Two-Electron Transfer Reactions in Electrochemistry for Solution

Article pubs.acs.org/JPCC

Two-Electron Transfer Reactions in Electrochemistry for SolutionSoluble and Surface-Confined Molecules: A Common Approach Manuela Lopez-Tenes, Joaquin Gonzalez, and Angela Molina* Departamento de Química Física, Facultad de Química, Regional Campus of International Excellence “Campus Mare Nostrum”, Universidad de Murcia, 30100 Murcia, Spain ABSTRACT: In this paper, the general characteristics of the normalized voltammetric response for reversible two-electron transfer reactions (EE mechanism) is analyzed and particularized to the application of derivative normal pulse voltammetry (dNPV) using electrodes of any geometry and size, and cyclic voltammetry (CV), when the molecule undergoing the process is soluble in solution and surface-confined, respectively. The analysis is based on the close relationships between the electrochemical response and the theoretical values of surface concentrations/excesses, and has led to the voltammetric signal of the EE mechanism being interpreted in terms of the percentage of E1e−E1e−, and E2e− character, as a function of the difference between the formal potentials of both electron transfers, ΔE0′ = E02′ − E01′. In line with the percentage of E2e− character, the term “effective electron number”, neff, has been introduced and related to the probability of the second electron being transferred in an apparently simultaneous way with the first one and a direct method to obtain the values of E01′ and E02′ for any ΔE0′ has been proposed. The key role of ΔE0′ (in mV, 25 °C) = −142.4, −71.2, −35.6, and 0 values in the behavior of the peak parameters of the voltammetric curves is explained in terms of the usual terminology (transition 2 peaks−1 peak, repulsive− attractive interactions, anticooperativity−cooperativity, and normal-inverted order of potentials). The EE mechanism is also compared with two independent E mechanisms (E+E).

1. INTRODUCTION Electrode processes consisting of two-electron transfers (EE mechanism) have been widely treated in the literature, both in their theoretical and applied aspects.1−9 This high productivity measures in some way the great presence and relevance of these processes in many fields, and hence the importance of understanding them. This behavior is very common in electrochemical reactions of alkylviologens and metallocenes, in the reductions of several metallic ions, of polyoxometallates and of a number of aromatic species like derivatives of tetraphenylethylene.1,2,4,8−13 In the specific case of biological molecules, such as oligonucleotides, metalloproteins, enzymes, etc., the application in recent years of techniques such as protein film voltammetry (PFV), combined with scanning probe microscopic techniques, has made it possible to characterize the biomolecule−electrode interface and electron-transfer processes in great detail, which is fundamental to exploit the naturally high efficiency of these biological systems in modern biotechnology (selective last-generation biosensors, environmentally sound biofuel cells, heterogeneous catalysts, biomolecular electronic components).3,5−7,9 Many of these systems present a reversible behavior (or it can be reached acting on the adequate experimental parameter in the particular electrochemical technique used), which simplifies the study of multielectron transfer processes by not having to consider the kinetics of the electron-transfer reactions. When studying the EE mechanism, two situations for the molecules undergoing the process can be encountered: when they are soluble in the solution and then need to be transported © 2014 American Chemical Society

to/from the electrode surface and when the molecules are surface-confined, so the transport is avoided. Traditionally, these two fundamental problems in electrochemical science are treated as completely different problems. In this paper, both possibilities are tackled in the case of a reversible EE mechanism, in order to establish the correspondences between their electrochemical responses and the conditions under which a common study of these situations can be made. Thus, it has been shown that, with the appropriate normalization in each case, the electrochemical signal obtained when normal pulse voltammetry (NPV) is applied in the case of solution soluble species for electrodes of any geometry and size is coincident with the charge transferredpotential one for surface-confined systems in any electrochemical technique. Thus, their derivatives lead to a common normalized peak-shaped voltagram in derivative normal pulse voltammetry (dNPV), for solution soluble molecules,1,2,4,14−30 and in cyclic voltammetry (CV), for surface-immobilized ones.1,3,5−9,31−36 These normalized responses are only potential-dependent as a consequence of the reversible behavior of the process, which is reflected in the independence of time of surface concentrations (solution phase case)15,24−26 and excesses (immobilized species).31−36 Thus, the parallel characteristics of the common normalized voltammetric curve in dNPV and CV and the surface Received: March 14, 2014 Revised: April 30, 2014 Published: May 12, 2014 12312

dx.doi.org/10.1021/jp5025763 | J. Phys. Chem. C 2014, 118, 12312−12324

The Journal of Physical Chemistry C

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Table 1. Expressions for Diffusion Mass Transport Operators, δĵ (j =O, I, or R), Functions f G(qG,t) and f G,micro for the Main Electrode Geometriesa diffusion operator δ̂j

f G(qG,t) (Dj = D) 1 (πDt )1/2

f G,micro

∂ ∂2 − Dj 2 ∂t ∂x

sphere (radius rS, AS = 4πr2s )

⎛ ∂2 ∂ 2 ∂⎞ − Dj⎜ 2 + ⎟ ∂t r ∂r ⎠ ⎝ ∂r

1 1 + rs (πDt )1/2

1 rs

disk (radius rd, Ad = πr2d)

⎛ ∂2 1 ∂ ∂ ∂2 ⎞ − Dj⎜ 2 + + 2⎟ r ∂r ∂t ⎝ ∂r ∂z ⎠

⎞⎞ ⎛ r r 4 1⎛ ⎜0.7854 + 0.44315 d1/2 +0.2146 exp⎜− 0.39115 d1/2 ⎟⎟⎟ π rd ⎝ (Dt ) (Dt ) ⎠⎠ ⎝

41 π rd

band (height w, length l, Aw = wl)

⎛ ∂2 ∂ ∂2 ⎞ − Dj⎜ 2 + 2 ⎟ ∂t ⎝ ∂x ∂z ⎠

1 1 + w (πDt )1/2

1 2π w ln[64Dt /w 2]

electrode plane

if Dt /w 2 < 0.4

⎛ π ⎞1/2 ⎛ (πDt )1/2 ⎞ ⎟+ 0.25⎜ ⎟ exp⎜−0.4 ⎝ Dt ⎠ w ⎠ ⎝

π ⎛ (Dt )1/2 ⎞ w ln 5.2945 + 5.9944 w ⎟ ⎝ ⎠ ⎜

if Dt /w 2 ≥ 0.4 cylinder (radius rC, length l, AC = 2πrcl)

a

⎛∂ 1 ∂⎞ ∂ − Dj⎜ 2 + ⎟ r ∂r ⎠ ∂t ⎝ ∂r 2

⎛ (πDt )1/2 ⎞ 1 1 ⎟+ exp⎜− 0.1 ⎛ rc (Dt )1/2 ⎞ (πDt )1/2 ⎠ ⎝ rc ln⎜5.2945 + 1.4986 r ⎟ ⎝ ⎠ c

