Application of a Power Time Current to the Study of a Catalytic Mechanism in Chronopotentiometry and Reciprocal Derivative Chronopotentiometry. Advantages of a Cyclic Stationary Response

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Application of a Power Time Current to the Study of a CatalyticMechanism in Chronopotentiometry and Reciprocal DerivativeChronopotentiometry. Advantages of a Cyclic Stationary ResponseAngela Molina,* Manuela Lopez-Tenes, Carmen M. Soto

Departamento de Qu mica F sica, Universidad de Murcia, Espinardo 30100, Murcia, Spain*e-mail: [email protected]

Received: November 27, 2007Accepted: January 17, 2008

AbstractThis paper develops the theory for a pseudo-first order catalytic mechanism in chronopotentiometry with a powertime current, I(t)¼ I0tu (u��1/2), applied to a spherical electrode of any size, and the advantages of the use of smallelectrodes and ultramicroelectrodes are discussed. The advantages of using a cyclic power time current to reach astationary response suitable for characterizing a catalytic mechanism easily and accurately are reported. Thereciprocal derivative (dt/dE�E) curves, which present peaks quantitatively related to the kinetic parameters of thechemical reaction, have been obtained from the potential-time responses. The influence of the homogeneous kinetic,the electrode radius and the power of time current in the achievement of a stationary response is analyzed. Methodsfor determining thermodynamic and kinetic parameters of the chemical reaction are proposed.

Keywords: Catalytic mechanism, Power time current, Reciprocal derivative chronopotentiometry, Sphericalelectrodes, Steady state

DOI: 10.1002/elan.200704143

1. Introduction

In a previous paper we discussed the different behaviorshown by a pseudo-first order catalytic process comparedwith other electrochemical processes which are complicatedby chemical reactions as are CE, EC, CEC, ..., mechanisms[1]. The characteristic behavior of a catalytic process is dueto the fact that the reaction product is simultaneouslyrequired for both the chemical and electrochemical reac-tions in order to regenerate and transform the depolarizer,respectively. This study is of great importance when theredox mediator is soluble in the electrolytic solution andwhen it is immobilized at the electrode surface, as in the caseof certain enzymes. Indeed, the catalytic action is veryfrequent in many reactions in Biochemistry, Biology andPharmacy, and in industrial and environmental processes [2,3].

In recent papers, the study of the behavior of a pseudo-first order catalytic process when a constant current isapplied to a spherical electrode of any size has been carriedout [1, 4]. The aim of the present work is to extend ourknowledge of such a process by analysis of the potential –time (E� t) and reciprocal derivative (dt/dE�E) curvesin chronopotentiometry with a programmed current of theform I(t)¼ I0tu (u��1/2), since the use of a power timecurrent presents important advantages over the applicationof a current step.

On the one hand, the possibility of varying the values of I0and/or u allow us to change the scan rate easily such that asuitable E� t response can be obtained in a wide range ofconcentrations. Moreover, by changing these parameters wecan find the propitious conditions to obtain a transient or astationary response. In general, a stationary response issimpler and easier to analyze and hence, it is desirable, ifpossible, to work under steady state conditions.

It is precisely under these conditions that the use of apower time current assumes greater importance since whena current step is applied, the stationary E� t response givesrise to a constant potential and so no information of theprocess is obtained. In contrast, the use of a power timecurrent always leads to a typical chronopotentiogram withtransition time, from which thermodynamic and kineticinformation of the catalytic process can be easily obtainedboth from the experimental E� t response or, moreaccurately, from the peaked response in reciprocal deriva-tive chronopotentiometry (dt/dE�E curves). In this tech-nique, the peaks are obtained from the central zone of thechronopotentiogram and thus they are scarcely affected bycapacitative effects [4 – 7]. Furthermore, the use of a cyclicpower time current under stationary conditions allows theeasy characterization of a catalytic process since for the mostusual case of an irreversible chemical reaction, identicalresponses are obtained for all odd currents whereas noresponse is obtained for the even applied currents.

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Electroanalysis 20, 2008, No. 11, 1175 – 1185 @ 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

A similar behavior to that above explained is observedwhen a variable time current is used to study a simple Emechanism in microelectrodes, that is, in this case it isnecessary to use very small electrodes to attain the steadystate conditions [6, 8]. However, for a catalytic process thedesirable stationary response can be obtained for sphericalelectrodes of any size, even for planar electrodes, if thehomogeneous kinetic is sufficiently fast, something which isfrequent in practical cases [1]. Furthermore, despite thewell-known advantages that the use of microelectrodesoffers, they can lead to the loss of the kinetic information inthe study of kinetic processes [1, 9 – 13].

Previous studies of this process with chronopotentiomet-ric techniques have been made by using planar electrodes[14] or a DME [15, 16], in such a way the behavior of thisprocess in small sized electrodes or under steady stateconditions has not been analyzed.

From the equations obtained in this paper for a sphericalelectrode of any size, from planar to ultramicroelectrodes,we demonstrate the advantages of a stationary responsewhen a power time current (such as a square root time, I0t

1/2,and a linear time one, I0t) is applied. We also proposemethods to obtain information for a catalytic process easilyand accurately from the stationary response, and in bothdirect and reciprocal derivative chronopotentiometry.

