On the influence of coupled homogeneous redox reactions on electrode processes in d.c. and a.c. polarography: II. The e.e. mechanism with a coupled homogeneous redox reaction

Electroanalytical Chemistry and Interfacial Electrochemistry, 58 (1975) 145-175 145 © Elsevier Sequoia S~A., Lausanne- Printed in The Netherlands

ON THE INFLUENCE OF COUPLED H O M O G E N E O U S REDOX REACTIONS ON ELECTRODE PROCESSES IN D.C. AND A.C. POLAROGRAPHY

II. THE E.E. MECHANISM WITH A COUPLED HOMOGENEOUS REDOX REACTION*

IVICA RUZIC** and DONALD E. SMITH*** Department of Chemistry, Northwestern University, Evanston, Ill. 60201 (U.S.A.)

(Received 28th June 1974)

INTRODUCTION

In the first paper of this series 1 a comprehensive treatment of the mechanism involving two independent electrode reactions coupled to a homogeneous redox reaction was presented for d.c. and a.c. polarography. Such a mechanism is closely related to another one where the homogeneous redox reaction is coupled, not with independent electrode reactions, but with individual steps of an overall multielectron process in which all species are solution-soluble (e.e, mechanism with "nuances"):

ks , l , ~1

A + e . ' B (electrode reaction, E °, E½,1)

ks,2, ~2

B + e " C (electrode reaction, E °, E~,z) (I)

kl A + C -,- ~ 2B (homogeneous redox reaction)

k2

The influence of the backward homogeneous redox reaction, 2 B ~ A + C, on the first reduction step has stimulated considerable interest in the context of most electrochemical relaxation techniques z-32. Very recently, the influence of this disproportionation process on the first reduction step in a.c. polarography has been treated for the case with a large d.c. potential separation between individual heterogeneous steps/. However, the influence of disproportionation on the second reduction step has not been considered up to now. These previous studies also have been limited to the "pure" (irreversible) disproportionation process which appears under the following conditions:

e ° ~> El ° (A)

E~, z ~ E~, 1 (B)

* Dedicated to Dr. J. E. B. Randles on the occasion of his retirement from the Chemistry Department, University of Birmingham.

** On leave from the Center for Marine Research, Ruder Boskovic Institute, Zagreb, Yugoslavia, 1972-75.

*** To whom correspondence should be addressed.

146 i. RUNIC, D. E. SMITH

The influence of the forward homogeneous redox reaction, A + C ~ 2 B (reproportionation), has received relatively much less attention. An elegant spectro- electrochemical investigation by Winograd and Kuwana 33 is the most notable contribution in this area. Their study was concerned with the irreversible reproportionation step which appears under the conditions:

e ° ~ e ° (C)

E~, 2 ¢ E~, ~ {D)

The inattention paid to effects of reproportionation in more conventional electrochemical techniques probably arises from the belief that these effects will be nonexistent. For example, with several techniques it has been proven for the case where both electrode reactions are nernstian, that there will be no effect of the homogeneous reproportionation reaction, even if it is reasonably rapid 34. However, the lessons provided by the theoretical study of the so-called Yamaoka mechanism ~ suggest that reproportionation might influence observables obtained with the. e.e. mechanism under appropriate conditions.

A comprehensive treatment of the e.e. mechanism without nuances has been presented for the d.c. 35 and a.c. polarography 36'37. In the case of a.c. polarography, only the theory for the stationary plane diffusion model has been given. Except for certain special cases (e.g., nernstian d.c. processes with all redox forms soluble in solution phase), this model is not appropriate for careful, quantitative interpretation of data obtained with the commonly-used dropping mercury electrode (DME). This observation, together with the uncertainties regarding the influence of the homogeneous redox reaction, suggested that the phenomenological a.c. polarographic rate law should be generalized to encompass the homogeneous reaction's effects within the framework of a more rigorous electrode model. This objective demands a cor- respondingly rigorous treatment of the d.c. polarographic response with nuances, which has not been presented previously. Consequently, a rigorous d.c. and a.c. polarographic theory has been developed for Mechanism I, and its predictions carefully surveyed. The theory encompasses the expanding plane and expanding sphere models of the DME, and accommodates differences in diffusion coefficients of the reacting species. The development is applicable to nearly the full range of variations of Mechanism I, from pure disproportionation, through the various intermediate stages to the pure reproportionation case. Results of this investigation which concern the effect of the homogeneous redox reaction are reported here. A separate report is given of the predicted effects of mercury drop growth and sphericity on the a.c. polarographic response with a "pure" e.e. mechanism (without nuances) 38.

THEORY

Basic assumptions used in the derivation presented here are the same as those used for the case of two independent electrode reactions ~. The reader is referred to this earlier publication for details. The derivation is confined to the case of two one- electron steps for simplicity and because any multi-electron step can be considered as a special case of a multi-step reaction with several very unstable intermediates. One can readily extend the procedure described here to the case of electrode reactions with more than two steps. One of the most important multi-step cases is the three-step

COUPLED HOMOGENEOUS REDOX REACTIONS 147

process with three possible homogeneous redox reactions, such as the U(VI)/U(V)/ U(IV)/U(III) system in acid perchlorate media 39. This case will be treated later.

Following the usual custom ~'2, the derivation will be illustrated in the context of the expanding plane electrode model. Adaption to the expanding sphere case involves appropriate modification of the d.c. boundary value problem, which is solved by the simulation method 4°. Because the derivational pattern is almost identical to the one presented for the case of two independent electrode reactions ~, this presentation here will be less detailed.

(I) Preliminary manipulations of boundary value problem The expanding plane boundary value problem for mechanism I may be

writ ten :

OCA - D t~2CA 2x O¢A lklCACC 1 2 Si n ~ + Yi ~X +~k~. (1)

de B (~2C B 2X OcB fit - DB ~x 2 + ~ -~x + klCACc-k2c2 (2)

~Cc ~2cc 2x dec ½klcaCc ~ 2 (3~ -Ot - Dc ~x 2 + 3t Ox +gk2cB

t = 0 , x>~O" CA=C* (4)

cB = Cc = 0 (5)

t > 0 , x ~ : CA--;C* (6)

c B -~ c c ~ 0 (7)

t > 0 , x = 0 " D g ( ~ f g ) _ i l(t) (8) \Ox/~= o FA

D. (3CB] _ 1 \dx/x=o FA [i2(t)--ix(t)] (9)

D c ( dec] = i2(t) (10) \OX/x=O FA

il (t) =.fACA(x=0)exp ~--cqF (E_EO) ] FAk~,I L RT

i2(t) ~ (E_EO) ] FAke,2 = faea(~=o) exp [_ RT

-fcCc(~=o)ex p I ( l ~ - ~ )F (E-E°~ (12)

Note that the homogeneous rate constants, kl and k 2 are not independent quantities. Their ratio is controlled by the difference in standard potentials by the well-known

148 I. RUZIC, D. E. SMITH

K

Introducing

~pl =

02 = where

thermodynamic relationship :

k!l

the variable transformations

(2DA CA + DB Ca)/D 1

( 2Dccc + DBCB)/ D 2

allows

where

and

D 1 = (D g + KD.)/(1 + K)

D2 = (Dc + KD.)/(1 + K)

one to reformulate the boundary value problem as follows~ :

001 602 I//1 2x 001 0t = D 1 ~ + 3 5 0-2

002 0202 2X 002 ~t = 0 2 ~ + 3 t 0-2

OCB = 0 2 cB 2x OcB Ot DB ~ + 3t Ox

2DA /

+ ~1~1

+ ~2~2

D202-DacB_] __ k2 c2 2Dc /

D B - D A ( 1 - (1+/()D1

D B - D c (2 - (1 + K) D z

0 (C B -- 2Kc A)

0 (c a - 2Kc c) ¢2 - &

2x O(CB--2KCA)

3t OX

2X 0(%-- 2Kcc) 3t Ox

t = O , x > / O : ~ b l = 2 D A c ~ / D 1

02 = cB= 0

t > O , x-+oo " Ot ~2DAc~/Dt

02 ~ c s ~ 0

t > O , x = O : Da k O x / x _ ° = - D 2 = - - - x:o FA [il(~)+i2(t)]

(oc,) DB \Ox/x=o = FA [i2(t)--it(t)]

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(2S)

(26)

(27)

(28)

(29)

(30)


As discussed in detail previously 1, the quantities ~1 41 and ~242 will be negligibly small under circumstances where small diffusion coefficient differences prevail (DA~ DB ~ Dc) and/or when the homogeneous redox reaction is sufficiently fast that steady-state ~1'42 conditions exist. These conditions are applicable under most circumstances of interest, so negligibility of ~ 1 ¢ 1 and ~2 42 is assumed in the remainder of this derivation. The latter assumption, together with standard procedures of Koutecky41,42 and Matsuda 43'44 are applied to eqns. (18) and (19) and the associated initial and boundary conditions (eqns. 25-29) to conclude that

fill -[- (D2/D 1) ½ if/2 = 2C~DA/D1 (31)

Equation (31) may be substituted in eqn. (20) to yield

where

OC~ = DB C32 CB 2X t?c B ¢3t ~ + 3t- ~W

+ 4DAD c(DI~kl-DBcB) c* D-DI~b:-DBCB]- k2c ~ {32)

D = (/91D2)~. (33)

The foregoing manipulations have reduced the boundary value problem to one involving two differential equations (eqn. 32 and the modified form of eqn. 18 where (1 41 =0) in the unknowns ~01 and cB. The relevant boundary conditions are provided by eqns. (25)-(30).