1 2 rc ln[4Dt /rc 2]

qG is the characteristic dimension of the electrode: rs for spheres or hemispheres; rd for discs; w for bands; rc for cylinders.28,29,37

inverted order of potentials.4 Also, it has been discussed the values of ΔE0′ for which two and one peaks appear in the voltammetric curve, and the different evolution of one-peak curves with ΔE0′, in terms of the intermediate stability. For completeness, the EE mechanism has been compared with the case of two independent one-electron-transfer reactions with identical initial concentrations/excesses, which has been called an E+E mechanism in this article.1,33 It has been shown that whenever two peaks are obtained in the voltammetric response of an EE mechanism (ΔE0′ < −71.2 mV) it can be considered practically indistinguishable from that for an E+E process.

concentrations/excesses have been highlighted. On the basis of this study, the variation of the voltagrams shape depending upon the relative values of the formal potentials of both chargetransfer reactions in the EE mechanism, as expressed by ΔE0′ = E02′ − E01′, has been interpreted in terms of the percentage of consecutive (E1e−E1e−) and apparent simultaneous (E2e−) characters of the electron-transfer process, which have been intrinsically related to the surface concentrations/excesses values of the intermediate and the sum of the two extreme oxidation states of the molecule, respectively, at the average formal potential, E̅ 0′ = (E01′ + E02′)/2, and, more practically, with the current value at E̅ 0′ (peak or valley) in the voltagram. From these concepts, the term “effective number of electrons transferred”, neff, has been introduced, which varies between 1 for ΔE0′ ≪ 0 (0% character E2e−, very stable intermediate) and 2 for ΔE0′ ≫ 0 (100% character E2e−, very unstable intermediate) and has been related to the probability of the second electron being transferred in an apparently simultaneous way with the first one. The values of neff for any ΔE0′ have been discussed and compared with the “apparent number of electron transferred”, napp, extensively used in bioelectrochemistry, and defined in the literature for ΔE0′ ≥ − 35.6 mV values, i.e., for cooperative behavior between the electron transfers.5,32 Related to the above ideas, a simple direct method for obtaining the individual formal potentials, E01′ and E02′, regardless of the ΔE0′ value, has been proposed. In this study of the two-electron transfer reactions the importance of the ΔE0′ (in mV, 25 °C) = −142.4, − 71.2, −35.6, and 0 values (K = 1/28, 1/24, 1/22, 1/20, respectively, with K being the disproportionation constant and 2 the number of electron transfers) in the behavior of the peak parameters of the voltammetric response (peak potentials, peak heights, and half-peak widths) is noteworthy. This capital role has also been explained in terms of the surface concentrations/excesses of the species, establishing the relation between them and the different terminologies that are used in the study of processes with two-electron transfers: 2−1 peaks in the response,14,18,31−36 negative−positive (repulsive−attractive) interactions,1 anticooperativity−cooperativity,5,32 and normal-

2. THEORY Consider an electrode process in which a molecule reduces reversibly involving two-electron transfers, according to the following reaction scheme (EE mechanism) O + e− ⇄ I E10 ′ I + e− ⇄ R E20 ′

(I)

in which O (oxidized), I (intermediate, or half-reduced) and R (reduced) refer to the different redox states of the molecule and E01′ and E02′ are the formal potentials of the first and second steps, respectively. The average formal potential, E̅0′, given by

E̅ 0′ =

E10 ′ + E20 ′ 2

(1)

plays an essential role in the study of the process since it is the formal potential for the reaction O + 2e− ⇄ R

(II)

i.e., the sum of both steps in scheme (I). Also, a key parameter is the difference between the formal potentials, ΔE0′, defined as ΔE 0 ′ = E20 ′ − E10 ′ 12313

(2)



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⎛ F ⎞ Ji = exp⎜ (E − Ei0 ′)⎟ ⎝ RT ⎠

which determines the stability of intermediate I in scheme (I). Indeed, species I is totally stable for ΔE0′ ≪ 0 and totally unstable for ΔE0′ ≫ 0.1,2 A brief outline follows of the most important insights involved in the development of the theory concerning the reversible EE mechanism in order to establish the correspondences between the voltammetric responses obtained when the species in scheme (I) are soluble in the solution and when they are immobilized at the electrode surface forming a monolayer, so that a common study of both physical situations is carried out in the Results and Discussion. 2.1. Solution-soluble Species. Normal Pulse Voltammetry at an Electrode of Any Geometry. The first case considered is that in which the molecule undergoing the process is soluble in solution, assuming that it is initially present only in its totally oxidized state, with concentration cO* (see scheme (I)). When a constant potential E is applied to an electrode of any geometry, the following solution-phase reaction related to process (I)1,4,14−16,18,23,24,27

COs = J ̅2 C Rs ⎛ F ⎞ J ̅ = exp⎜ (E − E ̅ 0 ′)⎟ ⎝ RT ⎠ COsC Rs (C Is)2

cj(q , t ) cO*

⎛ ∂C ⎞ I1 = FA GDOcO*⎜ O ⎟ ⎝ ∂q ⎠q

⎛ ∂C ⎞ I2 = −FA GDR cO*⎜ R ⎟ ⎝ ∂q ⎠q

(3)

where AG is the electrode area for the specific geometry considered (see Table 1). From eq 3, it is clear that the concentration profiles of species O, I, and R are dependent on the kinetic of the disproportionation/comproportionation reaction, so the current given by eqs 13−15 will also depend on it. To avoid this latter dependence, equal diffusion coefficients are assumed for the three species, i.e., DO = DI = DR = D, so, taking into account eq 6, it is fulfilled that NPV ⎛ ∂(2CO + C I) ⎞ IEE =⎜ ⎟ ∂q FA GDcO* ⎝ ⎠q

⎪

⎪

surface

C Is = J2 C Rs

surface

(16)

Note that for both lineal combinations of concentrations in eq 16, i.e., for W (q , t ) = 2CO(q , t ) + C I(q , t ) (a) ⎫ ⎪ ⎬ X(q , t ) = 2C R (q , t ) + C I(q , t ) (b) ⎪ ⎭

=0

surface

(6)

COs = J1 C Is

⎛ ∂(2C R + C I) ⎞ = −⎜ ⎟ ∂q ⎝ ⎠q

(5)

t > 0, q = qsurface: ⎛ ∂C ⎞ + DR ⎜ R ⎟ ⎝ ∂q ⎠q

(15)

surface

(4)

surface

(14)

surface

t = 0, q ≥ qsurface ⎫ ⎬CO = CO* = 1, C I = 0, C R = 0 t > 0, q → ∞ ⎭

surface

(13)

with

j = O, I, R

⎛ ∂C ⎞ + DI⎜ I ⎟ ⎝ ∂q ⎠q

(12)

NPV IEE = I1 + I2

where q and t refer to the values of a set of coordinates characteristic of the given geometry and of time, respectively. The boundary value problem is given by

⎛ ∂C ⎞ DO⎜ O ⎟ ⎝ ∂q ⎠q

⎛ F ⎞ = exp⎜ ΔE 0 ′⎟ = K ⎝ RT ⎠

where K gives the value of the equilibrium constant (= k1/k2) for the reaction in scheme (III), being 0 ≤ K ≤ ∞. The current, INPV EE , can be expressed as the sum of I1 and I2 for steps 1 and 2, respectively, in reaction I

where δ̂j is the mass transport operator for the geometry considered, given in Table 1, and the normalized concentration, Cj(q, t), is defined as Cj(q , t ) =

