Study of multistep electrode processes in triple potential step techniques at spherical electrodes

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Electrochemistry Communications 7 (2005) 751–761

Study of multistep electrode processes in triple potentialstep techniques at spherical electrodes

Angela Molina a,*, Manuela Lopez-Tenes a, Carmen Serna a,Marien M. Moreno a, Manuela Rueda b

a Departamento de Quımica Fısica, Facultad de Quımica, Universidad de Murcia, Espinardo, 30100 Murcia, Spainb Departamento de Quımica Fısica, Universidad de Sevilla, C/Profesor Garcıa Gonzalez no. 2, Sevilla 41012, Spain

Received 6 April 2005; accepted 25 April 2005

Available online 2 June 2005

Abstract

In this paper, a reversible multistep electrode process is studied by the most popular triple pulse techniques as Reverse Differen-

tial Pulse Voltammetry and Double Differential Pulse Voltammetry. The equations presented here are valid for spherical electrodes

of any size (including planar and ultramicrospherical electrodes as limit cases) and can be used to describe the electrochemical

behaviour of molecules with multiple redox centers, whether they are interacting or not. The comparison between results in triple

and double pulse techniques for reversible multistep processes allows us to draw important conclusions regarding the suitability of

such techniques and the characterization of reversible processes.

� 2005 Elsevier B.V. All rights reserved.

Keywords: Multistep electrode processes; Multicenter molecules; Triple pulse voltammetry; RDPV; DDPV; Spherical electrodes; Reversibility

criterion

1. Introduction

Electrochemical techniques based on a succession of

potential steps are widely used nowadays because they

offer great sensitivity and selectivity, a small contribu-

tion of capacitive current, and the possibility of avoiding

the influence of natural convection in the response [1,2].

Modern digital equipments allow the experimentalist toprogram a wide variety of potential step perturbations,

but, for the information pertaining to the electrode

reaction under study to be obtained, the appropriate

theoretical support to analyze the experimental response

must be available.

However, the analytical theoretical support is not al-

ways readily available because the inclusion of each new

1388-2481/$ - see front matter � 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.elecom.2005.04.032

* Corresponding author. Tel.: +34 968367524; fax: +34 968364148.

E-mail address: [email protected] (A. Molina).

pulse inevitably complicates the mathematical formula-

tion of techniques and, of course, the difficulty increases

when the electrode process implies more than one

electrochemical step, i.e., when a multistep process is

considered. These complex multielectron transfer mech-

anisms are very common in real electrochemical systems

and their response to several electrochemical perturba-

tions has been developed [1–10].Triple pulse techniques as are Reverse Differential

Pulse Voltammetry (RDPV) [11–15] and Double Differ-

ential Pulse Voltammetry (DDPV) [12,14–16] have re-

cently been developed. The advantages of these

techniques, among other important features, are that

they permit an easy discrimination between different pro-

cesses, they are very sensitive to the presence of amal-

gamation, they offer a great accuracy in thedetermination of the formal potential and they present

an insignificant charging current, which leads to low

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752 �A. Molina et al. / Electrochemistry Communications 7 (2005) 751–761

limits of detection. Initially the equations for reversible,

irreversible and quasi-reversible electrode processes

under planar diffusion conditions were deduced [12,13],

but still analytical solutions for other mechanisms are

lacking.

Thus, the aim of this paper is to analyze the behav-iour of a multistep process with reversible electrochem-

ical reactions when a sequence of three potential

pulses is applied to a spherical electrode of any size.

For that, the equation recently deduced for this mecha-

nism under multipotential step technique is assayed [10].

Only nernstians conditions are considered not only for

the sake of mathematic simplicity but also because many

real and relevant systems showing nernstian behaviourcan be encountered, namely: ferrocene (one step) [17],

alquil-metil viologens (two steps) [18], natural carote-

noids [19,20], fullerenes [21–23] and other multicenter

molecules [10,24–26] (three to many steps).

