t k c o a - I
E~l + ( 3 f ) t ln(n/la)K~kdc°O
E ° + ( 3 f ) - 1 In 0.432 D - t v~ K¢ ke c o co- 1
E ° - 1.268 f - 1 + ( 3 f ) - 1 In f - 1Kckdcov - 1
E~l + (3f) -1 ln(n/7)K¢KakpZ°O
E ° '- 1.021 f -1 + ( 3 f ) - 1 In f - ~ K ~ K d kpZ ° v EO + ( 3 f ) - 1 In 0.864D-~v~ K¢KakpZ°to -
E ° + (2f)-1 In (3n/7) k¢ 0
E°-0.78 f-t+(2f)-' In f - i k,:v-1 EO + ( 2 f ) - 1 In 2.592D-~v*k~to - 1
1
(E ° + E°)/2-O.563 f - ' + ( 4 f ) - ' In f - t D¢D-I K~kbZOv-i (E° + E°)/2 + (4f) -1 In 2.592 D~D- ~v ~ K~ k~,Z° to - ' (E ° + E°)/2 + (4f) '1 In (3n/7) D¢ D - I K~ kp Z ° 0
L S V peak potential E v = RDEV I CP E~ =
> < > Z
N
Z ~7
>
O
e-C-p-d-p
e--c- P - d - p
(u) ~
e-c-p-D-p
(B)°
e--c-p-D-p
e-C-p~-p
e~--P~-p
)
,U
v
~ I~V
1~
1
1
1 kc
E° + ~ f l n ~ - f ~
1
2 - 1 tp e x p ( _ ~ ) = 0.992, 0.78, 1.85
E0 + ~-fln~K
-l
;po Z
~rn-
~
:-
~
c:rn
:-0 ;po Z U
o
tv
- ..l
-
ELECTROCYCLIZATION. FORMAL KINETICS. I
173
electron transfer between two species A and C 2 - having unequal diffusion coefficients with a standard potential (E ° + E°)/2 + (2f)-1 In Kc and followed by an irreversible first-order reaction the rate constant of which is kp Z,°. The solution of such a problem in "pure kinetic" conditions can be arrived at by a slight modification (unequality of the diffusion coefficients) of previous treatments (see e.g. ref. 19). This leads to the wave equations and the Ep, E i expressions given in Table 3. e-C-e-p--p: coupling (1) is the rate-determining step. The derivative equation system is now: M (CA) = 0 M(CA-) = k ~ c A Mc(cc-) = --k¢eAwith the initial and boundary conditions: t=~
x/>0and/orx=oo,
x = 0 , t ~>0:
t>/0: CA=C o, CA-=Cc- = 0 CA = CA- e x p f ( E - - E ° ) , Cc- = 0
i = FS {D (t3CA/dx)o+ D¢ (ticc ./t3X)o } This is a typical first order ECE reaction scheme (see ref. 20 and refs. therein). As far as "pure kinetic" conditions are concerned (see ref. 21, appendix A):
i = 2FSD (SCA/dX)o Thus the wave equations and the Ep, E& expressions are those given in Table 3. The only difference with a standard ECE reaction scheme is in the transition between "pure kinetic" conditions and pure diffusion control. In this last situation a reversible 2e-wave may be obtained instead ofa le-wave since the standard potentials E ° and E ° may now be close to each other (in the case where AE°=(RT/F) In 4 the diffusion-controlled'wave is exactly twice the one-electron reversible wave16).