(11)

and

can take place in the vicinity of the electrode. k1 and k2 are the rate constants of the disproportionation and comproportionation reactions, respectively. If diffusion is the only transport mechanism, the differential equation system to be solved for process (I), taking into account the reaction scheme (III) is given by δÔ CO = k1C I2 − k 2COC R ⎫ ⎪ ⎪ 2 ̂ δ IC I = −2k1C I + 2k 2COC R ⎬ ⎪ δ R̂ C R = k1C I2 − k 2COC R ⎪ ⎭

(10)

with

(III)

k2

(9)

where qsurface is the value of the normal coordinate to the electrode at the surface, Dj is the diffusion coefficient of species j (O, I, or R), the superscript “s” refers to the value of concentrations at the electrode surface, and F, R, and T have their usual meaning. From eqs 7 and 8 it follows that

k1

2I ⇄ O + R

i = 1, 2

(17)

and thus for Y (q , t ) = CO(q , t ) + C I(q , t ) + C R (q , t )

(7)

(18)

it is fulfilled that (see eq 3) ̂ = δX ̂ =0 δ Ŷ = δ W

(8)

(19)

The boundary value problem for δ̂Y = 0 in eq 19 is given by (see eqs 5 and 6)

with 12314



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t = 0, q ≥ qsurface ⎫ ⎬Y = Y * = 1 t > 0, q → ∞ ⎭

(20)

⎛ ∂Y ⎞ t > 0, q = qsurface : ⎜ ⎟ ⎝ ∂q ⎠q

(21)

where the function fG(qG,t), which only depends on time and on the particular electrode geometry, is given in Table 1 and is the same as that obtained for a one-electron transfer reaction (E mechanism).28,29,37 Thus, eq 28 shows that the current is not influenced by the homogeneous reaction in scheme (III). By taking into account the expression for the diffusion37 controlled limiting current for a simple E mechanism, INPV E,lim:

⎪

⎪

=0

surface

NPV IE,lim = FA GDcO*fG (qG , t )

and the solution of this problem leads easily to Y (q , t ) = CO(q , t ) + C I(q , t ) + C R (q , t ) = 1

(22)

Equation 28 can be written as (see eqs 17b and 23)

24,28

regardless of the electrode geometry considered. Therefore, from eq 22 for q = qsurface, and the nernstian conditions 7 and 8 (or their equivalents, eqs 10 and 12), the following expressions for the surface concentrations, Csj (j = O, I, R), written in terms of the average formal potential, E̅0′ (eqs 1 and 11), and ΔE0′ (eqs 2 and 12), are obtained: ⎫ ⎪ K1/2 + J ̅ + K1/2J ̅ 2 ⎪ ⎪ ⎪ J̅ s C I = 1/2 ⎬ 1/2 2 K + J̅ + K J̅ ⎪ ⎪ K1/2 ⎪ C Rs = 1/2 1/2 2 ⎪ K + J̅ + K J̅ ⎭

NPV IEE,N =

(C Rs )E10′ = (COs)E20′ =

(23)

=2

K 2+K

⎪

(24)

−K3/2J ̅ + K3/2J ̅ 3 (K + K1/2J ̅ + KJ ̅ 2 )2

(31)

0

(26)

dNPV (ψEE )E10′ =

s RT ⎛ dC R ⎞ 2K1/2 = 2 ⎜ ⎟ = 2(C Rs )E̅ 0′ F ⎝ dE ⎠ E ̅ 0 ′ 1 + 2K1/2

(32)

CsI

s dC Is ⎞ RT ⎛ dC R 1 + 5K = 2 + ⎜ ⎟ F ⎝ dE dE ⎠ E 0 ′ (2 + K )2 1

3K 1−K =2 + 2 (2 + K ) (2 + K )2 (27)

fG (qG , t )

+

Note that = 0 since is also an even function of (E − E̅ 0′) (see eq 23) and presents a maximum at E = E̅ 0′, as can be expected, whose value is given by eq 24 (see dotted black curves for CsI in Figure 2 below). For E = E01′ (J ̅ = 1/K1/2) and E = E02′ (J ̅ = K1/2), ψdNPV takes EE the same value; thus, from eq 31

dNPV (ψEE )E20′ =

(33)

s dC Is ⎞ RT ⎛ dC R 1 + 5K = + ⎜2 ⎟ 2 F ⎝ dE dE ⎠ E 0′ (2 + K ) 2

1 + 2K −1 + K =2 + (2 + K )2 (2 + K )2

NPV IEE = (W * − W s)fG (qG , t ) FA GDcO*

K1/2 + J ̅ + K1/2J ̅ 2

(K + K1/2J ̅ + KJ ̅ 2 )2

(dCsI/dE)E̅ 0′

leads to the following expression for the current as a consequence of Ws only depending on the applied potential24,28 (see eqs 16, 26, and 27)

2K1/2 + J ̅

J̅

= (COs + C Rs )E̅ 0′

(25)

2K1/2J ̅ 2 + J ̅ + J ̅ + K1/2J ̅ 2

K3/2J ̅ + 2K 2J ̅ 2

dNPV (ψEE )E ̅ 0 ′ =

t > 0, q = qsurface : W s = 2COs + C Is

=

+ J̅ + K

= X s = 2C Rs + C Is

is an even function of (E − E̅ ′); i.e., the function takes the same values by changing J ̅ for 1/J.̅ Thus, at E = E̅ 0′ the function presents a maximum (peak) or a minimum (valley) depending on the value of K (see the Results and Discussion), which has the value (see eq 31 with J ̅ = 1 and eq 24)

⎪

K

K

1/2 2

ψdNPV EE

1 ; 2+K

t = 0, q ≥ qsurface ⎫ ⎬W = W * = 2 t > 0, q → ∞ ⎭

1/2

2K1/2 + J ̅ 1/2

NPV K3/2J ̅ + 4K 2J ̅ 2 + K3/2J ̅ 3 RT dIEE,N = F dE (K + K1/2J ̅ + KJ ̅ 2 )2 s dC Is ⎞ RT ⎛ dC R = + ⎜2 ⎟ F ⎝ dE dE ⎠

Once the surface concentrations are known, the expression for the current can be obtained. Indeed, the solution of the differential equation δ̂W = 0 (eq 19), with the boundary conditions (see eqs 17a, 5, and 23)

=

=

dNPV ψEE =

K1/2 1 ; (C Is)E̅ 0′ = 1/2 1 + 2K 1 + 2K1/2

(COs = C Is)E10′ = (C Is = C Rs )E20′ =

IENPV ,lim

(30)

which are independent of the existence of the disproportionation/comproportionation reaction. From eq 23, the surface concentrations at the particular values of the potential E = E̅0′ (J ̅ = 1), E = E01′ (J ̅ = 1/K1/2), and E = E02′ (J ̅ = K1/2) have the form (COs = C Rs )E̅ 0′ =

NPV IEE

Thus, the normalized current, INPV EE,N, given by eq 30, is independent of time and electrode geometry. Since the INPV EE,N − E response is in form of waves, it is more appropriate to use the derivative of the current in order to have a peak-shaped response. So, in derivative normal pulse voltammetry (dNPV), the normalized ψdNPV is given by (see eq 30) EE

K1/2J ̅ 2

COs =

(29)

(34)

