On the influence of coupled homogeneous redox reactions on electrode processes in d.c. and a.c. polarography: I. Theory for two independent electrode reactions coupled with a homogeneous

Electroanalytical Chemistry and ln terfacial Electrochemistry, 52 (1974) 157-192 157 © Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands o

ON THE INFLUENCE OF C O U P L E D H O M O G E N E O U S REDOX REAC- TIONS ON ELECTRODE PROCESSES IN D.C. AND A.C. POLARO- GRAPHY

I. THEORY FOR TWO I N D E P E N D E N T ELECTRODE REACTIONS C O U P L E D WITH A H O M O G E N E O U S REDOX REACTION

IVICA RUNIC* and DONALD E. SMITH**

Department of Chemistry, Northwestern University, Evanston, Ill. 60201 (U.S.A.)

STEPHEN W. FELDBERG***

Brookhaven National Laboratory, Upton, Long Island, N.Y. 11973 (U.S.A.)

Received 16th January 1974)

INTRODUCTION

Miller and Orlemann were the first to discover the influence of homogeneous redox reactions on electrode processes in d.c. polarography through the so-called "nonadditivity of diffusion currents ''x. This effect can be observed when two or more depolarizers of nonequal diffusion coefficients are present in the electrolyte solution and when the products of the corresponding electrode reactions are soluble in the solution phase. Recently, Yamaoka reported a significant effect in a.c. polarography whereby an a.c. polarographic wave is enhanced by addition of a second, more easily reduced depolarizer and where all electroactive species are solution-soluble 2. On the basis of convincing empirical evidence, Yamaoka attributed this effect to the same mechanistic pathway proposed by Miller and Orlemann. The reaction scheme may be written

A + e - ~-~ B (electrode reaction, E °, E~. x)

C + e ~ D (electrode reaction, E °, E~. 2) kl

A + D ~ C + B (homogeneous redox reaction) (I) k2

where species A and C are initially present in the solution. The conditions extant in the Miller-Orlemann-Yamaoka investigations were characterized by well- resolved polarographic waves in most instances, where the observed half-wave potentials, E½. t and E~. 2, and the thermodynamic standard potentials, E ° and E °, fell in the same order. Specifically, the applicable relationships were

E ° >> E ° (A)

E~, 1 >> E~. 2 (B)

* On leave from the Center for Marine Research, Rudjer Bo~skovie Institute, Zagreb, Yugoslavia, 1972-74.

** To whom correspondence should be addressed. *** Part of this work was carried out under the auspices of the Atomic Energy Commission.

158: I. RUZI(2, D. E. SMITH, S. W. FELDBERC

Of course, with these conditions the homogeneous redox step can proceed and manifest itself on the polarographic response only at potentials where the second wave ensues. Significantly, the a.c. polarographic effects reported by Yamaoka are much larger than in d.c. polarography and did not seem to require nonequal diffusion coefficients for their appearance.

Mechanism (I) has been largely ignored in quantitative theoretical treatments of modern voltammetric and amperometric techniques. However, this mechanism is closely related to two other mechanistic schemes which can be influenced by corresponding homogeneous redox reactions. Inthese cases this influence has been subject to detailed theoretical study in the cgs~text of several common electrochemical techniques. One of these is the stepwise electrode reaction (e.e. mechanism with "nuances")

A + e ~ B (electrode reaction)

B + e ~ C (electrode reaction)

A + C ~ 2B (homogeneous redox reaction) (II)

Jacq 3 applied a steady-state approach to evaluate the expression for the current under convective diffusion conditions with mechanism (II). His solution was limited to the case of a totally irreversible homogeneous reaction, i.e., where there was a large difference in the standard potentials. Jacq also derived the equation for the case of an infinitely fast homogeneous reaction and found that it had no influence if the corresponding electrode reactions are reversible. Bonnaterre and Cauquis 4 derived the equation for the first wave with mechanism (II) and convective diffusion, considering the situation where the standard potentials were comparable. The effect of the homogeneous redox reaction was discussed also for the case where species C is insoluble in the solution phase 5-s. Feldberg 9 has described several alternatives to mechanism (II), all with reversible electrode reactions. For conditions where the homogeneous redox process proceeds in the disproportionation direction, theories by Kouteck~¢ and Koryta 1°, Orlemann and Kern 11, Holub12, Booman, Delmastro and Pence 13 15, Olmstead and Nicholson 16, and Sav6ant and coworkers 17-19 are applicable to several measurement schemes. Andrieux and Sav6ant 2° have applied e.p.r, spectroscopy to measure the homogeneous redox reaction rate in the case where the electrode reactions are reversible and there is no effect of the homogeneous reaction on the electrochemical responses. In the same situation a spectroelectrochemical measurement approach using optically transparent electrodes has been applied with notable success by Winograd and Kuwana 21. The second related mechanism is the well-known e.c.e, mechanism where the term "nuance" was first introduced in reference to the coupled homogeneous redox reaction 2z 23. The simplest case involves a first-order intermediate chemical step (non-redox" and is represented by the sequence

A + e ~ - B

B.~_~ C

C + e ~ D

A + D . ~- B+C

(electrode reaction)

(intermediate homogeneous chemical step)


(homogeneous redox reaction) (m)

COUPLED H O M O G E N E O U S REDOX REACTIONS 159

This case has been treated by Feldberg et a/. 22 -26 using the digital simulation method for stationary electrode voltammetry and the rotating disk electrode. Sav6ant et al. 27" 28 also have considered this mechanism and the second-order e.c.e. mechanism

A + e ~ - B

A+B ~ C

C + e --* D

A + D ~ - - B + C


(intermediate homogeneous chemical reaction)


(homogeneous redox reaction) (iv)

Nelson and Feldberg 29 have described a second-order e.c.e, mechanism involving dimerization of the initial electrode reaction product.

Theoretical treatment of the a.c. polarographic response (faradaic admittance with d.c. polarization) has not been reported for any of these reaction schemes, although the e.e. and first-order e.c.e, mechanisms have been treated 3°-33 for the situation where the homogeneous redox reaction is inoperative. This gap in a.c. polarographic theory is unfortunate because of the established importance of these mechanistic schemes in the general framework of electrochemical kinetics, because the attendant data provide definitive kinetic-mechanistic insights into rapid homogeneous redox reactions, and because the a.c. polarographic method may provide a sensitive data base. Consequently, we have undertaken a program to develop the theoretical a.c. polarographic rate laws for mechanisms (I)-(III), to examine their predictions, and to test experimentally these predictions. Treatment of the boundary value problems associated with these mechanisms, which include nonlinear partial differential equations, is accomplished successfully by applying mathematical techniques which were recently applied to the simpler mechanisms involving irreversible disproportionation and dimerization following the heterogeneous charge transfer step 34.

In the present paper we report results of our treatment of mechanism (I). The rate laws derived include effects of heterogeneous charge transfer kinetics, encompassing both the quasi-reversible and irreversible realms. Consequently, the observed polarographic half-wave potentials, E~.I and E~,2 are not subject to thermodynamic constraints imposed by the standard potentials, E ° and E °. The report emphasizes conditions characteristic of the Miller-Orlemann-Yamaoka studies (eqns. A and B), and those applicable to the second-order "catalytic" process 35 where the applicable relationships are either

E ° ~> E2 ° (C)

E~, 1 ~ E~. 2 (D) o r

E? E ° (E)

E~, 1 >~ E½. 2 (F) However, situations are also considered where the observed polarographic half-wave potentials and/or the thermodynamic standard potentials are in close proximity, such as cases where

160 I. RUZIC, D. E. SMITH, S. W. FELDBERG

o r

E ° ~ E ° (G)

E~, 1 > E~, 2 (H)

E ° > E2 ° (I)

E~. 1 ~ E~. 2 (J)

etc. Results presented are based on the expanding plane and expanding sphere electrode models, but equations are readily adapted to other electrode models. The notation used is summarized in an appended glossary on p. 190.

THEORY

L Assumptions The derivation to follow utilizes most of the usual assumptions which are

invoked when deriving d.c. and a.c. polarographic rate laws ae 3v. However, the • development includes several special characteristics which are worth noting at the outset.

First, because the two redox reactions represented by mechanism (I) may involve species with somewhat dissimilar transport properties, an attempt has been made to waive the usual assumption of equal diffusion coefficients 3a37. An approach suggested by Kouteck~ 3s is adopted to develop equations which can account for diffusion coefficient differences. Some approximations are necessary in treating the a.c. part of the boundary value problem. However, it is felt that the final expressions can accommodate diffusion coefficient differences for solution- soluble species which are normally encountered under polarographic conditions. This is particularly true when the homogeneous redox reaction is sufficiently rapid to produce the effects of interest to the present study.

Second, the derivation is confined to the one-electron transfer mechanism represented by mechanism (I). This is done because: (a) it simplifies the theoretical presentation; (b) heterogeneous and homogeneous multi-electron processes are likely to follow more complex mechanisms than mechanism (I); (c) experimental examples have been confined primarily to the one-electron situation 2. Of course, adaptation of the equations given below to the multi-electron case where n, = n 2 = n is trivial, requiring only replacement of F by nF in the theoretical relationships.

Finally, the main portion of the derivation is presented within the framework of the expanding plane electrode model. As usual, adaptation to other electrode geometries simply involves modification of the d.c. part of the boundary value problem3,~. 36. The latter operation is briefly discussed for the case of the expanding sphere model.