Solution electron transfer The system to be" solved is now:
M (CA) = -- kd CA- CO-'+ k'~ CA Cc2M (cA-) = k~cA -k'ccc- +kdcA- Cc- --k'dCACc2-
M(Cc ) = --kcCA +kjCc- +kdCA.Cc---kdCACc2M(Cc2 )= --kdCA-C c +k'dCACc2 +kpZ°cc 2with the initial and boundary conditions: t = 0 , x > ~ 0 a n d / o r x = o o , t~>0: c A = c o, CA =Cc x = 0 , t >~0:
=0
cA = CA- e x p f ( E - E °) (~c ~/,~x) + (,~cA- /Ox) = 0 ~)ec-/~x = t)Cc2 / # x = 0
the current is now:
i = FSD (~CA/OX)o
174
c . P . A N D R I E U X , J. M. S A V E A N T
The stationary state assumption is made for C- and C 2- : kdC A CC- --k'dCnCc2- = k c c A _ - k ~ c
c- = kpZ°cc 2_
The original system thus becomes: M
(2C A + CA . ) = 0
M(CA.- ) = - _
2kckdkp z ° c 2 -
(
(kdCA + k p Z °) k~ + ~CA ~ - k ~ O / e - c - d - P - p : protonation (3) is the rate-determining step, i.e. : k, Z ° >K d kp Z ° CA-/CA The last derivative equation then becomes: M(CA ) = 2 K c K d k p Z ° c 2-/CA In other words letting kap be the apparent rate constant: k,,p = 2 K c K d k p Z ° the polarization problem can be formulated as: M(2CA+CA ) = 0 M (CA-) = ka, c 2-/CA t = 0 , x ~ > 0 a n d / o r x = ~ - , ~ , t / > 0 : c A = c o, cA _ = 0 x =0, t>~O"
CA/CA _ = e x p f ( E - E
o)
i = FSD (OCA/OX)o = -- FSD (0c A_/OX)o Such a system has not been solved previously. It will be found, in the following, to be formally applicable to other reaction schemes, kap having eventually a different expression. Its resolution is presented in Appendix 1 for pure kinetic conditions in the case of LSV, RDEV and CP. This leads in the present case to the wave equations and Ep, E l expressions shown in Table 3. e-c-D--p-p: electron solution transfer (2) is the rate-determining step, i.e., kpZ ° ~ k ' d C A and k'¢ ">kdCAThe system to be solved is thus now: M (2C A + CA - ) = 0 M(CA- ) = 2Kckd c2_ t = 0 , x > ~ 0 a n d / o r x = o ~ , t ~ > 0 : c A = c o, CA = 0 x = O, t >/0:
CA/C A . =
expf(E-E
°)
i = FSD (OCA/OX)o = -- FSO (OcA-/OX)o This corresponds exactly to a DISP2 reaction scheme 2° with an apparent disproportionation rate constant of 2K~ kd. The wave equations and Ep, E~ expressions can thus be readily derived from previous work 2°.
ELECTROCYCLIZATION. FORMAL KINETICS. I
175
e - C - d - p - p : cyclization (1) is the r a t e - d e t e r m i n i n g step, i.e. : either kp Z ° ,~ k~ CA a n d k'~ 4, kp Z ° k d CA-/k'o CA or
kp Z ° > k~ CA a n d k'¢ ,~ k d c A
The p o l a r i z a t i o n p r o b l e m is now f o r m u l a t e d as follows: M (2c A + c A ) = 0 M (c A- ) = 2k c c A with the same initial a n d b o u n d a r y c o n d i t i o n as above. Fhis is a typical D I S P I reaction scheme 23, 2k¢ being the rate c o n s t a n t of the r a t e - d e t e r m i n i n g step. Previous results z3 on this scheme are readily a d a p t e d to the present case a n d given in T a b l e 3. CYCLIZATION OF THE ANION-RADICAL: PROTONATION OF THE CYCLIZED ANIONRADICAL
Electrode electron transfer The derivative e q u a t i o n system to be solved is:
M(cA)
=0
M (c A . ) = k¢ c A • - k'¢c c
M¢(Cc- ) = --k~CA- +k'~cc. + k p Z ° cc Mc (Cc.) = - kp Z ° Cc t=0,
x/>0and/orx=~,t/>0:
X = 0, t />0:
C A = C o, CA- = C c - = C c . = 0
CA~CA- = e x p f ( E - - E °) CCH = 0
( & # a x ) + (&A- lax) = o acc. / & = 0 T h e c u r r e n t is:
i = FS {O (aCA/aX)o + O c (aCca/aX)o } T h e s t a t i o n a r y state a s s u m p t i o n is m a d e as c o n c e r n s C k~ c A- - k~ Co- = kp Z ° CcIt follows that the m e c h a n i s m c o r r e s p o n d s to a t y p i c a l E C E reaction scheme. As far as pure kinetic c o n d i t i o n s a r e c o n c e r n e d :
i = 2 F S D (aCA/aX)o T h e initial system can thus be r e d u c e d t o : M(CA) = 0 M (CA-) = k.p CAwith
k~ kp Z ° k~p - k, +kpZO CA-
176 t=0,
C. P. A N D R I E U X , J. M. SAVI~ANT
x>~0and/orx=oo,
t>~0: c A = c o, C A - = 0
CA/CA - = exp f ( E -
x = 0, t >/0:
E °)
(acAlax) + (acA- lax) = o
e--c-P-e-p: p r o t o n a t i o n (2) is the r a t e - d e t e r m i n i n g s t e p ; k~ >>k v Z ° a n d thus k a p ----K
e k p Z °. This leads to the wave e q u a t i o n s a n d the E p , E~ expressions given in T a b l e 3. e-C-p-e-p: cyclization (l) is the r a t e - d e t e r m i n i n g step, k ' e ~ k p Z °, thus
kap = k c.