For K = 1 (ΔE0′ = 0 mV), eqs 32−34 are coincident, being 0 0 0 (ψdNPV EE )E1′ = E2′ = E̅ ′ = 2/3. If ultramicroelectrodes are used, when the stationary state is attained the current is independent of time, and thus, eqs

(28) 12315



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RT dQ EE,N dE v F dE dt K3/2J ̅ + 2K 2J ̅ 2 −K3/2J ̅ + K3/2J ̅ 3 =2 + (K + K1/2J ̅ + KJ ̅ 2 )2 (K + K1/2J ̅ + KJ ̅ 2 )2

28−34 are fulfilled for any potential-time waveform applied.28,29 In this case, the function fG(qG,t) (eqs 28 and 29) becomes f G,micro (see Table 1). 2.2. Surface-Immobilized Species. Cyclic Voltammetry. In this case, it will be assumed that there are no interactions between the immobilized molecules, that heterogeneity of the electroactive monolayer can be ignored, and that no desorption takes place in the time scale of the experiment, so the total excess ΓT is constant and independent of the potential during the whole experiment for any potential-time waveform applied to the electrode. Thus, if the molecule is initially present only in its totally oxidized state (see eq I), with excess Γ*O = ΓT, the surface coverages, f j (j = O, I, or R), defined as f j = Γj/ΓT, where Γj is the surface excess of species j, fulfill that

fO + fI + fR = 1

CV ψEE =

(41)

ψCV EE

where v = dE/dt is the sweep rate. Equation 41 for for surface-immobilized molecules has, logically, the same expression as that for ψdNPV for solution-soluble species (eq 31). EE Thus, parallel expressions to eqs 32−34 are also fulfilled for CV 0 CV 0 0 (ψCV EE )E̅ ′, (ψEE )E1′, and (ψEE )E2′, respectively.

3. RESULTS AND DISCUSSION After discussion of electrochemical techniques for a reversible EE mechanism with species soluble in solution for electrodes of any geometry and size, and immobilized at the electrode surface, the characteristics of the common normalized peakshaped response for CV, ψCV EE , in surface-confined processes (eq 41) and for dNPV, ψdNPV EE , in diffusion-controlled processes (eq 31), are now revised and summarized. Also, several aspects of the reversible behavior have been analyzed. In the following dNPV discussion, the superscript CV and dNPV in ψCV EE and ψEE have been removed, and we refer only to ψEE. It is well known that the EE mechanism behaves as two independent one-electron E mechanisms for ΔE0′ ≪ 0 (K → 0) − E1e−E1e−, see scheme (I), with two peaks centered at the individual formal potentials, E01′ and E02′ −, and as a twoelectron E mechanism for ΔE0′ ≫ 0 (K → ∞) − E2e−, see scheme (II), with a single peak centered at the average formal potential, E̅ 0′ −.1,2 Hence, in this paper, the case of two simple independent E mechanisms has also been considered for comparison, i.e.

(35)

Equation 35 is equivalent to eq 22, obtained for solutionsoluble species. Thus, by combining eq 35 with the nernstian conditions f O/f I = J1 and f I/f R = J2 (see eqs 7 and 8), the expressions for surface coverages f O, f I, and f R are obtained, which are obviously identical to CsO, CsI, and CsR, respectively, in eq 23. From this result, the charge QEE transferred for the EE mechanism is straightforwardly obtained as the sum of Q1 and Q2 for steps 1 and 2, respectively, in the scheme (I)33 Q EE = Q 1 + Q 2

(36)

with Q 1 = −FA ΓT

Q 2 = FA ΓT

∫0

∫0

τ

τ

⎛ dfO ⎞ ⎜ ⎟dt = FA ΓT (1 − fO ) ⎝ dt ⎠

(37)

O1 + e− ⇄ R1 E10 ′

⎛ dfR ⎞ ⎜ ⎟dt = FA ΓT fR ⎝ dt ⎠

O2 + e− ⇄ R 2 E20 ′ k1

O2 + R1 ⇄ O1 + R 2

where A is the area of the electrode and τ is the duration of the potential pulse applied. Therefore (see eqs 23 and 35) Q EE, N =

Q EE QF

=

2K1/2 + J ̅ K1/2 + J ̅ + K1/2J ̅ 2

= 2fR + fI

(V)

k2

with the equilibrium constant, K (= k1/k2), for reaction V given by (see eq 12): s s ⎛ F ⎞ CO C R ΔE 0 ′⎟ = s 1 s 2 K = exp⎜ ⎝ RT ⎠ CO2C R1

(39)

where Q F = FA ΓT

(IV)

(38)

(42)

Note that in this case, due to the independence of the two electrochemical steps in reaction IV, only the values ΔE0′ ≤ 0 (eq 2) must be considered, that is (0 ≤ K ≤ 1, eq 42). In the following we refer to process (IV) as E+E mechanism. Thus, if O1 and O2 are initially present with identical excesses/concentrations values, and all species have equal diffusion coefficients in the solution-soluble case (such that reaction V has no effect on the current),1 the expression for the normalized current, ψE+E, is (see eqs 31 and 41)1,33

(40)

Note that the QEE − E curve for a reversible EE process taking place in a monolayer is independent of time (i.e., has a stationary character), and therefore, eq 39 is fulfilled when any potential−time waveform is applied to the electrode. It is also important to highlight that the normalized charge, QEE,N, has an identical expression to that for the normalized transient current INPV EE,N obtained for solution-soluble species when the NPV technique is applied to an electrode with any geometry (eq 30) and also to the normalized stationary current obtained for solution-soluble species when any potential−time waveform is applied for ultramicroelectrodes with any geometry. The normalized current−potential curve in CV, ψCV EE , can be easily obtained from eq 39, being31−36

ψE + E =

K1/2J ̅ (1 + K1/2J ̅ )2

+

K3/2J ̅ (K + K1/2J ̅ )2

(43)

with J ̅ and K given by eqs 11 and 42, respectively. In eq 43, the first and second addends refer to the contributions to the total current of O1/R1 and O2/R2 reactions in eq IV, respectively. As 12316



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Table 2. Analytical Expressions for the Roots with Physical Meaning (Epeak, Evalley) of the Derivative of the Normalized Current, ψ, for the EE and E+E Mechanisms (eqs 31 or 41, and 43, Respectively)33,36,a

a

K is given by eqs 12 and 42 for the EE and the E+E mechanisms, respectively. Note that peak potentials in dNPV (solution-soluble species) and CV (surface-confined species) correspond to the half-wave potentials in NPV1,14 and charge-based techniques,33 respectively. T = 298 K.

for an EE mechanism, ψE+E is an even function of (E − E̅0′), and at E = E̅0′ takes the form (see eq 43 with J ̅ = 1 and ref 1.) (ψE + E)E̅

0′

⎛ K1/2C s + C s ⎞ K1/2 R1 R2 ⎟ =2 = ⎜⎜ ⎟ 1/2 2 1/2 (1 + K ) ⎝ 1+K ⎠ E̅ 0′

Figure 1c). In Figure 1a, the peak potentials are referred to the average formal potential value, E̅0′. Thus, since the (ψ − E) curves for a reversible process are symmetrical with respect to the vertical axis located at E = E̅ 0′ (see Figure 2 below), it is fulfilled that

