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Two-Electron Transfer Reactions in Electrochemistry for Solution Article pubs.acs.org/JPCC Two-Electron Transfer Reactions in Electrochemistry for Solution- Soluble 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 Δ 0′ 0′ − 0′ − “ ff transfers, E = E2 E1 . In line with the percentage of E2e character, the term e ective ” 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 0′ 0′ Δ 0′ method to obtain the values of E1 and E2 for any E 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 to/from the electrode surface and when the molecules are fi Electrode processes consisting of two-electron transfers (EE surface-con ned, so the transport is avoided. Traditionally, mechanism) have been widely treated in the literature, both in these two fundamental problems in electrochemical science are − ff their theoretical and applied aspects.1 9 This high productivity treated as completely di erent problems. measures in some way the great presence and relevance of these In this paper, both possibilities are tackled in the case of a processes in many fields, and hence the importance of reversible EE mechanism, in order to establish the correspond- understanding them. This behavior is very common in ences between their electrochemical responses and the electrochemical reactions of alkylviologens and metallocenes, conditions under which a common study of these situations in the reductions of several metallic ions, of polyoxometallates can be made. Thus, it has been shown that, with the and of a number of aromatic species like derivatives of appropriate normalization in each case, the electrochemical − tetraphenylethylene.1,2,4,8 13 In the specific case of biological signal obtained when normal pulse voltammetry (NPV) is molecules, such as oligonucleotides, metalloproteins, enzymes, applied in the case of solution soluble species for electrodes of any geometry and size is coincident with the charge transferred- etc., the application in recent years of techniques such as fi protein film voltammetry (PFV), combined with scanning potential one for surface-con ned systems in any electro- probe microscopic techniques, has made it possible to chemical technique. Thus, their derivatives lead to a common − normalized peak-shaped voltagram in derivative normal pulse characterize the biomolecule electrode interface and elec- 1,2,4,14−30 tron-transfer processes in great detail, which is fundamental to voltammetry (dNPV), for solution soluble molecules, ffi and in cyclic voltammetry (CV), for surface-immobilized exploit the naturally high e ciency of these biological systems 1,3,5−9,31−36 in modern biotechnology (selective last-generation biosensors, ones. These normalized responses are only environmentally sound biofuel cells, heterogeneous catalysts, potential-dependent as a consequence of the reversible 3,5−7,9 behavior of the process, which is reflected in the independence biomolecular electronic components). Many of these − of time of surface concentrations (solution phase case)15,24 26 systems present a reversible behavior (or it can be reached 31−36 acting on the adequate experimental parameter in the particular and excesses (immobilized species). electrochemical technique used), which simplifies the study of Thus, the parallel characteristics of the common normalized multielectron transfer processes by not having to consider the voltammetric curve in dNPV and CV and the surface kinetics of the electron-transfer reactions. When studying the EE mechanism, two situations for the Received: March 14, 2014 molecules undergoing the process can be encountered: when Revised: April 30, 2014 they are soluble in the solution and then need to be transported Published: May 12, 2014 © 2014 American Chemical Society 12312 dx.doi.org/10.1021/jp5025763 | J. Phys. Chem. C 2014, 118, 12312−12324 The Journal of Physical Chemistry C Article ̂ Table 1. Expressions for Diffusion Mass Transport Operators, δj (j =O, I, or R), Functions f (q ,t) and f for the Main a G G G,micro Electrode Geometries ff δ̂ electrode di usion operator j f G(qG,t)(Dj = D) f G,micro plane ∂ ∂2 1 − D 1/2 ∂t j ∂x2 ()πDt π 2 