The c o r r e s p o n d i n g wave e q u a t i o n s a n d E p , E i values are shown in T a b l e 3.
Solution electron transfer T h e system to be solved is n o w : M(CA)
= --koCA-CCu
M(c A-) = k c c A - k ~ c c . + k dc A- C c . Me(co ) = --kcCA + k ~ c c + k p Z ° c c - - k ' p c T _ c c "
M¢ (c¢.) = - kp Z ° c¢- + k'o C T - CCH 4- k d c A - CCH Me ( c c . - ) = - kd CA Con + k Z ° Cc._ - k'c.r- Cc.2 M T (c T - ) : - - k p Z 0 c C _ 4 - k'p c T - CCH - - k Z °Ccn_ + k' C-r. Cc. 2 T h e s t a t i o n a r y state a s s u m p t i o n here c o n c e r n s C - , C H and C H - : kc c A _ - k~ c c_ = kp Z ° c c _ - k'p c T C c n = = kd CA- CC. = k Z °Ccn ._ - k' c T - Ccu 2 It follows that
M(CA- ) = 2kdCA. Con and k¢ kp Z ° c A _
Ccu
p t k¢kpCT+ k d ( k ¢t + k p Z 0 )CA-
T h e system can thus be r e d u c e d t o : M (2CA + CA- ) = 0
M(CA. ) = t=0,
2kekpkd Z ° c 2 k '~k'p c-r - + k d ( k c' + k p Z o)CA-
x~>0and/orx-=~,t>/0:
cA=c o CA- = 0
x = 0, t > / 0 :
(gCA/~X)+(~CA-/aX) = 0
CA/CA - = exp f ( E - E °) In buffered m e d i u m : c T - = Z ° KT/C.. (KT: acid d i s s o c i a t i o n c o n s t a n t o f the p r o t o n d o n o r ) whereas in unbuffered m e d i u m : M ( c A - ) + M~(cx-)=0. e-c-p--D-p: s o l u t i o n electron transfer (3) is the r a t e - d e t e r m i n i n g step, i.e., k'p cT- >>kd CA- a n d k~ >>k. Z °.
177
ELECTROCYCLIZATION. FORMAL KINETICS. I
Thus:
M(CA ) = 2Kc(kpZ o/kpcT. , )kdc 2In buffered medium:M( cA ) = (2Kc CH"/Kp)k a c 2 , Kp being the acid dissociation constant of the C H ' / C ~ couple. This is a typical DISP2 mechanism 2° leading to the wave equation and Ep, E½ values shown in Table 3. In unbuffered medium: M (cA ) = 2K~ (K T Z°/Kp CT- )ko c~ ... The system to be solved is very closely similar to one considered in the analysis of the electrohydrodimerization formal kinetics, i.e. the e - p - R R C - p reaction scheme in unbuffered medium (see ref. 15, eqns. (1)-(3) p. 340 and the resolution on pp. 358, 359). The only difference is that now the equation M (2CA+ CA- ) = 0 replaces M (CA)= 0. The consequence of this is that the current is now simply the double of what it was in the cited case of ref. 15. The results obtained are thus readily transposed which leads to the wave equations and characteristics shown in Table 3. e-c-P-d-p: protonation (2) is the rate-determining step, i.e. k'~ >~kpZ ° M(CA ) = 2K~kpZ°cA_
kdC A- ~ k'pCT - and
e-C-p-d-p: cyclization (l) is the rate-determining step, i.e. k d c A - ~ k ' p c T . and k'¢0and/orx=o%t/>0:
c A - - c o, c A . = c A ~ _ = c c ~
x = 0 , t >~0:
c A = c A_ e x p f ( E - E
°)
cA = cA~- exp f ( E -
E °)
(Oc,,/Ox) + (Oc,,_ IOx)+
=0
= o
178
C. P. A N D R I E U X ,
J. M. S A V E A N T
i = F S D { (8CA/OX)o -- (t3CA~-/OX)o} = F S D {2(OCA/OX)o + (Oe A_/OX)o }
Z ° being a constant the resolution of the system is the same when either the protonation of the cyclized dianion is rate determining : e - e - c - P - p , protonation (3) being the rate-determining step this means: k~ >>kp Z ° or when the cyclization step itself is rate determining: e - e - C - p - p , cyclization (2) being the rate-determining step: k~ 0and/orx-~,t>/0:
cA = c o C A - ~--- CA2- ~ CC 2- ---~ 0
x = 0 , t/>0:
CA/C A _ = e x p f ( E - E
°)
OCA2-/OX = 0 2 CC2-/X = 0 i = F S D (OCA/~X)o
=
- -
F S D (0CA-/OX)o
The stationary state assumption concerns both A 2- and C 2- : ½(kd c 2- -- k~ CACA2- ) = k¢ CA~- -- k; Cc~- = kp Z ° Cc,-
E L E C T R O C Y C L I Z A T I O N . F O R M A L KINETICS. 1
