Derivation of a general theory for reversible multistep electrode processes in voltammetry with constant potential at spherical electrodes

www.elsevier.nl/locate/elecom Electrochemistry Communications 2 (2000) 267–271

Derivation of a general theory for reversible multistep electrode processes in voltammetry with constant potential at spherical electrodes ´ ´ a, Rafael Chicon ´ b Angela Molina a,*, Carmen Serna a, Manuela Lopez-Tenes a

´ ´ ´ Departamento de Quımica Fısica. Facultad de Quımica, Universidad de Murcia, Espinardo, 30100 Murcia, Spain b ´ ´ Departamento de Fısica, Facultad de Quımica, Universidad de Murcia, Espinardo, 30100 Murcia, Spain Received 25 January 2000; accepted 31 January 2000

Abstract The rigorous theory for a reversible multistep electrode process in voltammetry with constant potential at spherical electrodes is derived. The expressions for the surface concentrations, the concentration profiles of species participating in the different steps and the current as a function of time have been deduced. These expressions are valid for any number of steps in which the initial species is oxidized/reduced, whatever the relative values of the formal potentials of the different steps and for any electrode radius value, including plane electrodes and ultra-microelectrodes. q2000 Elsevier Science S.A. All rights reserved. Keywords: Multistep electrode processes; Voltammetry; Reversible processes; Spherical electrodes; Microelectrodes

1. Introduction The study of electrode processes in which oxidation or reduction takes place in several steps is of great interest and very topical [1–7]. Thus, for example, fullerenes in general and C60 in particular can present up to five reversible oneelectron reductions [8–12]. Nevertheless, the theoretical treatment of this type of process in voltammetry is not easy. Indeed, while the case of a two-step electron transfer (EE mechanism) has been widely dealt with in the literature [1,13–15], the theory corresponding to electrode processes with three or more steps is, to date, not very advanced [3,16,17]. The aim of this paper is to find a general analytical expression for the response of multistep electrode processes in voltammetry with constant potential at spherical electrodes. The use of spherical electrodes, such as the SMDE, for the study of multistep reactions, is very advantageous compared with solid plane electrodes (e.g. a glassy carbon or carbon electrode, a platinum or gold electrode). On the one hand, this electrode has the advantages afforded by its smooth and homogeneous surface and, on the other, which is fundamental in our case, the large overpotential for hydrogen evolution makes mercury the material of choice for cathodic processes [10]. As indicated in Ref. [10]: ‘‘the reduction of fullerenes, * Corresponding author. Tel.: q34-9683-67524; fax: q34-9683-64148; e-mail: [email protected]

especially C60, should be performed at sufficiently negative potentials if one wants to generate highly charged anions (for example C606y and C706y). Therefore, the mercury electrode is very suitable for investigating the electrochemical behavior of fullerenes’’. As indicated above, although the solution for this type of problem is difficult in general, certain situations do exist where it is easy to obtain such expressions for these cases. In these situations, a problem of k steps can be treated as k independent problems of only one step. To this end, we must demonstrate that the boundary value problem can be expressed in the same general form for any step; i.e. the superposition principle is fulfilled [18,19]. This is the case when all steps are reversible and the diffusion coefficients of all species are equal when spherical diffusion is considered [20]. Taking into account the above considerations, we have deduced general expressions for the surface concentrations of species participating in the different steps, concentrations in terms of distance and time (concentration profiles) and current as a function of time for the spherical stationary electrode model. These expressions are valid for any number of steps considered in the multistep electrode process and whatever the relative values of the formal potentials of the different steps. The equations obtained are also valid for any electrode radius value, including plane electrodes (r0™`) and ultramicroelectrodes (r0™0). Thus, the equations previously

1388-2481/00/$ - see front matter q2000 Elsevier Science S.A. All rights reserved. PII S 1 3 8 8 - 2 4 8 1 ( 0 0 ) 0 0 0 1 9 - 9

Thursday Mar 02 11:57 AM

StyleTag -- Journal: ELECOM (Electrochemistry Communications)

Article: 196

268

A. Molina et al. / Electrochemistry Communications 2 (2000) 267–271

obtained in the literature for an EE mechanism at plane electrodes [13] and spherical microelectrodes [15] and for an EEE mechanism in plane electrodes [16] are easily deduced from our general equations.

