Special DOI:10.1002/elan.201400639 Normal Pulse ......the thickness of the reaction and diffusion...

DOI: 10.1002/elan.201400639

Normal Pulse Voltammetry and Steady State Voltammetryof the Square Mechanism at Spherical MicroelectrodesA. Molina,*[a] E. Laborda,[a] F. Martnez-Ortiz,[a] and J. M. Gýmez-Gil[a]

Dedicated to Professor R. G. Compton on the Occasion of his 60th Birthday

1 Introduction

The square mechanism given in Scheme 1 describes manyexperimental systems where electron transfer processesare coupled to structural changes (isomerisation [1–4],conformation modifications [5,6]), complexation andligand exchange [7], encapsulation [8], (de)protonationand substitution [9–12], ion pairing [13–15], etc. This isalso the case of ion transfer processes across liquid j liquidinterfaces where complexation of the ion may take placein the organic and/or aqueous phases [16]. Moreover,more simple situations, such as the CE, EC and ECEmechanisms, can be studied as particular cases. Therefore,the theoretical and practical interest of the squarescheme is well justified.

The approach to the square mechanism is not straight-forward given the number of thermodynamic and kineticvariables involved, such that the complete understanding,identification and quantitative analysis are, at least, chal-

lenging. In this paper we aim to assist the above tasks byreporting a simple (yet accurate) analytical solution forthe study of the square mechanism via normal pulse vol-tammetry (NPV) at microelectrodes and also steady statevoltammetry. The diffusive-kinetic steady state (dkss) ap-proximation will be employed, which has led to accuratesolutions for the CE and EC mechanisms [17–20] and itenables the derivation of closed-form expressions for thecurrent response and the concentration profiles. As men-tioned above, many different situations can be consideredas particular cases of the square mechanism depending onthe values of the formal potentials and the rate and equi-librium constants. In this paper we are going to focus onthe situations where the two chemical processes takeplace with similar kinetics as well as on the CE and ECmechanisms. These last two mechanisms correspond tothe particular cases of the square mechanism where onlyone of the electron transfers and only one of the chemicalreactions occur. Other cases where one of the chemicalreactions is significantly slower or even it does not takeplace (the ECE mechanism) will be tackled in the future.

From the analytical solution deduced here, the effectsof the rate and equilibrium constants of the chemical re-actions, the difference between the formal potentials andthe electrode radius on the electrochemical response will

Abstract : A simple analytical solution is presented forthe study of the square mechanism in normal pulse andsteady state voltammetries at spherical (ultra)microelectr-odes. The analytical expression deduced enables the de-scription of a broad range of situations depending on thechemical kinetics and the difference between the half-wave potentials of the electron transfers. Regarding the

chemical kinetics, the electrochemical response is ana-lysed from the limit of slow kinetics, where the thicknessof the reaction layer tends to coincide with that of the dif-fusion layer, up to the limit of very fast kinetics, wherethe reaction layer vanishes. Finally, the conditions underwhich the total steady state response is sensitive to thehomogeneous kinetics are studied.

Keywords: Square mechanism · Spherical microelectrode · Normal pulse voltammetry · Steady state voltammetry · Diffusive-kineticsteady state approximation

Scheme 1. Square mechanism where k1, k2, k3 and k4 are (pseu-do)first-order rate constants (species L is assumed to be presentin a large excess) and E8’A/B and E8’AL/BL are the formal potential ofthe redox couples A/B and AL/BL, respectively.