The electrochemical behaviour of these kind of sys-

tems in the well known triple pulse techniques RDPV

and DDPV is described. The results have been com-

pared with those in several double pulse potential tech-niques. The study reveals important conclusions in

relation to the criteria of reversibility and, on the other

hand, the comparison with the double pulse techniques

allows the selection of the simplest technique which pro-

vides the thermodynamic and kinetic information con-

cerning the process under study.

2. Expressions of the current for first, second and third

potential pulses

Consider the reduction of a molecule containing n

electroactive redox centers according to Scheme 1, in

such a way that there are n redox couples On� j + 1Rj� 1/

On� jRj with formal potentials E00

j ðj ¼ 1; 2; . . . ; nÞ and(n + 1) possible redox states On� jRj (j = 0, 1, 2, . . .,n).Note that subscript j denotes the number of reduced

sites in the molecule.

In this paper, we study the multistep process given in

Scheme 1 in triple pulse techniques, i.e., when three suc-

cessive potential pulses E1, E2 and E3 are applied to a

spherical electrode of radius r0 during t1, t2 and t3,

respectively. By using an equation for the current corre-

sponding to the application of any potential pulse p

Scheme 1.

(p = 1, 2, 3, . . .), deduced in a recent paper (Eq. (1) in

[10]), we can write:

I1 ¼ FA

ffiffiffiffiDp

rUðt1Þ

Xn�1

j¼0

ðn� jÞ c�j � c1j ðr0Þh i

; ð1Þ

I2 ¼ FA

ffiffiffiffiDp

rUðt1 þ t2Þ

Xn�1

j¼0

ðn� jÞ c�j � c1j ðr0Þh i(

þUðt2ÞXn�1

j¼0

ðn� jÞ c1j ðr0Þ � c2j ðr0Þh i)

; ð2Þ

I3 ¼ FA

ffiffiffiffiDp

rUðt1 þ t2 þ t3Þ

Xn�1

j¼0

ðn� jÞ c�j � c1j ðr0Þh i(

þ Uðt2 þ t3ÞXn�1

j¼0

ðn� jÞ c1j ðr0Þ � c2j ðr0Þh i

þUðt3ÞXn�1

j¼0

ðn� jÞ c2j ðr0Þ � c3j ðr0Þh i)

; ð3Þ

where F, A and D have their usual meaning and U(t) isgiven by

UðtÞ ¼ 1ffiffit

p þffiffiffiffiffiffiffipD

p

r0; ð4Þ

and where cpj ðr0Þ ðp ¼ 1; 2; 3Þ denotes the surface con-

centrations of the different oxidation states of the mole-cule corresponding to the applied potential Ep [10]:

cpj�1ðr0Þ ¼c�Qn

f¼jJpf

1þPn

j¼1

Qnf¼jJ

pf

� � ; j ¼ 1; 2; . . . n; ð5Þ

cpnðr0Þ ¼c�

1þPn

j¼1

Qnf¼jJ

pf

� � ð6Þ

with

Jpj ¼ exp

F ðEp � E00

j ÞRT

!ð7Þ

and c* is the initial concentration of the multicenter

molecule. In Eqs. (1)–(3), c�j ¼ 0 for j 6¼ 0 if the molecule

On is initially present in solution (i.e., c� ¼ c�0), and for

j 6¼ n when the molecule Rn is initially present (i.e.,c� ¼ c�n).

Eqs. (1)–(3) present the following features:

� They are applicable when: (a) all the electrode reac-

tions in Scheme 1 are reversible; (b) the diffusion coef-

ficients (=D) for the (n + 1) possible oxidation states

are equal; (c) no other chemical reactions between the

different oxidation states than those of dis-compropor-tionation take place. If conditions a and b hold, these

dis-comproportionation reactions have absolutely no

effect on the I/E curves for any single or multipotential

step voltammetric technique when linear, spherical or

cylindrical semi-infinite diffusion is considered [9].

�A. Molina et al. / Electrochemistry Communications 7 (2005) 751–761 753

� They are applicable without restrictions on the dura-

tion of the first (t1), second (t2) and third (t3) potential

pulses and hence, they can be used for any designed

triple pulse technique.