(II) Separation of a.c. and d.c. parts of boundary value problem

Following the previously described procedure 1'2, the unknowns ca and ~O 1 are expressed as sums of d.c. and a.c. components,

cB = CB + CB (34)

01 = (35)

and substituted into eqns. (18) and (32). The latter are then simplified by ignoring higher-order terms in the a.c. components, such as ~2, ~2 and ~B~I- The latter ap- proximation carries the implicit assumption that the applied a.c. perturbation has a negligible effect on the d.c. process, so that the boundary value problem may be separated into d.c. and a.c. parts. One obtains for the d.c. part:

~3~B = DB ~2 CB 2x t?~ B ~t ~Tx2 + 3 t O~-

kl (DI~I_DBOB) [2c.(~)D_D-~I_DBOB]_ k2(~B)2 (36) + 4DAD ~

~ 1 02~1 2x ~ 1 (37) & - = 9 1 ~ + 3 t O--x-

t=O, x>~O : ~l --2OAC] (38) D1

?B = 0 (39)

150 I. RUZI~, D. E. SMITH

t > 0 , X"O'GO : ~1 "--+ 2DAc~ (40) . D1

~B ~ 0 (41)

t > 0 , x = 0 " D~ /dO-~) 1 = o = F A [ i , (t) + i2 (t) ] (42)

DB (0CB~ 1 \ Ox /~ = o = F ~ [i2 ( t ) - l l (t)] (43)

The a.c. part of the boundary value problem one obtains may be written"

where

and

0~x = D1 02~1 2x ~i~1 (44) +

~CB OZc B 2X ~cB Ot = DB ~ + 3t 0-~ + k*~l -k~cB (45)

I

kl [2DDAc~,-2DDx~I +DB (D-- Da)~,] k T - 4DAD c

k * = 4DAD ck~II2DDB(~-~A1) c ~ - D B ( D - D 1 ) ~ I - 2 D ~ B ] + 2 k 2 c B

t = 0 , x>~0" f f l = ~ B = 0

t > 0 , x ~ o o : ~ l ~ B ~ 0

t > 0 , x = 0 " Dl \ OX/x=O = F~[rl( t)+fz(t)]

( ~ B t 1 DB ~-X/x= ° = F--A [/'2(t)-/'I (t)]

(46)

(47)

(48) (49)

(50)

(51)

(III) Treatment of a.c. part of boundary value problem The foregoing manipulations have reduced the a.c. part of.the boundary value

problem to one where the homogeneous chemical kinetic terms in the Fick's law expression (eqn. 45) are pseudo first-order. Solving this problem for unequal diffusion coefficients is effected using a method which has been described in detail 1. Only the main steps in the derivation will be surveyed here.

Introducing the new variable

0--- K * ~ I - ~ . (52) where

= k l / k 2 (53) allows one to replace eqn. (45) with the expression

- + ~'~ (54) dX 2 D B

C O U P L E D H O M O G E N E O U S REDOX REACTIONS 151

where "( = D1-DB (55)

D1 DB

= 0~1 2x ~ 1 (56) Ot 3t Ox

7 , - D, ~--CB (57) D1-D B

As in the previous application of this procedure 1, eqn. (54) assumes simple, analytically- soluble forms in two circumstances which together encompass the majority of situations of interest. First, if D: ~ DB, only the first term in eqn. (57) is important, which leads to the special form of eqn. (54):

O~ 02 0 2x O0 Ot = D1 O-~ + 3t 8x k*0 (58)

where k* = (D1/DB} k* (59)

Second, if steady-state conditions prevail, ~ becomes negligible and (~ may be ignored regardless of the status of D1 and DB. In this case eqn. (54) becomes

0 2 0 k*O 0X 2 -- DB" (60)

The boundary value problem comprised of eqns. (44) and (58), and suitably-modified forms of eqns. (48)-(51) (substitute for CB using eqn. (52)), is readily solved by standard procedures 41-44. Expressions for 0x=O and ff1(~=o~ are obtained x which may be converted to the following surface concentration expressions with the aid of eqns. (14), (15), (31) and (52)"

c* {(DI_K~dDB) ~ I' u}[O'(u)+O2(u)]du CACx= O) -- 2D A o ( t} --u}) ½

+ (DBK*+DI) , , , o (U--U:) ½

V/-~T~ Ii u~e-k~"-"'Q-'2(u)du~ (61) + (DBK*-D1) 3n (t~--u:) ~ J

CB ( x = O) - - c * D B K , ~ ,_, DB 3n o (t:--u~):

)V/~I '''~'-~'-"'~ , , , - ~ 1 , ~ . , ~ - , . a a . , - (DBK*+D 1

o (t1-u:):

u~ e-k~t'-") ~ 2 (u)du [ (62)

152

CC (x = 0) --

I. RU2IC, D. E. SMITH

C~t 12 [(D)'i-DBK~]vI~ ~ f t U~-[OI(U)"i-O2(U)] du 2D c 3n o (t~-u~) ½

+ (DBK,+Dt)~f-~ fluke-k;U-u)O-'l(U) du (t ~ -u~) ~

where

fro u'l-e-~°-u, O" :(u)du'~ + (D.tiK'~-Ol) ( t ' -u ' ) ~ J (63)

~(t) (64) Qi (t) = FADi c*

* (65) K * - K~= o

k~ - k*= o (66)

One substitutes into the absolute rate equations (eqns. (11), (12)), eqns. (61)-(63), together with the relationships

i~(t) = ii(t ) + ?i(t) (67) E = Ed.~.- AE sin cot (68)

and employs well-documented procedures 36'37'4s to extract the pair of integral equations representing the small amplitude fundamental harmonic a.c. polarographic response. One obtains

~l,a (t) = F 1 (t) sin cot+ 21 [~1,o (11 +/2) + Z1,1 Ik,1 + Z1,2 Ik,2 ] (69)

~2,1 (t) ----- F 2 (t) sin cot + )]'2 [Z2,O (lx + I2)+ Xz,x/kA + Z2,2 lk,2] (70)

where ~1,1 (t) and ~2.1 (t) are related to the fundamental harmonic current associated with the first and second charge transfer steps, 11 (tot) and 12 (cot), by the expression 36'37,

Ii(cot ) - F2 Ac] D~ AE RT ~i,1(t) (i = 1, 2) (71)

Other new quantities in eqns. (69) and (70) are defined as follows:

q33~ f~o u~',, l(u)du 4 = (t~-u~)~ (72)

V / ~ I* u%-k;('-u) ~'li'l (u)du " o (t'-u~)~

1 (D. ? eJ'] Zl.o = ( l+e~' ) [2~-~ ( D 1 - K * D " ) - \ O A /

(O,+K~O.) (OA]~e,,] XI,, = Dn(l_.l_eJ,) [~--I-\DBJ

o,g zl,2 = oA(1 +eJ') + t, o U

(73)

(74)

(75)

(76)


1 IK * (O+O.K~)e~2] X z o - ( l + e j2) _ 2 - ~ c D - ~ _1 (77)

)~2,1 -- DB(1 +ej~ ) + \ D c j

--D~K~d--D1 [1 /DB ]~ ~ ] (79) •22 D~(I+ ej2) k + \DcJ

(e -~'1jl + e p'j') (80) 2t )

k s z A . ~ . . . . 22 = v . . . . le- J2+eo2J~) (81) DT, 2 " -- ,

F l ( t ) = c~(l+e ~') 1CA(x=°} + \ D A ] flleJ'e~(x=°)

F2(t) - c]( l+eJ~ ) 2~m~=o) + \DB] fl2eJ~c(x=o) (83)

f~ = f ~ f]' {84) f2 = f~2 f~2 (85)

DT, ~ = D 1 (D./DA) ~' (86)

DT, 2 = D l (Dc/DB) ~ (87)