(44)

|Epeak,1 − E ̅ 0 ′| = |Epeak,2 − E ̅ 0 ′| =

that is, at the average formal potential both single-electron transfers have the same contribution to the current, as expected. Table 2 shows the analytical expressions for the roots with physical meaning of the derivative of the normalized current, ψ, for the EE and E+E mechanisms (eqs 31 or 41 and 43, respectively), which correspond to the peaks and valley potentials in the response, given as a function of K (eqs 12 and 42 for the EE and the E+E mechanisms, respectively).14,33,36 The values of K (i.e., of ΔE0′) for the transition 2 peaks−1 peak in each mechanism, and the limit value of K for a response with two peaks centered at the individual formal potentials are also indicated. These results can be seen in Figure 1, which shows the evolution with ΔE0′ of the peak parameters of the (ψ − E) response, for the EE and E+E mechanisms (calculated from eqs 31 or 41 and 43, respectively): peak potentials (Epeak, Figure 1a), peak height (ψpeak, Figure 1b), and half-peak width (W1/2,

|ΔEpeak | 2

(45)

with ΔEpeak (= − being the difference between peak potentials. The horizontal black line at E = E̅0′ and the two symmetrical red branches represent, respectively, that only one peak and two peaks are obtained in the corresponding range of ΔE0′ values in abscissas, as is indicated in Figure 1. The two oblique dashed black lines with slope |0.5| represent that E2peak

E1peak)

|E10 ′ − E ̅ 0 ′| = |E20 ′ − E ̅ 0 ′| =

|ΔE 0 ′| 2

(46)

Hence, for the values of ΔE0′ at which the red and the dashed black lines are overlapped, the peak potentials coincide with the individual formal potentials. Figure 1b shows that the values of the peak height, ψpeak, for an EE process, change with ΔE0′ between 1/4 (for ΔE0′ ≪ 0) and 1 (for ΔE0′ ≫ 0), as expected, since the normalized current for an E mechanism 12317



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excess ΓT, for immobilized surface molecules, or the bulk concentration, c*O, for solution-soluble species.31−33,35,36 The most singular characteristic exhibited by Figure 1 is that the values of ΔE0′ (25 °C) = −142.4, −71.2, −35.6, 0 mV (K = 1/28, 1/24, 1/22, 1/20, respectively, see eq 12, where the base “2” refers to the number of electrons transferred), play an essential role in the behavior of the peak parameters. Indeed, the following are observed (see also Table 2): (1) For ΔE0′ ≤ − 142.4 mV (K ≤ 1/28) the two peak potentials in the voltagram (for both EE and E+E mechanisms) correspond to the individual formal potentials, E01′ and E02′ (see the superposition of red and dashed black lines in Figure 1a), and ψpeak = 1/4 (see Figure 1b and eqs 33 and 34), both peak parameters being characteristic of a simple E mechanism, although the value W1/2 = 90 mV is only obtained for ΔE0′ ≤ −200 mV (see Figure 1c).1,33 (2) For ΔE0′ = −71.2 mV (K = 1/24) in the EE mechanism and ΔE0′ = −67.7 mV (K = (7−4√3)) in the E+E mechanism (i.e., with a difference of 3.56 mV), the transition 2 peaks−1 peak takes place (see Figure 1a), obtaining in both cases identical ψ − E curves, with ψpeak = 1/3 (see Figure 1b and eqs 32 and 44) and W1/2 = 143.8 mV ≈ 2|ΔE0′|EE (Figure 1c).14,18,32,33,36 For more positive values of ΔE0′, only one peak appears in the response, whose peak potential corresponds to E̅ 0′ (see Figure 1a) and whose peak heights are given by eqs 32 for the EE mechanism and 44 for the E+E one. (3) When two peaks appear in the response for the EE mechanism (i.e., ΔE0′ < −71.2 mV, see Figure 1a-c), it can be considered as practically indistinguishable from the E+E one and, therefore, can be treated in a simpler manner in this case. (4) For ΔE0′ = −35.6 mV (K = 1/22) in the EE and ΔE0′ = 0 mV (K = 1) in the E+E mechanisms, both processes present the same response, whose height is double that for a simple E mechanism, that is, ψpeak = 1/2 (Figure 1b), but W1/2 = 90 mV (Figure 1c).1,17,19,25,32 This is an obvious result in the case of an E+E mechanism. (5) For any value of ΔE0′ for the EE mechanism in the interval −71.2 ≤ ΔE0′ ≤ − 35.6 mV, there is always a higher ΔE0′ value for the E+E one in the interval −67.7 ≤ ΔE0′ ≤ 0 mV for which the voltammetric responses in both mechanisms are identical (see Figures 1b,c). This correspondence can be seen in Figure 1d; for example, an EE process with ΔE0′ = −48.0 mV will present the same voltagram as an E+E one for ΔE0′ = −37.0 mV. (6) The peak height for an EE process (eq 32) presents a linear behavior with ΔE0′ in the interval (see the dashed black line in Figure 1b), − 71.2 mV ≤ ΔE0′ ≤ 0 mV: ψEE,peak = ((ΔE0′)/(3(71.2))) + 2/3, i.e., 1/3 ≤ ψEE,peak ≤ 2/3 that is (see Figure 1a), between ΔE0′ = −71.2 mV, for which it is fulfilled that Epeak,1 = Epeak,2 = E̅ 0′ (transition 2 peaks−1 peak), and ΔE0′ = 0 mV, for which E01′ = E02′ = E̅0′ (transition normal ordering-inverted potentials4). (7) For ΔE0′ ≥ 200 mV (K → ∞) for the EE mechanism: ψpeak = 1 (eq 32, Figure 1b), W1/2 = 45 mV (Figure 1c); therefore, the EE process behaves as an E mechanism of two electrons. Obviously, this response cannot be obtained for an E+E process.

Figure 1. Evolution with ΔE0′ of the peak potentials, Epeak − E̅ 0′ (a), the normalized peak-height, ψpeak (b), and the half-peak width, W1/2 (c), of the ψ − E response, for the reversible EE and E+E mechanisms (calculated from eqs 31 or 41 and 43, respectively). The appearance of one or two peaks in the ψ − E response is indicated by the arrows in (a) (the red lines correspond to the apparition of two peaks). The curve in Figure 1d shows the correspondence (obtained from (b,c)) between the ΔE0′ values for the EE (in ordinates) and E+E (in abscissas) mechanisms, in order to obtain an identical voltammetric response. T = 298 K.

with n electrons transferred is ψE,peak = n2/4;1,2 thus, 1/4 for n = 1 and 1 for n = 2. In the same way, Figure 1c shows that the half-peak width, W1/2, changes in the extreme negative and positive values of ΔE0′ between 90 mV (E1e−) and 45 mV (E2e−) for an EE process, but a sharp jump in the half-peak width, W1/2, is observed at ΔE0′ ≈ −135 mV (∼2 × (−67.7 mV), see below), which corresponds to a ψ − E curve with two peaks, whose central trough is situated just at the half-peak height. Curves a−c of Figure 1 can be used as working curves to accurately determine the values of ΔE0′ and the total surface 12318



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Figure 2. ψEE − (E − E̅ 0′) curves (red lines) and f j (= Csj ) − (E − E̅ 0′) curves (j = O, I, or R: solid, thick dotted and dashed black lines, respectively) obtained for a reversible EE mechanism from eqs 31 or 41 and 23, respectively, for six representative values of ΔE0′ (= E02′ − E01′), shown in parts a− f. The cross points PE01′, PE02′, and P E̅ 0 ′ correspond to the points where the surface coverages/concentrations take the same values, according to the nernstian conditions: f O = f I at PE01′ (eq 7), f I = f R at PE02′, (eq 8), and f O = f R at P E̅ 0 ′ (eq 10). T = 298 K.

