Análisis de Problemas, Árbol de Problemas

For a reaction occurring at the solution-electrode interface, O + 𝑛𝑒− ⇌ R, there are several steps involved, i.e. mass transport, chemical reactions before or after the electron transfer (ET), other surface-bound reactions (such as adsorption and desorption), and electron transfer at the surface of electrode (Figure 1.10).60 _{The slowest step can be a limiting factor for the} overall electrochemical processes.

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There are three types of mass transport in electrochemical systems, i.e. migration, diffusion and convection. Migration occurs when charged species move under the influence of an electric field; diffusion accounts for movement of species driven by a gradient of chemical potential (e.g. concentration gradient); convection movement of species is triggered by mechanical stirring or density gradients. Diffusion is the most commonly seen mass transfer mode and the diffusion profiles are different on electrodes of different dimensions. As shown in Figure 1.11, planar diffusion is observed for macroscale electrodes and spherical diffusion is found for micro/nanoscale electrodes under steady-state conditions.61

Figure 1.11 Profiles for (a) planar diffusion and (b) spherical diffusion at the electrode-solution interface, subject to the size of electrode area.

1.2.2 Electron Transfer Kinetics 1.2.2.1 Overpotential

An overpotential, η, is often needed to drive electrochemical reactions, where ET occurs. The driving force is defined as the difference of the applied potential (E) and the equilibrium potential (Eeq):

𝜂 = 𝐸 − 𝐸_𝑒𝑞 (eq. 1.1)

For a system (one–electron reaction) in a state of equilibrium, it is governed by the Nernst equation and Eeq can be derived from eq.1.2

𝐸 = 𝐸0′₊𝑅𝑇 𝐹 ln

𝐶_𝑂∗

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where E0_{′ is the formal potential, R is the gas constant, T is temperature, F is} Faraday’s constant, CO* and CR* are the bulk concentrations of the oxidised and reduced species, respectively.

1.2.2.2 Butler–Volmer Equation

The current as a result of applied overpotential in an electrochemical system can be predicted using Butler-Volmer equation.60 _{For a simple one-electron} reaction process,

O + 𝑒−𝑘_⇌𝑓

𝑘𝑏

R (eq. 1.3)

where kf and kb are the heterogeneous rate constants for the forward (reduction) and backward (oxidation) reactions, respectively. The overall current, i, is the difference of cathodic current (ic) and anodic current (ia), expressed as

𝑖 = 𝑖𝑐− 𝑖𝑎 (eq. 1.4)

While the current for either direction is dependent on the corresponding heterogeneous rate constant (kb or kf),

𝑖_𝑐 = 𝐹𝐴𝑘_𝑓𝐶_𝑂 (eq. 1.5)

𝑖_𝑎 = 𝐹𝐴𝑘_𝑏𝐶_𝑅 (eq. 1.6)

where A is area, Ci is the concentration of species i of the redox couple (O or

R). Then the net current is

𝑖 = 𝐹𝐴(𝑘_𝑓𝐶_𝑜− 𝑘_𝑏𝐶_𝑅) (eq. 1.7)

While kf and kb can be deduced as a function of standard heterogeneous rate constant (k0) following the Arrhenius equation:

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𝑘𝑏 = 𝑘0𝑒(1−𝛼)𝑓(𝐸−𝐸0′) (eq. 1.9)

where the coefficient f = F/RT and the transfer coefficient, α is a dimensionless parameter (often assumed as 0.5).