� They are valid whatever the values of the formal

potentials E00

j ðj ¼ 1; 2; . . . ; nÞ (see Scheme 1), i.e.,they can be used to describe the electrochemical

behaviour of molecules with multiple redox centers,

whether they are interacting or not [10].

� In Eqs. (1)–(3), Ip (for p = 1,2,3) is transformed into

that corresponding to a planar electrode, Iplanep , by

making r0 ! 1, i.e., rejecting the termffiffiffiffiffiffiffipD

p=r0, and

into that corresponding to an ultramicrospherical

electrode, Imicrop , when r0 ! 0, i.e., neglecting the term

1=ffiffit

p. In this last case Eqs. (1)–(3) are heavily simpli-

fied to

Imicrop ¼ FAD

r0

Xn�1

j¼0

ðn� jÞ c�j � cpj ðr0Þh i

; p ¼ 1;2;3. ð8Þ

Note that the current for a plane electrode Iplanep de-

pends on all the potentials applied up to the poten-

tial pulse p. However, the current for small

electrodes Imicrop is independent of time and hence,

it is only dependent on the potential step which is

being applied, Ep.� A final and very important characteristic of Eqs. (1)–

(3) is that the current Ip (p = 1,2,3) consists of p

addends, each one being the product of two contribu-

tions, namely, a factor U(t) depending on the time

and the electrode radius (Eq. (4)), identical to that

obtained for a simple charge transfer reaction (E pro-

cess) [14], and another factor containing the surface

concentrations of the different oxidation states ofthe molecule, which depend on the potentials applied

(Eqs. (5)–(7)). This special feature concerning to

Technique RDPV

Perturbation

Time

Pote

ntia

l

E1

E2

t1t2

E3

t3

E

1 2 3t t t , 2 3,t t

Response RDPV 3 2I I I

2 2,E tEquivalences

3 3,E t

∆

+ ∀

= −

>>

Scheme 2

reversible multistep processes has repercussions on

the behaviour of such processes in differential tech-

niques (see below).

3. Triple pulse electrochemical techniques

In this section, we will apply the general Eqs. (1)–(3)

to deduce the I/E response for a reversible multistep pro-

cess in the particular cases of the well known triple pulse

techniques Reverse Differential Pulse Voltammetry

(RDPV) [11–15] and Double Differential Pulse Voltam-

metry (DDPV) [12,14–16].

3.1. Reverse Differential Pulse Voltammetry

In this technique, the condition t1 � (t2 + t3) is fulfilled

(see Scheme 2). The value of the first applied potential E1

is kept constant at a value for which the process is con-

trolled by diffusion and E2 varies from E1 as in the RP

technique [1,2,27]. The difference DE = E3 � E2 is keptconstant and the RDP current I3 � I2 is recorded versus

E2, with I3 and I2 beingmeasured at times t3 and t2, respec-

tively [11–15]. The response obtained may present from

one to n peaks, depending on the relative values of

E00

j ðj ¼ 1; 2; . . . ; nÞ (see Fig. 1).Thus, by subtracting Eqs. (3) and (2) we obtain

w ¼ ½Uðt2 þ t3Þ � Uðt2Þ�Xn�1

j¼0


þ Uðt3ÞXn�1

j¼0


; ð9Þ

with

DNPV

Time

Pote

ntia

l

t1

E2

t2

EE1

1 2,t t

DNPV 2 1I I I

1 1,E t

2 2,E t

∆

∀

= −

.


w ¼ I3 � I2FA

ffiffiffiffiffiffiffiffiffiD=p

p . ð10Þ

According to the above explanation for the RDP tech-

nique, the following cases are possible:

(a) E1 ! �1, when initially all the electroactive

centers in the molecule are oxidized (On species

in Scheme 1). From Eqs. (9) and (5)–(7) weobtain

E

wRDPVðOnÞ ¼ � ½Uðt2 þ t3Þ � Uðt2Þ�

�Xn�1

j¼0

ðn� jÞc2j ðr0Þ þ Uðt3Þ

�Xn�1

j¼0


. ð11Þ

(b) E1 ! 1, when initially all the electroactive centers

in the molecule are reduced (Rn species in Scheme1). From Eqs. (9) and (5)–(7) we have now

wRDPVðRnÞ ¼ ½Uðt2 þ t3Þ

� Uðt2Þ� nc� �Xn�1

j¼0

ðn� jÞc2j ðr0Þ !