II. Derivation A. Preliminary manipulations of boundary value problem. The expanding plane

boundary value problem for mechanism (I) may be written:

0C A 0 2 C A 2x 0C A ot = D,, + k,c cD+g c . (1)

COUPLED HOMOGENEOUS REDOX REACTIONS 161

~¢B (~2 CB

~Cc Dc 0 2 Cc o t =

~CD = D D 632 c D

~t

t = 0 , x>~O ;

t > 0 , X--*ct3 ;

2x dcB + - - _ _ + k~CACD--k2Cc%

3t OX

2X ~Cc + - - _ _ + kxCACD--kzCcCB

3t dx

2x ~C D + kl CACD+k2CcCB

3t ~x

C A -~- C~

C~ = 0

CC= C~

CD= 0

C A--~ C ~

cH -*O

C C ~ C~

CD~O

t > O , x = O ;

C3CB i 1 (t) ~CA _ D s - DA ~ = ~X FA

_ 0 C D i 2 ( t ) t~Cc DD -- DA ~X ~X FA

il(t) -- fACA(x=o, eXp [ - -~IF (E_EO)I FA k~ 1

(2)

(3)

(4)

(5)

(6)

(7)

(S)

(9)

(10)

(11)

(12)

(13)

(14)

exp[tl- l)FRT/ -E° ] t15/ i2 (t) ~ F

FAk~2 - fcCc(~=o, exp [ - ( E - E ° ) ] , T ~

_ fDCD,x=0)exp [(1-~2)FRT ( E - E ° ) ] (16)

Here we accompany the usual Fick's law relationships, modified to account for the homogeneous redox process (eqns. 1-4), with initial conditions which consider standard polarographic conditions where only the oxidized species are initially present (eqns. 5-8), with boundary conditions for semi-infinite diffusion (eqns. 9-12) in which adsorption plays a negligible role (eqns. 13, 14) and with the absolute rate theory for the heterogeneous charge transfer processes (eqns. 15, 16).

It should be noted that the rate constants of the homogeneous reaction, kl and k:, are not totally independent quantities. Their ratio is controlled by the difference in standard potentials of the electrode reactions through the well-known relationship:

162 I. RUZIC, D. E, SMITH, S. W. FELDBERG

K = k,/k2 = exp [(F/R T)(E ° - E°)] (17)

Introducing the transformations suggested by KouteckS ,38 for treating kinetic currents with nonequal diffusion coefficients;

where

01 =(DA CA + DBcn)/D,

¢S ~ = (De Cc + D~ co)/D i

(18)

(19)

D, = (Da + KDB)/(1 + K) (20)

DE = (Dc +KDD)/(1 +K) (21)

allows one to reformulate eqns. (1)-(14) in terms of the variables CA, CC, 01, and ~1. One obtains

where

and

0C A a2CA 2X &A 0t - D A ~ ~-+ 3t 0X

ac__5_c= 02Cc 2x &c at Oc ~ + 3t a~

601 _ D1 6201 at 7 ~ x 2 + - - - -

002 0202

at - ° 2 7 7 x ~ + - -

(Dz~2-Dcccl (D101--DACA] (23) + k l C A \ DD 7 -- k2cc • D. /

2x ~01 + (~ ~ (24) 3t 0x

2x 0~2 + ~2~2 (25)

3t ax

¢, = ( o . - D~)/(1 + K)D,

~2 = (Dc-DA)/(1 + K)Dz

0 2x a ~ ' = & ( c B - - K C A ) " 3t Ox ( c a - - K C A )

0 2x 0 32 = ~ (Co-- KCc) 3t ax (co- Kcc)

t = O , x >O ; CA=C*

C C = C~

01 = DACe~D1

02 = Dcc~/D2

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

t > O , x ~ o o ; C A ---+ C~

C C ~ C~

01--*DA c*/D1

02-,Dcc*/Dz

(34)

(3s)

(36)

(37)


t > 0 , x = 0 ; (gCA] il (t) (38) DA \ ~ x & = o -- FA

( ~ - ] iz(t) (39) Dc \ gx / x=O - FA

__o 9x ] ~=o x=O

(40)

At this point attention should be directed to the quantities ~1 {1 and ~2{2 on the right sides of eqns. (24) and (25). It is obvious that when the so-called steady-state conditions prevail, where the terms Di (aq ~at) and (2x/3t)(aq/ax) are negligible 3 s. 39,

gl = ~2 = 0 (41)

so that the quantities in question are zero. At the same time, one must recognize that the coefficients of 41 and 42 in eqns. (24) and (25), ~1 and (2, usually, will have magnitudes less than unity because they represent relative differences in diffusion coefficients (approximately). Indeed, with the majority of systems, the statement

~L 2 ~ 1 (42)

will be appropriate. Consequently, the terms {1 ~1 and ~2~2 will be significantly smaller than the remaining terms on the right sides of eqns. (24) and (25) in the case of small differences in diffusion coefficients (DA~DB, Dc~Do) and will approach zero under steady-state or near steady-state conditions, which is where the chemical kinetic effects of interest will be largest. For such circumstances, eqns. (24) and (25) reduce to:

a01 = D1 aZOa 2x 90~ (43) 9t ~ + 3t 8-x-

al[12 -- D2 92 02 2x 902 (44) at ~ + 3t a-x

Standard methods of Kouteck~ ,3s 39 and Matsuda 4a 41 may be applied to eqns. (43) and (44) and the associated initial and boundary conditions (eqns. 32, 33, 36, 37, 40) to conclude that 01 and 02 are constants given by

01 = DAC*/D1 (45)

02 = Dc c*/D2 (46)

Equations (45) and (46) may be substituted in eqns. (22) and (23) to obtain

OCA 92CA 2X aCA kl (DC)cA(c~_Cc)+k2 (DA) at = DA ~x 2 + 3t ax _ . _ . ~ ~ Cc(C*--CA) (47)

aCe 92Cc2X8Cc {Dc~ / D A \ , = Dc ~ + 3t 9--x + kl \ /I Ca(C~-Cc)-k2[~-} cc(ca-cA) (48) O-7- g \ u a /

164 I. RU~I(2, D. E. SMITH, S. W. FELDBERG

The foregoing manipulations have reduced the boundary value problem to that of solving eqns. (47) and (48), given the initial and boundary conditions of eqns. (30), (31), (34), (35), (38), and (39).

One may question certain aspects of the procedure leading to the analytical solutions for Ot and ~2 represented by eqns. (45) and (46). For example, the validity of this procedure with regard to the a.c. concentration components might be of concern, particularly because the steady-state argument is much less likely to be applicable on the a.c. time scale. However, the prediction of eqns. (45) and (46) that the a.c. components of ~1 and q*2 are zero (~1 and ~2 are constants), is negligibly dependent on the approximations invoked. This conclusion is reached by recalling that the a.c. concentration waves are highly damped 36, so the behavior of ~ and ~z regarding a.c. terms is relevant only in close proximity of the electrode (x ~ d.c. diffusion layer thickness). Combining this observation with the fact that Ot and ~'2 have zero surface gradients (eqn. 40), indicating that they have constant values (zero a.c. terms) in the vicinity of the electrode, regardless, leads one to recognize that the approximations implicit in eqns. (45) and (46) are of negligible concern for the a.c. terms. Of course, in the case of significant differences between diffusion coefficients, combined with a relatively slow homogeneous redox reaction, eqns. (45) and (46) will be inaccurate regarding the d.c. terms and a rigorous analytical treatment is unavailable. However, a digital simulation procedure involving time and space incrementation and the finite difference approximation can be used to obtain sufficiently accurate assessments of ~1 and ~2 in such situations.

B. Separation of a.c. and d.c. parts of boundary value problem. At this point it becomes convenient to resolve the boundary value problem into its a.c. and d.c. portions. This separation may be effected by the recently described 34 variant of the Gerischer 42 approach. The concentrations, ca and Cc, are expressed as sums of a.c. and d.c. terms, i.e.,

CA = ~A + ~A (49)

cc = ~c + ~c (50)

and substituted in eqns. (47) and (48). This is followed by invoking the approximation suggested by Gerischer which, in the present case, may be written

lea ~c I ~ I CA ( 4 -- ec ) - CA Cc "~- eA (C~ -- ~-C ) I (51)

leA?el ~ I?-c (c*-gA)--eC ?g + ?c ( c* - ?g)l (52) Inequalities (51) and (52) amount to stating that higher-order effects due to the a.c. perturbation (rectification effects, higher harmonics) are negligible, which is appropriate with the small amplitude a.c. perturbations normally employed in a.c. polarography3e~ 3v. Among other things, this small amplitude approximation also is equivalent to assuming that the d.c. process is uninfluenced by the applied a.c. perturbation. Upon applying the Gerischer approximation and some algebraic rearrangement, one obtains the relationships

O~'A 02t~A 2X ~C'A (De) (DA) ~7 - = DA~-TX2 + 37~x k, ~ ea(C~-ec)+k2 ~ 0c(C~.--eA) (53)

() (°A) , Oec 02ec 2x Oe c Dc eA(Ct_~.c)_k 2 ec(CA--eA) (54)


where

and

(~-'A (~2C A 2x agA

0~ c ~2 CC 2X C~fi c ~t - oc ~ + 3 t c3-~- + k* CA - k~ e c

k~ = k 1 (Dc/ DD) (c~ - cc) + k2 (D A/ Dn) cc

k~ = k, (Dc/ DD) CA+ k2 (DA/ DB)(c*-- CA)

t = 0 , x~>0 ;

(55)

(56)

(57)

(58)

CA = C* (59)

CC = C* (60)

CA = Cc = 0 (61)

t > 0 , x--+oo ; CA ~ C* (62)

gC --'C~ (63)

CA ~ Cc --' 0 (64)

(?~Al _ i 1F_A(t) (65) t > O , x = O ; DA\t~X/x= 0

Dc(OCc~ - i~A) (66) \~x /~=o

I)A (~A~ --71(t) (67)

(0 cj 72(t) (68) Dc \~X/x=O - FA

Equations (53)-(68) represent a separation of the a.c. and d.c. parts of the boundary value problem for CA and Cc. Eqns. (53), (54), (59), (60), (62), (63), (65), and (66) represent the d.c. part, while the a.c. part is given by eqns. (55)-(58), (61), (64), (67), and (68).