3.1. Analysis of the (ψEE − E) Response from the Study of the Surface Concentrations/Coverages. In Figure 2, the ψEE − (E−E̅ 0′) curves (red lines) for an EE mechanism (eqs 31 or 41) and the surface coverages/concentrations, f j/Csj (j = O (solid), I (dotted), R (dashed) black curves) (eq 23) have been plotted for six representative values of ΔE0′ (in mV), in accordance with the results in Figure 1: (a) −200, (b) −142.4, (c) −71.2, (d) −35.6, (e) 0, (f) 200. For simplicity in the nomenclature, in the following discussion we will refer to the surface coverages and all the comments are valid for the surface concentrations, since both are given by eq 23. In Figure 2, three characteristic cross points for the surface coverages can be observed: PE01′ (f O = f I, eq 7), PE02′ (f I = f R, eq 8), and P E̅ 0′ ( f O = f R, eq 10), located, respectively (see dotted vertical lines), at E = E10′ (E − E̅ 0′ = −ΔE0′/2), E = E02′ (E − E̅0′ = ΔE0′/2), and E = E̅ 0′, regardless of the difference between the formal potentials, ΔE0′ (see Figure 2a−f). Note that when ΔE0′ increases from −200 mV (Figure 2a) to 200 mV (Figure 2f), the values of the surface coverages at PE01′ and PE02′ vary between 0.5 (i.e., as corresponds to a simple E mechanism) and 0, and contrarily, between 0 and 0.5 at P E̅ 0′. The species O and I, at PE01′, and I and R, at PE02′, are formally related as in two separated E mechanisms through the nernstian conditions given by eqs 7 and 8, although only if the species I is stable enough for (f I)E01′ = ( f I)E01′ ≈ 0.5 (see Figure 2a−c), the process will actually behave as separated E mechanisms. Additionally, the species O and R, at E̅0′, are formally related

as in a reversible E mechanism of two electrons (eq 10) although only in the absence of species I (f O)E̅0′ = (f R)E̅0′ = 0.5) will the process really behave as an E2e− mechanism (see Figure 2f). In brief, the gradual conversion of a 100% E1e− E1e − mechanism into a 100% E2e− one when increasing the ΔE0′ value can be quantified in terms of the presence of the intermediate. As explained in section 2, f I presents, logically, a maximum at E = E̅ 0′ and consequently (f O + f R) has a minimum at this potential value (eq 24). Thus, the evolution of the process with ΔE0′ can be regarded as a “two against one contest”, so the values of (fI )E̅ 0′ and (fO + fR )E̅ 0′ are an indicative of the % character E1e−E1e− and % character E2e−, respectively. For ΔE0′ ≤ −142.4 mV (K ≤ 1/256) (see Figure 2a,b), from eq 25 it is fulfilled that (f O = f I)E01′ = ( f I = f R)E02′ ≈ 0.5 and ( f R)E01′ = ( f O)E02′ ≈ 0 ( f O + f I + f R = 1, eq 35). In these conditions, the intermediate I is very stable; i.e., the value of ( f I)E̅0′ is near unity and the peak potentials in the ΨEE − E response (red curve) correspond to the individual formal potentials; that is, they are coincident with those at PE01′ and PE02′ (see also EE-red line in Figure 1a). So, the peak height, Ψpeak = 1/4 (Figure 1b), coincides with that obtained from eq 33 or 34. Indeed, from eq 33, (ΨEE)E01′ = ΨEE,peak,1 ≈ 2 × 0 + 1/4, a result that corroborates the stability of the intermediate. Note that, strictly, (fI )E̅ 0′ ≈ 1 for ΔE0′ ≤ − 200 mV (see Figure 2a and eq 24), in this case, W1/2 = 90 mV in the ΨEE − E response (see also Figure 1c), as indicated at point 1 in the 12319



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the process changes from 66.7% character E1e−E1e− to 66.7% character E2e−. However, for ΔE0′ > 0 (K > 1), also with one peak in the voltagram, Figure 1a, the intermediate is unstable ((fI )E̅ 0′ < (fO = fR )E̅ 0′). For ΔE0′ ≥ 200 mV (Figure 2f, see points PE01′ and PE02′, which have interchanged their relative left−right positions with relation to those for ΔE0′ < 0; see dashed black lines in Figure 1a) the intermediate practically disappears, and (fO )E̅ 0′ = (fR )E̅ 0′ = 0.5. Thus, the typical behavior of surface coverages for a two-electron E mechanism is observed (100% character E2e−), as indicated in point 7 in the discussion of Figure 1. In the previous discussion, the percentage of character E1e−E1e− and E2e− in the ψEE − E response has been discussed in terms of the surface coverage values at the average formal potential, E̅ 0′. Moreover, the relation between (ψEE)E̅ 0′ and (fO + fR )E̅ 0′ given by eq 32 allow us to write

discussion of Figure 1. This limit situation corresponds to a genuine (100%) E1e−E1e− mechanism with a totally stable intermediate (0% character E2e−). For ΔE0′ = −142.4 mV, (fI )E̅ 0′ = 8(fO + fR )E̅ 0′ = 8/9 (see eq 24 and Figure 2b); thus, the process has 89% character E1e−E1e− and 11% character E2e−, which is manifested in the fact that the value of W1/2 in the ΨEE − E response is larger than 90 mV (Figure 1c). For −142 mV < ΔE0′ < −71.2 mV (Figure 2b,c), the intermediate remains stable ((fI )E̅ 0′ > 2(fO + fR )E̅ 0′, eq 24), and the cross points PE01′ and PE02′ are still near the value 0.5 corresponding to simple E processes (0.485 for ΔE0′ = −71.2 mV; see Figure 2c and Figure 4). Therefore, for ΔE0′ < −71.2 mV the EE mechanism behaves practically like the E+E process, as indicated in points 2 and 3 in the discussion of Figure 1. In this interval of ΔE0′ values, two peaks are still obtained in the ψEE − E signal, although, as shown in Figure 1a, the peak potentials do not correspond to the individual formal potentials (unlike that obtained for ΔE0′ ≤ −142.4 mV). Note that a visual inspection of the curves of surface coverages in Figure 2b, and due to the axial symmetry of the curves at E = E̅0′, allows us to detect that the curve of f I between PE01′ and PE02′ reproduces exactly that of peak potentials given in Figure 1a in the interval of ΔE0′ considered (−142.4 mV ≤ ΔE0′ ≤ −71.2 mV); therefore, the transition 2 peaks−1 peak in the ψEE − E response must take place at ΔE0′ = −142.4/2 mV = −71.2 mV. Indeed, for ΔE0′ = −71.2 mV (Figure 2c), one peak centered at E = E̅ 0′ is obtained in the ψEE − E curves; thus, from the electrochemical response, it could be inferred that the intermediate has “lost” the contest, however it is still stable (according to eq 24 it is fulfilled that (fI )E0̅ ′ = 2(fO + fR )E̅ 0′ = 2/3 (see also Figure 2c); i.e., the process has 66.7% of the character of E1e−E1e−. Nevertheless, from this value of ΔE0′, the intermediate does not contribute to the peak height, since dfI