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[a] A. Molina, E. Laborda, F. Martnez-Ortiz, J. M. Gýmez-GilDepartamento de Qumica Fsica, Facultad de Qumica,Regional Campus of International Excellence “CampusMare Nostrum”, Universidad de Murcia30100 Murcia, Spaintel: +34868887524; fax: +34868884148*e-mail: [email protected] Information for this article is available on theWWW under http://dx.doi.org/10.1002/elan.201400639

www.electroanalysis.wiley-vch.de Ó 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Electroanalysis 2015, 27, 970 – 979 970

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be studied. The conditions under which two waves ora single wave appear are discussed, as well as thosewhere the electrochemical response is useful for quantita-tive studies of the homogeneous kinetics. For the latter,the electrode size must be carefully considered in orderto ensure adequate sensitivity. Particular attention will bepaid to the analysis of steady state voltammetry [21–23]that can be described as a particular case of the analyticalsolution here deduced. Also, the connection between theelectrochemical response and the relative magnitude ofthe thickness of the reaction and diffusion layers is dis-cussed.

2 Theory

In the square mechanism considered in Scheme 1 the twoelectron transfers are reversible whereas finite-rate kinet-ics are considered for the two coupled chemical reactions.Upon the application of a potential pulse E such that spe-cies A and/or AL are reduced at a working electrode ofspherical geometry, the response of the system underfully-supported conditions will be described by the fol-lowing differential equation system that accounts for themass transport by diffusion and the chemical kinetics insolution:

@cAðr; tÞ@t

¼ D @2cAðr; tÞ@r2

þ 2r@cAðr; tÞ@r

¢k1cAðr; tÞ þ k2cALðr; tÞ

@cALðr; tÞ@t

¼ D @2cALðr; tÞ@r2

þ 2r@cALðr; tÞ

@r

þk1cAðr; tÞ ¢ k2cALðr; tÞ

@cBðr; tÞ@t

¼ D @2cBðr; tÞ@r2

þ 2r@cBðr; tÞ@r

¢k3cBðr; tÞ þ k4cBLðr; tÞ

@cBLðr; tÞ@t

¼ D @2cBLðr; tÞ@r2

þ 2r@cBLðr; tÞ

@r

þk3cBðr; tÞ ¢ k4cBLðr; tÞ

ð1Þ

and by the associated boundary value problem:

t ¼ 0; r r0

t 0; r!1

9=; cA ¼c*KA

1þKA ; cAL ¼c*

1þKAcB ¼ 0; cBL ¼ 0

t > 0; r ¼ r0

ð2Þ

D@cAðr; tÞ@r

r¼r0¼ ¢D @cBðr; tÞ

@r

r¼r0

D@cALðr; tÞ

@r

r¼r0¼ ¢D @cBLðr; tÞ

@r

r¼r0

ð3Þ

cAðr0; tÞ ¼ ehA=B cBðr0; tÞcALðr0; tÞ ¼ ehAL=BL cBLðr0; tÞ

ð4Þ

where all the species are assumed to have the same diffu-sion coefficient D and:

hA=B ¼F

RTðE¢ E00A=BÞ

hAL=BL ¼F

RFðE¢ E00AL=BLÞ

ð5Þ

In Condition 2 it is assumed that only species A andAL are initially present in solution at the equilibriumconcentrations established by the total concentration,c*=c*A +c*AL, and the effective equilibrium constant:

KA ¼ k2=k1 ¼ c*A=c*AL ð6Þ

Note that the effective equilibrium constant of theequilibrium B/BL is given by:

KB ¼ k4=k3 ¼ c*B=c*BL ¼ KA expf¢FDE0=RTg ð7Þ

where ~E8’=E0AL=BL¢E0A=B. The ÐeffectiveÏ equilibriumconstants are related to the ÐtrueÏ constants (KA’ and KB’)according to the following relationships:

KA ¼ K0A=c*L, KB ¼ K0B=c*L ð8Þ

where c*L is the concentration of species L that is assumedto be present in a large excess. For a given system, the KAand KB values can be modified experimentally by chang-ing c*L to make evident the presence of chemical processescoupled to the electron transfers as will be discussed inthe following sections.