C. Treatment of a.c. part of boundary value problem. It should be noted that the foregoing manipulations have succeeded in reducing the a.c. boundary value problem to one where the homogeneous chemical kinetic terms in the Fick's law expressions are pseudo-first order. The methods for treating such problems for equal diffusion coefficients are well-documented s~3v'43. For unequal diffusion coefficients, it is helpful to again follow the suggestion of Kouteck) ~s8 and introduce the new variables

01 = (DA CA q'-Dc Cc)/D (69)

02 = ?-c - K*aa (70) where

166 I. RUZlC, D. E. SMITH, S. W. FELDBERG

K* = kl/k2 (71)

D = (DA + K*Dc )/(i + K*) (72)

Equations (55) and (56) can be reformulated in terms of the new variables as follows

001 0201 2x 001 Ot - D-~Tx2 + 3 t 8~- + (1 ~1 (73)

0202 k* OX 2 -- 5 02-{-~1 ~2 (74)

where

(1 = (Dc-DA)/(1 + K*)D (75)

~1 = 002 2x 002 at 3t Ox (76)

2,, Ot 3t ax (77)

DADc L \- D~-cZc_N / 02

k* = (k~[ + k'~)D2/DAOc (79)

By the same reasoning which was applied earlier in regard to eqns. (24) and (25), one may ignore the ~1 ~1 term in eqn. (73). The treatment of eqn. (74) and its subsidiaries (eqns. 75, 77, and 78) is somewhat more complicated, because the quantity Dc --DA appears both in the numerator and denominator of the ~1 ~2 term. Two situations lead to useful analytical results. First, if the diffusion coefficients are not too dissimilar (no restriction on homogeneous rate constant), i.e., if

DA ~ Dc (80) then

1 (002 2x e02) t81)

and eqn. (74) becomes

002 02 02 2x 002 k* 02 (82) Ot = D ~ x 2 + 3t Ox

Combining eqn. (82) with the simplified form of eqn. (73),

001 0201 2X 001 (83) a t - = D O~T- + 3 t a-x-

and the initial and boundary conditions;

t = O , x>~O ; 0 1 = 0 2 = 0 (84)

t > 0 , X-'-+OO ; 01-+02--+0 (85)


where

t>O, x = O aO,] ~,(t) ~(t)

~x /~= o FA-Dc FADA

(86)

(87)

go* .~ K~:o (88) and applying standard methods of Kouteck3~ 38.39 and Matsuda 4°,41 to eqns. (82)-(88), one obtains

O~ o= ( 7 ~ ft u~['i~(u)+'i2(u)] du x= - , ~ / o FAD~(t~-u~)½ (89)

where

2x:O = -- ( D ) { 7 ] ~ [t u~e,~i2(u)d u D-c \37r/ Jo FAD½(t~-u~)½

f u e q (u) du \ O A~ \3re/ o FAD~(t ~-u~) ~ (90)

k~ - kx=o (91)

Substituting eqns. (89) and (90) in eqns. (69) and (70), respectively, and solving for CAtx=o) and Cct~=o) yields,

{ (77f, aA(x=O)-- I+K~ _ - \3n/ )o FAD~(t~-u~) ~

( Z ~ ! ' u~e-k~"-"'i2(u)du + \3rt/ o FAD~(t~-u~) ~

' u~e-k;('-")i I (u)du (92)

. 1 { ( 7 ~½ f t u~['il(u)+'iE(U)]du Cc,~=o, - I+K~ - K * \ ~ / Jo ,VAO~(t~-.~)~

(DA~(7~ ½ f' u~e-~t,-u, i2(u)d u l

\Dc/ \3re/ Jo FAD½(t~-u~)~

( 7 ) ½ (' u~e-k3"-'~'t (u)du } + K* \3rc/ )o FAD½(t~-u~)~ . (93)

If D A and D c are significantly dissimilar, then an analytical expression is obtainable only if the homogeneous redox reaction is fast enough to make the time derivatives in ~2 negligible. Then eqn. (74) becomes

0202 k* OX 2 = 5 02 (94)


Combining eqn. (94) with eqns. (82), (84)-(87) and proceeding as before (standard methods 3s-41) yields

CA(x=O)- I + K * -- \37rl o FAD~(t~-uk) ½

+ FA(k~D)~ ~ F~T(~o~ j (95)

Cc(x=°) - I + K * \ 3 n / o FAD~(t~-u~) ~

(DA) ~2(t) K*ii(u) - \ FA(k~D) ½ + F-~(k~))½J

(96)

It should be recognized that eqns. (95) and (96) also are limiting forms assumed by eqns. (92) and (93) for large values of k* t 43. In other words, no restrictions on DA and Dc are necessary in the large rate constant ("a.c. steady-state") limit. How- ever, for smaller k~t-values it is necessary that DA ~Dc to validate analytical results deduced from eqns. (92) and (93) (given below). Regardless of which forms are used for CA(x=0) and ?c(x=o), the surface concentrations gB(x=o) and gO(~=O) can be easily obtained from the foregoing results by recognizing that the consequences ofeqns. (45) and (46) for the a.c. concentration components are given by the expressions (applicable for all x)

DA gA + DBg B ----0 (97)

Dc~c +DD~v = 0 (98)

Thus, one may apply the special cases of eqns. (97) and (98)

cB(x = o~ = - (DA/DB) EA ~x= o) (99)

CD(x= 0) = -- (Dc/DD)Cc (x= o) (100)

Equations (49), (50), (92), (93), (99), and (100) are substituted in the absolute rate expressions (eqns. 15 and 16), together with the relationships

ii(t ) = i i( t ) + ~i(t) (101)

E = Ed.~. - A E sin ~ot (102)

to obtain a set of two integral equations relating the d.c. and a.c. components of i l(t ) and i2(t) to the experimental and redox system parameters 3°'31. Standard, well-documented procedures 3°' 31. 36 are then applied to extract the pair of integral equations representing the small amplitude fundamental harmonic a.c. polarographic response. One obtains (using notation similar to that of ref. 31)

2, ( _ i _RcI2_K~Rolk,,+Rclk2) (103) gJ,,, (t) = F 1 (t) sin cot + ~

22 , ~ 2.1 (t) = Fz (t) sin o~t + ~ ( - Ko R21 I1 - K* 12 + I,;~ R2 ' Ik, , - e ~ 1 I~,2)

(104)


where I~1. l(t) and 7*=,~(t) are related to the fundamental harmonic current associated with the first and second charge transfer steps, Ii(o0t) and I2(ot), by the usual type of expression (for n= 1) 31, i.e.,

l,(cot) = ( F 2 Ac~ O ~ AE/R T) ~L x (t) (105) 12(mt) = (F2Ac~ D ~ AE/RT) t//2, 1 (t) (106)

The other previously-undefined quantities in eqns. (103) and (104) are given by the expressions:

u ~l(U)du (107)

(7 )½ f ' u}e-k~'t-")~i"(u)du (108) Ik'i= ~ o ( t~-u~) ~

(where i= 1,2)

R~ = c_ff /c,~

e , = D~ /D,,

).1 ~-- (/¢s. 1 fl/D', I'. 1)[ e-=' j~ + (DA/DB? ep' ~*]

)'2 = (/g. 2f2/D}. 2)[ e-=~ j= + (Dc/DD) ~ea~a~]

rl( t ) = [I+(DB/DA)~e_j, ] 1 + \DA]

F2(t ) = [l+(Do/Dc)½e_i~ ] 1 + f12 \ D c /

(109) (110) (111) (112)

(CA(x=O}~t (113) e-~'-ll \- c* M

e - h - - l ] (Cc'x. ° ' - - ~ t (114) \ Cc /J

(The parameters c8¢~=o) and CD¢~=O) were eliminated from eqns. (113) and (114) by introducing relationships between the d.c. concentration components obtained from eqns. (45) and (46).)

f l = f~A' f{ '

f2 = fPc2 f ~ 2

DT. , = D(DB/D,) ~'

Dx. z = D(DD/Dc )~

Ji = (F/RT)(Ed.c. - U~, ~)

E~. I = E ° - ( R T / F ) In (fB/fA)(DA/DB) ~

~. 2 = E° 5(RT/F) In ( f~/A. )(Dc /DD?

fli = 1 - oq

(115) (116) (117) (118) (119) (12o) (121) (122)

As in previous work 3a 31.36 , the solution of eqns. (103) and (104) is effected by first recognizing that the kui.a(t ) contain only fundamental harmonic components. Thus, one may write


~1, 1( t ) ----- a sin cot+b cos ogt (123)

7Jz 1(0 = c sin cot+ d.cos cot (124)

Substituting eqns. (123) and (124) in eqns. (103) and (104) (and the subsidiary expressions, eqns. (107) and {108)) yields a form which is integrable by neglecting the slight time dependence of a, b, c, and d 31, in which case one can apply the known relationships 43

where

( 7 ) ~ ! ' u*" sincoudu _ 1

3n! o (t~-u~) ½ (2°) ~

f' u, cos udu 3 / o (t]-u]) ½

( 7 ) ~ f t u~e-ka"-"}sincoudu 0

(7~½ f' u}e-kac'-"'coscoudu 3=! o u;)

(sin cot- cos cot) (125)

1 (sin ogt+cos cot) (126) (2co) ~

1

(2co) ~ (G+ sin cot- G_ cos cot) (127)

1 t z )'~co'½ (G_ sin cot+G+ cos cot) (128)

I 2-}+ } G+ ( l + g ) " g l {129)

_ = l + g 2 ]

g = k * / c o (130)

In the large k*t case, where DA and Dc are unrestricted (eqns. (94)-(96) are applied), one obtains for G,

G+ =(2/9) ~ (131)

G_ = 0 (132)

Equations (131) and (132) are the well-known limiting forms of eqn. (129) for g > 10 (error < 1~o) 59 which establishes quantitatively the conditions required to validate this "a.c. steady-state approximation".