above initial and limiting conditions for ks1 and ks2, it can easily be demonstrated that (see Eqs. (A6), (A13) and (A14) in Appendix A): kq1

U

8 ckO (r0 ,t)scO i

;k

1

(7)

is1

2. Theory The following mechanism represents the reduction of species O1 in k steps step 1

O1qn1 eylO 2

E019

2

O2qn 2 eylO 3

E029

n

n

j

Ojqnj eylOjq1 E0j 9

n

n

and taking into account the surface conditions given by Eqs. (5) and (7) we obtain (;k) the following expressions, which are dependent only on the applied potential E, for the surface concentrations of species Oi (is1, 2,«, kq1) (see Scheme (I)). cU O1

k

k

(iskq1)

ž /

1q 8 2 Jf

(I)

ps1 f sp k

2 Jf

0 k

cU O1

k

k

(is1, 2,«, k)

ž /

1q 8 2 Jf

Okqnk e lOkq1 E 9

ps1 f sp

0 j

W

T T

(8)

X

f si

ckOi(r0)s y

k

1

ckOkq1(r0)s

Y

where E 9 (js1 to k) is the formal potential for the corresponding step, all of which are totally independent, and nj is the number of electrons transferred in step j. If we suppose that (a) only species O1 is initially present in solution, (b) diffusion coefficients of all species are equal (sD), (c) all steps are reversible and (d) spherical semiinfinite diffusion to and from the surface of the electrode takes place, then we have:

The independence of time of the surface concentrations shown in these expressions allows us to carry out the generalization of the solution for this problem to any number of electrochemical steps, k. Thus, by applying the superposition principle (see Appendix A) we find the following expressions for the concentration profiles:

dˆ ckOi(r,t)s0

U ckO1(r,t)scO qF1k 1

(is1, 2,«, kq1)

(1)

2

≠ ≠ 2 ≠ with dˆ s yD q 2 ≠t ≠r r ≠r

ž

/

(2)

with the superindex k being the number of steps of the electrode process and the subindex i referring to the species considered. It can be appreciated that the boundary value problem can be expressed by the following general equations for any value of k (see Appendix A) ts0, rGr0 t)0, r™`

∂

8 is1

ž /

T X is2, 3,«kq1T

(9)

Y

k

k

ž / 8ž /

ps2 f sp

k 1

F sy

k

k

cU O1

(10)

2 Jf

ps1 f sp

(3)

k

2 Jf cU O1

f si

Fiks

k

k

is2, 3«k

(11)

ž /

1q 8 2 Jf s0

ps1 f sp

(4)

rsr0

k Oj

c (r0 ,t)sJj ckOjq1(r0 ,t)

(js1, 2,«, k)

(5)

Fkkq1s

1 k

k

cU O1

(12)

ž /

1q 8 2 Jf

with Jjsexp

/

W

1q 8 2 Jf

k cO s0 i

t)0, rsr0: ≠ckOi ≠r

ž

/

where

(is2, 3,«, kq1)

kq1

ž

r ryr0 c (r,t)sF erfc r 2(Dt)1/2 k 0 i

k Oi

1q U ckO1scO , 1

r0 ryr0 erfc r 2(Dt)1/2

ps1 f sp

ž

nj F (EyE0j 9) RT

/

(6)

By solving the differential equations system (1) with the

Thursday Mar 02 11:57 AM

Once the concentration profiles are available, the expression for the current can be deduced. So, for mechanism (I) it holds that

StyleTag -- Journal: ELECOM (Electrochemistry Communications)

Article: 196

A. Molina et al. / Electrochemistry Communications 2 (2000) 267–271

Ik1 ≠ckO1 sD n1 FA ≠r

ž /

Ik s FA

W rsr0

T ž / T /

Ik2 Ik1 ≠ckO 2 y sD n2 FA n1 FA ≠r

ž /

n k ky1

rsr0

= (13)

y

k k

k Okq1

I ≠c sD nk FA ≠r

ž

rsr0

1/2

I D sy nj FA pt

1/2

ž/ž

1q

(pDt) r0

/8

Fpk

ps1

(14)

ps1

D is1

G

∂

1/2

(n1qn 2qn1 J2)