The above problem can be tackled by introducing thefollowing variables:

zAðr; tÞ ¼ cAðr; tÞ þ cALðr; tÞzBðr; tÞ ¼ cBðr; tÞ þ cBLðr; tÞ

ð9Þ

Aðr; tÞ ¼ cAðr; tÞ ¢KAcALðr; tÞBðr; tÞ ¼ cBðr; tÞ ¢KBcBLðr; tÞ

ð10Þ

where fA and fB refer to the perturbation of the chemicalequilibria A/AL and B/BL, respectively, such that the f-value is zero under equilibrium conditions. The Differen-tial Equation System 1 in terms of the new variables 9and 10 becomes:

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@zAðr; tÞ@t

¼ D @2zAðr; tÞ@r2

þ 2r@zAðr; tÞ@r

@zBðr; tÞ

@t¼ D @

2zBðr; tÞ@r2

þ 2r@zBðr; tÞ@r

@Aðr; tÞ

@t¼ D @

2Aðr; tÞ@r2

þ 2r@Aðr; tÞ

@r

¢ kAAðr; tÞ

@Bðr; tÞ@t

¼ D @2Bðr; tÞ@r2

þ 2r@Bðr; tÞ

@r

¢ kBBðr; tÞ

9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;

ð11Þ

where:

kA ¼ k1 þ k2kB ¼ k3 þ k4

ð12Þ

2.1 Diffusive-Kinetic Steady State Approximation (dkss)

Within the dkss theoretical approach, kinetic steady stateconditions are assumed such that the perturbations of thechemical equilibrium are not time-dependent:

@Aðr; tÞ@t

¼ 0@Bðr; tÞ

@t¼ 0

ð13Þ

and the form of the solutions for fA and fB is given by:

AðrÞ ¼r0rAðr0Þ exp ¢

ffiffiffiffiffiffikAD

rðr ¢ r0Þ

BðrÞ ¼

r0rBðr0Þ exp ¢

ffiffiffiffiffiffikBD

rðr ¢ r0Þ

ð14Þ

The second assumption employed in the dkss approxi-mation is that the concentration profiles of the pseudo-species zA and zB have the form of a species only suffer-ing diffusion, which in a semi-infinite, spherical diffusionfield is given by:

zAðr; tÞ ¼ z*A ¢r0r

z*A ¢ zAðr0; tÞ£ ¡

erfcr ¢ r02ffiffiffiffiffiffiDtp

zBðr; tÞ ¼

r0r

zBðr0; tÞerfcr ¢ r02ffiffiffiffiffiffiDtp

ð15Þ

where z*A ¼ c*A þ c*AL ¼ c*.The resolution of the problem involves determining the

form of the surface values (fA(r0), fB(r0), zA(r0,t) andzB(r0,t)) by application of the surface boundary condi-tions. Taking into account the form of the solutions givenby Equations 14 and 15, as well as the following relation-ships between the ÐrealÏ concentrations and the variablesz and f:

cAðr; tÞ ¼KAzAðr; tÞ þ AðrÞ

1þKA ; cALðr; tÞ ¼zAðr; tÞ ¢ AðrÞ

1þKAcBðr; tÞ ¼

KBzBðr; tÞ þ BðrÞ1þKB ; cBLðr; tÞ ¼

zBðr; tÞ ¢ BðrÞ1þKB

ð16Þ

the conditions of Equations 3 and 4 turn into:

t > 0; r ¼ r0KA z*A ¢ zAðr0; tÞ¨ ¦¢ dd

dr;AAðr0Þ ¼

1þKA1þKB

KBzBðr0; tÞ þ

dddr;B

Bðr0Þ ð17Þ

ðz*A ¢ zAðr0; tÞÞ þdd

dr;AAðr0Þ ¼

1þKA1þKB

zBðr; tÞ ¢ Bðr0Þ

dddr;B

ð18Þ

KAzAðr0; tÞ þ Aðr0Þ ¼ ehA=B1þKA1þKB

KBzBðr0; tÞ þ Bðr0Þ½ ¤

ð19Þ

zAðr0; tÞ ¢ Aðr0Þ ¼ ehAL=BL1þKA1þKB

zBðr0; tÞ ¢ Bðr0Þ½ ¤

ð20Þ

where dd refers to the thickness of the linear diffusionlayer:

dd ¼1ffiffiffiffiffiffiffiffiffipDtp þ 1

r0

¢1ð21Þ

and dr to the thickness of the linear reaction layers:

dr;A ¼ffiffiffiffiffiffikAD

rþ 1

r0

¢1dr;B ¼


rþ 1

r0

¢1 ð22Þ

Note that in the limit of macroelectrodes (linear diffu-sion), the linear diffusion layer thickness takes the valuedmacrod ¼

ffiffiffiffiffiffiffiffiffipDtp

(by making r0!1 in Equation 21) andthe linear reaction layer dmacror ¼

ffiffiffiffiffiffiffiffiffiffiD=k

p(by making r0!

1 in Equations 22).From Equations 17–20 one can conclude that, under

the conditions where the dkss assumptions (Equations 13and 15) are appropriate (see Results and Discussion), theresponse of the system is determined by the thermody-namic parameters (formal potentials and ÐeffectiveÏ equi-librium constants) and by the following relationships:

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dr;Add¼

1ffiffiffipp þ 1

R0

ffiffiffiffiffifficAp þ 1

R0

; dr;Bdd¼

1ffiffiffipp þ 1

R0

ffiffiffiffifficBp þ 1

R0

ð23Þwhere the dimensionless parameters of the electrodesphericity (R0) and the chemical kinetics (cA, cB) are de-fined as:

R0 ¼r0ffiffiffiffiffiffiDtp ; cA ¼ kAt ; cB ¼ kBt ð24Þ

Note that the dd value gives an estimation of the thick-ness of the region where the concentration profiles aredisturbed as a result of the application of the potentialpulse: zi(r,t)¼6 zi* (iA,B). Regarding the dr,A and dr,Bvalues, these inform about the thickness of the regionwhere the corresponding chemical equilibrium is broken,that is, where fi(r,t)¼6 0 (iA,B) and so ci(r,t)/ciL(r,t)¼6 Ki.In both cases, the diffusion and reaction layers increase asthe associated process (either mass transport or chemicalreaction) is slower. Thus, dr !dd when the chemical kinet-ics is very fast (cA, cB @1) such that the chemical equili-bria are broken only in a very narrow region next to theelectrode surface: dr!0. As the chemical reaction isslower, the thickness of the reaction layer (the regionwhere the chemical equilibrium is broken) is larger and drtends to dd in the limit of very slow chemical reactions(ÐfrozenÏ/inert equilibria).

The resolution of the system of Equations 17–20 leadsto the following expression for the current response:

I ¼ FAD @zAðr; tÞ@r

r¼r0¼ FAD z*A ¢ zAðr0; tÞ

dd

¼

¼ FADz*Add

1þKA þKA ehAL=BL ¢ ehA=Bð Þ. dr;A

dr;Bþ ehA=B

1þKAð Þ 1þ

1þKA1þKB

ehAL=BL

þ e

hAL=BL ¢ ehA=Bdr;Adr;Bþ ehA=B

!KA þ

dr;Add

þ 1þKA

1þKB

KAehAL=BL ¢

dr;Add

8>>>>>>>>>:

9>>>>>=>>>>>;ð25Þ

Equation 25 enables us to obtain the current-potentialresponse in the NPV technique where single potentialpulses of the same duration and different potential valuesare applied, restoring the initial conditions before eachpulse [22]. A Mathcad program is included as SupportingInformation where the above solution is implemented tocalculate the voltammetric response of the square mecha-nism at spherical microelectrodes.