þ Uðt3ÞXn�1

j¼0


.

ð12Þ

To deduce the cathodic and anodic limiting currents

from Eqs. (11) and (12) we make E2, E3 ! �1 and

E2, E3 ! 1, respectively. Thus, we obtain:

Technique DDPV

Perturbation

Time

Pote

ntia

l

t1t2

E2

E3

t3

2E

1EE1

1 2 3t t t , 33 22 /t t

Response DDPV 3 2 12I I I I

1 3,E E

2Equivalences

3t

∆

∆

>> + =

= − +

Scheme 3

wl.a.RDPVðOnÞ ¼ wl.c.

RDPVðRnÞ ¼ 0; ð13Þwl.a

RDPVðOnÞ ¼ �wl.c.RDPVðRnÞ

¼ �½Uðt2 þ t3Þ � Uðt2Þ�nc�. ð14Þ

Moreover, from Eqs. (11) and (12) we deduce that

wRDPVðRnÞ � wRDPVðOnÞ ¼ ½Uðt2 þ t3Þ � Uðt2Þ�nc� ð15Þand therefore, for given values of t2, t3 and c*, the differ-

ence between the responses is constant at any potential,

i.e., the response wRDPV(Rn) is exactly the same as theRDPV wave for On, only displaced in the w-axis, as

established by Eq. (15) (see Fig. 4). This finding is of

great interest since it signifies that the curves corre-

sponding to the RDP technique when molecule Rn is ini-

tially present in the solution can be easily obtained from

those corresponding to when molecule On is initially in

solution.

Fig. 1 illustrates the form of the response for a six-step process in the RDP technique, both with DE < 0

and DE > 0, when species On is initially present in solu-

tion (wRDPVðOnÞ=ðE2 � E00

1 Þ, Eq. (11)). Fig. 1(a) corre-

sponds to the reduction of fullerene C60, and has been

plotted using the experimental values for

E00

j ðj ¼ 1; 2; . . . ; 6Þ and D given in [21]. Owing to the

particular values of the formal potentials, six well-

defined peaks are registered, with each one correspond-ing to a reversible charge transfer of one electron. In

contrast, Fig. 1(b), which corresponds to the reduction

of a hypothetical molecule with six noninteracting cen-

ters, shows only one peak with the shape expected for

a nernstian one-electron transfer, but with the current

amplified by a factor of six [10,24]. In this case, the val-

ues of the formal potentials have been calculated with

Eq. (1) in [24] or Eq. (13) in [10] (see legend in Fig. 1).

ADPV

Time

Pote

ntia

l

t1

E1

E2,c

t2

0E

0E

E1

t2

E2,a

t1

1 2t t

ADPV 2, 1 2,2c aI I I I

2,aE , 2,cE

1E

2t

∆

∆

>>

= − +

.

Fig. 1. Reverse Differential Pulse Voltammograms wRDPVðOnÞ=ðE2 � E00

1 Þ (Eq. (11)) obtained for a spherical electrode of radius r0 = 0.025 cm and

|DE| = 50 mV. t1 = 3 s, t2 = 0.1 s, t3 = 0.05 s, c* = 1 mM, T = 298 K: (a) reversible six-step process corresponding to reduction of fullerene C60, whose

values of formal potentials (in mV) relative to AgQRE and the diffusion coefficient [21] are: E00

1 ¼ �620; E00

2 ¼ �1020; E00

3 ¼ �1520;

E00

4 ¼ �2010; E00

5 ¼ �2490; E00

6 ¼ �2860, D = 6.6 · 10�6 cm2 s�1; (b) reversible reduction of a molecule with six noninteracting redox centers.