F Ji = ~ (Ed.c.--ERA) (88)

r ~.~_.~) (DA ~ ~ (89) E½"i=E° F T ln ~,b-BB J

R T l n ( ~ ) ( D B ~ (90) Er,2 = E o _ ~ - \ D c /

]~i = 1--~i (91) The integral equations defining ~i, 1 (t) (eqns. (69), (70)) are solved by the same method used previouslyk The following solution for the fundamental harmonic a.c. polarographic response associated with mechanism I is obtained:

F 2 Ac* O~ AE [~o Aa "~ Ac 1 I(eot) -- R T A [(Aa+Ac)Z+(Ab+Ad)2] ~ x sin t+c° t -1 A ~ - ~ d J

where A is the determinant (92)

(1 +y~) 71 6~ 61

-~,1 0+~,~-) -6? 6~ A = 7~- 72 (1 +5~-) 52 (93)

-~; ~ -6; 0+61)

154 i. RUZI(~, D. E. SMITH

and Aa, Ab, Ae, and A d are determinants obtained by replacing the first, second, third, and fourth columns in A, respectively, by the column

Other definitions are

21 7; = (2o) ~ 0(1,o+XI,IG+) (94)

21 6~ - (2xo)~ (ZI,o+Zl,2G±) (95)

22 Y~ - (2o) ~ (Zz,o+X2,1G±) (96)

22 ~ -- (2(D)½ (Zz,o+Z2,zG±) (97)

I( _~_ 2½- F ½ G + = 1 9 ) - 9 ] (98)

- l + g Z J

g = k~/co (99)

Here we recall that the foregoing solution applies when eqn. (58) is applicable (i.e., when Da ~ DB). If the alternative, eqn. (60), is used in place of eqn. (58)(a.c. steady-state, large homogeneous rate constants), the same derivational approach leads to a solution which is identical to the foregoing, except that eqn. (98) is replaced by the limiting expressions which are applicable when kS ~> ~o, namely,

(100) a _ = 0 (101)

When eqns. (100) and (101) are applicable, no restrictions are placed on the relevant diffusion coefficient values, as explained previously ~.

As before 1, no effort is made to expand the determinants in eqn. (92) manually and express the theoretical rate law in a conventional algebraic format. Determinant manipulation is handled by the FORTRAN program developed to calculate the a.c. response from eqn. (92) and its subsidiaries.

(IV) Treatment of d.c. part of the boundary value problem The d.c. surface concentration components, ~a(x= 0), ~Btx= 0), and ~c(x=o), which

are required to complete the above solution, are obtained by a digital simulation procedure 4°'46. The FORTRAN simulation routine is applicable to the expanding plane and expanding sphere electrode models. Establishment of d.c. boundary conditions is made by the method specially elaborated for a two-step process ~6.

In the case of extremely fast homogeneous redox reactions, the usual simulation


procedure is not effective and a special treatment is invoked using suggestions recently made by Ru~i6 and Feldberg 47. Unfortunately, these suggestions can be used in a straightforward manner with mechanism I only when the disproportionation reaction predominates (k 2 ~ kl). In this case, ifk 2 is large enough, eqn. (2) reduces to 48

d 2 c a DB d x 2 -- k2 c2 (102)

and

CB F I"5DB (dCB~ 21½ = L k2 \d~-x] J (103)

This expression can be used in the simulation procedure 4°,47 in the following form

B(0) = FF(2). TERM (104)

where

TERM = [1.5/k 2 DB FF(2)] ~ (105)

where FF (2) represents the flux of intermediate B and B (0) is the surface concentration. The usual procedure for simulation of the d.c. process is based on the following

set of equations 4°'47 :

FF(1) = RHF(1) 'A(0) -RHB(1) .B(0) (106)

FF(3) = RHB (2)" C (0)- RHF (2)- B (0) (107)

FF(1) = 2DA(A(1)-A(0)) (108)

FF(2) = 2Da(B(1)-B(0)) (109)

FF(3) = 2Dc(C(0)-C(1)) (110)

from which one obtains

FF (1).(1 + RHF (1)/2D A + RHB (1)/2DB) + FF (3)" RHB (1)/2 DR

= RHF(1) .A(1) -RHB(1) 'B(1) (111) and

FF (1)" RHF(2)/2DB + FF(3)" (1 + RHF(2)/2D B + R H B (2)/2Dc)

= RHB(2). C (1 ) - RHF(2)- B(1) (112)

The quantities RHF(i) and RHB(i) represent the quantities t. o-JaJt/r~½ and "~s, i ~ /~'A ks, i ePi~t/DA, respectively. The final results for FF(1) and FF(3) have been reported by Feldberg 46 as:

RHF(1).A(1)-RHB(1).B(1) RHB(1) [RHB(2).C(1)-RHF(2)-B(1).] FF(1) = 2Oa L 1 +RHB(2)/EDc +RHF(2)/2DB_J (113)

RHB(1) [ 1 +RHB(2)/2D C ] I+RHF(1)/2DA + 2-TD~B 'LI+RHB(2)/2Dc+RHF(2)/2DB

and d

RHB(2)-C(1)-RHF(2).B(1) RHE(2) r RHF(1)-A(1)-RHB(1)'B(I) _] FF(2) = 2DB [_1 + RHF(I)/2DA +RUB (1)/2DB_] (114)

• RHF(2) [ I+RHF(1)/ED A ] 1 +ana(E)/2O c + ~ - a " [1 +RHF(1)/2DA+RHB(1)/2DBJ

156 i. RUZIC, D. E. SMITH

The treatment of the homogeneous reaction is given in a special subroutine by which the correction of concentration profiles can be performed. In the case of extremely fast disproportionation of the intermediate B, the system of equations represented by eqns. (106)-(110) cannot be used 47, and the following procedure is then applied. First, the 01 and 02 functions are introduced in the expressions for FF(1) and FF(3) as follows:

D101 (0)-DBB(0) _ RHB(1)'B(0) FF(1) = RHF(1). 2D A

= RHF(1). D, 01 (0) __ 21B(0) (115)

where

2DA - FF (3) = RHF (2)- B (0)- RHB (2)" D2 02 (0)- DB B (0)

2Dc

= 22B(0 ) = RHB(2)'D202(0) 2D c (116)

21 = RHF(1)'DB + RHB(1) (117) 2DA

RHB (2)" DB 22 = RHF(2) +

2Dc These expressions are combined with the following:

B(0) = FF(2)" TERM FFAR = FF(1)- FF(3) FF(2) = -FF(1 ) -FF(3 ) FFAR = 2D1 (0~ (1)- 0~ (0)) FEAR = 2D~ (02 (0)- 0~ (1))

to obtain

FFAR = RHF(1)2DA D1 i//1 (1) _ RHB(2)" D 2 0 2 ( 1 ) + 2 D c (22- 21)" FF (2)" TERM

(118)

(119) (120) (121) (122) (123)

(124)

and

- FF(2) =

RHF(1) RHB(2) l + - - + - -

4D A 4Dc

RHF(1). D~ ~,(1) + RHB(2) 2DA 2Dc

(RHB(2) RHF(1) D202(2 ) + \ 4Dc ~ / ' F F A R

1--(/]'1 + 2 2 ) " T E R M

Solving eqns. (123) and (124) for FFAR and FF(2) gives:

- FF(2) =

RHF(1) ( 24 ) 2D A "D1¢1(1)" 1 + 1 - - ~ 3 +

RHB(2) (1 2D c Dz~bz(1)" - _ _

(125)

1+2 3

L 1

- L "~'+22 i ¥ ~ j TERM (126)

COUPLED HOMOGENEOUS REDO)( REACTIONS

RHF(1) Dt~bl(1).( l_222TERM ) _ _ _ FFAR = 2DA

RHB(2)

2Dc

157

• Dz~k2(1)(1 - 221 TERM)

( 1 + 2 3 ) { 1 - [21+2 z 24 (1~-321) 1 • TERM } (127)'

where 23 RHB(2) RHF(1) = - - + - - (128)

2Dc 2DA

24 - RHB(2) RHF(1) (129) 2Dc 2DA

Equation (126) is nonlinear with respect to FF(2). It can be solved by an iterative procedure whereby one first calculates FF(2) by neglecting TERM, then calculates TERM from this initial FF(2) estimate, using eqn. (105), then recalculates FF(2), etc. The FF(2) value obtained by this procedure then allows one to calculate FFAR from eqn. (127)• Once FFAR and FF(2) are obtained, the FF(1) and FF(3) values needed for the simulation are obtained from the relationships:

FF(1) = I(FFAR - FF(2)) (130)

FF(3) = ½(FFAR+ FF(2)) (131)

At potentials where E1 ° ~ E ~ E½, 2 (relevant to the case where the intermediate, B, is very unstable), eqns. (126) and (127) reduce to the simpler forms:

FF(2) = -RHF(1) '~bI(1) '2D1/[4DA+RHF(1)(1-ZDBTERM)] (132) and

FFAR = - FF(2) (133)

Therefore, for these conditions FF(1)~ FF(3)= FF(1)+ FF(3), so FF(3) must equal zero. Consequently,

FFAR = FF(1) = - FF(2). (134)

The latter is the relationship for the one-step process, which one expects to apply under the conditions in question. These simplified relationships have been used successfully in treatment of the first wave with follow-up disproportionation 2.