179
It follows that the above system can be simplified leading to: M(2c A+cA ) = 0
M (CA- ) --
2kdkckpZ° C2A-_ _ (k 2k~kpZ°~ (k'c + kpZ °) 'tiCA + k ~ k ~ 6 J
e-d--c-P-p: protonation (3) is therate-determining step, i.e., kpZ ° ~ k'~ and k'dCA ~ 2kpZ°kc/k'c M(CA- ) = 2KdKckpZ°C2A /CA The kinetics are thus of the same type as already treated in Appendix I with an apparent rate constant kap =
2KdK¢kpZ °
e-d-C-p-p: cyclization (2) is the rate-determining step, i.e., krZ ° >>k'~ and kdC A >>2k c The formal kinetics are again of the same type:
M(CA- ) = 2KakcC2A . /CA Here: k~p= 2K d k c. The only difference with the preceding case is the lack of variation of the characteristic potentials with the concentration of proton donor. e-D c-p-p: solution electron transfer (1) is the rate-determining step i.e., either
kpZ ° >>k~ and k'oCA ~ 2k~ or
kpZ ° ,~ k'~ and k'dCA ~ 2KckpZ ° Then :
M(2CA+C A-) = 0 U ( c A ) = kd
This is a typical DISP2 reaction scheme 2° the formal kinetics of which have already been analysed 2°. D I A G N O S T I C CRITERIA
The variations of the peak potential in LSV with the sweep rate, the initial concentration and the proton donor or the pH provide a means for discriminating between the various possible mechanisms. This can be done also by using the halfwave potential in RDEV, the sweep rate being replaced by the rotation speed. CP is of much less value since the drop-time can be varied over a much more restricted range than the sweep rate in LSV or the rotation speed in RDEV. For RDEV and CP the logarithmic analysis of the wave can also serve for the mechanism analysis. This is also true for the waves obtained in convolution potential sweep voltammetry 24, i.e. the curves obtained by performing the convolution of the LSV wave with the time
180
C. P. ANDRIEUX, J. M. SAVI~ANT
TABLE 4 DIAGNOSTIC CRITERIA
Reaction scheme
-~:Ep(E i) loglo ~,'(w. 1/0)
dEp(Ei) /~ Ioglo c°
oe~(E_,) ? Iog,o Z ° ( - p l l )
/raV per decade at 25°C Cyclization of the anion-radical. Protonation of the cyclized dianion e--c-e -P-p 14.8 0 14.8 e-C-e-p -p 29.6 0 0 e--c~l -P-p 19.7 0 19.7 e--c -D-p-p 19.7 19.7 0 e C~cl--p-p 29.6 0 0 Cyclization of the anion-radical. Protonation of the cyclized anion-radical e--c-P-e-p 29.6 0 29.6 e-C -p-e-p 29.6 0 0 e - c - p - D - p (B) 19.7 19.7 19.7 e - c - p - D - p (U) 19.7 0 19.7 e-c-P~l-p 29.6 0 29.6 e--C -p-d-p 29.6 0 0
C yclization of the dianion e--e--c- P-p e--e-C-p-p e-d -c-P-p e~d-C -p-p e--D-c-p-p
14.8 14.8 19.7 19.7 19.7
0 0 0 0 19.7
14.8 0 19.7 0 0
function t- ~ in linear diffusion. However, it may occur that various formally different logarithmic analyses fit the data equally well within the range of experimental errors. It must thus be emphasized that an accurate mechanism determination requires these logarithmic analyses to be performed over a large range of sweep rates (LSV) or rotation speeds (RDEV). This is the equivalent of the peak or half-wave potential shift method. On the whole, the accuracy is somewhat better using the logarithmic analysis method over a large range of sweep rates or rotation speeds rather than the potential shift method since more of the available experimental information is used. The rates of variation of the peak (or half-wave) potential with log v (or log to), log c o and log Z ° (or - pH) are summarized in Table 4 for each of the cyclization reaction schemes. Once the mechanism and the rate-determining step have been established the overall rate constant can be