Thus, according to Eq. (13) in [10]: DE00

j ¼ E00

jþ1 � E00

j ¼ RTF ln jðn�jÞ

ðjþ1Þðn�jþ1Þ

h i; j ¼ 1; 2; . . . ; n� 1, the values of DE00

j (in mV) are:

DE00

1 ¼ DE00

5 ¼ �22.43; DE00

2 ¼ DE00

4 ¼ �16.11; DE00

3 ¼ �14.74, D = 10�5 cm2 s�1.


3.2. Double Differential Pulse Voltammetry (DDPV)

In this technique we successively apply three potential

steps, E1, E2 and E3, during t1, t2 and t3, respectively,

with the condition t1 � (t2 + t3), and the differences

DE1 = E2 � E1 and DE2 = E3 � E2 are of equal sign

(see Scheme 3). Thus, DI1 = I2 � I1 and DI2 = I3 � I2

also result of the same sign. The signal registered in

DDPV is [12,14–16]:

IDDPV ¼ DI2 � DI1 ¼ I3 � 2I2 þ I1; ð16Þwhich is plotted versus E2.

From Eqs. (1)–(3), with t1 � (t2 + t3), Eq. (16) can be

written:

Fig. 2. Double Differential Pulse Voltammograms wDDPV=ðE2 � E00

1 Þ (Eq. (17)) obtained for the systems considered in Fig. 1. DE1 = DE2 = �50 mV.

All conditions as in Fig. 1.


wDDPV ¼ ½Uðt2 þ t3Þ � 2Uðt2Þ�Xn�1

j¼0

ðn� jÞ

� c1j ðr0Þ � c2j ðr0Þh i

þ Uðt3ÞXn�1

j¼0

ðn� jÞ

� c2j ðr0Þ � c3j ðr0Þh i

; ð17Þ

where the current wDDPV is given by

wDDPV ¼ IDDPV

FAffiffiffiffiffiffiffiffiffiD=p

p . ð18Þ

For the particular case t3/t2 = 2/3, the following mathe-

matical relation is fulfilled:

Uðt2 þ t3Þ � 2Uðt2Þ ¼ �Uðt3Þ ð19Þ

and Eq. (17) simplifies to:

wDDPVðt3=t2 ¼ 2=3Þ ¼ Uðt3ÞXn�1

j¼0

ðn� jÞ

� 2c2j ðr0Þ � c1j ðr0Þ � c3j ðr0Þh i

.

ð20Þ

Fig. 3. Influence of t3/t2 on the Double Differential Pulse Voltammograms wDDPV=ðE2 � E00

1 Þ (Eq. (17)) obtained for a reversible two-step process

with not well-separated steps in a spherical electrode with r0 = 0.01 cm. DE1 = DE2 = �50 mV, t1 = 3 s, t2 + t3 = 0.15 s, c* = 1 mM, T = 298 K. The

values of formal potentials (in mV) relative to Ag/AgCl and the diffusion coefficient have been taken from results in [31] for pyrazine in aqueous acid

media: E00

1 ¼ �308.5; E00

2 ¼ �425.5, D = 7.8 · 10�6 cm2 s�1. The values of t3/t2 are on the curves.

Fig. 4. Comparison between the responses in RDPV (Eqs. (11) and (12)) and DNPV (Eq. (22)) when the equivalences given in Scheme 2 for both

techniques are accomplished, and with molecules On or Rn being initially in solution. The curves have been calculated using the experimental data for

pyrazine in aqueous acid media given in Fig. 3. The parameters for RDPV are: |DE| = 50 mV, t1 = 3 s, t2 = 0.5 s, t3 is on the curves. Other conditions

as in Fig. 3. Curve (a): t3 � t2 in RDPV, t2 � t1 in DNPV. In these conditions: wRDPV(On) = wRDPV(Rn) = wDPV(On) = wDPV(Rn) (Eqs. (24)–(26)).