2. Theory: Chronopotentiometry with a Power TimeCurrent I(t)¼ I0t

u (u��1/2)

The reaction scheme for the catalytic mechanism can bewritten as (Scheme 1):

where C and B are the electroactivespecies and Z and Y are electro-inactive species in the whole rangeof potentials measured. Assumingthat the bulk concentrations ofspecies Z and Y (cZ*, cY*) aremuch larger than those for C andB (cC*, cB), we define the pseudo-

first order rate constants k1¼ k1’cZ* and k2¼ k2’cY* andconsequently:

K¼k2/k1¼ cB*/cC* (1)

k¼ k1þ k2 (2)

whereK is the equilibrium constant andkf andkb in Scheme 1are the rate constants of forward (reduction) and backward(oxidation) heterogeneous processes, respectively.

In the Appendix (Sec. 5) we have deduced the expressionscorresponding to the concentration profiles for a pseudo-first order catalytic mechanism (Scheme 1), when a pro-grammed current of the form I(t)¼ I0tu (u��1/2) is appliedto a spherical electrode of any radius, r0, including as limitcases a planar electrode (r0!1 ) and an ultramicrospher-ical electrode (r0! 0). Tables 1 (transient response) and 2(stationary response) give the expressions for the surfaceconcentrations and the E� t response in nernstian con-ditions.

2.1. Transient Response

The expressions for the surface concentrations are given inTable 1 by Equations T1.1 and T1.2 for a spherical electrode,and Equations T1.3 and T1.4 for the limit case of a planarelectrode.

In Equations T1.1 – T1.4 I(t)¼ I0tu is the applied currentwhereas I(tss) is the current corresponding to the transitiontime in the stationary state (see Eqs. T2.3 in Table 2 and 14).Tu(x, c) in Table 1 is a function which is dependent on the

electrode sphericity (x ¼2p__Dt/r0, Eq. A.16) and the homo-

geneous kinetic (c ¼kt, Eq. A.17), and also on the power oftime, u, in the applied current, according to the expression:8

Tuðx; cÞ ¼ xe�xX1j¼0

X1i¼0

ð�1Þixi

Qil¼0

p2uþ2jþlþ1

0BB@

1CCA cj

j!ð3Þ

Scheme 1.

Table 1. Equations for the surface concentrations and the corresponding transient potential-time response for a catalytic mechanismwhen a power time current I(t)¼ I0tu (u��1/2) is applied to spherical and planar electrodes. Functions Tss and Tss

plan are given byEquations 9 and 10, respectively, and I(tss) is given by Equation T2.3 in Table 2 and Equation 14.

Surface concentrations E� t response

Sphericalelectrodeof radius r0

cCðr0; tÞe

¼ 11þK

1� IðtÞIðtssÞ

Tuðx; cÞTss

(T1.1) EðtÞ ¼ E �’þ RT

nFln

IðtÞTuðxt; ctÞ � IðtÞTuðx; cÞIðtÞTuðxt; ctÞK þ IðtÞTuðx; cÞ

(T1.5)

cBðr0; tÞe

¼ 11þK

K þ IðtÞIðtssÞ

Tuðx; cÞTss

(T1.2)

Planarelectrode(r0!1 )

cCð0; tÞe

¼ 11þK

1� IðtÞIðtssÞ

Tplanu ðcÞTplan

ss

(T1.3) EðtÞ ¼ E �’þ RT

nFln

IðtÞTplanu ðctÞ � IðtÞTplan

u ðcÞIðtÞTplan

u ðctÞK þ IðtÞTplanu ðcÞ

(T1.6)

cBð0; tÞe

¼ 11þK

K þ IðtÞIðtssÞ

Tplanu ðcÞTplan

ss

(T1.4)

1176 A. Molina et al.

Electroanalysis 20, 2008, No. 11, 1175 – 1185 www.electroanalysis.wiley-vch.de @ 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

www.electroanalysis.wiley-vch.de

where the exponent u can take any real value u��1/2 forthe surface concentration to have physical meaning [17].

The transition time, t, can be deduced by making cC

(r0,t)¼ 0 in Equation T1.1. Thus, we obtain the followingnonexplicit expression:

tuTuðxt; ctÞ ¼4pr0nFDe

I0ð1þKÞ ð4Þ

where Tu(xt,ct) is the value of this function at t¼ t (Eq. 3).By including the expressions for the surface concentra-

tions in Table 1 in the Butler – Volmer equation, theexpression for the E� t response is obtained for any degreeof reversibility of the charge transfer. For a reversiblereaction, from Nernst equation we obtain directly EquationT1.5 for spherical electrodes and Equation T1.6 for planarelectrodes. Equation T1.5 can also be written as:

IðtÞTuðx; cÞIðtÞTuðxt; ctÞ

¼ 1�KehðtÞ

1þ ehðtÞ ð5Þ

where

hðtÞ ¼ nFRT

EðtÞ � E �’½ ð6Þ

withE8’being the formal potential of the charge transfer andI(t) is the current applied at t¼ t (I(t)¼ I0tu).

From Equation 5 it is clear that a plot of I(t)Tu(x,c)/I(t)Tu(xt,ct) vs. E for any power time current applied isidentical to the cathodic zone of the plot of I/Ilim vs. E involtammetry (see Eq. 27 in [1]).

2.1.1. Particular Cases

a) Current Step at Spherical ElectrodesFor the particular case of a current step, u¼ 0, Equation 3

can be written in a compact form [1, 4]

T0ðx; cÞ ¼x2

4c� x2 e�x ð7Þ

� exp ðx=2Þ2erfc ðx=2Þ � exp ðcÞ 1�2ffiffifficp

xerf

ffiffifficp� � ��

b) Planar ElectrodeFor a planar electrode r0!1 , functionTplan

u (c) in Table 1has the following form:

Tplanu ðcÞ ¼ t1=2e�x

X1j¼0

1p2uþ2jþ1

cj

j!ð8Þ

This expression can be written in the form of Besselfunctions [14].