(18)

I 1qn1 J2 r(n1qn 2) 2 s Ilim 1qJ2qJ1 J2

(19)

where I2lim is the current for E approaching y` (the limiting current), which is given by Eq. (18) with J1sJ2s0:

k

I ksIk1qIk2q«qIkks 8 Ikj

(15)

js1

Thus, from Eqs. (14) and (15) we obtain the following expression for the current corresponding to an electrode process with k steps at spherical electrodes: 1/2

ž/ž

1q

(pDt)1/2 r0

cU O1

/ 8ž / k

k

2 Jf

1q

k

ky1

B

ps1

ps1

D is1

p

k

E

2 JfF 8 n pq 8 C 8 n if spq1 G

D I sAFcO1(n1qn 2) pt

1/2

ž /

U

2 lim

(20)

Eq. (19) is coincident with Eq. (11) in Ref. [13]. (b) From Eq. (17) with ks3 (EEE mechanism) we obtain: I3 cU D O1 s FA 1qJ3qJ2 J3qJ1 J2 J3 pt

1/2

ž /

(n1qn 2qn 3 (21)

q(n1qn 2)J3qn1 J2 J3)

ps1 f sp

µ

js1

µ

E

k

p

Eq. (18) can be rewritten as

where the functions Fpk are given by expressions (10)–(12). The total current is

=

B

2

j

js1, 2,«, k

Ik D s FA pt

ky1

ž /

Taking into account Eqs. (9) and (13), we obtain: k j

k

(17)

2 JfF 8 njq 8 C 8 n i f spq1

I2 cU D O1 s FA 1qJ2qJ1 J2 pt

Y

rsr0

ž / ž / k

1q 8 2 Jf

This expression is valid for any number of electrochemical steps k at a plane electrode. For multistep electrode processes, two particular cases of Eq. (17) have been treated in the literature: (a) For ks2 (EE mechanism) Eq. (17) simplifies to

k Ok

I I ≠c y sD nk FA nky1 FA ≠r

k

1/2

D pt

ps1 f sp

X

k k

cU O1

269

(16)

∂

This equation, for n1sn2sn3s1, becomes coincident with that previously derived by Lovric in Ref. [16] by making the diffusion coefficients of all species equal in this reference. 3.2. Ultramicrospherical electrode

Eq. (16) is valid for any value of k and whatever the relative values of the formal potentials E0j 9 (js1 to k) (see Eq. (6)) and of the electrode radius.

The expression of current for a multistep electrode process in ultramicrospheres, when the steady state is reached, can be deduced from Eq. (16) setting r0™0: Ik s FA

cU O1 k

k

ž /

1q 8 2 Jf

D r0

ps1 f sp

3. Results and discussion = We have analysed the above current expression in the following particular cases.

(22)

k

ky1

B

js1

ps1

D is1

µ

p

k

E

2 JfF 8 njq 8 C 8 n if spq1 G

∂

The particular case of Eq. (22) for an EE mechanism (ks2) has been studied in the literature. In this case:

3.1. Planar electrode

I2 cU D O1 s (n1qn 2qn1 J2) FA 1qJ1qJ1 J2 r0

From Eq. (16) with r0™` we obtain the following expression for the current:

Eq. (23), with n1sn2sn and J1sJ2 (i.e. E019(E029) becomes coincident with Eq. (8) in Ref. [15].

Thursday Mar 02 11:57 AM

StyleTag -- Journal: ELECOM (Electrochemistry Communications)

(23)

Article: 196

270

A. Molina et al. / Electrochemistry Communications 2 (2000) 267–271

From the general expression for the current (Eq. (16)) we have analysed the influence of electrode sphericity [(Dt)1/2/r0] on the response obtained for a multistep process. 1/2 Fig. 1 shows the I/E curves (I krFAcU versus Ey O (D/p) 1

E019) for a process with six successive electrochemical steps (ks6). The values of formal potentials for the successive reductions and the diffusion coefficient have been taken from the experimental results in Ref. [10] for the reduction of fullerene C60. In this case E0194E0294E0394E0494E0594E069 and well-separated one-electron waves are obtained. From this figure it is clear that the current increases strongly when r0 decreases and it is also shown how the steady state is approached when r0 assumes values lower than 5=10y4 cm for tG0.1 s. So for r0s10y4 cm, the relative difference between the limit currents corresponding to any step is less than 5% when the time varies between 0.1 and 1 s (see plot c).