Equations (125)-(128) allow one to conclude that, given eqns. (123) and (124), one obtains

11 = (2co)- } [(a + b) sin cot- ( a - b)cos cot] (133)

12 = (2o)-~[(c+d) sin cot-(c-d) cos cot] (134)

Ik.1 =(2co) -½ [(aG+ +bG_) sin cot-(aG_-bG+) cos cot] (135)

lk,2 =(2co) -~ [(cG++dG_) sin cot-(cG_-dG+) cos cot] (136)

Substituting eqns. (123), (124), (133)-(136) in eqns. (103) and (104), and equating coefficients of sin ~ot and cos cot on each side of the resulting equations yields a system of four linear algebraic equations in the four unknowns, a, b, c, d. Solution of the system of algebraic equations is effected by the method of determinants. The determinant of the system is


1 (1+7?) ~; 6? 6;

= - 7 ; (1+7?) -6; 6? 7; 7; (1+6~) 6~

- 7 ; 7f - 6 2 (1 +6~-) where

~ =

~ =

The special obtained by

V, (o t)

F~t)

21(l+K*RoG+_) (1 + Kt)(2~o) ~

21Re(I-G+) (1 + Ko)(2~o) 2

(1 + K~)(2co) ½

2 2 (KtR~ ~ G+) (1 + K*)(2co) ~

(137)

(138)

(139)

(140)

(141)

determinants for a, b, c, and d (A., Ab, Ac, and Ad, respectively) are replacing one of the columns of the system determinant by

(replace first column for a, second for b, etc.). With the aid of eqns. (105), (106), (123), and (124) and algebraic manipulation, the total fundamental harmonic current, l(oot) may be related to the system determinant and the special determinants, A,, Ab, Ac, and Ad by the relationship

I(o)0 - f 2 ac~,O~aE [(A. + R~A~) 2 + (~tb + n~&)2] *

RT4

x sin[cot+cot-l(A~+R~A~l (142)

In previous derivational efforts of this type 31, it has been customary to expand "'manually" the determinants and express the theoretical rate law in a conventional algebraic format. The latter approach often has the advantages of revealing key functional relationships and reducing computational time. However, these advantages are diminished as the mechanistic scheme and attendant rate law become more complex. Eventually, simple functional relationships, if existent at all, tend to be obscured by algebraic complexity, making them difficult to recognize except in special limiting cases. The fact that in the present case the d.c. boundary value problem must be solved by numerical methods (see below) further diminishes the likelihood that useful relationships can be recognized by determinant expansion. Likewise, the computational advantage also is lost because the predominant


fraction of computer time will be devoted to the d.c. problem. Consequently, except for some special limiting cases considered below, we terminate algebraic manipulation of the a.c. response relationships at eqn. (142) and its subsidiaries. The FORTRAN program we have developed to calculate the a.c. response includes subroutines for evaluation of the determinants represented in eqn. (142).

D. Treatment of d.c. part of boundary value problem. The foregoing equations for the a.c. polarographic response are incomplete because explicit expressions for the d.c. surface concentration components. ~A(x=0) and ~c{~=o), have not been provided. These quantities are obtained by solving the d.c. boundary value problem which was identified above. It has been pointed out 34" 36 that the relevant electrode geometry associated with the a.c. polarographic rate law is determined solely by the geometry model used to calculate these d.c. surface concentration components. Other aspects of the theoretical formalism given above are geometry-independent. With mechanism (I), an approach to obtaining a general analytical solution to the d.c. boundary value problem for any electrode geometry is unknown to us. The problem is best treated by the digital simulation technique (explicit finite difference method) 44. Procedures are well-documente& 4 for treating by the simulation approach boundary value problems of the type under consideration, including use of the expanding plane model of the dropping mercury electrode. However, since effects of sphericity can be expected to be important, particularly with coupled homogeneous chemical reactions of second-order ~ 3-15. 34, a simulation procedure for the d.c. boundary value problem using the expanding sphere model was developed for the present case. It includes some recently devised procedures for handling the sometimes troublesome situation (for digital simulation) associated with very rapid coupled chemical reactions and compact reaction layers that are contiguous to the electrode surface 45. Simulation of diffusion to an expanding sphere is a modification of previously described procedures for the expanding plane and stationary sphere models 44. The results for simple diffusion agree well with the theory of Newman 46. Details will be published at a later date.

DISCUSSION OF PREDICTIONS FOR MECHANISM (I)

I. The case where E ° ~> E ° and E.~. 1 >> E~. 2 A. The electrode reaction, C+ e ~ D, is irreversible. The studies of Yamaoka 2,

which stimulated this work, involved a number of systems (a majority of those studied) in which the two electrode processes are well-separated on the d.c. potential axis, and the second electrode process is irreversible. We have carefully scrutinized the predictions of the equations derived above for this situation because Yamaoka's data indicated that the associated a.c. polarographic-response provided a previously-unrecognized, versatile approach to studying homogeneous redox reactions. In this case, the forward redox reaction predominates (k~ ~> k2), the polarographic waves are completely resolved, and there is no influence of the homogeneous redox reaction at potentials of the first wave. However, the theory presented here predicts that, at potentials of the second wave, both the d.c. and a.c. polarograms are strongly influenced by the forward homogeneous redox reaction if it is sufficiently rapid. Some results are shown in Fig. 1 for the case where e* =c~. For rate constant values lower than 105 M-1 s- l , only a shift to negative

CO U PL E D H O M O G E N E O U S REDOX REACTIONS 173

1,5

1,0

;~ 5.00

8 2.00 o

1.00 LJ

<[

2.0

i -0.25 0.50 -o.75

Ed c . / V

t k ~ o 2 k 2,0 ~ 104 MIS -I 5 k ~ I.O x I05 M-IS '

0 .00 -0 .25 - 0 , 5 0 -0.75

E d . c . / V

I

- 1.00

- 1.00

Fig. 1. Predicted d.c. and a.c. polarograms at second wave with mechanism(I) where kl >> k2. Parameter values: T=298 K, cA* -- Cc* --1.00 x10 - 3 M , D A = D ~ = D c = D D = 4 x l 0 6cm 2s l , t = 4 . 0 0 s , ~ o = 2 5 0 s -~ E ° = l . 0 0 V, E ~ . 2 = - 0 . 5 0 V (when k=0), /~ 1=oc, /~ 2~< 10 -6 cm s 1, c~2=0.50 ' k-values shown in Figure (k=kl +kz~-kl), expanding plane model, direct current magnitude given in apparent n-value units, i.e., ordinate =(3rot/7) ~ i a.~./FAc~ D o. A.c. current is given in units of [102R T l(e)t)/FZA (2e)O)½c*].

potentials occurs in d.c. polarography (up to about 40 mV). With rate constants in the range, 105< k I < 106 M - t S - t , a drastic change in the shape of the first two-thirds of the d.c. wave appears, reaching a limit close to 10 6 M-1 s-1, beyond which rate constant increases have little effect, in the a.c. polarogram the redox

• reaction increases the wave height with a maximum enhancement occurring at about kl = l 0 6 M- t s- t for the conditions of Fig. 1. Beyond this, the enhancement in wave height becomes independent of kt. The homogeneous redox reaction also induces a shift in the peak potential• The peak moves to negative values for kt ~< l0 s M -1 s -1 (up to about 25 mV), following the same trend exhibited by the d.c. wave. The peak potential shifts in a positive direction as kl is increased beyond 105 'M-1 s-1, reaching its original value for kl = 0 (approximately) before kt reaches the magnitude 106 m - 1 s -1 The trends illustrated in Fig. 1 are representative of those obtained from calculations for other circumstances within the context of the "Yamaoka case", such as at different frequencies, with varying degrees of electrode curvature, with unequal diffusion coefficients, with unequal bulk concentrations of species A and C, etc. In general one observes that the effect of the homogeneous redox reaction ranges from a negligible value with small k 1, to a maximum value at sufficiently large kl. Of course, the associated kl-limits and the magnitudes of the effects differ quantitatively as the diffusion coefficient and concentration parameters are varied. For example, the maximum increase of the a.c. peak magnitude, relative to the kt --0 value, is approximately proportional to c* (c~ held constant). It should be noted that the characteristics of the predicted enhancement of

174 1. RUZ1C, D. E. SMITH, S. W. FELDBERG

the a.c. peak shown in Fig. I are remarkably similar to the experimental results of Yamaoka 2. Consequently, we believe calculational results such as shown in Fig. 1 represent the first confirmation based on a quantitative a.c. polarographic rate law of Yamaoka's interpretation.