Unfortunately, for the case where an extremely rapid reproportionation step prevails (kl >~k2, klc~,t>lO 3, approx•), the problem of simulation has not been solved to date. Difficulties with the "Heterogeneous Equivalent "47 scheme appear when more than one electroactive species influence the kinetic term• A special simulation of the reaction layer is probably the only method that can be potentially useful for this purpose, but it requires development 49.

DISCUSSION OF PREDICTIONS FOR MECHANISM I

A FORTRAN program was developed to calculate the d.c. and d.C. polarographic responses predicted by the theoretical rate laws discussed above. It invokes the expanding sphere electrode model (expanding plane predictions obtained in the limit of small sphericity) in treating the d.c. process, it accommodates inequalities in diffusion coefficients of the reactants, and is applicable to all variations of mechanism

158 I. RUZIC, D. E. SMITH

I, except the extremely rapid reproportionation process, and a very slow homogeneous redox reaction with large differences in diffusion coefficients for species A, B, and C (the latter should be rare). The predictions discussed subsequently were obtained with this FORTRAN program. It is available from the authors on request.

(I) Case of ihe stable intermediate (E ° >> E °, E~,I >> E~,2) In this important special case, the polarographic response reveals two resolved

waves, and the "nuance" reaction corresponds to irreversible reproportionation

2 k l = 0 k I = 2 D x 10 4 M -z S - I

• 3 k I = 1.2 x I 0 5 M "~ S - I

g

~ 1.5 o

c5

I.O

5.OO

2.00

L)

"- I.OO

2.0

, , ! ' - 0 2 5 - 0 5 0 - 0 7 5 - I . 0 0

E D c / V O I t s

I k I = 0 2 k= = 1.0 x 104 M-IS -I 3 k = 2 0 x 104 M -z S -I

I - 0 . 2 5 - 0 5 0 - 0 . 7 5 - LO0

E o c / V O l t S

Fig. 1. Examples of predicted d.c. and a.c. polarograms at second wave with mechanism I where k a >~k 2. Parameter values : T=298°K,c*=l.Ox 10 -3 M, DA=DB=Dc=4.0x 10-6 cm2 s-~ , t = 1.00s, m = 2 5 0 s -1, El° = 1.00 V, E ~ = - 0 . 2 5 V, ks,1 = 2 . 0 cm s -1, ks.2 = 2 . 0 x 10 -3 cm s - I , e l =c~2 =0-50, kl values shown in Figure, expanding plane model, direct current magnitude given in apparent n-value units, i.e., d.c. o rd ina te = (3nt/7)~id.c./FAc*D~A and a.c. current is given in units of [10 2 RTI(o~tl/F2A(2O~OA)½C~].

COUPLED HOMOGENEOUS REDOX REACTIONS 159 kl

(A+ C--~ 2B). The homogeneous redox reaction can occur only at d.c. potentials corresponding to the second (more negative) reduction wave (Ed.c. ~ E~,2), so only these conditions are considered in the present discussion.

Results of calculations for this case are qualitatively completely analogous to those obtained for corresponding conditions with two independentelectrode reactions ~. Thus, for example, the predicted polarographic response shows no influence of homogeneous reproportionation when the second reduction process (B + e ~ C) is nernstian (reversible) in the d.c. sense. However, if this process is quasi-reversible or irreversible in the d.c. sense, both the d.c. and a.c. responses at the second wave are strongly influenced by the reproportionation process. Some typical predicted effects on the polarograms are shown in Fig. 1 for the case of nearly irreversible charge transfer. Figure 2 provides plots of the ratio of kinetic and nonkinetic a.c. currents

2.00

n,- 1.50

1.00

k I =120 x 105 M "1S -I

k l = 4 0 x lO 4 M ' I s -I

k I = 2 0 x l O 4 M ' IS "~

k I =1.0 x 104 M'=S -I

k i = 6 0 x lO ~ M ' i s "t

k~ = 2 0 xlO 3 M-IS -I k~ = LO x 103 M "= S -~ kl =0

-o .25 -o . o - 0 . 7 5 -LOO

E D c / V O I t S

Fig. 2. Examples of predicted a.c. polarographic kinetic-nonkinetic current ratio (R) at second wave with mechanism I and k 1 >>k2. Parameter values: same as Fig. 1.

[R = I (oOt)k/I (COOk = 0] versus d.c. potential, which was proposed as a useful observable for estimation of the homogeneous rate constants in the context of independent electrode reactions 1. The R-value profiles shown in Fig. 2 bear the following characteristics which are shared by the independent electrode reaction case 1 : (a) the profiles are sigmoidal; (b) with equal diffusion coefficients for all participants in the homogeneous reaction, the plateau R-value ranges between unity (small kl) and two (large kx); (c) the R-value profile is frequency-independent; (d) the dependence of the profile on sphericity is significant, but small and can be handled in the manner suggested for independent electrode reactionsL These qualitative similarities attending the homogeneous reaction's effects on the independent electrode reaction case and the presently- considered e.e. mechanism are striking, but are only qualitative. Quantitatively, for a given kx-value, the effects of the homogeneous step are somewhat smaller in the present case. This is understandable because, with the e.e. mechanism, only one reactant (species A) is present in the solution bulk (for conditions considered here), and the second component required for the homogeneous reaction (species C) must be generated electrolytically via the intermediate (species B). This leads to somewhat

160 I. RUT.IC, D. E. SMITH

lower concentrations of species C in the reaction layer at the relevant d.c. potentials than if it were generated from a species initially present in the solution bulk, as in the case of independent electrode reactions 1. This conclusion is not altered by using known alternative definitions for the rate constant, which differ by a factor of two from that employed here (eqns. (1)-(3)). For example, the disproportionation rate constant has been defined such that it is a factor of two smaller than the one given by our definition z°. However, even with this redefinition, the effects of the homogeneous reaction are still larger for the case of independent electrode reactions. This is illustrated in Fig. 3 which plots k t v e r s u s log(Rmax-R), where R is the kinetic-nonkinetic a.c, current ratio (Fig. 2) and Rma x is its maximum value. The latter type of plot has been suggested as a useful working curve from which the homogeneous rate constant can be obtained for the case of independent electrode reactions t. Figure 3 shows dearly that the same observation is applicable in the present case.

IO0

~ 8o

0

x 6.0

,20 l/ 2040/ j

0 lI.O 21o 31o ""

-log (RMAx-R)

Fig. 3. Examples of predicted log (Rma x - R) vs. k j profiles at second wave with mechanism I and independent electrode reaction case, where k 1 ~ k z. (1) Profile at plateau of R vs, Ed.~. plot for mechanism I with kl defined as in present paper. (2) Same as (1), except k 1 defined as one-half value used in this paper 2°. (3) Profile at plateau of R vs. Ed.c. plot for two independent electrode reactions case a. Other parameters as in Fig. 1.

The influence of the heterogeneous rate constant on the R - E d . e . profiles is similar to that found with independent electrode reactions. Increasing ks,2 from the irreversible limit causes changes in position and form of the profile, while the plateau R-value remains constant. When ks becomes sufficiently large (d.c, reversibility), the rising portion of the R - E d . c . profile is shifted to negative potentials outside the range of measurable currents, so that R = 1 where the response is measurable; i.e., no homogeneous reaction effect is observable. Figure 4 illustrates these remarks.