2.2. Stationary Response

The stationary response can be obtained from transient oneunder the condition c� 1 (i.e., for a fast chemical reaction).Under this condition the general functions Tu(x,c) (Eqs. 3and 7) and T plan

u (c) (Eq. 8) notably simplify to functions Tss

and T planss , respectively, given by

Tss ¼ffiffiffiffiffiDpffiffiffiffiffi

Dpþ r0

ffiffiffikp ¼ dr

r0ð9Þ

Tplanss ¼ (2k1/2)�1 (10)

which are independent of the value of exponent u in theapplied current.

Note that the only requirement, c� 1, makes it possiblefor the stationary state to be attained in a planar electrode.This behavior is characteristic of a catalytic mechanism [1].

In Equation 9 dr is the thickness of the linear reaction layerin spherical diffusion, whose expression is given by [12, 13]

dr ¼1

1r0þ

ffiffiffikD

p ð11Þ

In the limit case r0!1 , from Equation 11 we obtain:

dplanr ¼

ffiffiffiffiffiDk

rð12Þ

which is the classical expression of the thickness of the linearreaction layer in planar diffusion [18].

In the limit case of an ultramicroelectrode (r0! 0), fromEquation 11 we have [12, 13]

dru�micro¼ r0 (13)

Note that the Equations T2.1 and T2.2 point out that thedependence of the surface concentrations with the timeunder steady state conditions is only due to the appliedcurrent time function (I0t

u).The transition time under steady state conditions, tss, can

be obtained by making cC(r0, tss)¼ 0 in Equation T2.1 ofTable 2:

tuss ¼

nFADe

I0ð1þKÞdrð14Þ

Note that the current necessary for reaching the transitiontime in chronopotentiometry, I(tss) (Eq. T2.3 in Table 2), is

1177Catalytic Mechanism in Chronopotentiometry



identical to the voltammetric cathodic limit current understeady state conditions, that is I(tss)¼ Iss

lim;c [1].In the limit case of an ultramicroelectrode (r0! 0), from

Equations 13 and 14 we have:

tuss ¼

4pr0nFDe

I0ð1þKÞ ¼Idð1Þ

I0ð15Þ

where Id(1 ) is the voltammetric steady-state limitingcurrent for a simple charge transfer process [1] . Therefore,no kinetic information can be obtained for the catalyticprocess when using very small electrodes [1, 9 – 11].

The E� t response for a reversible process is given byEquation T2.4 in Table 2. This equation can be rewritten inthe form (see Eq. 5):

IðtÞIðtssÞ

¼ 1�KehðtÞ

1þ ehðtÞ ð16Þ

which implies that the relation between current andpotential at the steady state is unique, that is, one value ofcurrent corresponds to one value of potential and vice-versa.Thus, it is clear that the stationary I/E response is commonfor any applied power time current or any variable with timepotential.

The expressions obtained for the E� t response, be ittransient or stationary (see Tables 1 and 2), are valid in arange of potentials which depends on the value of equilib-rium constant, K. Thus,

Eðt! 0Þ ¼ E�’þ RTnF

ln1K¼ Eeq ð17Þ

In other words, if a cathodic current is applied, the responsedevelops only for E�Eeq. For the most usual value ofequilibrium constant, K¼ 0, the E� t response develops inthe whole range of potential values [1, 4]. The startingpotential, according to Equation 17, is independent ofexponent u and r0. Note that this equation can be used todetermine the value ofK from the experimentalE� t curve.

2.3. Reciprocal Derivative Chronopotentiometry

The reciprocal derivative chronopotentiometry technique,also known as potentiometric stripping analysis (PSA) [19,20], consists of plotting the dt/dE versus E curves, whichpresent characteristic peaks that are related to the kineticparameters of the chemical reaction. These peaks are barelyaffected by capacitative effects since they are obtained fromthe central zone of theE� t curves. This fact, together with ahigh reproducibility in the measure of peak parameters,among other advantages, makes this technique a powerfultool in the study of electrochemical processes [4 – 7, 20 – 22].

Thus, once the expressions for the E� t response areavailable, the corresponding reciprocal derivative responsecan be obtained. In the case of a transient response (seeTable 1), the E� t curves are numerically differentiated byusing a finite-difference formula of the fifth degree [23].However, in the case of a stationary response, the simpleexpression obtained for the E� t response (see Table 2)makes it possible to carry out the analytical derivation dt/dE.

Thus, from Equation 16 it is obtained:

dtdE¼ � nF

RTtu

ss

uð1þKÞð1�KehÞ1�u

ueh

ð1þ ehÞ1þuu

ð18Þ

Alternatively, from Equation 16 we can obtain the expres-sion for dI(t)/dE in stationary state conditions, which is givenby:

dIðtÞdE¼ � nF

RTI0tu

ssð1þKÞ eh

ð1þ ehÞ2 ð19Þ

The characteristic peak parameters, peak potential,Ep, peakheight, hp, and semipeak width,w1/2, corresponding to the dt/dE and dI(t)/dE versus E curves (for the most usual case ofK¼ 0), are summarized in Table 3. It is interesting to notethat the expressions in Table 3 are valid for any size of theelectrode.

Table 2. Equations for the surface concentrations and the potential-time response for a catalytic mechanism when a power time currentI(t)¼ I0tu (u��1/2) is applied to a spherical electrode of any radius under steady state conditions (c� 1).