2.2 Total Steady State. Steady-State Voltammetry

At (sub)micrometric electrodes the steady-state voltam-metric response can be attained, which retains kinetic in-formation about the coupled chemical reactions providedthat the electrode size is carefully chosen (see Sec-tion 3.4). To obtain the mathematical solution for the

square mechanism, steady state conditions for thepseudo-species zA and zB are imposed (along with the ki-netic steady state conditions (Eq. 13)):

@zAðr; tÞ@t

¼ 0@zBðr; tÞ

@t¼ 0

9>>=>>; ð26Þsuch that the form of the solutions are given by:

zAðr; tÞ ¼ z*A ¢r0r

z*A ¢ zAðr0; tÞ¨ ¦

zBðr; tÞ ¼r0r

zBðr0; tÞð27Þ

These lead to a linear equation system analogous toEquations 17–20 where the thickness of the linear diffu-sion layer is time-independent, that is, r0

ffiffiffiffiffiffiffiffiffipDtp

and so:

dssd ¼ r0 ð28Þ

Therefore, the total steady state response can be ob-tained from (25) by making dd =d

ssd = r0 such that:

dr;Adssd¼ 1þ cmicroA¨ ¦¢1

;dr;Bdssd¼ 1þ cmicroB¨ ¦¢1 ð29Þ

Note that the steady state voltammetry is defined bythe dimensionless parameters:

cmicroA ¼ r0ffiffiffiffiffiffikAD

r; cmicroB ¼ r0


rð30Þ

3 Results and Discussion

3.1 Evolution of the NPV Response with ~E88’, c and R0

Figure 1 shows the different NPV responses of the squaremechanism with KA =1 (i.e., c*A =c*AL) as a function of thedifference between the formal potentials (DE8’=E0AL=BL¢E0A=B), the chemical kinetics (c=cA =cB) and theelectrode sphericity. As expected, in all cases the limitingcurrent for E !E0A=B, E

0AL=BL corresponds to the mass

transport-controlled electro-reduction of an electroactivespecies of concentration c*=c*A +c*AL at a spherical elec-trode, that is: Ilim(c*)=FADc*/dd. The results obtainedfrom the analytical solution obtained (Equation 25) hasbeen found to yield accurate results (less than 5 % error)for r0

ffiffiffiffiffiffiDtp

(i.e. , R01).

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When the absolute value of the difference betweenthe formal potentials is larger than 100 mV in Figure 1(jDE8’ j >100 mV), the voltammogram shows two waveswhen the chemical kinetics is slow compared with themass transport such that the interconversion A/AL (andB/BL) is kinetic-controlled, with the current response in-cluding a kinetic contribution. As the chemical kineticsare faster (c increases) the magnitude of the first wave,corresponding to the electro-reduction of species A in thecase ~E8’=¢300 mV, increases since the species A con-sumed at the electrode surface is more rapidly replacedby the chemical reaction. When the chemical kinetics isvery fast with respect to the mass transport, a single waveis obtained independently of the difference between theformal potentials, the electrode size and the equilibriumconstants (see Figure 2). Under such conditions, the inter-conversion A/AL (and B/BL) takes place instantaneously.Therefore, equilibrium conditions are maintained at anypoint in solution and at any time of the potential pulse(fully labile equilibria). From Equation 25 the followingexpression is obtained for the current response in thefully labile limit:

I c!1ð Þ ¼ FADc* 1dd

1

1þ 1þKA1þKB

ehAL=BL

8

Let us consider now the partially-labile situation, whichis the most interesting case given that the electrochemicalresponse is sensitive to the chemical kinetics. A singlewave is obtained when the absolute value of the differ-ence between the formal potentials is smaller than100 mV in Figure 1. As can be observed in Figures 1 and2, the slope of the wave depends on the equilibrium con-

stants and chemical kinetics in such a way that the effectof the coupled homogeneous reactions can be mistakenby a deviation from reversibility of the electron transferprocesses [11]. For example, taking the Tomes criterion[22] it may be that jE3/4¢E1/4 j >56.4 mV (where Em/4 isthe potential at which I= Ilim ×m/4 with m=1, 3). The dis-crimination between both mechanisms (the quasi-reversi-

Fig. 2. Influence of the ÐeffectiveÏ equilibrium constant of the equilibrium A/AL (KA) on the normal pulse voltammogram (Equa-tion 25) for R0 =1 and for R0 =0.1 (steady-state voltammetry) for different ~E8’ values. c=kAt =kBt=10, KB =KA exp(¢(F/RT) DE8’),T=298 K.