sphere (radius rS, AS =4 rs ) ⎛ 2 ⎞ 11 1 ∂ ⎜ ∂ 2 ∂ ⎟ + − Dj + 1/2 ∂t ⎝ ∂r2 rr∂ ⎠ rs ()πDt rs π 2 ⎛ ⎞ ⎞ disk (radius rd, Ad = rd) ∂ ∂2 ∂ ∂2 ⎛ ⎛ ⎞ 41 ⎜ 1 ⎟ 41 rd rd ⎟ − Dj + + ⎜0.7854++− 0.44315 0.2146 exp⎜ 0.39115 ⎟⎟ ∂ ⎝ ∂ 2 ∂ ∂ 2 ⎠ 1/2 1/2 π rd t r rr z π rd ⎝ ()Dt ⎝ ()Dt ⎠⎠ band (height w, length l, A = wl) ⎛ 2 2 ⎞ 11 12π w ∂ ∂ ∂ +<2 − D⎜ + ⎟ 1/2 ifDt / w 0.4 2 ∂t j⎝ ∂xz2 ∂ 2 ⎠ w ()πDt w ln[64Dt / w ] ⎛ ⎞ ⎛ π ⎞1/2 ()πDt 1/2 π 0.25⎜⎟ exp⎜−+ 0.4 ⎟ ⎝ ⎠ ⎛ 1/2 ⎞ Dt ⎝ w ⎠ ()Dt wln⎜⎟ 5.2945+ 5.9944 ⎝ w ⎠ ifDt / w2 ≥ 0.4 cylinder (radius rC, length l, AC = ⎛ 2 ⎞ ⎛ 1/2 ⎞ 12 π ∂ ⎜ ∂ 1 ∂ ⎟ 1 ⎜ ()πDt ⎟ 1 2 rcl) − Dj + exp−+ 0.1 2 ∂ ⎝ 2 ∂ ⎠ 1/2 ⎝ ⎠ ⎛ 1/2 ⎞ rc ln[4Dt / rc ] t ∂r rr ()πDt rc ⎜⎟()Dt rc ln 5.2945+ 1.4986 ⎝ rc ⎠ a 28,29,37 qG is the characteristic dimension of the electrode: rs for spheres or hemispheres; rd for discs; w for bands; rc for cylinders. concentrations/excesses have been highlighted. On the basis of inverted order of potentials.4 Also, it has been discussed the this study, the variation of the voltagrams shape depending values of ΔE0′ for which two and one peaks appear in the upon the relative values of the formal potentials of both charge- voltammetric curve, and the different evolution of one-peak transfer reactions in the EE mechanism, as expressed by ΔE0′ = curves with ΔE0′, in terms of the intermediate stability. 0′ − 0′ E2 E1 , has been interpreted in terms of the percentage of For completeness, the EE mechanism has been compared − − − consecutive (E1e E1e ) and apparent simultaneous (E2e ) with the case of two independent one-electron-transfer characters of the electron-transfer process, which have been reactions with identical initial concentrations/excesses, which intrinsically related to the surface concentrations/excesses has been called an E+E mechanism in this article.1,33 It has been values of the intermediate and the sum of the two extreme shown that whenever two peaks are obtained in the oxidation states of the molecule, respectively, at the average voltammetric response of an EE mechanism (ΔE0′ < −71.2 0′ 0′ 0′ formal potential, E̅ =(E1 + E2 )/2, and, more practically, with mV) it can be considered practically indistinguishable from that the current value at E̅0′ (peak or valley) in the voltagram. From for an E+E process. these concepts, the term “effective number of electrons ” transferred , neff, has been introduced, which varies between 1 2. THEORY Δ 0′ ≪ − for E 0 (0% character E2e , very stable intermediate) and Consider an electrode process in which a molecule reduces Δ 0′ ≫ − 2for E 0 (100% character E2e , very unstable reversibly involving two-electron transfers, according to the intermediate) and has been related to the probability of the following reaction scheme (EE mechanism) second electron being transferred in an apparently simulta- 0 neous way with the first one. The values of n for any ΔE ′ −′ eff Oe+⇄ IE 0 have been discussed and compared with the “apparent number 1 of electron transferred”, n , extensively used in bioelectro- −′0 app Ie+⇄ RE2 (I) chemistry, and defined in the literature for ΔE0′ ≥−35.6 mV values, i.e., for cooperative behavior between the electron 5,32 in which O (oxidized), I (intermediate, or half-reduced) and R transfers. Related to the above ideas, a simple direct method (reduced) refer to the different redox states of the molecule and 0′ 0′ for obtaining the individual formal potentials, E1 and E2 , E0′ and E0′ are the formal potentials of the first and second Δ 0′ 1 2 regardless of the E value, has been proposed. steps, respectively. The average formal potential, E̅0′, given by In this study of the two-electron transfer reactions the importance of the ΔE0′ (in mV, 25 °C) = −142.4, − 71.2, 0′′0 0′ EE1 + 2 −35.6, and 0 values (K = 1/28, 1/24, 1/22, 1/20, respectively, E̅ = 2 (1) with K being the disproportionation constant and 2 the number of electron transfers) in the behavior of the peak parameters of plays an essential role in the study of the process since it is the the voltammetric response (peak potentials, peak heights, and formal potential for the reaction half-peak widths) is noteworthy.
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