Surface concentrations E� t response

Spherical electrode 8 r0cCðr0; tÞ

e¼ 1

1þK1� IðtÞ

IðtssÞ

(T2.1)

Planar electrode (r0!1 )cBðr0; tÞ

e¼ 1

1þKK þ IðtÞ

IðtssÞ

(T2.2) EðtÞ ¼ E�’þ RT

nFln

IðtssÞ � IðtÞIðtssÞK þ IðtÞ (T2.4)

Ultramicro electrode (r0! 0)(E mechanism)

I(tss)¼ I0 tssu (T2.3)




2.4. Cyclic Chronopotentiometry with Power TimeCurrents I(tj)¼ (�1) jþ1I0t u

j , j¼ 1, 2, 3...

When j successive and alternating in sing power timecurrents of the form (�1)jþ1I0tuj are applied, if the catalyticrate constants are higher enough for the steady state to havebeen reached, the expressions of the surface concentrationscorresponding to any j applied current have an identicalform to that in Table 2 for a single power current (see Sect.5.2):

cC;jðr0; tjÞe

¼ 11þK

1�ð�1Þjþ1I0tu

j

IðtssÞ

" #ð20Þ

cB;jðr0; tjÞe

¼ 11þK

K þð�1Þjþ1I0tu

j

IðtssÞ

" #ð21Þ

and the expression for the E� tj response for any j currentapplied is given by:

EðtjÞ ¼ E�’þ RTnF

lnIðtssÞ � IðtjÞ

IðtssÞK þ IðtjÞð22Þ

where I(tss) is given by Equation T2.3 in Table 2 andEquation 14.

3. Results and Discussion

3.1. Study of the Behavior of Function Tu

From the equations of surface concentrations in Tables 1and 2 we can easily understand the key role of functionsTu(x,c) (spherical electrode) and Tplan

u (c) (planar electrode)in the attainment of the stationary state. In this section westudy the behaviour of these functions and the influence ofthe homogeneous kinetic, the electrode sphericity and thepower of time in the applied current in attaining thestationary limit.

In Figure 1a the function T ’planu (c) (¼T plan

u /t1/2 planarelectrode, Equation 8) is plotted versus c(¼ kt), for three

selected values of the exponent u in the applied current (u¼0, 1/2, 1). The dotted line to which the three curves convergecorresponds to the function T ’plan

ss (¼T planss /t1/2, Eq. 10) which

is independent of exponent u in the applied current. As canbe seen, the function T ’plan

u (c) reaches the stationary limit(c� 1) at relatively small values of c, and smaller ones as theexponent u diminishes. Indeed, as is shown in Figure 1a, foru¼ 0, the error of the function T ’plan

u (c) with respect to thestationary limit is less than 5% for c� 2, for u¼ 1/2 thishappens for c� 5.5 and for u¼ 1 it must be fulfilled that c�10.

According to these results, it is possible to obtain astationary response for a catalytic mechanism in a planarelectrode (at short times), provided the homogeneouskinetic constants are sufficiently high [1].

Thus, for very fast catalytic processes, as many in practiceare, the response should be stationary for any value of u, butfor moderately fast processes, the response would changefrom stationary to transient by increasing the value ofexponent u.

In Figures 1b and 1c the same type of plots as inFigure 1a for a planar electrode are represented, this timefor a spherical electrode at different values of thesphericity parameter, x (¼ 2

p__Dt/r0). In these figures,

the function Tu(x, c) (Eq. 3) is plotted in solid lines versusc, for u¼ 1/2 (Fig. 1b) and u¼ 1 (Fig. 1c). The dotted linescorrespond to the function at the stationary state Tss

(Eq. 9), which is independent of the exponent u in theapplied current. The vertical lines are traced, for eachvalue of x, at the value of c for which the differencebetween the values of functions Tu(x,c) and Tss is less than5%. Furthermore, the limit of the stationary function, Tss

(Eq. 9), when the homogeneous catalytic reaction doesnot take place (i.e., k¼ 0, c! 0) or for ultramicroelectr-odes (r0! 0, x!1 ) behaves as an E simple mechanismbeing in these conditions Tss¼ 1.

As can be seen, the kinetic stationary limit is reached(with an error of less than 5%) at smaller values of c when x

increases (i.e., r0 decreases).The series in function Tu(x, c) (Eq. 3) is slowly

convergent for x> 17 in such a way that the function Tss

(Eq. 9) can be used under these conditions if c> 0.6 forthe values of the exponent u considered. Thus, for x> 17and c< 0.6 an asymptotic solution should be used, andTasymp

u (x,c) is calculated from the following expression(see Eq. 3):

Table 3. Characteristic peak parameters for the dt/dE�E curves (Eq. 18) and dI(t)/dE�E (Eq. 19) curves under steady stateconditions: peak potential, Ep, peak height, hp, and semi-peak width, w1/2. h’ is given by Equation 6 with the values of E(t) correspondingto w1/2.

Ep j hp j w1/2

dt/dE vs. E E8’þ (RT/nF) ln unFRT

tss

ð1þ uÞ1þ1u

1u

eh0

ð1þ eh0 Þ1uþ1¼ 1

21

ð1þ uÞ1uþ1

dI(t)/dE vs. E E8’nFRT

I0tuss

43.53 RT/nF




Tasympu ðx; cÞ ¼ xe�x

X1j¼0

X1i¼0

ð�1Þix�ðiþ1ÞYi�1

l¼0

p2uþ2j�1

!cj

j!ð23Þ

3.2. Influence of the Applied Current on the ResponsesObtained

3.2.1. Potential – Time Curves

In Figures 2a – c we have plotted the E� t response for areversible catalytic process withK¼ 0 (Eq. T1.5 in Table 1),obtained at a spherical electrode of radius r0¼ 0.01 cm whenvarying the value of the kinetic constant k1. Figures 2a – ccorrespond, respectively, to values u¼ 0 (current step), u¼

1/2 (square root time current) and u¼ 1 (linear timecurrent), in the applied current, I(t)¼ I0tu.