Curve (b): wRDPV(On) = wDNPV(Rn) (Eq. (24)). Curve (c): wRDPV(Rn) = wDNPV(On) (Eq. (25)).



This specific value of the ratio t3/t2 is the same as that

obtained for a simple E process [28]. This coincidence is

due to the fact, previously highlighted in Section 2 of

this paper, that the temporal factors of the current are

independent of the number of steps implied in the mech-

anism, which are reflected in the terms containing thesurface concentrations of the different oxidation states

in the molecule.

Eqs. (17) and (20) are valid irrespective of whether

the molecule On or Rn is initially present in solution.

From these equations we deduce the cathodic and ano-

dic limiting currents by making E1, E2, E3 ! �1 and

E1, E2, E3 ! 1. Hence, we obtain:

wl.a.DDPV ¼ wl.c.

DDPV ¼ 0. ð21ÞFig. 2 shows the typical response in the DDP technique

(wDDPV=ðE2 � E00

1 Þ, Eq. (17)) with DE1 = DE2 =

DE = �50 mV for the same molecules considered inFig. 1, i.e., fullerene C60 in Fig. 2(a) and a hypothetical

molecule with six noninteracting centers in Fig. 2(b).

Thus, in the case of well separated steps (Fig. 2(a)), we

can observe two perfectly defined peaks with currents

of different sign for each one-electron charge transfer,

whereas in Fig. 2(b), the response displayed consists of

only one pair of peaks with current of different sign,

again with the current values being six times those ex-pected for a molecule with a single center.

In Fig. 3 it is shown the current wDDPV versus

E2 � E00

1 (Eq. (17)) for a reversible electrode process pro-

ceeding in two successive steps, at different values of the

Fig. 5. Comparison between the responses in DDPV with t3 = 2t2/3 (Eq. (20

both techniques are accomplished. The curves have been calculated using the

The parameters for DDPV are: |DE1| = |DE2| = 50 mV, t1 = 3 s, t2 = 3t3/2, t3potential ð¼ ðE00

1 þ E00

2 Þ=2Þ typical of the ADP voltammograms [31].

relation t3/t2. In this case, an intermediate situation be-

tween those considered in Figs. 1 and 2 is presented,

namely that in which the existence of an EE mechanism

is evidenced, but both steps partially overlap. This situ-

ation corresponds to the behaviour of many real sys-

tems, as, for example, pyrazine in acidic aqueousmedia [29,30], and in fact, the values of the formal

potentials and the diffusion coefficient in Fig. 3 have

been taken from experimental results on this system in

[31]. It can be observed that, for low values of the rela-

tion t3/t2 the peaks for the first charge transfer step are

both of positive current, whereas for the second transfer,

the first peak�s current is positive and the second is neg-

ative, and in the limit for extremely low t3/t2 values, thispeak actually disappear. This behaviour is exactly the

contrary for high values of the quotient t3/t2, for which

is the first peak, of positive current, that gets to disap-

pear as t3/t2 increases. The particular case t3/t2 = 2/3 will

be considered later since it presents interesting particular

features (see Fig. 5).

4. Comparison between triple and double potential pulse

techniques

4.1. RDPV versus DNPV

In the DNPV technique the difference DE = E2 � E1

is kept constant, and the currents corresponding to both

potential steps I2 and I1 are measured at times t2 and t1,

)) and ADPV (Eq. (30)) when the equivalences given in Scheme 3 for

experimental data for pyrazine in aqueous acid media given in Fig. 3.