determined provided it is not too high in respect of the available range of sweep rates or rotation speeds. In such conditions indeed a pure diffusion-controlled situation can be met at high sweep rate (or rotation speed) that allows the determination of the standard potential of the system. In "pure kinetic" conditions the characteristic potential is a known function of the standard potential, the rate constant and various known factors. The rate constant can thus be deduced from the measurement of this characteristic potential since the standard potential is known from the pure diffusion-controlled experiments.
ELECTROCYCLIZATION. FORMAL KINETICS. I
181
ACKNOWLEDGEMENTS
Work was supported in part by the C.N.R.S. (Equipe de Recherche Associ6e no. 309: "Eiectrochimie Organique"). Numerical calculations were performed on the CDC 3600 of the C.I.R.C.E. (C.N.R.S., Orsay). SUMMARY
The formal kinetics of electrochemical cyclization reactions is discussed in the cases where the cyclization steps follow the electron-transfer steps, i.e. when they are not preceded by any irreversible chemical steps. Emphasis is laid on the discrimination between the mechanisms involving a cyclization of the initially formed anion-radical or a cyclization of the dianion produced either by a further reduction step at the electrode or by a solution electron transfer reaction. Transposition to oxidation and cations is immediate. Wave equations and characteristics are derived for linear sweep voltammetry, rotating disc electrode voltammetry and classical polarography. Diagnostic criteria to be used when observing the peak (LSV) or halfwave (RDEV) potential shifts with the sweep rate or the rotation speed, the initial concentration and the acidity of the medium are presented. APPENDIX I
Resolution of the system : M(2CA+C A-) = 0
m (cA- ) = ka~c~,-/CA t = 0 , X~>0 and/or x = ~ , t 1 > 0 :
cA = c o, CA-=0
X----0, t>~0:
CA/CA = exp f (E-- E °)
(dcA/dx) = (c~cA /dx) = i/FSO T.he system can be formulated in a dimensionless form in LSV and RDEV by introducing the following variables and parameters: a=CA/C ° , a - = C A /Co Z = f v t (LSV) y = x ( f v / D ) ~ (LSV),
y=6-'
exp D - ' 0
V(~)d~ dz(RDEV)
•
4 ' = - I ( E - E ~ ) , u' = f ( E i - E ~ ) (LSV, ei: initial potential) ~ = i / f S c ° D ~ f ~ v ½ (LSV), ~P = i6/FSDc ° (RDEV)
;.ap = kap/fv (LSV),
=
(RDEV)
LSV: in "pure kinetic" conditions d(2a + a- )/ dz = d2 (2a + a- )/ dy 0 = ~2 a-/~y2 _ 2~ (a-)2/a "t" ~ 0 ,
y>~O and y = ~ , z > ~ O : a = l , a - = O
y = 0 , z >~0:
a/a- = e x p ( - ¢ ' ) , (daiSy) = - ( a a - l d y ) =
182
C. P. ANDRIEUX, J. M. SAVI~ANT
According to the fact that the chemical process is assumed to be fast: a o ,~ a o and a-~ ao (the subscript 0 indicates that the concentrations are taken at the electrode surface). Integration of the differential equations taking into account the initial and boundary conditions leads to: ao = 1 - I I U / 2 U ~, = (a o )(ao)- ~ (2),ap/3)* and thus: (U/2) exp { - ( 3 / 2 ) ~ 1 - ( I / 2 ) I n ).ap/6} = 1 - I '
U/2
Introducing the new potential variable and parameter ~* = (3/2)~ I +(1/2)In 2~p-ln 3 u* = (3/2)u I - ( 1 / 2 ) I n 2ap+ln 3 It follows that: (U/6 ~) e x p ( - ~*) = 1 - I* W 6 ½ The integral equation without the factor 6 i has already been computed. The peak values are thus in the present case: Up = 1.215,
3" = 0.78,
~ - - ~ p * / 2 = 1.85
Introducing the same kind of ~ variable as in the other cases : = -f(E-Eo)
with : E°=E°
1 k.p + ~ f l n 6/" t,
i.e.