From Figure 1 it can be noticed that by increasing thevalue of k1 sufficiently (that is, increasing c) the function Tureaches the stationary limit (i.e., a stationary response isobtained), for any value of u. However, Figures 2a – c showthat an increase of k1 has a very different effect on the E� tcurves for a current step (Fig. 2a) and a variable-with-timecurrent (Figs. 2b and 2c).

For a current step, I0, the transient curve (with transitiontime) and the stationary one (constant potential) can bedetected at sight [1, 4]. The condition for no transition timeto be obtained is I0< nFADcC*((1/r0)þ

ffiffiffiffiffiffiffiffiffiffiffik1=D

p) (see

Eq. T2.3 in Table 2 and Eq. 14) and therefore, the steadystate will be more easily reached the higher the k1 and thesmaller the r0 values are.

For a power time current, a typical chronopotentiogramwith transition time is always obtained, regardless ofwhether the steady state has been attained or not. In fact,curves for k1¼ 10 s�1 and k1¼ 20 s�1 in Figures 2b and 2ccorrespond to stationary responses. In this case, fromEquation T2.4 in Table 2, it is possible to detect that thesteady state has been reached by using a plot of E vs. ln((tu

ss� tu)/tu), which must be linear for a stationary curve. Theformal potential E8’ can be obtained from the ordinate ofthis plot, and the slope corresponds to RT/nF. The value ofthe rate constant k1 can be obtained from the value of tss

(Eqs. 11 and 14). This is valid for planar and sphericalelectrodes provided the electrode radius is not too small, inwhich case the kinetic of the process can not be detected (seeSec. 2.2).

Thus, the use of a power current time function (u= 0) is ofspecial interest in order to obtain kinetic information in thistechnique (and also in reciprocal derivative chronopotenti-ometry, see below), since a stationary response showingtransition time is obtained.

3.2.2 Reciprocal Derivative Curves

Figures 2d – f show the reciprocal derivative curves (dt/dEvs.E) corresponding to the E� t responses in Figures 2a – c.Hence, the curves have been plotted at several values of thekinetic constant k1 and the same values of exponent u in theapplied current: u¼ 0, 1/2 and 1 in Figures 2d, 2e and 2f,respectively.

As can be seen in these figures, for any value of exponentu, when the steady state has not been reached both the peakpotentials and the peak heights depend on the rate constantk1 in such a way that the peak heights increase as higher thevalue of k1 is.

In contrast, under steady state conditions the behavior ofthe derivative curves obtained with a power time currentdiffers from that observed for a current step. Indeed,whereas in the last case these curves do not present a peak(curves for k1¼ 10 s�1 and k1¼ 20 s�1 in Fig. 2d) and hencethey are not suitable to obtain kinetic information, thestationary curves corresponding to a square root timecurrent (curves for k1¼ 7 s�1, k1¼ 10 s�1 and k1¼ 20 s�1 in

Fig. 1. Dependence of the functions a) Tu’plan(c) (¼Tuplan/t1/2, Eq

8), b) and c) Tu(x,c) (Eq. 3), on the kinetic parameter c (¼ kt), insolid lines. In (a) the values of exponent u (I0t

u) are on the curves.In (b) (u¼ 1/2) and (c) (u¼ 1), the values of the sphericityparameter x (¼2

ffiffiffiffiffiffiDtp

=r0Þ are also on the curves. The correspond-ing stationary functions a) Tss’

plan (¼Tssplan/t1/2, Eq. 10), b) and c) Tss

(Eq. 9) are plotted in dotted lines. Dashed lines indicate the valuesof c for which the error of function Tu’

plan(c) and Tu(x,c) withrespect to the stationary limit is less than 5%.




Fig. 2e) and a linear time current (curves for k1¼ 10 s�1 andk1¼ 20 s�1 in Fig. 2f) present well defined peaks, whoseheight increases with the catalytic kinetic constant k1.However, the position of peaks remains unchanged, inde-pendently of the value of k1. According to the equations inTable 3, this behavior informs us that the stationary state hasbeen reached, since in these conditions the peak potential,Ep, does not depend on k1.

Similar plots, with the same qualitative behavior, as thosein Figure 2 (which have been obtained by increasing thevalue of the homogeneous rate constant), can be registeredby decreasing the radius of the electrode.

From the equations in Table 3 we can propose easymethods to obtain the characteristic parameters of theprocess from the stationary dt/dE vs.E curves. Indeed, fromthe peak potentials, and by using two different u values, wecan obtainE8’ and n, and from the peak height the value ofk,included in the expression of tss (Eqs. 11 and 14) can bedetermined, for planar and spherical electrodes. Theseparameters can be also obtained from the dI(t)/dE vs. Ecurves (see Table 3) [24].

To illustrate the above results, in Figure 3 the variation ofthe peak potential (Figs. 3a and 3b) and the peak height(Figs. 3c and 3d) with the catalytic kinetic constant has beenplotted for K¼ 0 and three values of the electrode radius,when power currents of the form I(t)¼ I0t1/2 and I(t)¼ I0t(the values of u are shown in the figures) are used. It isobserved in Figures 3a and 3b that the peak potentialchanges with k1 and r0 until a value k1� 10 s�1 (for thespecific conditions of Fig. 2). From this value, the peak

potential does not change either with k1 or r0, whichindicates that the steady state has been reached. Thus, asexplained above, the formal potentialE8N and the number ofelectrons transferred can be determined.