Practical use of working curves of the types shown in Figs. 2-4 to assay the reproportionation rate constant kt demands, of course, experimental characterization of the kinetic-nonkinetic current ratio. In the present context this should be a reasonably straightforward task. I (eOt)k = o is obtained by observing the second polarographic

COUPLED HOMOGENEOUS REDOX REACTIONS

- - ks, 2 = 4 0 xlO'Scm, s -I . . . . . . ks, 2 = 20 X 10-2era. $ -I . . . . . . . . . . . . ks, 2 = 2.0 x IO "~ cm. s -i

t 40 . . " ' " " " ..... 5 . ~

..:" / / / / " /

.:" / / :' /

~ 1.2o .: ii 1

::" / / :" /

=~ ~ k s , = > > 2~xlO Zcm s I ~ oo .......... 7:._? Z..~. ~

161

0 8 0 ~ _0~50 L -0 .30 - 0 7 0 - 0 .90

EDc/VOltS Fig. 4. Examples of predicted effect of heterogeneous charge transfer rate on a.c. polarographic R-values at second wave with mechanism I and k I >> k 2. Parameter values: as in Fig. 1, except k I = 1.0 x 104 M- 1 s- 1, and k~.2-values shown in Figure.

wave with only the stable intermediate, species B, present in the solution bulk. Under these conditions species A's presence is excluded, and reproportionation cannot occur. I (O~t)k is then measured by observing the second polarographic wave with only species A present in the solution bulk, as considered in the foregoing theory. Thus,

I ( ( . 0 t ) k I ((7L)t') only species A in solution b u l k R - - - - ( 1 3 5 )

I ( ° t ) k : o / ( ( ~ 0 t ) o n l y s p e c i e s B in solution bulk

(II) Case of the thermodynamically unstable intermediate (E° < E °, E½,1 >> E½,s) In this special case, the polarographic response may yield a single wave at

potentials of the first reduction step (first wave), or two resolved waves, depending on the rate of the homogeneous redox reaction (see below). Here the homogeneous redox reaction corresponds to irreversible disproportionation (2B-*A+C). Within the restrictions of mechanism I, this situation is possible only if the second electrode reaction (B + e--*C) is sufficiently irreversible that it ensues at d.c. potentials negative of the first step, despite its more positive E°-value 2°. For these conditions, careful theoretical studies have been made of the effect of disproportionation on the first waves in d.c. 3- 5,19,22,23,29 and a.c. 2 polarography, so this aspect of the predictions of the above rate law will not be discussed here. The present remarks will be confined to the effects of the homogeneous reaction on the second wave.

In d.c. and a.c. polarography, increasing disproportionation rate constants lead to a diminution of the second wave, and eventually its complete disappearance for sufficiently large k 2. Growth of the first wave at the expense of the second accom- panies this effect in the d.c. polarography. However, in a.c. polarography it has been noted that increasing k s leads also to suppression of the first wave which approaches a non-zero limiting amplitude s as k 2 ~ 00. Figure 5 illustrates some of these effects. A qualitative behavior difference between the e.e. mechanism in this situation and corresponding conditions with independent electrode reactions (catalytic case) 1 is seen clearly from the plot of the a.c. ratio, R, versus Ed.,. Only one non-unity plateau

162 I. RU~I(~, D. E. S M I T H

2.0

1.5

1.0

1.50

1.00

'~ 0.50

I . k2 = 0 M "l S "1

2. k z = LO x I0 :~ M "I S "1

3. ke = 2.0 x lO 3 M- IS -I

4. k2 = 6 . 0 x l O 3 M ' I S "1

5. kz = 2 - 0 x 1 0 4 M "= S -=

6. kz = 2 . 0 x lO r' M' )S -~

7. k= =2 .0 x l O 6 M ' I s "1

I - 0 . 2 5

E3 5 / ~ s I i //~\ 2 k. : 2 o . , o . - 5-

I i f~ \ \ .. k, = 2 o . , o . . - ~ ~-,

III A ,. ,,[,.o.,o,,-,,-,

I . 0 . 175 - 0 . 5 0 -1 .00

E OC /volts

I. k 2 = 2 . 0 x l O s M "l S "1

- 0 . 2 5 - 0 . 5 0 - 0 . 7 5 - I . 0 0

E o c / v o l t s

Fig. 5. Examples of predicted d.c. and a.c. polarograms at second wave with mechanism I where k 1 ~ k 2. Parameter values: same as Fig. 1, except k2-values shown in Figure.

appears at positive potentials, and at negative potentials the a.c. response of the second wave is not influenced by the homogeneous reaction in the case of the e.e. mechanism (Fig. 6). This contrasts with the two-plateau appearance of the R-Ed~c. profiles and significant homogeneous kinetic influence over the entire wave which one finds in the independent electrode reaction case 1. Here also the magnitudes of the homogeneous reaction's effects are smaller for the e.e. mechanism, relative to the independent electrode reaction situation. Also worth noting is the fact that the effect ofdispropor- tionation on the second a.c. wave is of opposite direction and smaller magnitude than that produced by the reproportionation reaction. This becomes evident upon comparing Figs. 2 and 6. A more quantitative illustration of this effect in terms of the log (Rma x -R) vs. k profiles is given in Fig. 7.


1,50

I.O0

2 5

4

5

6 O50

I k,~ =0 M -= S -=

2, k z = l .OxlO a M ' IS *1

3 k= = 2 0 x l O 3 M -= S "= x

4 k z = 6.0 x 103 M-I S -I

5, kz = l -Ox lO 4 M -= S "=

6 k= = 2,0 x 104 M "= S "=

-0 )25 -0 )50 -0175 I

I00

5 0

/ - 1.00 ' l lO

E ,~ / ,,o,t~ -log [mM,x-F~]

Fig. 6. Examples, of predicted a.c: polarographic R-values at second wave with mechanism I and k 1 4 k 2.

Parameter values: same as Fig. 5.

Fig. 7. Comparison of predicted log (Rr~x-R) vs. k profiles at second wave with mechanism I for cases with predominant disproportionation and predominant reproportionation. (1) Predominant disproportionation, ordinate=k 2 x 10 -4 M- 1 s- 1. (2) Predominant reproportionation, ordinate=k1 x 10 -~ M- 1 s- 1 Other parameters as in Fig. 1. Rma x for disproportionation is unity; i.e., the value in absence of chemical kinetic effects.

For this situation involving predominant disproportionation, there are problems connected with experimental evaluation of the second wave's R-value for purposes of rate constant evaluation. If the disproportionation reaction is sufficiently facile to influence the polarographic response, then there is no way of preparing a solution of the same composition where the chemical reaction is nonexistent. Thus, I(oOt)k=O is not accessible to direct experimental measurement as when reproportionation predominates. Consequently, I (COOk = 0 can only be calculated from know- 'ledge of diffusion coefficients, a-values, etc. and the theory for the irreversible a.c. polarographic wave. Since the parameters necessary for the calculation may not always be readily available, one may be reduced to estimation and the associated greater uncertainty regarding measurement fidelity. Fortunately, one need not rely on the irreversible second wave for characterization of the homogeneous reaction when disproport ionation predominates. Current amplitude and phase angle measure- ments involving the first wave have been shown to be well-suited to this purpose 2.

( I I I ) Case involving reversible homogeneous redox reaction-resolved waves (e° e °, %,, e ,2)

1-iere we consider the case where the E°-values for the two heterogeneous steps are comparable, so that bo th disproportionation and reproport ionation occur to a significant extent, yet resolved waves still are obtained by virtue of irreversibility of the second step, B + e - ,C . This example encompasses the extremes between the above- considered examples of pure reproport ionat ion (kl ~> k2) and pure disproportionation (kt < k2), within the context of resolved polarographic waves. "Reversible dispropor-

164 I. RUZ IC , D. E. S M I T H

tionation" (or "reversible reproportionation") has received negligible attention in the literature to date, for either a.c. or d.c. polarography.

Calculations for this case show that, as with independent electrode reactions 1, introducing homogeneous reversibility into the rate law normally suppresses the effects of the homogeneous reaction, relative to what is observed with pure disproportionation or reproportionation. At the same time, these interactions between the forward and reverse homogeneous process are not as strong as in the case of independent electrode reactions ~. Nevertheless, the effects are significant and the possibility of homogeneous reaction reversibility must be addressed in any data analysis undertaking. Figure 8 provides a d.c. polarographic example, where the effects of varying k I with fixed k2 are shown for two k2-values. It is evident in this example that the enhancement by the disproportionation step of the first (most positive) d.c. wave at the expense of the second is retarded by the existence of significant reproportionation.

2 . 0 0

1.50

1.013

/ z 7

i =o M'~S "1 /i/7 kn =2D~r / / / ,~ "

k:~o M-,s-,~ __ . - ' . j r - - k~ =60,,0" M"S-' k,=2,0xlO 4 M " 1 S ' I ~ % / . . . . k z = I 0 • 104 M -O S "n

. . . . - _ -_ - - -S .~

_ 0.125 I L I - 0 5 0 - 0 7 5 - I . 0 0

Eoc//volts Fig. 8. Examples of predicted d.c. po la rograms with mechanism I showing effect of homogeneous redox reaction reversibility on second wave, and first wave plateau. Parameter values : same as Fig. 1, except kl and k z values shown in Figure, E ° =0.00 V, E~, 2 = - 0 . 5 0 V (when k 1 = k 2 = 0), ks. 2 ~ 1.0 x 10-3 cm s - l.