Figure 1 makes it evident that the effects of the homogeneous redox reaction on the polarographic responses can provide the basis for measurement of k~. Figure 2 shows that when k 1 is in the proper range, the standard d.c. polarographic logarithmic diagram is significantly influenced by the homogeneous reaction and can be used for estimation of k~. However, the a.c. polarograms exhibit a larger

kO

1.0 / I k = 0 k = 2 .0× i0 4 M- Is -I

5 k : 2,5x105 M IS-I

4 k ~ i .0×106 M IS-I

0 ' '

-o:4o - o:~o o:8o

Ed.c. / V Fig. 2. Predicted d.c. polarographic log [(i.-i)/iJ plots at second wave with mechanism (I) and kl "> k2. Parameter values: Same as Fig. 1.

effect for a given rate constant and, therefore, provide a more precise basis for achieving this objective. The ratio of the a.c. polarographic currents with and without the homogeneous redox reaction kinetic effect appears to provide a remarkably simple basis for obtaining kl from experimental data. The experimental value of this ratio, which we define as R=l(~ot)k/I(oot)k=o, is obtained simply by observing the a.c. polarographic wave due to the second depolarizer (species C) in the presence and absence of the first depolarizer (species A). Figure 3 shows the predicted behavior for this ratio as a function of d.c. potential for the conditions considered in Fig. 1. The current ratio-d.c, potential profile is sigmoidal. The profile's magnitude increases with increasing kl until k~ ~ 105 M 1 s- i where the plateau reaches a maximum value proportional to c*. The current ratio along the rising portion of the sigmoidal curve continues to increase until kl ~ 106 M 1 s -1 Thus, one immediately concludes that the plateau magnitude in Fig. 3 provides a basis for calculating kl up to k~=105 M - t s -1 and, for the range 105<k<106, the

C O U P L E D H O M O G E N E O U S REDOX REACTIONS 175

2 . 0 k~ L06 M - I s - I ~

= , - - i

1.5

1.0 - 0 . 2 5 - 0 . 5 0 - 0 . 7 5

Ed.c, / V

/ . k ~ O

k = 6 D x l O 4 M 4 S i

- - k= 2.0x lO4M-IS a

k = 6 . 0 x IO~M'IS'I

k = 2.0 x 103 M-IS'~

-[.00

Fig. 3. Predicted a.c. polarographic kinetic nonkinetic current ratio (R) at second wave with mechanism(I) and k I ,> k z. Parameter values: Same as Fig. 1.

current ratio along the rising portion of the profile, such as at the d.c. half-wave potential (when c*---0), can be used. A preferable variant to using absolute magnitudes of the current ratios versus kl as a working curve is to plot kl vs. log (R~,ax-R) where R stands for the kinetic nonkinetic a.c. current ratios plotted in Fig. 3, and Rm, x is their maximum value. The corresponding semilogarithmic diagrams are presented in Fig. 4 for the plateau and irreversible d.c. half-wave potentials. The linearity of these semilog diagrams over a significant range makes such diagrams somewhat ideal working curves for evaluating the homogeneous redox reaction rate constants under the Yamaoka conditions.

A factor which makes the working curves shown in Figs. 3 and 4 of more general applicability than might normally be expected for a.c. polarography is that

I

L• 3.0

i

Lo

I.O 2:0 - log (RMA × -R )

Fig. 4. Predicted log (R ,~ ,~ -R) vs. k profiles at second wave with mechanism (I) and kl>>k2. Parameter values: Same as Fig. 1. (A) Profile at d.c. half-wave potential (where E~ is value observed when chemical reaction is non-existent, i.e., c~ =0) . (B) Profile at plateau of R veraus Ea.~. plot.

176 I. RU~It~, D. E. SMITH, S. W. FELDBERG

they are frequency-independent. Because the electrochemically irreversible process under consideration has a nearly frequency-independent response, one might at first interpret this observation to be applicable only when the process, C + ~ D , is irreversible. However, the behavior is more general than this as can be deduced from eqns. (137)-(142) as follows. If

E ° > E ° (143)

so that

kl >> k2 (144) and

E d.~. ~ E~. 1 (145)

Ed.c. ~ E~. 2 (146)

(i.e., one is at the potential of the second wave) then one has

k~ = k 1 (Dc/Do) (c~ - gc(x= o)) ~- kT. o >> k~. o (147)

K~ > 1 (148)

21 > 2z (149)

The consequences of eqns. (147)-(149) are that

7~ >> 1 (150)

(151) + + (152) H ~ H

6~ = 22/(209) ½ = 82 (153)

The determinant of the system is then

%- o o

o 0

A -~

0 0 (1+62) 62

0 0 - -6 2 (1+~2)

= [ (7 / )z+(~ , l )Z] [ ( l+a2)z+ bz 2] (154)

The special determinants are given by

A~ = (1 +~5~] An= [(y~-)2 +(y~)/] (1 +62)Fz(t)> A.,b (155) \ 3 2 //

Thus, the a.c. polarographic response at the second wave assumes the familiar form 36

12(o9 0 = I,~vF'2(t)Gz((2oJ)~/22) sin [~ot+cot- l (1 + (2co)~/22)] (156)

where


I, ov = F2 (coO) AE 4 R T cosh 2 (½J2) (157}

\ ,h / = - (158)

F'2(t) = (1 +e/:)(1 +e -s~) 22 Fz(t) (159)

Equations (156)-(159) are nearly identical to the well-known expression for a simple, quasi-reversible a.c. polarographic wave 36. Indeed, the only term which contains contributions of the coupled homogeneous redox reaction is F'z(t). The frequency dependence and phase angle are independent of the existence of the homogeneous redox reaction and, consequently, are of no use in determining kl under the Yamaoka conditions (eqns. 143-146). The effects of the homogeneous redox reaction on the a.c. polarographic response in this important special case are simply a manifestation of the status of the d.c. process, reflected in the F'2(t) term. The alternating current ratios considered in Figs. 3 and 4 are given simply by the ratio of F'z(t) with and without the chemical kinetic effect, i.e.,

R = [F'2(t)]k/[F'z(t)]k= o = [F2(t)]k/[Fz(t)]k= o (160)

An important implication of this result is that the R-value under these conditions will be relatively insensitive to assumptions used in obtaining analytical solutions to the a.c. problem with dissimilar diffusion coefficients (eqn. 94). Only the much less restrictive assumptions regarding the d.c. process will be relevant and these can be eliminated completely in digital simulation of the d.c. process, if itseems necessary. Also, the fact that the product lr~, G((2o9)~/22) in eqn. (156) is nearly frequency- independent in the special case of irreversible charge transfer presently being considered is only incidental. It is not responsible for the frequency-independence of R, which exists when eqns. (143)-(146) are applicable regardless of the kinetic status of the heterogeneous charge transfer step, C + ~ D (see below for discussion of quasi-reversible situation). We believe that the qualitative physical interpretation of the result embodied in eqn. (156) is quite simple. When eqns. (143)-(146) are applicable, the second wave occurs at d.c. potentials where the surface concentration of species A is zero. As a result, the rate of the forward homogeneous redox reaction, kl CACD, is zero at the electrode surface and remains small until some distance from the electrode surface where CA becomes appreciable. The reaction rate versus distance profile exhibits a maximum at some distance from the interface, beyond which the decrease in Co suppresses the rate. One concludes that the predominant effect of the homogeneous reaction on the concentration profiles occurs in a "band" located sufficiently far from the interface that it is not penetrated by the highly-damped a.c. concentration wave. Consequently, the homogeneous redox reaction cannot directly alter the a.c. concentration wave, as is required to influence the frequency dependence. Rather, the homogeneous reaction's existence can only be manifested on the a.c. response through its influence on the d.c. concentration components.

The calculations used to generate Figs. 1-4 employed h very small sphericity factor (=(Dt)~/ro) and essentially represent predictions of the expanding plane


2,0 ~ ~ . . ~

~ 1.5 5

kO ' -025 -0.50 -0.75

E d c / V

s~ O5 30

~ 2.0

~ I.O

-0.25 0.50 -0.75 Ed,c./V

Fig. 5. Predicted effect of electrode sphericity on d.c. and a.c. polarograms at second wave with mechanism(I) and k~ ~> k z. Parameter values: Same as Fig. 1, except for k = 2 . 0 x 10 4 M -1 s 1, and sphericity values (S=(Dt)~/ro) shown on Figure.

model. The effect of electrode curvature has been examined carefully for the case under consideration and it is found that effects of this experimental perturbation are conveniently handled. The effects of sphericity on the second wave polarographic current magnitudes can be quite substantial as shown in Fig. 5. Clearly, significant error could result from ignoring sphericity in relating absolute currents to rate parameters. Fortunately, the sphericity effect on the alternating current ratio, R, is much smaller and, more important, a simple linear relationship between R and electrode sphericity is predicted. This is illustrated in Fig. 6 which also shows that the linear profile's slope is a function of ka and Eo.~.. To make use of this result, it is only necessary to recognize that the profile is linear. One need not know the quantitative relationship between slope, kl, and Ed.~. Since electrode sphericity is proportional to m - ~ (ro~m~) , one simply obtains experimental values of the a.c. ratio, R, at several known mercury flow rates, plots the observed R versus m -~ and extrapolates the linear profile to the R intercept. The latter value is the R-value at zero sphericity and can be used to obtain kl from current ratio working curves based on the expanding plane model (e.9., Figs. 3 and 4), thus avoiding the requirement of calculating and utilizing working curves for various sphericities.