From the equation for the peak height in the dt/dE�Ecurves, j hp j , in Table 3 and Eqs. 11 and 14 , we can write:

hp

�� N¼

hp

�� nFRT nFA

ffiffiffiffiffiDp

cC� 1=uð1þ uÞ1þ1=u

¼ffiffiffiffiffiDp

r0þ

ffiffiffikp

!1=u

ð24Þ

Thus, it is clear that for a linear time current (u¼ 1), a plot ofj hp j N vs.

p_k will be linear with the ordinate given byD1/2/r0

and, hence, parallel lines are obtained at different values ofthe electrode radius once the steady state is reached, as seenin Figure 3d. In these conditions, from this plot the value ofthe homogeneous kinetic constant can be accurately ob-tained. Once more it can be noticed that the steady state isreached at smaller values of k1 when the electrode radiusdecreases.

Similar plots are obtained for a square root time current(Fig. 3c). In this case a plot of j hp j N vs. k is linear when thesteady state is reached.

3.2.3 Cyclic Curves

Figure 4 shows the stationary E-t response for a catalyticmechanism, when ten consecutive and alternating in signsquare root time currents are applied to a spherical electrodeof radius r0¼0.01 cm, for two values of equilibrium constant

Fig. 2. Reversible potential – time curves (a – c) (Eq. T1.5 in Table 1) and reciprocal derivative curves (d – f) obtained for a catalyticprocess with K¼ 0 when a power time current I(t)¼ I0tu is applied to a spherical electrode of radius r0¼ 0.01 cm. The values of exponentu are indicated in the figures. The influence of the catalytic rate constant k1 on E� t and dt/dE�E curves is shown.D¼ 10�5 cm2/s, n¼ 1,T¼ 298 K, e*¼ 1 mM, I0/A¼ 103 mA/cm2. The values of k1(s�1) are on the curves.




K (Figs. 4a and 4b) and the corresponding reciprocalderivative curves (Figs. 4c and 4d). As can be seen, andaccording to Equations 20 –22, whenK¼0 (Fig. 4a), identicalresponses are obtained for all odd currents applied (j¼1, 3,5,...), with the transition time given by Equation 14, whereasno response is obtained for the even applied currents (j¼2, 4,6, ...), as expected for an irreversible chemical reaction. Incontrast, forK¼0.5 (Fig. 4b), the initial presence of species Bin solution gives rise to a reversal response, which is identicalfor all even currents, with the transition time, tss’, given by (cB,j

(r0,tss’)¼0, see Equation 21):

tss’¼K1/u tss (25)

where tss is the transition time for the odd responses, givenby Equation 14. Note that under steady state conditions, allcurves (Figs. 4a and 4b) start from the equilibrium potential(see Eq. 17).

According to the above results, the derivative responseswill give rise to an unique curve (in reality five curvessuperposed) for K¼ 0 (Fig. 4c), whose peak height is givenin Table 3. If K= 0 (Fig. 4d) an only cathodic-anodicresponse (due to the odd (j¼1, 3 ...) and even (j¼2, 4...)currents respectively) will be obtained, with the cathodic,hp, c, and anodic, hp, a, peak heights given, respectively, by:

hp; c ¼dtj

dE¼ � nF

RTtss

uð1þKÞð1�KaÞ1�u

ua

ð1þ aÞ1þuu

ð26Þ

and

hp; a ¼dtj

dE¼ nF

RTtss

uð1þKÞðKa0 � 1Þ1�u

ua0

ð1þ a0Þ1þuu

ð27Þ

where

a ¼ ehp;c ¼ ð1þKÞ2uK

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þKÞ2 � 4Ku2

p2uK

ð28Þ

a0 ¼ ehp;a ¼ ð1þKÞ2uK

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þKÞ2 � 4Ku2

p2uK

ð29Þ

and

hp;c ¼nFRTðEp;c � E�’Þ ð30Þ

hp;a ¼nFRTðEp;a � E�’Þ ð31Þ

with Ep, c and Ep, a being the cathodic and anodic peakpotential values, respectively.

For the particular case of a linear time current (u¼ 1),Equations 26 and 27 are notably simplified and the relationbetween the peak heights is given by:

Fig. 3. Variation of a), b) peak potential, Ep, (see Table 3), and c), d) normalized peak height, j hp j N (Eq. 25), with the kinetic constantk1, obtained from the dt/dE�E response for different electrode radius under conditions of Figure 2, for a square root time current (aand c) and a linear time current (b and d). The values of r0 (cm) are on the curves.




hp;c

�� hp;a

¼ ð1þKÞ24K

ð32Þ

and

Ep, c¼E8’ (33)

Ep;a ¼ E�’þ nFRT

ln1K

ð34Þ

Thus, the value ofK can be determined easily and accuratelyfrom the peak height relation. Note that it could also bedetermined from Equation 25. Nevertheless, the use ofEquation 32 allows a more accurate determination ofequilibrium constant since the peaks proceed from thecentral zone of the chronopotentiogram, where the capaci-tative effects are minima.