Acknowledgements The authors greatly appreciate the financial support pro´ General de Investigacion ´ Cientıfica ´ vided by the Direccion y ´ ´ Tecnica (Project No. PB96-1095) and by the Fundacion SENECA (Expedient No. 00696/CV/99).

Appendix A The theory presented in this work for a multistep electrode process is a generalization for any number of steps k of the equations obtained for ks1 (E mechanism) and ks2 (EE mechanism).

A.1. E mechanism (k51)

O1qn1 eylO 2

E019

Eq. (1) for ks1 is

dˆ c1O1sdˆ c1O 2s0

(A1)

and the boundary value problem is given by (Eqs. (3)–(6) with ks1) ts0, rGr0 t)0, r™`

∂

U c1O1scO , 1

1 cO s0 2

(A2)

t)0, rsr0: ≠c1O1 ≠r

≠c1O 2 ≠r

ž / ž / q

rsr0

s0

(A3)

rsr0

c1O1(r0 ,t)sJ1 c1O 2(r0 ,t)

(A4)

The resolution of the differential equations system (A1) with the above conditions (A2–A4) [21] gives the following expressions for the concentration profiles of species O1 and O 2: 1 cO (r,t) 1

s1y

1 r0 ryr0 W erfc 1qJ1 r 2(Dt)1/2

ž

c1O 2(r,t) 1 r0 ryr0 s erfc U cO1 1qJ1 r 2(Dt)1/2

ž

/T X

/

(A5)

T Y

From these equations it is fulfilled that k

U

1/2

0 1

Fig. 1. Curves I/E (I rFAcO1(Drp) vs. EyE 9) for a multistep electrode process with six successive reductions (ks6) obtained for a spherical electrode (Eqs. (6) and (16)). Ts291 K, njs1, Ds6.6=10y6 cm2 sy1. The formal potentials (in mV) relative to AgQRE [10] are: E019sy620, E029sy1020, E039sy1520, E049sy2010, E059sy2490, E069sy2860. The values of r0 (in cm) are: (a) 10y3, (b) 5=10y4, (c) 10y4. The values of time are on the curves.

Thursday Mar 02 11:57 AM

c1O1(r,t)qc1O 2(r,t)scU O1

(rGr0)

(A6)

For the particular case rsr0 (electrode surface), from Eq. (A5) we obtain the expressions for the surface concentrations (see Eq. (8)). The expression for the current is given by Eq. (16) with ks1 [21].

StyleTag -- Journal: ELECOM (Electrochemistry Communications)

Article: 196

A. Molina et al. / Electrochemistry Communications 2 (2000) 267–271

(8) with ks2), and the expression obtained for the current is given by Eq. (16) for ks2. Thus, from the results of Eqs. (A6) and (A13) and by applying the simple induction method, it is easily demonstrated that for any number of steps k:

A.2. EE mechanism (k52)

O1qn1 eylO 2

E019 E029

O 2qn 2 eylO 3

kq1 i

(A7)

and the boundary value problem is now (Eqs. (3)–(6) with ks2)

∂

U c2O1scO , 1

2 2 cO scO s0 2 3

≠c ≠r

≠c ≠r

q

rsr0

2 O3

q

≠c ≠r

rsr0

s0

(A9)

rsr0

c2O1(r0 ,t)sJ1 c2O 2(r0 ,t)

(A10)

2 c2O 2(r0 ,t)sJ2 cO (r0 ,t) 3

(A11)

We now have a system of differential equations (Eq. (A7)) and a set of boundary conditions (Eqs. (A8), (A9), (A10) and (A11)) with a new unknown, but they are formally identical to those of the E mechanism (ks1) (Eqs. (A1), (A2), (A3) and (A4)). Thus, the resolution of differential equations system (A7) is analogous to that of Eq. (A1) and we have deduced (by applying the dimensionless parameters method [22]) the following expressions for the concentration profiles of species O1, O2 and O3: c2O1(r,t) 1qJ2 r0 ryr0 W s1y erfc U cO1 1qJ2qJ1 J2 r 2(Dt)1/2