is on the curves. Other conditions as in Fig. 3. Ec is the central cross


respectively. The DNP current I2 � I1 is recorded versus

E1 with the pulse amplitude, DE, being scanned either in

the negative direction (DE < 0) or in the positive one

(DE > 0) (see Scheme 2). This current is easily obtained

without restriction on the times of the application of the

two potential steps [1,2,27]. Thus, by subtracting Eqs.(2) and (1), we obtain:

wDNPV ¼ ½Uðt1 þ t2Þ � Uðt1Þ�

�Xn�1

j¼0

ðn� jÞ c�j � c1j ðr0Þh i

þ Uðt2Þ

�Xn�1

j¼0


; ð22Þ

with

wDNPV ¼ I2 � I1FA

ffiffiffiffiffiffiffiffiffiD=p

p . ð23Þ

If Eqs. (22) and (11) for wRDPV(On) are compared, and

the equivalence between t2, t3, E2, E3 in RDPV and t1,t2, E1, E2, respectively, in DNPV (see Scheme 2), is

established, it can be concluded that both equations

are identical if c�j in Eq. (22) is zero for j = 0 to

(n � 1). This condition implies that the molecule with

all centers reduced, Rn, should be initially present in

solution in order to compare with wRDPV(On). Thus, it

can be written that

wRDPVðOnÞ ¼ wDNPVðRnÞ. ð24ÞConversely, if now Eqs. (22) and (12) for wRDPV(Rn) are

compared it can be concluded that both equations are

formally identical if the molecule with all centers oxi-

dized, On, is initially present in solution (c�j ¼ 0 except

for j = 0 in Eq. (22)), i.e.,

wRDPVðRnÞ ¼ wDNPVðOnÞ. ð25ÞThis behaviour is easily understandable if the conditions

of application of the RDP technique, i.e., t1 � (t2 + t3),

are considered. In fact, when a first pulse potential is

applied over a long time under limiting current condi-

tions, all redox centers in the molecule will be reduced

if E1 ! �1, or oxidized if E1 ! 1, i.e., the moleculeRn or On, respectively, exist at the electrode surface,

and a layer of it extends outwards. Thus, the concentra-

tion profile near the electrode is dominated by the pres-

ence of these species and therefore, when the second and

third potentials are applied, a reversible system replies as

in the case of DNPV when Rn or On are initially present

in solution, respectively.

From Eqs. (15) and (25), we can write:

wRDPVðOnÞ ¼ wDNPVðOnÞ � ½Uðt2 þ t3Þ� Uðt2Þ�nc�. ð26Þ

Thus, it can be concluded that it is sufficient to obtain

the current wDNPV(On) and, automatically, wRDPV(Rn)

(Eq. (25)), wRDPV(On) (Eq. (26)) and wDNPV(Rn) (Eq.

(24)) are available. Obviously, in the case when t3 � t2(i.e., t2 � t1 in DNPV), the four responses tend to be

coincident. This is quite a nice result showing that, for

single or multiple reversible charge transfer reactions,

the same information can be obtained from a triplepulse technique like RDPV and with a simpler double

pulse technique like DNPV. This peaked output tech-

nique is very useful in studying multistep electrode pro-

cesses, and advantageous with respect to other

techniques with sigmoidal wave-like response, such as

NPV or RPV [27].

These important results are illustrated in Fig. 4,

where we have plotted together the responses calculatedin the RDP (Eqs. (11) and (12)) and DNP (Eq. (22))

techniques for the same experimental data of pyrazine

in Fig. 3, considering that the molecule On or Rn is ini-

tially present in solution. As can be observed in curve

(a), the same response is obtained in both techniques if

t3 � t2 in RDPV and t2 � t1 in DNPV, whatever the

species that is initially in solution (On or Rn). Thus,

curve (a) is really made of four superposable responses.Moreover, curves (b) and (c) in this figure correspond to

the results shown by Eqs. (24) and (25), respectively, i.e.,

each of these curves include two superposable responses.

The fulfilment of Eqs. (15) and (26) is clearly noted in

this figure.

4.2. DDPV versus ADPV

The technique ADPV (additive differential pulse

voltammetry) [28,31] is a double pulse technique based

on obtaining the two following differential signals, cor-

responding to the same first pulse potential E1 (see

Scheme 3)

DIc ¼ I2;cðE2;cÞ � I1ðE1Þ; ð27ÞDIa ¼ I2;aðE2;aÞ � I1ðE1Þ ð28Þ

with |DE| = �(E2,c�E1) = E2,a�E1 being DE < 0 in Eq.