= ~-¢*+ ~ In 3 The wave equation becomes: 2 -1U exp(-3~/2) = 1-2 -IIU It follows that the peak values are: Up=1.215,
¢p=0.655,
~ p - ~p/2 = 1.23
RDEV-CP
d 2 ( 2 a + a - )/dy E = 0 d E( a - ) / d y E = 2.p(a- )2/a y=~:
a=l
a-=O
y = O:
a/a- = e x p f ( E - E
°)
U = (da/dy)o = - ( d a - / d y ) o
Integration of the. first differential equation leads : ao = 1 - U/2 and of the second one to:
183
ELECTROCYCLIZATION. FORMAL KINETICS. I
71= (22.0/3)(a o )3 a o ,
i.e.: 71/(2 - ~u) = (2.p/6)~ exp - ( 3 f / 2 ) ( E - E °) when E ~ - o o , i ~ i I = 2FSDc°6 -1 The logarithmic equation of the wave is thus:
E = E~ + ~ l n ij-i •
with
/
l
]Cap ~2
E~ = E°a + ~-rln ~j 6 APPENDIX
D
2
e-e-c-p-p MECHANISMS
LSV In dimensionless form the polarization p r o b l e m can be formulated as follows :
Oa/~z = 0 z a/Oy z da-/Oz = 02 a- IOy 2
Oa2- /Oz = OZaZ- /OY2-- 2apazz = 0 , y>~0 and y = o o ,
z>~0: a =
(J'av = k.o/fv)
l,a-=a
2- =0
a/a- = exp f(E--E~1)
y = 0, Z ~>0:
a - / a 2 - = exp T ( E - E °) (OalOy) + (Oa- IOy) + (Oa2- IOy) = 0 The dimensionless current ~ is the sum of two terms: ~u = ~x + ~2 with ~ , = (Oa/Oy)o and I//2
=
-(Oa2-/Oy)o
let the dimensionless potential ~' be defined as: ~' = - f { E - (E ° + E ° ) / 2 } , the potential separation between the two successive electron transfers be represented by: A = exp ( - f A E ° / 2 ) and the initial potential factor by:
u' = f { E i - ( E ° + E°)/2}
I
(in the case where AE ° = f In 4: A = 1/2) F r o m the integration of the three partial derivative equations it follows that :
184
C . P . ANDRIEUX, J. M. SAVEANT
ao = 1-I'~ t ao
= I ~ 1 --
I' ~2
aoz- = ~2 ;t.-~½ (in "pure kinetic conditions") Taking into account the two Nernst equations I'~pl
I'~2A e x p ( - ~ ' ) + 1 =
A exp(-¢')+ 1
and ~2). -~ e x p ( - 2 ~ ' ) { 1 +(2A) -t exp(~')} = 1 - ½ I ' ~ u Introducing the new dimensionless potential variable and parameter = ¢'+¼ In 2.p =
-f(E-Eo)
with E o = E ° + E ° + 1 Ink~p and u = u ' - ¼ in l~Ut =
'~ap
17~z A ).~p e x p ( - ~)+ 1 A2~p e x p ( - ~ ) + 1
1.4
J_
1.2
_~
(1)
1.2 1.1 ]
l
. . . . . .
t
-1--
Qg -5
0
o
"~
_
,"5
I
I
,
-2
0
+2
I
log (~-ao)
-1 -2
Fig.