ž

2 O2

c (r,t) J2 r0 ryr0 s erfc U cO1 1qJ2qJ1 J2 r 2(Dt)1/2

ž ž

c2O 3(r,t) 1 r0 ryr0 s erfc cU 1qJ2qJ1 J2 r 2(Dt)1/2 O1

/T

/ /T X

(A12)

Y

Once again we see that U

8 c2O (r,t)scO i

(A14)

and, therefore, as a particular case of this equation for rsr0, Eq. (7) in this paper is fulfilled. The introduction of equation (7) as boundary condition considerably simplifies the resolution of the problem, which can be performed in a general way following the procedure outlined in the theory.

References 2 O2

ž / ž / ž /

3

(rGr0)

(A8)

t)0, rsr0: 2 O1

1

is1

dˆ c2O1sdˆ c2O 2sdˆ c2O 3s0

t)0, r™`

U

8 ckO (r,t)scO

In this case, Eq. (1) is

ts0, rGr0

271

1

(rGr0)

(A13)

is1

is fulfilled. At the electrode surface (rsr0), Eq. (A12) simplifies to those corresponding to surface concentrations (Eq.

Thursday Mar 02 11:57 AM

[1] Z. Galus, Fundamentals of Electrochemical Analysis, 2nd ed., Ellis Horwood, Chichester, 1994, Ch. 6 and Refs. therein. ´ [2] A. Molina, J. Gonzalez, C. Serna, F. Balibrea, J. Math. Chem. 20 (1996) 151. [3] C. Amatore, S.C. Paulson, H.S. White, J. Electroanal. Chem. 439 (1997) 173. [4] M. Rueda, in: R.G. Compton, G. Hancock (Eds.), Research in Chemical Kinetics, vol. 4, Blackwell, Oxford, 1997, p. 31. [5] D.A. Harrington, J. Electroanal. Chem. 439 (1997) 173. [6] D.A. Harrington, J. Electroanal. Chem. 449 (1998) 29. [7] F. Prieto, M. Rueda, R.G. Compton, J. Electroanal. Chem. 474 (1999) 60. [8] D. Dubois, G. Moninot, W. Kutner, M.T.J. Jones, K.M. Kadish, J. Phys. Chem. 96 (1992) 7137. [9] M.M. Khaled, R.T. Carlin, P.C. Trulove, G.R. Eaton, S.S. Eaton, J. Am. Chem. Soc. 116 (1994) 3465. [10] G. Diao, Z. Zhang, J. Electroanal. Chem. 414 (1996) 177. ´ Szucs, ´ A. ¨ ´ J. Elec¨ [11] M. Csiszar, M. Tolgyesi, J.B. Nagy, M. Novak, troanal. Chem. 441 (1998) 287. [12] M.R. Anderson, H.C. Dorn, S.A. Stevenson, S.M. Dana, J. Electroanal. Chem. 444 (1998) 151. [13] I. Ruzic, J. Electroanal. Chem. 52 (1974) 331. ´ [14] J. Galvez, R. Saura, A. Molina, T. Fuente, J. Electroanal. Chem. 139 (1982) 15. [15] Q. Zhuang, H. Chen, Chin. Chem. Lett. 3 (1992) 217. [16] M. Lovric, J. Electroanal. Chem. 102 (1979) 143. [17] S. Komorsky, M. Lovric, J. Electroanal. Chem. 112 (1980) 169. [18] F.J. Fritz, Partial Differential Equations Applied Mathematical Sciences, vol. 1, Springer, New York, 1982. [19] H.F. Weinberger, Ecuaciones Diferenciales en Derivadas Parciales, ´ Barcelona, 1992. Reverte, [20] A. Molina, C. Serna, L. Camacho, J. Electroanal. Chem. 394 (1995) 1. [21] A.M. Bond, K.B. Oldham, J. Electroanal. Chem. 158 (1983) 193. ´ Czech. J. Phys. 2 (1953) 50. [22] J. Koutecky,

StyleTag -- Journal: ELECOM (Electrochemistry Communications)

Article: 196