So, the reaction current can be obtained as

𝑖 = 𝐹𝐴𝑘₀[𝐶_𝑂𝑒−𝛼𝑓(𝐸−𝐸0′)− 𝐶_𝑅𝑒(1−𝛼)𝑓(𝐸−𝐸0′)] (eq. 1.10)

1.2.3 Redox Couples

Redox couples can be classified into outer-sphere and inner-sphere groups. Outer-sphere redox species do not interact strongly with the electrode surfaces (during electron transfer) and are normally separated from the electrodes by a distance of at least a solvent layer. In contrast, the inner- sphere species interact with the electrode surface strongly, where specific adsorption processes are often involved. Thus, the redox reactions of outer- sphere species are less dependent on the electrode materials compared with inner-sphere species.60

1.2.4 Energy View from Marcus Microscopic Model

When a heterogeneous outer-sphere reduction process at an electrode, involving one-electron transfer (see eq.1.3), or a homogeneous electron transfer process (in which О is reduced to R by another reactant in solution), is considered, there are two fundamental aspects60_{: (i) since electron transfer} process is radiationless, the electron must move from an initial state to a receiving state of the same energy, known as isoenergetic electron transfer; (ii) the reactants and products do not change their configurations during the transfer, based on the Franck-Condon principle, i.e. О and R share the same nuclear configuration at the moment of transfer.

As a potential is applied on the electrode, standard free energy of activation is changed to overcome the barrier for oxidation or reduction, but electron transfer would only occur at the transition state, where О and R have the same configuration. Here, the reorganization energy λ, which defines the

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energy required to transform the nuclear configurations in the reactant and the solvent to those of the product state, plays an important role.

𝜆 = 𝜆_𝑖+ 𝜆_𝑠 (eq. 1.11)

where λi represents the inner component from (bond) reorganization of species O, and λs the outer component from reorganization of the solvent.

1.2.5 Cyclic Voltammetry

Cyclic voltammetry (CV) is the most used potential-sweeping technique in the field of electrochemistry.61 _{The potential of the electrode is changed} linearly with time between two chosen values, E1 and E2, while the electrochemical current is recorded as a function of applied potential, generating a cyclic voltammogram (Figure 1.12).

Figure 1.12 (a) Profile of potential applied with time in cyclic voltammetry. A typical waveform of current response for (b) diffusion- and (c) adsorption- controlled processes as a function of potential. E1 and E2 are the starting and reversal potentials, respectively.

The observed current is due to both faradaic and nonfaradaic processes. Nonfaradaic current (inf) derives from double layer charging of the electrode, affected by scan rate (v) and double layer capacitance (Cd).

𝑖𝑛𝑓 = 𝑣𝐶𝑑 (eq. 1.12)

The faradaic current needs to be considered in two cases. For a planar diffusion-controlled (reversible) reaction (Figure 1.12b), the peak current is

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governed by Randles–Sevcikequation, which indicates that the peak current

(ip) is proportional to the square root of the scan rate.

𝑖𝑝 = 0.4463𝑛

3/2_𝐹3/2

𝑅1/2_𝑇1/2𝐴𝐷𝑂1/2𝐶𝑂∗𝑣1/2 (eq. 1.13)

where n is the electrons transferred, DO is the diffusion coefficient of the oxidised species.

For a reaction involving adsorbed species (Figure 1.12c), the peak current increase linearly with scan rate.

𝑖_𝑝 =𝑛_4𝑅𝑇2𝐹2𝑣𝐴𝛤_𝑂∗ _{(eq. 1.14)}

where ΓO* is the maximum surface coverage of species O. For an ideal adsorption peak, the full width at half-maximum (FWHM) should be 90.6/n mV.60

It should be noted that the peak-to-peak separation ΔEp obtained from CV can be an indication of reaction reversibility and a measure of electron transfer kinetics.60,62 _{For a reversible process (with high k0), the redox} species can be rapidly adjusted to those required by the Nernst equation and an electrochemical equilibrium is maintained at electrode surface, resulting in a small ΔEp (59/n mV for a diffusion process; 0 for an adsorption process). However, for quasi-reversible and irreversible processes associated with sluggish kinetics (low k0), Nernstian concentrations cannot be achieved and a significant activation overpotential (driving force) is need to motivate electron transfer, leading to shifts of peak potentials and an increase in ΔEp.