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ble and irreversible E mechanism and the square mecha-nism) will be considered in Section 3.2.

According to the above discussion, the size of theworking electrode needs to be carefully selected whenemploying electrochemical methods for quantitative ki-netic analysis:

– for the study of slow chemical kinetics macroelectrodesare recommended since microelectrodes are insensitiveto the chemical kinetics given that the kinetic contribu-tion to the current may be hidden by the very fast masstransport by diffusion

– in the case of fast chemical kinetics microelectrodes arerecommended since at large electrodes equilibriumconditions hold and, as has been discussed above, thevoltammogram is only sensitive to the equilibrium con-stants (Equation 32), and not to the chemical kinetics

Particular case ~E8’=0

In Figure 1 it has been shown that the case ~E8’=0cannot be distinguished from a simple E mechanism pro-vided that all the species have the same diffusion coeffi-cients and both reversible redox processes have the sameformal potential. Under such conditions, the A/B andAL/BL processes cannot be distinguished with voltam-metric techniques; that is, both the electroreduction of Aand AL are ÐidenticalÏ in practice.

It is worth highlighting that the above conclusion doesnot hold when the ratio between the diffusion coefficientsof the redox couples are different, i.e. , when DA/DB¼6DAL/DBL. This is shown in Figure 3 for a macroelectrodeand an ultramicroelectrode where DA/DB =1 and the DAL/DBL value varies between 0.04–25. As can be observed,the shape and position of the wave differs from that ofa simple E mechanism (black solid line). The divergencedepends significantly on the electrode sphericity such thatthis is more remarkable at ultramicroelectrodes where

the influence of the difference between diffusion coeffi-cients on the half-wave potential is larger [24].

3.2 Influence of the ÐEffectiveÏ Equilibrium Constants

The shape of the voltammograms is determined, apartfrom the chemical kinetics (c) and the electrode size (R0)(as shown in Figure 1), by the ÐeffectiveÏ equilibrium con-stant KA that establishes the values of c*A and c*AL.Figure 2 shows the effect of the equilibrium constants ofthe equilibrium A/AL in normal pulse voltammetry (R0 =1) and steady state voltammetry (R0 =0.1). As references,the voltammograms of the simple E mechanisms corre-sponding to the redox couples A/B and AL/BL are alsoplotted (black solid lines).

Given that the KA value defines the initial (equilibri-um) concentrations of species A and AL, KA also deter-mines the contribution to the voltammetric response ofthe electron transfer processes ÐA to BÏ and ÐAL to BLÏ.Thus, as can be observed in Figure 2, the voltammogramtransits from the response corresponding to the electrore-duction of A (E(A/B) in the graphs) to that for the elec-troreduction of AL (E(AL/BL) in the figure) as KA issmaller and so the equilibrium is displaced towards spe-cies AL.

Note that, attending to the influence of the KA (andKB) values on the voltammograms, it is possible to detectthe occurrence of coupled chemical reactions by varyingthe concentration of species L (when this is possible).This is particularly important when the voltammogramsof the square mechanism shows a single wave that can bemistaken by a reversible E mechanism (in the case offully-labile equilibria) or by a quasireversible/irreversibleE mechanism (in the case of partially-labile equilibria).As shown in Figure 2, the variation of c*L within the ap-propriate range, and so of KA (and KB), can lead tochanges in the shape (for partially-labile equilibria) andposition (always) of the voltammograms when the system

Fig. 3. Influence of the divergence of the diffusion coefficients of the electroactive species on the normal pulse voltammograms atmacroelectrodes and steady state voltammograms at ultramicroelectrodes when ~E8’=0 (curves obtained by numerical simulations[27]). DA =DB =DAL, c=kAt=kBt=10, KA =1, ~E8’=0, T =298 K.