Finally, it should be mentioned that the quantitative characteristics of working cu rve s such as shown in Figs. 3 and 4 are dependent on the charge transfer

coefficient, ~z, and the diffusion coefficient magnitudes and ratios, as one would expect. Although the effects on observables are not particularly large, accurate evaluation of kl requires use of working curves for D~ and ~2 values associated

C O U P L E D H O M O G E N E O U S REDOX REACTIONS 1'79

2.0

L5

LO U 0.10 0.20

Ed.C /" A A 0.75

B 02"5 B

C 0,75

O 058

E -0,75 C V 050

_D

-F

0.50 0.40 0.50 0.60

SPHERICITY

k/t mo~ I s-I

2.5 ~ I0 5

6.O xlO 4

2£) x 10 4

2,0 × I04

6O x ~0 ~ 2.0 x IO 4

Fig. 6. Predicted effect of electrode sphericity on a.c, polarographic R-values at second wave with mechanism(I) and k 1 ~> kx. Parameter values: Same as Fig. 1, except for E,, , and sphericity values shown on Figure,

with the system under investigation. This is not a major problem as these parameters can be determined from experimental a.c. and d.c. polarographic observations in absence of the heterogeneous redox reaction (i.e., measurements with species A and C individually), prior to calculation of the working curves.

B. The electrode reaction, C + e ~ - D , is quasi-reversible or reversible. We proceed now to consider the situation where /~ 2 is sufficiently large that the second polarographic wave can be characterized as quasi-reversible or reversible 36. Clearly, many redox couple pairs which might undergo mechanism (I) under the constraints of eqns. (143)-(146) will fall in this classification. Before continuing, it should be recognized that under these conditions the kinetic status of the first charge transfer step, A + e ~ B , has no influence on the second wave, which is why this variable is not considered here.

Calculations as a function of ~. z show that the influence of the homogeneous redox reaction on the d.c. and a.c. polarograms begins to change when /q. 2 is increased to the point where the electrode process becomes quasi-reversible on the d.c. time scale. The effect's magnitude begins to diminish and there are some noticeable variations in its characteristics, e.g., at some potentials the a.c. polarographic current is diminished, rather than enhanced, by the homogeneous reaction. As ,k z is increased further, the homogeneous reaction's influence is continually suppressed until it disappears completely at about the point where the electrode process can be considered reversible in the d.c. sense (e.g., k~ 2/> 10-2 cm s - l ) 36. Figure 7 illustrates these remarks in terms of the kinetic-nonkinetic a.c. current ratio, R, versus d.c. potential. At first, increasing k~. 2 simply shifts the profile in a positive direction without altering its shape or magnitude. This reflects the corresponding shift in the position of the irreversible a.c. polarogram with k~. 2. When k, 2 is made large enough that the d.c. process is quasi-reversible, the R-Ea.c. profile is predicted to undergo significant distortion, with the most noticeable characteristic being the appearance of a minimum at the profile's positive side. Under the latter


circumstances the relationship between the position of the a.c. polarographic peak and the R-Ed.~. profile remains favorable, so that experimental characterization of the profile's shape and plateau magnitude seems to be possible. Further increase in /~. 2 eventually results in a suppression of the profile's minimum and the shift in position with /~. 2 reverses direction (toward negative potentials). The plateau magnitude remains unchanged, and the position of the a.c. peak becomes roughly constant, while the a.c. peak shape becomes sharper. The profile's shift toward negative potentials continues until its rising portion occurs negative of the a.c. polarographic peak and, under the a.c. peak, R = 1. This disappearance of the homogeneous redox reaction's effects occurs when k~ reaches approximately 10- 2 cm s-1, which represents the lower limit for a reversible d.c. process. These results are consistent with Yamaoka's observation that the second wave must be nonreversible to obtain an effect of the homogeneous reaction.

2 . 0 "" . . . . . . . . ks,z = I,OxlO-~ c m S I

. . . . . . ks,2= I.OxlO -5 cm.S -I

. . . . . . . ks,2= I,OxlO -4 crn. S q

ks,2= LOx 10 -3 cm S -I . . ; -~"~"

. . . . . . . ks2= I,OxlO -2 cmS q / ' / " / ' / / . . ' ' / z z . . 1.5 - - . . . . . . . ks, 2 = ]'Ox I0-1 Cm S-I i" / ' I I i i "

/ / / "

l/ s t / / : • p •

/" , / ; - ' .

/ / I . '

1,0 . . . . . . . . . . . . . . . . . . . .

0.5

-o:2s -o'.so -ot75 -,.co

E d . c , / V

Fig. 7. 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 Fig. 1, except for /~. 2-values shown on Figure.

For the realm where the d.c. process is quasi-reversible, calculations support the notion that kl will be calculable from experimental a.c. polarographic data with reasonable accuracy and with only a little more effort than required for the irreversible case. For the quasi-reversible state, working curves such as given in Figs. 3 and 4 must be generated for the Di, ks.2 and ~z values characterizing the electrode reaction under investigation. Standard d.c. and a.c. polarographic methods are applicable to acquiring /%2, D~, and ~2 using measurements performed in absence of the homogeneous redox reaction, preliminary to calculating the working curves. As eqn. (156)suggests, it even would be convenient to evaluate the/~.2 and ~2 parameters from cot ~ measurements in the presence of the homogeneous redox reaction, since the quasi-reversible d.c. state is characterized by a sufficiently large ks. 2-value that cot ~b is a useful observable (not so in irreversible realm) 36.


II. The case where E ° ~ E ° and E{. x >> E{. 2 We consider now the classical "catalytic" case 3s'3s'sl where, despite a

more positive E°-value, the electrode reaction, C + e- ~-~ D occurs at a distinctly more negative potential than A + ~ B (i.e., the polarographic waves are resolved) because C + e ~ - D is highly irreversible. Thus, the situation is the same as that just considered with regard to the relative positions of the polarographic waves due to the reduction of species A and C. However, the direction of the homogeneous redox reaction is reversed (k2~/¢1)' The necessity that C + e ~ - D be highly irreversible to achieve the catalytic mechanism precludes consideration of quasi-reversible and reversible states for this electrode reaction. However, the status of the "first" electrode reaction, A + ~ B, now becomes relevant because in the catalytic situation the first polarographic wave is influenced by the homogeneous redox reaction, as is well known 51. The theory given above encompasses in the context of the catalytic case all conceivable situations, including second-order and pseudo-first-order homogeneous reactions, the case of the reversible homogeneous reaction (discussed in the next section), and all possible kinetic states for the first heterogeneous charge transfer step. These facts, together with the theory's accommodation of the expanding sphere electrode model, makes it a considerably more general treatment of a.c. and d.c. polarography with the catalytic mechanism than previously available.

Some typical calculated d.c. and a.c. polarograms for the case where c~ =c~ are shown in Figs. 8 and 9. The effects of homogeneous reaction on the d.c. polarogram are as expected. Increasing k2 enhances the first wave at the expense of the second until only the first wave is observed. Of course, disappearance of the second wave requires that c* ~>c~. For the case depicted in Fig. 8 where /~. 1 is large, the half-wave potential and shape of the first wave remain relatively constant, with the former shifting slightly in a positive direction for rate constants higher than 5 x 104 M -~ s-~. Also relatively unchanging are the second d.c. polarographic wave shape characteristics, as is made more evident in the log plot

2.0

L)

1.0

0 . 0 0 -0 .25 - 0 . 5 0 - 0 . 7 5

Ed.c. / V

Fig. 8. Predicted d.c. polarograms with mechanism(I) where k t .~ k 2. Parameter values: T=298 K, c * = c ( * = l . 0 0 x l 0 -3 M, D A = D B = D c = D D = 4 . 0 × I O -6 cm 2 s 1, t= l .00 s, 09=250 s -1, E°=0.00 V, E~ .2=-0 .50 V (when k=0), k~.~=oo, k~.2~<10 6 cmz s - i c~2=0.50 ' k-values shown on Figure (k= kl + k2 ~ k2), expanding plane model. Direct and alternating current units same as Figure 1.


shown in Fig. 10. If the first d.c. wave is quasi-reversible, its shape is more sensitive to the homogeneous redox reaction rate. The second d.c. polarographic wave moves to more negative potentials as k2 is increased. Figure 9 shows that the second a.c. polarographic wave is suppressed by the homogeneous redox reaction under catalytic conditions. This result is quite general and is in opposition to the trend shown above where kl >> k2. It is also interesting to note that the first a.c. wave in Fig. 9 is totally uninfluenced by the homogeneous reaction. This turns out to be a result which arises when certain special (but probably not unusual) conditions are applicable, as in Fig. 9. This can be seen from the following considerations.

At the potential of the first wave under catalytic conditions one may write

kz>> kl (161)

Ed.c. ~E¢. I >> E~,.2 (162)

ks. 2"~ 10 -3 cm s -1 (163)

In addition, in Fig. 9 the following conditions apply:

/g.l>> 10 -2 cm s -1 (164)

(i.e., the first d.c. process is Nernstian)

c ~ c 7 , (or < c*) (165)

9 < 1 (t66)

Equations (162) and (163) imply that, in the general theory given above + +

75 = 6~ = 0 (167)

o.5o

I 0

I k = O M - I S ~1

2 k : L O x 1 0 5 M - I S - I ~ "

5 k : 1,0 x 1 0 4 M - I S -I

- 0 . 2 5 - 0 , 5 0 - 0 . 7 5

A B C D

A k : O M I S " j

B k = 2 . 0 X 1 0 3 M - = S " l

C k : 6 0 x I O ~ M I S - I

x [04 M S I = , i -

Fig. 9. Predicted a.c. polarograms with mechanism(I) with kl < k2. Parameter values: Same as Fig. 8.