Effects of homogeneous reaction reversibility on the two a.c. polarographic waves are exemplified in Fig. 9A, together with the corresponding d.c. polarograms (Fig. 9B) for the conditions in question. As illustrated in this example, homogeneous redox reaction reversibility influences appreciably the characteristics of both a.c. polarographic waves, relative to the pure disproportionation case (kl =0). As shown in Fig. 8, monotonic increase in the reproportionation rate constant, ka, first leads to a suppression of the height of the first wave, but eventually a reversal in trend occurs whereby the wave height increases with increasing kl (i.e., a plot of height of first a.c. wave versus kl, k 2 held constant, will exhibit a minimum). No such reversal is seen in the second wave characteristics. Figure 10 shows the behavior of the kinetic- nonkinetic ratio, R, versus Ed.¢. at the second a.c. wave. The R-value diagrams show two non-unity plateaus at opposite sides of the second wave. The first (more positive) plateau is predominantly influenced by the disproportionation rate constant and, consequently, is characterized by a less-than-unity value as with pure dispropor-


1.50

E

._o, 1.00

0.50 / i

+0 20

I

: i

0

- - k t = 0 M ' t S "1

. . . . k I = LOx 104 M-I S -I

. . . . . . . k I = 6 0 x l O a M -I S-I

. . . . . k= = 2 . 0 x l O s M " 1 S -I

I

- 0 . 2 0

E ~ ) c / v o l t s

- 0 . 4 0 - 0 . 6 0 -O.BO

I I I ~. ~, . , .o . ,o . . - , , - , (.~ 1 .00 " = " " -

~5

0 5 0

• o.~o o -o'~o -o.'.o -o.'60

E D C / v o l t s

Fig. 9. Examples of predicted d.c. and a.c. polarograms encompassing first and second waves showing effects of homogeneous reaction reversibility. Parameter values: same as Fig. 8, except k 2 = 4.0 × 104 M - z s 1, and k] values shown in Figure.

tionation (Fig. 6). The second (negative) plateau is most sensitive to the reproportionation process and is attended by a greater-than-unity value, as with pure reproportionation (Fig. 2). Figure 11 gives a diagram of the magnitude of the two R-value plateaus of the second wave for different combinations of disproportionation and reproportionation rate constants. As in the corresponding situation with independent electrode reactions 1, one finds that the constant kz-value "contours" are nearly linear, while those for c o n s t a n t k I are somewhat nonlinear. The separation of the various contours in Fig. 11 is sufficient that this type of diagram could serve as the basis for evaluating 1 k~ and k 2. However, for situations of interest (facile disproportionation

166 L RUZI(~, D. E. S M I T H

160

140

120

I~ 1,00

0 8 0

0 ~ 0

0 4 0

k = = 6 D x l O a M °1 S -I

k= =l.OxlO 4 M "1S-I

/ k I = k 2 = 0

I i i I i i - 0 . 2 0 - 0 . 4 0 - 0 . 6 0 - 0 8 0 - I . 0 0

E oc / vo l ts

Fig. 10. Examples of predicted effects of homogeneous redox reaction reversibility on a.c. polarographic R-values at second wave with mechanism I. Parameter values: same as Fig. 8, except k2 = 4.0 × 104 M - 1 s - 1, and kl values shown in Figure.

2 0

1,8

1.6

o 14 O E "o r2

g~ 1.0

0 8

i ~ 6 o ~

Repropor t mnotlon Rote Constant x I0 "~ (k 0 x I0 "3]

M-I S-r

,o 6 4 ~ , o

Disproportionahon Rate Constant x 10-3(kz x IO -3) M -I S ~i i i i I

0 2 0 0 . 4 0 0 6 0 0 .80 I O0

First Ploteau Magnitude

Fig. 11. Examples of working curves of first v s . second plateau magnitudes in a.c. polarographic R-Ed.e. profiles at second wave with mechanism I. Parameter values: same as Fig. 8. ( - - ) Contours for constant k2 ; ( - - ) contours for constant kl.

"step), there is no direct empirical means of evaluating I(~ot)k=0. Thus, R-values are accessible only by combining observed I(oOt)k magnitudes with theoretical estimates of I(~ot)k=o. Here again, knowledge of the parameters necessary for calculation of I(~Ot)k= 0 may not be available, so use of the second wave's R-value profiles to calculate kl and kz often will not be applicable, in contrast to the more favorable prognosis obtained with independent electrode reactions 1. Fortunately, as with irreversible disproportionation, recourse to the characteristics of the first wave appears to provide

• a viable solution to the kinetic parameter measurement problem with reversible homogeneous reactions. Before considering options available, it is important to note

COUPLED HOMOGENEOUS REDO)( REACTIONS 167

that, unlike the second wave, the heterogeneous charge transfer characteristics of the first a.c. wave are not confined to the irreversible realm for the special conditions in question. Thus, the first wave often will be characterized by facile charge transfer kinetics and the full range of a.c. polarographic observables will be available for system diagnosis 2, including various frequency response profiles, and the phase angle, which are not highly informative in the irreversible case 5°. Figure 12 provides an example of the effects of homogeneous reaction reversibility on the frequency response profiles of the current amplitude and phase angle cotangent of the first a.c. wave. This example epitomizes calculational results which show that, at appropriate frequencies, cot ~ and I(~ot) are significantly sensitive to the status of the homogeneous redox reaction with regard to its reversibility. Thus, these observables should be useful as the basis

15

8 . ~ I 0

<~

~_ 5 . 0

1 .3

(3-

"6

'6 I . I

k I = 6 . 0 x l O 4 M - I S - '

k I = l . O x l O 4 M ' I s "1

k I = 0 M - I S - I

2 o , o 6 o

oJ /

I

I O 0

I k I = 0

2. k I = 1.0 x 10 4 M -I S "1

3. k I = 6 . 0 x l O 4 M -I S - t

1.0 i I i I i I ~ I i I i I , I ~ 0 4 O 6 O 8 0 I 0 0 1 2 0

oJ l l2 / S - i f=

Fig. 12. Examples of predicted effects of homogeneous redox reaction reversibility on a.c. polarographic frequency response profiles at first wave with mechanism I. Parameter values: same as Fig. 9.

168 I. RUZIC~, D. E. SMITH

~0 = = o

=

CD

8.0

~ e b

.,~ so g,

~ 6.0 ~ b o - -

4,0

0 I o

t

60 I

i10

Oispropor tionation Rate Constant x fO -3 M "t S -I

\

r.i2 LI4 1!6

\ , D'RPart °P ° rc~'°nl~ t/~m

tO ~ 20

oo . . . . . . . . . . . - ~ -_ ' -~ .~

~!o 1~2 1!4 1!6 Cot ~b at Current Peak

Fig. 13. Examples of working curves of first and second a.c. peak height ratios vs. cot q5 at first wave's peak with mechanism I. Parameter values : same as Fig. 8, except ~o = 250 (A) and 125 (B). ( ) Contours for constant k2; ( - - - - ) contours for constant kl.

for empirically obtaining kl and k2. Sample calculations have confirmed this expec- tation. Figure 13 shows one example of a diagram in which constant kl and k2 value contours are plotted in a coordinate system defined by two convenient experimental observables: (a) the cot q~ value at the first wave's current amplitude peak, and (b) the ratio of the first and second a.c. peak currents. Diagrams such as shown in Fig. 13 appear to represent a more viable basis for estimating k~ and kz than the R value profile of the second wave. Details of the contours are somewhat sensitive to values of~l, ~2, ks, t and the diffusion coefficients, but the latter are more readily acquired than the ks,2 and E~ values needed to invoke the R value diagram of Fig. 9. The diagram of Fig. 13 will vary with frequency as shown, so the possibility exists for utilizing a number of such diagrams, each corresponding to a different frequency, for purposes of mechanistic confirmation, cross-checking rate constants obtained from data at other frequencies, and/or as part of an iterative data analysis procedure. In addition to the observables utilized in Fig. 13, other plausible combinations exist which include pairs of observables selected from the following: (a) ratios of first and second d.c. wave limiting currents ; (b) ratios of d.c. and a.c. currents at first and second waves; (c) various second harmonic a.c. polarographic wave characteristics (if the theory is made available); (d) either of the parameters used in Fig. 13. The many possible options suggest good prospects for preliminary estimation followed by extensive refinement and rechecking of ka and kz values for the situation in question, provided that the disproportionation and reproportionation rate processes are kinetically significant.

C O U P L E D H O M O G E N E O U S REDOX REACTIONS 169

Of course, if kx and k 2 are acquired, then characterization of the standard potential (E2 °) and the heterogeneous rate c o n s t a n t (ks,z) for the irreversible second wave become possible, provided E1 ° is known (ef eqn (13) and discussion in ref. 1).

(IV) Cases where E~,I ~Ek,2 We have considered above the three main cases which can arise in the diagnos-

tically simple and convenient situation where the polarographic waves associated with the two heterogeneous charge transfer steps are totally resolved. Situations where the polarographic waves overlap also are encompassed by the rate law derived here. Within this framework, the three special situations considered above, E ° >>E °, EO..~ ~-o and E°~ E ° also arise, given appropriate heterogeneous charge transfer 1 ~ L ' 2 , kinetic characteristics. As one would expect, the nature of the results under these circumstances is much more complex and varied than when the waves are well- resolved. The possible cases to be considered are too numerous to attempt anything approaching a complete discussion. Consequently, we will limit our remarks on the case of unresolved waves to the following.