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follows a square mechanism, whereas this does nothappen when no coupled chemical reactions occur.

3.3 Limit Cases: the CE and EC Mechanisms

The limit cases of the CE and EC mechanisms can bestudied with the solution obtained for the SQ mechanismby setting the appropriate values for the equilibrium con-stants and the formal potentials. As indicated inScheme 2, the CE mechanism would correspond to thecase where species AL is not reduced in the potentialregion under consideration (Eapp) by making E

0AL=BL !

E0A=B; Eapp (~E8’!0) and species BL does not exist effec-tively since KB @1 (see Equation 7).

Figure 4 shows the voltammograms obtained fromEquation (25) with ~E8’=¢500 mV and from the expres-sion deduced for the CE mechanism [25]. In all cases theresults obtained with both solutions coincide showing thebehaviour reported for the CE mechanism such that thelimiting current increases with the chemical kinetics andthe voltammogram shifts towards more negative poten-tials.

Regarding the EC mechanism, as shown in Scheme 3this is a particular case of the square mechanism given inScheme 1 when E0AL=BL @E

0A=B; Eapp in such a way that

KA @ 1.The comparison of the results obtained with Equa-

tion 25 for ~E8’= +500 mV and from the solution avail-able for the EC mechanism [26] are also plotted inFigure 4. It can be confirmed that under the conditionsgiven in Scheme III the square mechanism behaves as theEC one such that the increase of the chemical kineticsleads to the shift of the voltammograms towards less neg-ative potentials whereas it does not affect the value ofthe limiting current.

3.4 Steady State Voltammetry

Under steady-state conditions the concentration profilesand current response are time-independent (i.e., thelinear diffusion layer thickness takes the value dssd = r0)and, consequently, the steady state voltammogram doesnot depend on the form of the potential-time perturba-tion applied. Thus, the curves obtained in “NPV-mode”

Scheme 2. CE mechanism.

Fig. 4. Steady-state voltammograms of the CE and EC mechanisms obtained from the solution obtained in this work for the squaremechanism (Equation 25) by setting ~E8’=¢500 mV (CE mechanism) and ~E8’= +500 mV (EC mechanisms) (dotted lines) andfrom the analytical solutions of each mechanism (solid lines) [25, 26]. R0 =0.1, T=298 K. Other conditions indicated on the graphs.~E8’.

Scheme 3. EC mechanism.

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experiments are equivalent to those obtained in “CV-mode”. For any chemical kinetics and equilibrium con-stants, it is found that steady state voltammograms areobtained provided that R00.1. This upper limit value forR0 ensures that the difference between the normal pulseand cyclic voltammograms is smaller than 5 %.

An important question in steady state voltammetry iswhether this contains kinetic information or not. It hasbeen reported that the use of very small electrodes for ki-netic studies is not adequate given that the high diffusiveflux can mask the kinetic contribution to the responsesuch that the voltammetry becomes “blind” to the chemi-cal kinetics. This issue can be tackled by determining theconditions under which the voltammogram differ morethan 10 % from that corresponding to c=0; that is, theconditions under which the kinetic contribution to thesteady state voltammetry is more than 10 % and so kinet-ic information can be extracted. For example, for ~E8’=¢300 mV and KA =1, it is found that the range of elec-trode size where a steady state voltammogram is obtainedand this is affected by the coupled chemical kinetics is0:2=ð0:8 ffiffifficp ¢ 1= ffiffiffipp Þ < R0 < 0:1. The lower limit would besmaller for lower KA values (considering that ~E8’

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Received: October 28, 2014Accepted: November 14, 2014

Published online: March 12, 2015

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Special DOI:10.1002/elan.201400639 Normal Pulse ......the thickness of the reaction and diffusion...

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