( ) dE

E̅ 0′

%character E2e− = (ψEE)E̅ 0′ × 100

(47)

Thus, by inserting eq 32 in eq 47 both the percentage of E2e− character and the ΔE0′ can be calculated from the value of (ψEE)E̅ 0′, independently of there being a valley or a peak at E = E̅ 0′ in the ψEE − E response (see Figures 2a-f). Indeed, taking into account eq 12 ΔE 0 ′ =

(ψEE)E̅ 0′ RT ln 2nF 2(1 − (ψEE)E̅ 0′ )

(48)

Thus, the individual formal potentials, E01′ and E02′, can be determined in any case from the average formal potential, E̅ 0′, directly located in the response and the ΔE0′ value calculated from eq 48 (see eqs 1 and 2). In line with the above, it can be introduced the term “effective electron number”, neff, defined as (1 + ((% character E2e−)/100)), i.e.,

= 0 (see eq 32).

The value ΔE0′ = −35.6 mV (Figure 2d, ((fI )E̅ 0′ = 2(fO )E̅ 0′ = 2(fR )E̅ 0′, i.e., (fI )E̅ 0′ = (fO + fR )E̅ 0′ = 1/ 2) corresponds to the case for which the f I curve becomes coincident with the ψEE − E one and has the particular interest of corresponding to 50−50% of character E1e−E1e− and E2e−. Indeed, the intermediate reaches at E̅0′ the half of its maximum value and hence, this value of ΔE0′ would correspond to the equiprobability of the O/I and I/R conversions. Thus, this ΔE0′ could be considered as the boundary between anticooperative and cooperative behavior of both electron transfer reactions (see below). Note also that the set of curves for coverages of f I and (f O + f R) (dotted blue curve in Figure 2d) presents the shape of a “peculiar” E mechanism, similar to that for the E+E mechanism with ΔE0′ = 0 mV, as indicated in point 4 in the discussion of Figure 1. For ΔE0′ = 0 mV (Figure 2e), PE01′, PE02′, and P E̅ 0′ become coincident and (fO )E̅ 0′ = (fI )E̅ 0′ = (fR )E̅ 0′ = 1/3 (i.e., 1

(fI )E̅ 0′ = 2 (fO + fR )E̅ 0′, 33.3% character E1e−E1e−, 66.7% E2e−). This value of ΔE0′ (K = 1, see eqs 12 and scheme (III)) corresponds to the stable-unstable intermediate transition (normal-inverted order of potentials4). Thus, in the interval −71.2 mV ≤ ΔE0′ ≤ 0 mV, with only one peak in the voltagram, in which the peak height presents a linear behavior with ΔE0′ (see Figure 1b and point 6 in the discussion of Figure 1), the intermediate is stable (K < 1) and

Figure 3. Evolution with ΔE0′ of the “effective electron number”, neff (solid line, eqs 49 and 47), and napp (dashed line, eq (A.14) in ref 32), at E = E̅ 0′. T = 298 K. 12320



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Figure 4. Evolution with ΔE0′ of the surface coverages/concentrations, f j/Csj , obtained for the reversible EE (j = O, I, R (a)) and E + E (j = O1, R1, O2, R2 (b)) mechanisms from eq 23 and ref 1, respectively, for E = Epeak,1 and Epeak,2 (green and pink lines), E = E̅ 0′ (red and blue lines), and E = E01′ and E02′ (thick dotted black line), as detailed in the figure. Points A, A′, B, B′ ( = C′), and C are cross points at the characteristic values of ΔE0′ = −71.2, − 67.7, − 35.6, and 0 mV. T = 298 K.

neff = 1 + (ψEE)E̅ 0′ =

2e RT /2F ΔE

(eq 49), is not coincident with the “apparent number of electron transferred”, napp, which is used mainly in bioelectrochemistry. In fact, napp is defined in refs 5 and 32 for ΔE0′ ≥ −35.6 mV and obtained, according to eq A.14 in ref 32, from the half-peak width, W1/2 (see Figure 1c). In Figure 3 the evolution of neff and napp with ΔE0′ has been plotted. It can be observed that neff is defined for any value of ΔE0′ and takes the value 1.5 for ΔE0′ = −35.6 mV, that is, for a 50% character E2e− process or null cooperativity degree between the electron transfers. However, for this ΔE0′ value napp = 1. For ΔE0′ ≥ 200 mV, the 100% character E2e− corresponds to a 100% cooperativity degree, and neff = napp = 2. Note that only in this limit neff and napp take the same value.5,32 That is, napp is not related with a cooperativity or anticooperativity degree scale since napp only has a clear physical meaning in the upper limit ΔE0′ ≥ 200 mV. In Figure 4, the values of f j/Csj for E = Epeak,1, Epeak,2, E̅ 0′, E01′, and E02′ are plotted vs ΔE0′ for the EE mechanism (j = O, I, R, Figure 4a) and for the E+E one (j = O1, R1, O2, R2, Figure 4b)

0′

1 + 2e RT /2F ΔE

0′

(49)

in such a way that (ψEE)E̅ 0′ (which corresponds to the lost of stability of the intermediate, i.e., to 1 − (fI )E̅ 0′) is the probability of the second electron to be transferred in an apparently simultaneous way with the first one. So, for ΔE0′ ≤ −200 mV, from eq 47 and Figure 2 it is easily deduced that the probability of an apparent simultaneous transfer of the second electron is practically null; thus, neff = 1 (eq 49); in the transition 2 peaks−1 peak (ΔE0′ = −71.2 mV), neff = 1.3̑ that is, this probability is 1/3; for ΔE0′ = −35.6 mV, neff = 1.5 (the second electron has a probability equal to 0.5 for being transferred in an apparently simultaneous way with the first); for ΔE0′ = 0 mV, neff = 1.6̑; for ΔE0′ ≥ −200 mV, neff = 2, logically. It is important to highlight that neff, defined in this paper from the stability of the intermediate and the % character E2e− 12321



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for comparison. Thus, the behavior given in Figure 1 for the ψ − E response is reproduced in this figure in terms of surface coverage values (note that, according to eq 32, the curve for (fO + fR )E̅ 0′ (dashed red line) in Figure 4a is coincident with that for ΨEE,peak in Figure 1b for ΔE0′ ≥ −71.2 mV, corresponding to the valley height for ΔE0′ < −71.2 mV, not shown in Figure 1b). Also according to the above results, the curves for (f I)E̅0′ (blue line) and (fO + fR )E̅ 0′ (dashed red line) in Figure 4a represent the character E1e−E1e− and character E2e−, respectively, of the EE process, as a function of ΔE0′ values (eq 24). Thus, dashed red line is the same as that of neff in Figure 3, translated one unity (see eq 49). In Figure 4a it can be observed that there are three characteristic values of ΔE0′ for which different curves cross (see points A (double), B and C): For ΔE0′ = −71.2 mV (see points A) the curves for surface coverages of O and R at Epeak,1 and Epeak,2 (green curves) become coincident and cross the curve of (fO + fR )E̅ 0′ (red line). Also, the curves for (f I)Epeak,1 = (f I)Epeak,2 (pink line) and (fI )E̅ 0′ (blue line) cross. This corresponds to the transition 2 peaks−1 peak. Points A in Figure 4a are equivalent to points A′ in Figure 4b (for ΔE0′ = −67.7 mV); i.e., points A′ also correspond to the transition 2 peaks−1 peak. In this case (fR = fO )E̅ 0′ = 0.79 and (fO = fR )E̅ 0′ = 0.21. For these values 1

2

1

the behavior of an apparent simultaneous two-electron E mechanism being attained with ΔE0′ ≫ 0.