(a) In the case where Ex ° ~ E °, unresolved waves occur only when a sufficiently irreversible first heterogeneous step occurs in combination with a relatively facile second heterogeneous process. Under these conditions, the direction of the homogeneous redox reaction corresponds to reproportionation, and calculations show that it can influence the d.c. and a.c. polarographic responses in a rather appreciable and interesting manner. Some typical results are shown in Fig. 14. The effect of the homogeneous redox reaction is to sharpen notably the rising portion of the d.c. polarographic wave, while the a.c. polarogram is enhanced and narrowed. This prediction is readily understood by recognizing that homogeneous reproportionation operates in parallel with the first heterogeneous step, providing an alternative mode for conversion of the depolarizer (species A) to the intermediate (species B), the latter being the active species for the second heterogeneous step. Consequently, at d.c. potentials where the concentration of species C in the diffusion layer is appreciable, the homogeneous step "substitutes for" the relatively sluggish heterogeneous reduction of A to B, the current due to the second step is increased at the expense of the first, and the polarographic wave properties become more characteristic of the facile second step, and less influenced by the rather sluggish first step. Figure 15 illustrates these points by showing the calculated direct currents associated with the first and second electrode reactions (theoretically calculable, but not experimentally observable). Note that in the limiting current region, for example, the homogeneous reaction's effect is to enhance the current due to the second step by a factor of approximately 1.75, at the expense of the first step.

Shapes of the R value vs. Ed.c. profiles assume unusual non-sigmoidal characteristics under these conditions, as illustrated in Fig. 16. Figure 16 also shows that the R value profiles are frequency-dependent, unlike the cases involving resolved waves (Figs. 2, 4, 10).

(b) In the .opposite extreme where E°~ E °, and disproportionation is the prevailing tendency for the homogeneous step, calculations show relatively small effects of the homogeneous process under most circumstances involving unresolved waves. As the second heterogeneous step is made increasingly irreversible, shifting E~, 2 toward more negative potentials, the small effects of the homogeneous nuance

170 I. RUZ1C, D. E. SMITH

16

12

d .~. 8.0

4.0

2.0

c

~-~ 1.0

C~

k z = 0 " M "1 S "1

k I = 4 . O x l O 4 M - I s -I

I

I ,j i I

*0.20 0 - 0 . 2 0

E o c / v o l t s

i 0 .20

8.0

f t ! I

/

I 'l " k l = O I I t| . . . . kt = 4 O x l O 4 6 .0 f I

I

~ 2.0

o -o.2o Q2o ; ~ -o:~o Eoc/VOIts Eoc/volts

- - k I = 0 M °t S " t

. . . . k I =4.0x104 M -I S-I

Fig. 14. Examples of effects of irreversible reproportionation on d.c. and a.c. polarographic responses with mechanism I and unresolved waves. Parameter values: same as Fig. 1, except E~=0.00 V, E~. 1 =0.00 V, ks, 1 ~ 10 -3 cm s -1, ks,2=5.0 cm s -1, and k I values shown in Figure.

increase in magnitude and can become experimentally significant with proper k z values before the waves are totally resolved (resolved wave case already discussed). However, the predominant calculational result when waves associated with the two heterogeneous processes are unresolved is the prediction of small or totally negligible homogeneous reaction contributions. This result arises because, under the conditions in question, electrolytic reduction of the intermediate (B) competes effectively with the homogeneous disproportionation step, minimizing the latter's significance.

(c) When E ° ~ E °, the existence of unresolved waves demands either that both


50

f ~ - - - ~ . . . . . k I = 4.0 x I04 M "1 S - I

i I - - Current from A+e ~B

I . . . . . Current from B ÷ e ~ C 2.0 /

0 j .~ ~ ~ _ _ k= = 0 M - t S -I

c3 . / h ~ I0 I ~

! k~ =40x lO 4 M "~ S -t

j ,

0.20 0 - 0.20

E o c / v o l t s

Fig. 15. D.c. polarographic currents associated with first and second heterogeneous charge transfer steps in example shown in Fig. 14A.

1 6 0

/ , , \ -

1.40 : l ~ .

1 .20 ) I oJ = 250 S - I

• - . . . . . . oJ = l e O 0 s . i 1.00

. . . . . . . . . . . . . ¢o = I0,0OO s-=

0:: 080

06o ~ ". , \ 0 .40

0 2 O

0.'20 e - o'.2o

E o c / v o l t s

Fig. 16. Examples of a.c. polarographic R-value profile with irreversible reproportionation, and unresolved waves. Parameter values: same as Fig. 14, except ~-values shown in Figure.

heterogeneous steps are nernstian, or that their ks values are comparable• Small to negligible effects of the homogeneous step characterize predictions of the rate law under these conditions also. Presumably, the suppressing influence of the reversibility of the homogeneous reaction, mentioned earlier, together with the second heterogeneous reduction step's ability to compete effectively with the disproportionation pathway, represent factors which combine to minimize the importance of the homogeneous redox process under these conditions.

(11) Electrode curvature and diffusion coefficient effects The foregoing illustrations of the rate law's predictions for Mechanism I

invoked the expanding plane limit in the FORTRAN calculations to economize on computer time. Further, for convenience, the assumption of equal diffusion coefficients

172 I. RUZI(~, D. E. SMITH

was employed. The FORTRAN program used for this purpose does not confine calculations to these conditions, and some results were obtained which illustrate electrode curvature and diffusion coefficient effects. The results show that, although qualitative characteristics of the above-illustrated responses are not altered by introducing experimentally reasonable electrode curvature and diffusion coefficient differences, quantitatively significant effects are predicted in many cases.

Previous publications dealing with simple disproportionation 2, the Yamaoka mechanism 1, and the pure e.e. mechanism3 8 provide illustrations of electrode curvature effects which are similar to those predicted in the present case for k2 >> k l , k2 ~ k l , and small k2 and kl, respectively. Intermediate situations are not accompanied by drastically different curvature effects. Consequently, this topic will not be explored further.

TABLE 1

AN ILLUSTRATION OF EFFECTS OF DIFFUSION COEFFICIENT VARIATIONS ON D.C. AND A.C. POLAROGRAPHIC PROPERTIES WITH MECHANISM I"

Di f fus ion coe f f i c ien t s × l O S / c m s -1

DA DB D c R 1 b R2 ~ Aid.l/ia, m a

0.40 1.013 1.00 0.595 1.78 0.489 0.40 1.00 0.40 0.622 1.88 0.441 0.40 1.00 0.20 0.639 1.91 0.407 0.40 0.40 1.00 0.502 1.39 0.572 0.40 0.40 0.20 0.481 1.56 0.580 0.40 0.40 0.40 0.483 1.52 0.550 0.40 0.20 1.00 0.498 1.20 0.603 0.40 0.20 0.40 0.425 1.28 0.638 0.40 0.20 0.20 0.436 1.35 0.631

a Conditions same as Fig. 10, except diffusion coefficient values are given in Table, and k2 = 6.0 x 10" M - 1 s- 1. b RI =magnitude of positive (first) plateau in R value vs. Ea.c. profile (see Fig. 10). c R2 =magnitude of negative (second) plateau in R value vs. Ea.c. profile. a ia,1 =d.c. limiting current at first wave in absence of homogeneous redox reaction (kl =k2=0); AidA = magnitude of homogeneous redox reaction kinetic effect on d.c. limiting current = (id,~)k-- (ia,~)k = O.

Table 1 provides examples of the influence of diffusion coefficients on a.c. polarographic R values and d.c. polarographic plateau currents (first wave), for conditions where the homogeneous redox reaction is reversible. It is clear from this tabul- ation that significant error sometimes can be introduced by the frequently-invoked assumption that all reactants in an e.e. sequence possess identical diffusion coefficients equal in magnitude to that associated with the initially present depolarizer (species A). Results shown in Table 1 are reasonably typical.

CONCLUSIONS

Theoretical treatment of the expanding sphere boundary value problem attending d.c. and a.c. polarographic responses for the e.e. mechanism with nuances is found to be tractable for most conditions, using a combination of numerical and


analytical procedures. Only the situation involving a very rapid reproportionation step does not succumb to the theoretical strategies invoked. The need to consider possible effects of the homogeneous "nuance" reaction in kinetic-mechanistic investigations involving the e.e. mechanism with solution-soluble electroactive components, including conditions where they previously have been ignored, is made apparent from this study. The development extends the d.c. and a.c. polarographic rate laws for the well-known disproportionation mechanism to encompass reversible disproportionation, and the effect of disproportionation on the manifestations of the second charge transfer step in the context of resolved and unresolved waves. Consider- ation of the important case where the predominant direction of the homogeneous redox reaction is reproportionation reveals the previously-unrecognized fact that this homogeneous process can influence significantly d.c. and a.c. polarographic responses associated with the second heterogeneous charge transfer step, if the latter is non- nernstian in the d.c. sense. Prospects for quantitative d.c. and a.c. polarographic characterization of the rate constants of the homogeneous reaction are found to be quite promising under many experimentally-accessible situations.