Fig. 10. Predicted d.c. polarographic log[(ij-i)/i] plots at second wave with mechanism(1) and kl < k2. Parameter values: Same as Fig. 8.

0 . 4 0 - 0 . 6 0 - 0 . 8 0

E~.~./V Ed.~./V


Equation (167) is sufficient to lead to the result, upon evaluating the determinants A, A., etc.

I F2Ac*D~AE F a(t) sin o) t+cot -1 , ~ - ] j (168) J1 ( ot) = R r [(1 + 7 +)2 +

Equation (168), and its subsidiaries, represent the general equation for the first polarographic wave under catalytic conditions (resolved waves). If, in addition, one applies eqn. ( 166), which means G + = 1, and eqns. ( 161) and (165), which lead to the result,

K~ ~ 1 (169)

one concludes that

7+ = 7- = 21/(209) '~ (170)

Finally, invoking eqn. (164), one obtains

Fl(r) = 1 (171)

Substituting eqns. (170) and (171) in eqn. (168) leads to the standard a.c. polarographic wave equation for the simple quasi-reversible case where the d.c. process is Nernstian 36, i.e., there is no influence of the homogeneous redox reaction. However, if any of the special conditions represented by eqns. (164)-(166) are violated, then the simple behavior of the first wave in Fig. 9 is not obtained. With regard to eqn. (165), it should be mentioned that when c*>>c* (Re>> 1), one concludes that

K~ ~> 1 (172)

so that

7 + = 21RoG+/(2o9)" (173)

Substituting eqn. (173) in eqn. (168) and setting the diffusion coefficients equal leads to the well-known expression for the catalytic a.c. polarographic wave with a first-order homogeneous redox reaction 3~5z53 (which was restricted to equal diffusion coefficients).

Figure 11 shows some working curves of the kinetic-nonkinetic current ratio, R, versus d.c. potential for the second a.c. polarographic wave under catalytic conditions. In this case the ratios are less than unity when the homogeneous reaction is kinetically important and the profiles are distinctly more complex than when kl >> kz, showing two distinct plateaus. Clearly, from Figs. 1, 3, 9, and i1 it should be apparent that there is no difficulty in distinguishing between the cases where kl >> k 2 (Yamaoka case) and kl ¢ kz (catalytic case) from the effects of the homogeneous process on the second wave. Interestingly, for a given k-value, the plateau at more negative potentials in Fig. 11 deviates from unity by the same amount (opposite direction) as the plateau in Fig. 3 for k~ >> k2 with rate constants up to about 105 M -I s -~. For higher rate constants there is a small, but discernible, departure from this behavior. In any case, the diagrams of log (Rmax - R ) for the second plateau in Fig. 11 will be almost identical to the corresponding diagrams in Fig. 4.

184 i. RUZI(2, D. E. SMITH, S. W. FELDBERG

1.0

R

0 .5

= 0 M ' IS -I

k = I .OX IO~M-IS - t

K = 2 .0x 103M ' IS - I

k = 6,0 x IOaM-=S -L

k • 2 .0X I04M*Is -I

I~ = 4 . 0 x 104M' Is - I

k " 6 . 0 x 104M' lS - I

- 0 . 25 - 0 . 50 - 0 . 75

Ed.~. / V

Fig. 11. Predicted a.c. polarographic R'values at second wave with mechanism(I) and ks ~ k2. Parameter values: Same as Fig. 8.

The effects of electrode curvature in the catalytic situation have been examined by us in less detail than for the Yamaoka case. However, the following observations can be made.

(a) The effect of sphericity on the first wave is comparable to that found in the normal quasi-reversible a.c. polarogram 3~ 5z 54, 55 when c~ ~c~. The effect diminishes as c~ is increased and, as is known 5z, it becomes negligible under pseudo-first-order conditions where c~ >> c*, unless the product k2c~ t is rather small ( < 10).

(b) Sphericity effects on the second wave are roughly comparable to those found when kl >> k2 (Fig. 6) and it appears that a procedure similar to that recommended above for the Yamaoka case will be applicable.

I l L The case where E ° ~ E ° and E~. 1 ~ E~. 2 We now focus attention on the important case where the standard potentials

of the two redox couples are sufficiently close that both the forward and reverse homogeneous redox processes must be considered (kt ~ k2). At the same time we retain the simplifying conditions that the two polarographic waves are well-resolved. As in the catalytic case, resolved waves will be observed provided that the electrode process C + ~ D is sufficiently irreversible. This example essentially encompasses all possibilities between the extremes of the Yamaoka case (k 1 ~ k2)

and the catalytic case (kl ~ k2). In general terms the case of the reversible Homogeneous reaction is im-

portant, not only because it may be encountered in an electrochemical investigation, forcing one to consider its consequences, but also because its existence may provide some unique measurement opportunities. For example, there are many redox couples whose E°-values are not known because the electrode process is irreversible. The results presented above clearly show that, if one adds to the solution containing the oxidized form of the irreversible redox couple ( C + e ~ D ) a second reducible component whose E ° is known (and very different from the first), there is the possibility of polarographically establishing whether the unknown E ° is greater than or less than the known value, provided that the homogeneous redox reaction

C O U P L E D H O M O G E N E O U S REDOX REACTIONS 185

is facile. One simply observes whether the homogeneous process enhances the irreversible a.c. polarographic wave ( E ° > E °) or suppresses it (E °<E°) . If one extends this idea into the realm where the two standard potentials are comparable, the possibility of quantitatively measuring the unknown standard potential becomes evident, as we shall indicate.

In general, as the homogeneous redox reaction becomes more reversible, its effects on the polarographic response tend to be reduced, relative to the irreversible cases considered above. This is shown for the d.c. polarographic case in Figure 12 where two examples involving fixed k2-values and varying kl-values are depicted. In this case, all homogeneous reaction-induced effects, from the catalytic enhancement of the first wave to the half-wave potential shifts, are suppressed by increasing kl. The effect on the a.c. polarographic kinetic-nonkinetic current ratio at the second (irreversible) wave is illustrated in Fig. 13. The effect of increasing kl,

Z 2.C

LU

0 1.0

8 ~5

~ k I = 0 M-IS -I

k I = 0 M'Is -I ~ ~ / I / /

/ / f k l = 6.0 X 104M'IS "1 ~

SIt I - - k 2 = i , O x l O a l M ' l s - I

- - - k 2 = .Ox 0 M S t t

J , i

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

E d , c . / V

Fig. 12. Predicted d.c. polarograms with mechanism(I) showing effect of homogeneous redox reaction reversibility. Parameter values: Same as Fig. 8, except k~ and k z values shown on Figure.

i.5

(25

- 0 . 2 5 - 0 . 5 0 - 0 , 7 5

E ( J I c . / V

k I = 2 ,0x 104M-IS "~

k I = I . O x 104M-IS -=

k ,= 6 .0XlO3M i S '

k I = 2.OxIO3M-IS "1

k I = I .OXlO3M-Is "~

~l = 0 M * I s - I

Fig. 13. Predicted effect 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= 1.0 x 104 M -~ s -~ and kl-values shown on Figure.

186 I. RUNIC, D. E. SMITH, S. W. FELDBERG

while holding k2 constant, is to raise the R-values at both plateaus. The second (more negative) plateau increases more rapidly than the first, and the latter never exceeds unity. A unity value for the first plateau implies that the condition corresponding to kl ~> k2 has been achieved. Because the a.c. polarographic peak is located favorably with regard to both plateaus in Fig. 13, it should be possible to obtain reasonably accurate estimates of the plateau magnitudes from which the kinetic parameters, kl and k2, can be estimated. In Fig. 14 a diagram is presented of the magnitude of the first versus the second plateau for different combinations of forward and backward rate constants. On this diagram the constant kz-value "contours" are nearly linear, while those for constant ka-values are substantially nonlinear. It can be seen from Fig. 14 that under many conditions the plateau magnitudes are sufficiently sensitive to the rate constant values that diagrams of this type should enable reasonably accurate assessment of kl and k2. Of course, estimates of k~ and k2 obtained in this manner can always be refined by detailed com- parison of experimental profiles, such as shown in Fig. 13, with theoretical predictions. It must be kept in mind that diagrams of the type shown in Fig. 14 must be prepared for the s-value appropriate to the process, C + ~ D.

Recognizing that assessment of kl and k2 is clearly possible under the conditions in question, then evaluation of E2 ° is simply a matter of implementing eqn. (17), provided E ° is known (or vice versa). Having obtained by this approach E ° for an irreversible process, C + e~- ~- D. it is then a simple matter to calculate ks, z for this process (e.9., from the E°-E~.2 difference). Thus, it is possible to evaluate the standard heterogeneous charge transfer rate constant and E°-value for a highly irreversible electrode process, whose E ° is not obtainable by other means (e.g.,

t // / /

2 I I i i

I i i /

I I t 1

I q I I I l l s 0~ I t / / / 4 , 0 x i O

,ItIF,.: ,°. I 4"0 x I0 4

1.0 x I0 5

&OXlO , 4.0 x lO 4

, 2.0 XlO 4

, 1.0 X I0 4

6 .0x I0 s

, 4.0x I0 ~

2,0 x I0 3

1.0 x ~0 ~

0

BACKWARD RATE CONSTANT k 2 / 1 rao[ - I s - I

FORWARD RATE CONSTANT k~/L mot'|s "I

015 Ll.O FIRST PLATEAU MAGNITUDE

Fig. 14. Working curves of first versus second plateau magnitudes in a.c. polarographic R-Ed.c profiles at second wave with mechanism(I). Parameter values: Same as Fig. 8, except kl and k2 values shown in Figure. ( ) Contours for constant k2, ( . . . . . . ) contours for constant kl.


potentiometry), provided one can find a second redox couple of known E ° whose presence will induce a reversible homogeneous redox process in accord with mechanism(I). While the requirements are rather specific, this concept should be applicable in many situations. Of course, there are close analogs to this procedure in homogeneous redox chemistry 56, whereby one obtains E°-values for redox couples whose heterogeneous electrochemistry is intractable, by involving the couple in a facile homogeneous redox reaction with a second redox couple of known E °.