For the particular case of a square root time current (u¼ 1/2), the relation between peak heights in Figure 4c, obtainedfrom Equations 26 and 27, is j hp, c j /hp, a¼ 3.2 and cathodicand anodic peak potential values, obtained from Equations28 – 31, are Ep, c�E8’¼�27.21 mV and Ep, a�E8’¼36.86 mV, respectively.

This behavior can be used to detect that the steady statehas been attained. Note that under steady state conditions itis necessary to use a power of time u> 0 since a current step(u¼ 0) would lead to independent of time responses.

4. Acknowledgements

The authors greatly appreciate the financial support pro-vided by the Direccion General de Investigacion Cient ficay Tecnica (Project Number CTQ2006-12552/BQU), and theFundacion SENECA (Project Number 03079/PI/05). C. M.Soto thanks the Ministerio de Educacion y Ciencia for thegrant received.

5. Appendix

5.1. Transient Conditions

In order to obtain the expressions for the concentrationprofiles of the electroactive species involved in a catalyticprocess represented by Scheme 1 (ci(r,t), i¼C, B), when acurrent of the form I(t)¼ I0tu (u��1/2) is applied to aspherical electrode of any size, the following equationsystem must be solved:

qcCðr; tÞqt ¼ D q2cCðr; tÞ

qr2 þ 2rqcCðr; tÞ

qr

h iþ k1cBðr; tÞ � k2cCðr; tÞ

qcBðr; tÞqt ¼ D q2cBðr; tÞ

qr2 þ 2rqcBðr; tÞ

qr

h i� k1cBðr; tÞ þ k2cCðr; tÞ

9=;ðA:1Þ

where D is the diffusion coefficient, assumed equal forspecies C and B; and r the distance from the centre of theelectrode to any point in the solution.

Fig. 4. Reversible cyclic potential – time curves (a and b) (Eq. 23), and cyclic reciprocal derivative curves (c and d) (Eq. 18), obtainedfor a catalytic process when ten successive square root time currents I(tj)¼ (�1)jþ1 I0tj

1/2 (j¼ 1,2,3...) are applied to a spherical electrodeof radius r0¼ 0.01 cm. The values of equilibrium constant K, and the index j for the current applied are in the figures. Other conditions asin Figure 2.




The boundary value problem is given by:

t ¼ 0; r � r0

t > 0; r!1

�cCðr; tÞ ¼ cCcBðr; tÞ ¼ cB

�ðA:2Þ

t > 0; r ¼ r0qcCðr; tÞ

qr

r¼ r0

¼ � qcBðr; tÞqr

r¼ r0

ðA:3Þ

DqcCðr; tÞ

qr

r¼ r0

¼ I0tu

nFAðA:4Þ

where A is the area of the stationary spherical electrode(A¼ 4pr0

2, with r0 being the electrode radius) and cC* andcB* are the bulk concentrations of C and B species.

On inserting the variable:

uiðr; tÞ ¼ ciðr; tÞe

rr0

i ¼ C;B ðA:5Þ

where

e ¼ cC þ cB ðA:6Þ

the system (A.1) is simplified to the following:

DuC(r,t)¼� DuB(r,t)¼ k1uB(r,t)� k2uC(r,t) (A.7)

where:

D ¼ qqt�D

q2

qr2ðA:8Þ

Taking into account that in a catalytic process followingScheme 1 it is always fulfilled that [1, 25 – 27]

uCðr; tÞ þ uBðr; tÞ ¼ rr0

8 r; t ðA:9Þ

and defining the function

f(r,t)¼(uB(r,t)�KuC(r,t))ekt (A.10)

only one differential equation in the variable f(r,t) needs tobe solved:

q�ðr; tÞqt

¼ Dq2�ðr; tÞ

qr2ðA:11Þ

with the following boundary conditions:

t ¼ 0; r � r0

t > 0; r!1

��ðr; tÞ ¼ 0 ðA:12Þ

t > 0; r ¼ r0

�ðr0; tÞr0

� q�ðr; tÞqr

r¼r0

¼ I0tu

nFADeð1þKÞekt

ðA:13Þ

Equation A.11 can be solved by following the procedureindicated in [28]. The solution is given by the expression:

�ðr; tÞ ¼ Nsð1þKÞtuþ12

�X1j¼0

X1i¼0

ð�1Þixi

Qil¼0

p2uþ2jþlþ1

0BB@

1CCA cj

j!y2uþ2jþiþ1ðsÞ ðA:14Þ

with

Ns ¼2I0

nFAffiffiffiffiffiDp

eðA:15Þ

x ¼ 2ffiffiffiffiffiffiDtp

r0ðA:16Þ

c ¼ kt ðA:17Þ

s ¼ r � r0

2ffiffiffiffiffiffiDtp ðA:18Þ

p2uþ2jþlþ1 ¼2G 1þ 2uþ2jþlþ1

2

� Gð2uþ2jþlþ2

2

ðA:19Þ

where G(x) is the Euler Gamma function, and y(s) are theKouteckyNs functions [28].

From Equations A.5, A.9, A.10 and A.14 we obtain thefollowing expressions for the concentration profiles ofspecies C and B

cCðr; tÞe

¼ 1ð1þKÞ �

r20

rIðtÞ

nFADeTuðx; cÞ y2uþ2jþlþ1ðsÞ ðA:20Þ

cBðr; tÞe

¼ Kð1þKÞ þ

r20

rIðtÞ

nFADeTuðx; cÞ y2uþ2jþlþ1ðsÞ ðA:21Þ

where Tu(x, c) is defined by Equation 3.




The expressions for the surface concentrations (r¼ r0) ofspecies C and B are in Table 1. Note that I(tss)Tss¼I(t)Tu(xt,ct), where I(tss)¼ I0 tss

u and I(t)¼ I0 tu (see Eqs.4, 9, and 14)).