ACKNOWLEDGEMENTS

This work was supported by NSF Grant GP-28748X. The authors are grateful to Mr. R. Schwall for helpful discussions related to this work.

SUMMARY

A theoretical study of the d.c. and a.c. polarographic response with the mechanism

ks, I ,~ i

A + e . ~- B

ks,2~ct2 B + e . "C

kt A + C . ' 2B

k2

is presented. The rate law derivation encompasses the expanding sphere electrode model, systems where diffusion coefficients of the reacting species are unequal, all possible combinations of the heterogeneous charge transfer rate parameters, and most values of the homogeneous rate parameters (excluding very large kl values). Calculations performed with a FORTRAN representation of the derived rate laws reveal that the homogeneous reaction can substantially influence the d.c. and a.c. polarographic behavior, not only in the well-recognized situation where the disproportionation step is significant, but also under certain conditions when the direction of the homogeneous step corresponds to reproportionation. A number of experimentally-significant situations are examined with emphasis on obtaining guidelines regarding when the homogeneous process is kinetically-significant and how its rate parameters might be characterized using d.c. and a.c. polarographic observables.

NOTATION

c i --- concentration of species i

174 I. RUZI(~, D. E. SMITH

e,

ci (x = o)

Di

t

x

F R T A O~ i

ks , i

k l , k2

Eo

Ed.c. E½,i

AE o)

i(t) ii(t)

i,(t) r (t) I (o~t) q~

m

r 0

= initial concentration of species i --- d.c. component of species i concentration --- a.c. component of species i concentration = surface concentration of species i --- diffusion coefficient of species i --- activity coefficient of species i = time --- distance from electrode surface = Faraday's constant = gas constant = absolute, temperature = electrode area = charge transfer coefficient for ith heterogeneous charge transfer step = heterogeneous charge transfer rate constant for ith charge transfer step

(at E °) = second-order rate constants for forward and reverse homogeneous redox

reactions (reaction I) = standard redox potential for ith charge transfer step (IUPAC convention) = d.c. component of applied potential = observed d.c. polarographic half-wave potentials for ith charge transfer

step = reversible d.c. polarographic half-wave potential for ith charge transfer

step = amplitude of applied alternating potential = angular frequency of applied alternating potential = total faradaic current (cathodic current positive) = faradaic current associated with ith charge transfer step = d.c. faradaic current component for ith charge transfer step = a.c. faradaic current for i th charge transfer step = faradaic fundamental harmonic current = phase angle of faradaic fundamental harmonic alternating current relative

to applied alternating potential = mercury flow rate = dropping mercury electrode radius

REFERENCES

1 I. Ru~i6, D. E. Smith and S. W. Feldberg, J. Electroanal. Chem., 52 (1974) 157. 2 J. W. Hayes, I. Ru~i6, D. E. Smith, G. L. Booman and J. R. Delmastro, J. Electroanal. Chem., 51 (1974)

245. 3 D. M. H. Kern and E. F. Orlemann, J. Amer. Chem. Soc., 71 (1949) 2102; 75 (1953) 3058. 4 J. Koutecky and J. Koryta, Collect. Czech. Chem. Commun., 19 (1954) 845. 5 J. Koryta and J. Koutecky, Collect. Czech. Chem. Commun., 20 (1955) 423. 6 0 . Fischer and O. Dracka, Collect. Czech. Chem. Commun., 24 (1959) 3046. 7 H. Imai, Bull. Chem. Soc. Jap., 30 (1959) 873. 8 W. Kemula, E. Rakowska and K. Kublick, Collect. Czech. Chem. Commun., 25 (1960) 3105. 9 Ya. P. Gokhstein and T. S. Kao, Zh. Fiz. Khim. SSSR, 35 (1961) 1699.

10 Ya. P. Gokhstein and T. S. Kao, Zh. Neoroan. Khim., 6 (1961) 1821. 11 S. Feldberg and C. Auerbach, Anal Chem., 36 (1964) 505.


12 B. Kastening, Collect. Czech. Chem. Commun., 30 (1965) 4033. 13 R. S. Nicholson, Anal. Chem., 37 (1965) 667. 14 G. L. Booman and D. T. Pence, Anal. Chem., 37 (1965) 1366. 15 G. L. Booman and D. T. Pence, Anal. Chem., 38 (1966) 1112. 16 B. McDuffie and C. N. Reilley, Anal. Chem., 38 (1966) 1881. 17 M. D. Hawley and S. W. Feldber~ J. Phys. Chem., 70 (1966) 3459. 18 M. Mastragostino, L Nadjo and J. M. Saveant, Electrochim. Acta, 13 (1968) 721. 19 M. Mastragostino and J. M. Saveant, Electrochim. Acta, 13 (1968) 751. 20 S. Feldberg, J. Phys. Chem., 73 (1969) 1238. 21 M. L. Olmstead, R. G. Hamilton and R. S. Nicholson, Anal. Chem., 41 (1969) 260. 22 D. T. Pence, J. R. Delmastro and G. L. Booman, Anal. Chem., 41 (1969) 737. 23 J. R. Delmastro, Anal. Chem., 41 (1969) 747. 24 L. Nadjo and J. M. Saveant, Electrochim. Acta, 16 (1971) 887. 25 M. L. Olmstead and R. S. Nicholson, Anal. Chem., 41 (1969) 862. 26 P. J. Kudirka and R. S. Nicholson, Anal. Chem., 44 (1972) 1786. 27 V. J. Puglisi and A. J. Bard, J. Electrochem. Soc., 119 (1972) 829. 28 V. J. Puglisi and A. J. Bard, J. Electrochem. Soc., 120 (1973) 748. 29 L. Nadjo and J. M. Saveant, J. Electroanal. Chem., 44 (1973) 327. 30 J. Kuta and E. Yeager, J. Electroanal. Chem., 46 (1973) 233. 31 D. Cukman and V. Pravdic, J. Electroanal. Chem., 49 (1974) 415. 32 D. Cukman, M. Vukovic and V. Pravdic, J. Electroanal. Chem., 49 (1974) 421. 33 N. Winograd and T. Kuwana, J. Amer. Chem. Soc., 93 (1971) 4343. 34 J. Jacq, Electrochim. Acta, 12 (1967} 311. 35 I. Ru~i6, J. Electroanal. Chem., 52 (1974) 331. 36 H. L. Hung and D. E. Smith, J. Electroanal. Chem., 11 (1966) 237. 37 H. L. Hung and D. E. Smith, J. Electroanal. Chem., 11 (1966) 425. 38 I. Ru,~i6 and D. E. Smith, J. Electroanal. Chem., in press. 39 L. Sipos, L. Jeftic, M. Branica and Z. Galus, J. Electroanal. Chem., 32 (1971) 35. 40 S.W. Feldberg in A. J. Bard (Ed.), Electroanalytical Chemistry, Vol. 3, Marcel Dekker, New York, 1969,

pp. 19~296. 41 J. Koutecky, Collect. Czech. Chem. Commun., 19 (1954) 857. 42 J. Koutecky, Collect. Czech. Chem. Commun., 18 (1953) 3l l . 43 H. Matsuda, Z. Elektrochem., 61 (1957) 489. 44 H. Matsuda, Z. Elektrochem., 62 (1958) 977. 45 D. E. Smith in A. J. Bard (Ed.), Electroanalytical Chemistry, Vol. 1, Marcel Dekker, New York, 1966,

pp. 1-155. 46 S.W. Feldbergin J. S. Mattson, H. D. MacDonald, Jr. and H. B. Mark, Jr. (Eds.), Computers in Chemistry

and Instrumentation, Vol. 2, Marcel Dekker, New York, 1972, pp. 185-215. 47 I. Ru~i6 and S. W. Feldberg, J. Electroanal. Chem., 50 (1974) 153. 48 J. Koutecky and J. Koryta, Electrochim. Acta, 3 (1961) 318. 49 I. Ru~i6 and S. W. Feldberg, in preparation. 50 D. E. Smith and T. G. McCord, Anal. Chem., 40 (1968) 474.

Documents

On the influence of coupled homogeneous redox reactions on electrode processes in d.c. and a.c. polarography: II. The e.e. mechanism with a coupled homogeneous redox reaction