IV. Cases where E~. x ,~E~.2 We have considered above the three main cases which can arise in the

diagnostically convenient situation where the polarographic waves for the two redox couples are totally resolved. The situation where the polarographic waves overlap also is encompassed by the theory presented here. The same three special situations considered above, E ° ~> E °, E ° ~ E2 °, and E ° ~ E2 ° are possible in this situation, given appropriate heterogeneous charge transfer characteristics. Cal- culations for these cases present no special difficulties. However, as one would expect, the nature of the results under these circumstances is much more complex than when the waves are well-resolved and the possible behavioral situations are so numerous that a complete discussion would be unduly lengthy and will not be attempted. Rather, we will limit our remarks on the case of unresolved waves to the following observations.

(a) Calculations for cases involving unresolved waves show that the measurement possibilities found with resolved waves remain applicable. That is, the possibility of obtaining under many conditions the desired kinetic and thermodynamic parameters from d.c. and a.c. polarographic observables clearly exists since the experimental quantities remain sensitive to the various rate and thermodynamic parameters.

(b) The correlation of experimental results with the theoretical rate law for the purpose of calculating the homogeneous reaction's rate parameters will be more difficult than with resolved waves. However, this problem normally will be readily surmounted because many parameters characteristic of the two redox couples (/~ i, ~i, E°) can be ascertained by examining first the polarographic behavior of each redox couple independently. Then, when the polarographic response of the solution containing both couples is analyzed, only a few unknown parameters, such as kl and k2, will remain.

(c) Traditional a.c. polarographic observables such as the frequency response profile and cot q~, which are predicted to be of limited use in the resolved wave situation considered above, are found to be of greater diagnostic utility with unresolved waves.

(d) The homogeneous chemical reaction's effect does not exist when both overlapping polarographic waves are Nernstian.

EXPERIMENTAL REQUIREMENTS

Some remarks regarding experimental plausibility of certain diagnostic procedures were cited above, such as whether useful segments of theoretical working curves coincided with conditions where the polarographic response is significant.

t88 I. RU~;ICl, D. E. SMITH, S. W. FELDBERG

However, some other aspects of the measurement problem should be noted. In particular, it should be recognized that in the majority of instances in which measurement objectives include the parameters kl and kz, experimental observations will be based on an irreversible, or nearly irreversible, a.c. polarographic wave. Such waves are quite small, yet reasonable measurement accuracy is required to properly utilize the diagnostic guidelines given above. Consequently, it is essential that all available precautions be taken to minimize the potentially error-inducing contributions of the double-layer charging current. These precautions should include the standard electronic strategies such as phase-sensitive detection of the in-phase fundamental harmonic, which includes nearly all of the faradaic component with an irreversible wave 57, and/or subtractive a.c. polarography 58. Another simple, but highly useful expedient is to employ somewhat higher-than-normal electroactive species concentrations. Finally, because of its proven effectiveness in essentially totally suppressing double-layer charging current contributions with irreversible waves 59, second harmonic a.c. polarographic measurements represent an appealing experimental alternative to fundamental harmonic studies. For this reason we are considering the extension of the above theoretical approach to the second harmonic component.

CONCLUSIONS

The theoretical treatment of mechanism(I), using a combination of numerical and analytical procedures to solve the boundary value problem, is found to be tractable and to lead to predictions which are at least semi-quantitatively consistent with published experimental observations, including the recent data of Yamaoka 2, The possibility of a.c. polarographic applications to the kinetic characterization of homogeneous redox reactions is clearly indicated for circumstances where the possibility was previously ignored (noncatalytic conditions). The applica- tion requires that one of the two electroactive species undergo an irreversible or quasi-reversible electrode reaction, but numerous possibilities can be found in inorganic and organic systems, despite this restriction. Some rather unique applications are suggested, such as the a.c. polarographic measurement of E ° and /~-values associated with irreversible electrode processes. Although the general theoretical relationships are rather cumbersome, significant simplification results in certain special cases of considerable experimental interest, such as the Yamaoka and catalytic situations. In these instances, simplicity also applies to the theoretical working curves and recommended experimental procedures.

With regard to the electrochemical characterization of homogeneous redox reaction kinetics, it should be recalled that spectroelectrochemical methods pioneered by Winograd and Kuwana 21 have been applied with outstanding success in the context of mechanisms (I) and (II). Whereas the spectroelectrochemical approach can be applied even with reversible electrode reactions, which are intractable to the a.c. polarographic approach described here, the former has special requirements regarding electrode materials and the homogeneous reaction component's spectral response which are not imposed on the a.c. polarographic method. Consequently, we view these two methods as complementary.

In addition to the kinetic-thermodynamic implications, which have been


stressed up to now, there are obvious and important implications for electrochemical analysis, some of which have been expressed previously 2. It is evident that the homogeneous redox reaction can lead to "interferences" in the a.c. polarographic analysis of multi-component systems when an irreversible or quasi- reversible a.c. wave, on which an assay procedure is based, is accompanied by a polarographic wave due to a more easily reduced species. The present theory provides guidelines for correcting for this interference, if desired. It also suggests the advisability of relying on electrode processes which are at least reversible in the d.c. sense when devising a.c. polarographic assay procedures for multi-component systems. The theory also suggests the possibility of analytical applications of this interference, because of the simple proportionality between the concentration of the more easily reduced species, c~, and the magnitude of the homogeneous reaction's effect on the irreversible wave of species C. This effect might be useful as a basis for a.c. polarographic analysis of the more easily reduced depolarizer in special situations, such as when its wave is located at potentials positive of the mercury oxidation wave. It should be recognized that these consequences in analytical applications extend beyond the techniques of conventional d.c. and a.c. polarography, since similar effects are anticipated in other popular electroanalytical procedures, such as stationary electrode polarography and pulse polarography. Indeed, with small amplitude differential pulse polarography, whose theory is related to that presented here by a Fourier transformation and a sampling operator 6°, the observed effects will be very similar to those depicted here.

We are now engaging in experimental studies designed to test and apply the above theoretical rate law. The theoretical procedures invoked here are being extended to mechanisms (II) and (III) mentioned earlier and other closely-related cases, such as the one involving mixed currents 61" 62 in which one of the steps in mechanism (I) is an oxidation.

ACKNOWLEDGMENTS

This work was supported by NSF Grant GP-28748X. The authors are indebted to R. Schwall and H. Yamaoka for helpful suggestions and discussion.

SUMMARY

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

A + e . • B

ks , 2 , ct2

C + e • " D

kl A + D . • C + B

k2

is presented. It is found that the coupling of the two heterogeneous electrode processes by the homogeneous redox reaction under many conditions will lead to significant effects on the a.c. polarographic response which can be exploited to


enable measurement of the rate parameters kt and/or k 2 of the homogeneous redox reaction. Other unique measurement possibilities also arise, such as deduction of the E ° and ks-values for totally irreversible electrode processes. In the extreme where k I >~ k 2, the theoretical predictions confirm the conclusions of Yamaoka regarding the enhancement of an irreversible a.c. polarographic wave by a second, more easily reduced, redox couple. In the opposite extreme (k, ~ k2) , which corresponds to the classical catalytic mechanism, the theory provides for the first time a.c. polarographic rate laws for the case where the homogeneous process is second-order. The theoretical equations are applicable to all combinations of homogeneous and heterogeneous kinetic--thermodynamic parameters, encompassing situations involving both resolved and unresolved polarographic waves. The usual assumption of equal diffusion coefficients is relaxed and the equations developed should account for moderate differences in diffusion coefficient under most circumstances, and large differences if the homogeneous redox reaction is reasonably rapid. Computer programs developed invoke the expanding sphere electrode model. Guidelines for experimental evaluation of various rate and thermodynamic parameters are presented. It is found that some rather simple measurement procedures based on appropriate theoretical working curves will permit evaluation of the homogeneous kinetic parameters. Implications of the results for a.c. polarographic analysis of multi-component systems are considered.

NOTATION

q q, q

q ~x= o)

f t

x

F R T A

~.~

](1, k2

Ed .c. E½.i Eri AE (D

concentration of species i initial concentration of species i d.c. component of concentration of species i a.c. component of concentration of species i 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 i th heterogeneous charge transfer step heterogeneous charge transfer rate constant for i th charge transfer step (at .E °) second-order rate constants for forward and reverse homogeneous redox

reactions (reaction I) standard redox potential for i th charge transfer step (IUPAC Convention) d.c. component of applied potential observed d.c. polarographic half-wave potentials for i th charge transfer step reversible d.c. polarographic half-wave potential for i th charge transfer step amplitude of applied alternating potential angular frequency of applied alternating potential


i(t) i,(r)

7,(t) l(oot)

I l l

I" 0

total faradaic current (cathodic current positive) faradaic current associated with ith charge transfer step d.c. faradaic current component for i th 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

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192 I. RU~.IC, D, E. SMITH, S. W. FELDBERG

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Documents

On the influence of coupled homogeneous redox reactions on electrode processes in d.c. and a.c. polarography: I. Theory for two independent electrode reactions coupled with a homogeneous