+4 log (~.ao)
1. e - e - c - p - p mechanisms. Variations of the peak values with the kinetic parameter
~p=ip/FSD½c°(fv)~;~'o=-f(E ~
E~2E~2).
2,p=kap/fo.
ELECTROCYCLIZATION. F O R M A L KINETICS. I
185
and q', e x p ( - 2 ~ ) { I +(2A2~p) - t exp ~} = 1-½1~P
(2)
when 2ap --' ~ ("pure kinetic" conditions): I~P I --~ I~P 2
and thus ~PI ~ ~u2 = ~/2
and the wave equation becomes: ½7~ e x p ( - 2 4 ) = I - ½ I ~ Defining 4" = 2 4 - ½ In 2, u* = 2u + ½ In 2 (~P/2~/2) exp ( - ~*) = 1 - I* ~P/2~/2 According to previous calculations22 : tPo = 1.403
4"=0.78
4"-40"/,= 1.85
and thus : 4p=0.563
4p-%/2~ =0.925
In the general case, i.e. for intermediate values of the kinetic parameter 2ap, the wave equation can be numerically computed by replacing in eqns. (1) and (2) the convolution integrals by finite sums along an already described procedure '2. This has been done in the most interesting case in practice i.e. when A = 1/2. The results concerning the peak values are given in Fig. I. RDEV-CP
For "pure kinetic" conditions the treatment is strictly the same as for LSV and leads to the wave equation: E=E~+
1 lni~--i l zJ
With E~
E° + E°
1
6'
+ ~-f In kap ~ -
REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13
J. Armand, L. Boulares and P. Souchay, C. R. Acad. Sci. Paris, 276 (1973) 691. J. D. Anderson and M. M. Baizer, Tetrahedron Lett., (1966) 51 !. J. D. Anderson, M. M. Baizer and J. P. Petrovich, J. Org. Chem., 31 (1966) 3890. J. P. Petrovich, J. D. Anderson and M. M. Baizer, J. Org. Chem., 31 (1966) 3897. M. R. Rift-l, J. Amer. Chem. Soc., 69 (1967) 442. M. R. Riffi, Tetrahedron Lett., (1969) 1043. M. R. Riffi, J. Or#. Chem., 36 (1971) 2017. M. R. Riffi, Collect. Czech. Chem. Commun., 36 (1971) 932. A. J. Fry and W. E. Britton, Tetrahedron Lett., 0971) 4363. A. J. Fry and R. Scoggins, Tetrahedron Lett., (1972) 4079. Azizullah and J. Grimshaw, J. Chem. Soc., Perkin Transactions 1, (1973) 425. G. Porter and F. Wilkinson, Trans. Faraday Soc., 57 (1961) 1686. E. Lamy, L, Nadjo and J. M. SavSant, J. Electroanal. Chem., 42 (1973) 189.
186 14 15 16 17 18 19 20 21 22 23 24
C. P. ANDRIEUX, J. M. SAV~ANT
E. Lamy, L. Nadjo and J. M. Sav6ant, J. Electroanal. Chem., 50 (1974) 141. L. Nadjo and J. M. Sav6ant, J. Electroanal. Chem., 44 (1973) 327. F. Ammar and J. M. Sav6ant, J. Electroanal. Chem., 47 (1973) 115. R. Dietz and M. E. Peover, Trans. Faraday Soc., 62 (1966} 3535. C. P. Andrieux, L. Nadjo and J. M. Sav6ant, J. Electroanal. Chem., 42 (1973) 223. J. M. Sav6ant and E. Vianello, C. R. Acad. Sci. Paris, 256-(1963) 2597. M, Mastragostino, L. Nadjo and J. M. Sav6ant, Electrochim. Acta, 13 (1968) 721. C. P. Andrieux, L. Nadjo and J. M. Sav6ant, J. Electroanal. Chem., 26 (1970) 147. H. Matsuda and Y. Ayabe, Z. Elektrochem., 59 (1955) 494. L. Nadjo and J. M. Sav6ant, J. Electroanal. Chem., 33 (1971) 419. J. C. Imbeaux and J. M. Sav6ant, J. Electroanal. Chem., 44 (1973) 169.