(27) and DE > 0 in Eq. (28).

In order to obtain the additive response, both differ-

ential signals DIc and DIa are added, such that

IADPV ¼ I2;cðE2;cÞ � 2I1ðE1Þ þ I2;aðE2;aÞ ð29Þ

and IADPV is plotted versus E1. In this technique, the

time of application of the first potential step, t1, is much

greater than that corresponding to the second potential

step, t2. Thus, by introducing Eqs. (1) and (2) with the

condition t2 � t1 in Eq. (29), it is obtained:

wADPV ¼ Uðt2ÞXn�1

j¼0

ðn� jÞ

� 2c1j ðr0Þ � c2;cj ðr0Þ � c2;aj ðr0Þh i

; ð30Þ

with


wADPV ¼ IADPV

FAffiffiffiffiffiffiffiffiffiD=p

p . ð31Þ

A simple inspection of Eqs. (30) (wADPV) and (20)

(wDDPV with t3 = 2t2/3) allows us to realise that both

expressions are formally identical and therefore, on

establishing the equivalence between t3, E2 in DDPV

and t2, E1 in ADPV, respectively, as shown in Scheme

3, it can be concluded that the responses obtainedwith both techniques under these conditions are

identical.

This behaviour is shown in Fig. 5 where curves

wDDPV=ðE2 � E00

1 Þ for three values of t3 have been plot-

ted, for the condition t3/t2 = 2/3 (Eq. (20)), and the

corresponding wADPV=ðE1 � E00

1 Þ curves calculated with

the condition t2(ADPV) = t3(DDPV), for the same

pyrazine system considered in Figs. 3 and 4 [31],i.e., for a reversible EE process with partially over-

lapped steps. Clearly, the curves obtained with both

techniques are superimposed. As observed in the fig-

ure, the two peaks in the middle are of lower absolute

value heights than the two other peaks. This is due to

the particular value of the difference between the for-

mal potentials ðE00

2 � E00

1 Þ for the pyrazine system. In

these conditions the curves always present three crosspotentials, of which the central ðEc ¼ ðE00

1 þ E00

2 Þ=2Þ is

extremely useful in determining simultaneously the

formal potentials when both electrochemical steps

are not completely separated, as indicated in [31].

Note that this central cross point exists whatever

species (On or Rn) is initially present in the solution.

This represents a great advantage over other electro-

chemical techniques as, for example, dc, NP, RP,DP and RDP voltammetries, in which no characteris-

tic central point is observed, and which lead therefore

to a far less accurate simultaneous determination of

E00

1 and E00

2 .

5. Conclusions

The study carried out in this paper for triple pulse

techniques allow us to conclude that, when electrochem-

ical processes involving any number of reversible steps

are considered, an identical response can be obtained

by using double pulse techniques.

Thus, owing to the greater simplicity of double pulse

techniques DNPV and ADPV in comparison with triple

pulse techniques RDPV and DDPV, respectively, it is al-ways preferable to use the first ones in studying multi-

step electrode processes with reversible electrochemical

reactions. This behaviour is not exclusive to double

and triple pulse techniques [28], but it also can be found

between techniques of single and double pulse potential

[1,27,32] and, in general, the response obtained for a

multistep electrode process with reversible reactions in

a technique of p potential pulses can be reproduced with

a technique of (p � 1) pulses.

Furthermore, this coincidence turns out to be an

important criterion of reversibility, since if both re-

sponses agree, this is clearly indicative of the existence

of reversible processes.

Acknowledgements

The authors greatly appreciate the financial support

provided by the Direccion General Cientıfica y Tecnica

(Projects Nos. BQU2003-04172 and BQU2001-3197),

and by the Fundacion Seneca (Project No. PB/53/FS/02). M.M.M. also thanks Direccion General Cientıfica

y Tecnica for the grant received.

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Documents

Study of multistep electrode processes in triple potential step techniques at spherical electrodes