4. CONCLUSIONS In this paper, a comparative study of the reversible EE mechanism with surface-immobilized and solution-soluble species has been carried out. In the latter, it has been considered that the species are transported to/from the surface of an electrode with any geometry and size only by diffusion, with equal diffusion coefficients. The correspondences between NPV and dNPV techniques (for solution-soluble molecules) and charge-transferred based techniques and CV (for surface-confined molecules), respectively, have been established. Thus, a common study of the corresponding voltammetric response can be made for both situations of solution-soluble and immobilized molecules with the appropriate normalization in each case. A parallel analysis of the common normalized peak-shaped voltammetric signal in dNPV (solution-soluble species) and CV (surface-confined systems) and of the surface concentrations/ coverages has been carried out, highlighting the close relationship between the two behaviors. On the basis of the surface concentrations/coverages values at the average formal potential, E̅0′ = (E01′ + E02′)/2, the evolution of the voltammetric response with ΔE0′ = E02′−E01′, has been interpreted and quantified as a gradual conversion of the EE mechanism from 100% character E1e−E1e− response for ΔE0′ ≪ 0 and 100% character E2e− one for ΔE0′ ≫ 0, thus characterizing the most interesting intermediate situations, corresponding to transitions 2 peaks centered-not centered at the individual formal potentials (ΔE0′ = −142.4 mV, 11% character E2e−), 2 peaks−1 peak (ΔE0′ = −71.2 mV, 33.3% character E2e−), repulsive−attractive interactions, anticooperativity−cooperativity (ΔE0′ = −35.6 mV, 50% character E2e−), and normal− inverted order of potentials (ΔE0′ = 0 mV, 66.7% character E2e−). The percentages indicated can easily be obtained in practice from the current height at E̅0′ (peak or valley) in the voltammetric curve. Thus, the value of ΔE0′ can also be obtained in this simple and direct way, and hence, those of individual formal potentials, E01′ and E02′, regardless of the value of ΔE0′. As a consequence of this study, the term “effective electron number”, neff, has been introduced and related to the probability of the second electron being transferred in an apparently simultaneous way with the first one. Indeed, neff varies between 1 for ΔE0′ ≪ 0 (0% character E2e−, very stable intermediate) and 2 for ΔE0′ ≫ 0 (100% character E2e−, very unstable intermediate). Thus, in the transition 2 peaks−1 peak (ΔE0′ = −71.2 mV), neff = 1.3̑, that is, this probability is 1/3; for ΔE0′ = −35.6 mV, neff = 1.5 (the second electron has a probability equal to 0.5 for being transferred in an apparent simultaneous way with the first); for ΔE0′ = 0 mV, neff = 1.6̑. The “effective number of electrons transferred” is not coincident with the “apparent number of electrons transferred”, napp, extensively used in bioelectrochemistry (see Figure 3). It has been shown that, although the intermediate is stable in the interval −71.2 mV ≤ ΔE0′ ≤ 0 mV, only one peak is obtained in the voltammetric curve, whose peak height increases linearly with ΔE0′. For ΔE0′ > 0 mV, (unstable intermediate) the peak height increases nonlinearly with ΔE0′ until the process behaves as a 100% character E2e− when the intermediate practically disappear.

2

of surface coverages it is fulfilled from eq 44 that (ψE + E)E̅ 0′ = 1/3, characteristic of the transition 2 peaks−1 peak (see Figure 1b). For ΔE0′ = −35.6 mV (see point B) the curves for (f I)E̅0′ (blue line) and (fO + fR )E̅ 0′ (dashed red line) in Figure 4a cross. Note that, morphologically, this cross point is comparable (both correspond to the ordinate 0.5) with that for ΔE0′ = 0 mV for the E+E mechanism in Figure 4b (see point B′). The ψ − E response for both mechanisms is double of that for a simple E mechanism. The origin of this equivalence (which is in agreement with the well-known statistical behavior as “noninteracting centers” of an EE mechanism with ΔE0′ = −35.6 mV1,17,19,25), lies in the relation between the surface coverages at both values of ΔE0′. Indeed, for the EE mechanism (Figure 4a), (fI )E̅ 0′ = (fO + fR )E̅ 0′ = 0.5, that is, the surface coverages are in the relation f O:f I:f R = 1:1 + 1:1 (K = 1/4, eq 12); for the E+E mechanism (Figure 4b), (f j )E̅ 0′ = 0.5 and the relation is f O1:f R1:f O2:f R2 = 1:1:1:1 (K = 1, eq 42). Thus, the value ΔE0′ = −35.6 mV corresponds to the transition repulsive−attractive (negative−positive) interactions in ref 1 and to the transition anticooperative−cooperative behavior.5,32 For ΔE0′ = 0 mV (see point C) the curves for (fI )E̅ 0′ (blue line), (fO = fR )E̅ 0′ (red line) and (f j )Ei0′ (dotted sigmoidal black line) cross, so (fO )E̅ 0′ = (fI )E̅ 0′ = (fR )E̅ 0′ = 1/3. This corresponds to the transition normal ordering-inverted potential.4 Note that point C is also equivalent to point B′ (= C′) for ΔE0′ = 0 mV for the E+E mechanism in Figure 4b, in the sense that it is fulfilled that (f j )E̅ 0′ = (f j )Ei0′, but the indicated transition is not possible for the E+E process because normal and inverted order of potential correspond to the same situation in this case (reflected in that the thick dotted black line of ordinate 0.5 in Figure 4b is horizontal). The ligature between O, I, and R in the EE mechanism leads to the sigmoidal thick dotted black line in Figure 4a, and ultimately to 12322



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From the study of two independent E mechanisms (E+E in this paper), it has been concluded that if two peaks appear in the response both mechanisms behave as practically indistinguishable.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We greatly appreciate the financial support provided by the ́ Dirección General de Investigación Cientifica y Técnica (Project No. CTQ2012-35700 cofunded by the European Regional Development Fund).

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REFERENCES

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Polyoxometalate [PMo12O40]3‑ Immobilized at a Boron Doped Diamond Electrode. Anal. Chem. 2013, 85, 8764−8772. (37) Molina, A.; Gonzalez, J.; Henstridge, M.; Compton, R. G. Voltammetry of Electrochemically Reversible Systems at Electrodes of Any Geometry: a General, Explicit Analytical Characterisation. J. Phys. Chem. C 2011, 115, 4054−4062.

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Two-Electron Transfer Reactions in Electrochemistry for Solution

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