5.2. Cyclic Chronopotentiometry Under Steady StateConditions

To obtain the expressions for the concentration profiles ofthe electroactive species involved in a catalytic processrepresented by Scheme 1 under steady conditions, ci, j(r,tj),i¼C, B, when j successive currents alternating in sign in theway I(tj)¼ (�1)jþ1 I0tj

u (u��1/2, j¼ 1, 2, 3,...) are applied toa spherical electrode of any size, by proceeding in the sameway as in Section 5.1 with the additional assumption qcC,j

(r,tj)/qtj¼ qcB(r,tj)/qtj¼ 0 in Equations A.1, and introducingthe function

fj(r,tj)¼uB(r,tj)�KuC(r,tj) (B.1)

instead of A.10, we must solve the following differentialequation:

q2�jðr; tjÞqr2

� kD�jðr; tjÞ ¼ 0 ðB:2Þ

with the boundary conditions:

r!1 fj(r,tj)¼ 0 (B.3)

r ¼ r0

�jðr0; tjÞr0

� q�jðr; tjÞqr

r¼r0

¼ð�1Þjþ1I0tu

j

nFADeð1þKÞ

ðB:4Þ

Thus, the solution of Equation B.2 is given by the expression:

�jðr; tjÞ ¼IðtjÞ

nFADedrð1þKÞe�ðr�r0Þ

ffiffiffiffiffiffik=Dp

ðB:5Þ

From Equations A.5, A.9, B.1 and B.5 we obtain theexpressions for the concentrations profiles of species C andB:

cC;jðr; tjÞe

¼ 1ð1þKÞ �

r0

rdr

IðtjÞnFADe

e�ðr�r0Þffiffiffiffiffiffik=Dp

ðB:6Þ

cB;jðr; tjÞe

¼ Kð1þKÞ þ

r0

rdr

IðtjÞnFADe

e�ðr�r0Þffiffiffiffiffiffik=Dp

ðB:7Þ

where dr is defined by Equation 11.The expressions for the surface concentrations are given

by Equations 20 and 21 in Section 2.4.

6. References

[1] A. Molina, I. Morales, J. Electroanal. Chem. 2005, 583, 193.[2] A. J. Bard, L. R. Faulkner, Electrochemical Methods, Funda-mentals and Applications, 2nd ed., Wiley, New York 2001.

[3] J. M. Saveant, Elements of Molecular and BiomolecularElectrochemistry, Wiley-Interscience, New Jersey 2006.

[4] M. Lopez-Tenes, I. Morales, A. Molina, Electrochim. Acta2006, 51, 2851.

[5] J. Gonzalez, A. Molina, M. Lopez-Tenes, C. Serna, J.Electrochem. Soc. 2000, 147, 3429.

[6] A. Molina, J. Gonzalez, I. Morales, Electroanalysis 2005, 17,674.

[7] M. Lopez-Tenes, J. M. Molina, A. Molina, Electroanalysis2004, 16, 938.

[8] A. Molina, J. Gonzalez, I. Morales, J. Electroanal. Chem.2004, 569, 185.

[9] M. Fleischmann, F. Lasserre, J. Robinson, D. Swan, J.Electroanal. Chem. 1984, 177, 97.

[10] M. A. Dayton, A. G. Ewing, R. M. Wightman, Anal. Chem.1980, 52, 2392.

[11] J. Galceran, J. Puy, J. Salvador, J. Cec lia, H. P. v. Leeuwen, J.Electroanal. Chem. 2001, 505, 85.

[12] A. Molina, I. Morales, M. Lopez-Tenes, Electrochem. Com-mun. 2006, 8, 1062.

[13] I. Morales, A. Molina, Electrochem. Commun. 2006, 8, 1453.[14] K. Kontturi, M. Lindstrçm, G. Sundholm, J. Electroanal.

Chem. 1975, 63, 263.[15] A. Molina, M. Lopez-Tenes, C. Serna, J. Electroanal. Chem.

1993, 346, 53.[16] A. Molina, J. Gonzalez, M. Lopez-Tenes, J. Math. Chem.

1998, 23, 277.[17] A. Molina, F. Mart nez-Ortiz, J. Albadalejo, J. Electroanal.

Chem. 1988, 251, 249.[18] J. Koutecky, R. Brdicka, Coll. Czech. Chem. Commun. 1947,

12, 337.[19] A. Fogg, J. Wang, Pure Appl. Chem. 1999, 71, 891.[20] J. Wang, Analytical Electrochemistry, Wiley, New York 2000.[21] S. Bi, J. Yu, J. Electroanal. Chem. 1996, 405, 51.[22] J. Wang, B. Tian, Anal. Chem. 2000, 72, 3241.[23] M. J. Maron, Numerical Analysis: A Practical Approach,

Collier McMillan, London 1982.[24] A. Molina, I. Morales, Int. J. Electrochem. Sci. 2007, 2, 286.[25] A. Molina, C. Serna, J. Gonzalez, J. Electroanal. Chem. 1998,

454, 15.[26] J. Weber, Chem. Listy 1958, 52, 1888.[27] A. Molina, J. Electroanal. Chem. 1998, 443, 163.[28] J. Koutecky, Czech. J. Phys. 1953, 2, 50.




Documents

Application of a Power Time Current to the Study of a Catalytic Mechanism in Chronopotentiometry and Reciprocal Derivative Chronopotentiometry. Advantages of a Cyclic Stationary Response