Model of an Electrochemical System

So far in this chapter we have treated the concentration as if it varied with only one independent variable — position, X. As we saw in Chapter 2, modelling of an electrochemical system requires the solution of Pick s second law, a partial differential equation in which C varies as a function of both space, X, and time, T. We must now therefore consider how the system evolves in time. Thus, we are going to study the simulation of the reversible, one-electron reduction of species A at a macroelectrode ... 

In maldug electrochemical impedance measurements, one vec tor is examined, using the others as the frame of reference. The voltage vector is divided by the current vec tor, as in Ohm s law. Electrochemical impedance measures the impedance of an electrochemical system and then mathematically models the response using simple circuit elements such as resistors, capacitors, and inductors. In some cases, the circuit elements are used to yield information about the kinetics of the corrosion process. 

Impedance models are constructed according to the electrochemical phenomena. The total impedance of an electrochemical system can be expressed by different combinations of the electrical elements. This section covers the features of basic equivalent circuits commonly used in electrochemical systems. In Appendix D, the effect of an element parameter change on a spectrum related to a given equivalent circuit is described in detail. 

This is the simplest model of an electrocatalyst system where the single energy dissipation is caused by the ohmic drop of the electrolyte, with no influence of the charge transfer in the electrochemical reaction. Thus, fast electrochemical reactions occur at current densities that are far from the limiting current density. The partial differential equation governing the potential distribution in the solution can be derived from the Laplace Equation 13.5. This equation also governs the conduction of heat in solids, steady-state diffusion, and electrostatic fields. The electric potential immediately adjacent to the electrocatalyst is modeled as a constant potential surface, and the current density is proportional to its gradient ... 

EIS changed the ways electrochemists interpret the electrode-solution interface. With impedance analysis, a complete description of an electrochemical system can be achieved using equivalent circuits as the data contains aU necessary electrochemical information. The technique offers the most powerful analysis on the status of electrodes, monitors, and probes in many different processes that occur during electrochemical experiments, such as adsorption, charge and mass transport, and homogeneous reactions. EIS offers huge experimental efficiency, and the results that can be interpreted in terms of Linear Systems Theory, modeled as equivalent circuits, and checked for discrepancies by the Kramers-Kronig transformations [1]. 

The impedance spectrum of an electrochemical system is most often represented in Nyquist coordinates (3l(Z), S(Z)), whereZis the system impedance. In this representation, the spectrum intersections with the real axis 3l(Z) give the main resistivities in the system. Fitting model equations to experimental impedance spectra enables determining the kinetic and transport parameters of the electrochemical system. In recent years, this technique has been widely used to study fuel cells and cell components (Orazem and Tribollet, 2008 Yuan et al., 2009). ... 

Scale- Up of Electrochemical Reactors. The intermediate scale of the pilot plant is frequendy used in the scale-up of an electrochemical reactor or process to full scale. Dimensional analysis (qv) has been used in chemical engineering scale-up to simplify and generalize a multivariant system, and may be appHed to electrochemical systems, but has shown limitations. It is best used in conjunction with mathematical models. Scale-up often involves seeking a few critical parameters. Eor electrochemical cells, these parameters are generally current distribution and cell resistance. The characteristics of electrolytic process scale-up have been described (63—65). 

Figure 17.2 illustrates our model for splitting water by solar energy. I" is important that all the redox reactions involved in thf system be reversible. The quinone compound in the organic solvent combines the two photocatalytic reactions, and its function can be compared to the electron relaying molecules in thylakoid membranes of chloroplasts. Electron transfer reactions via quinone compouncs in artificia systems have been studied as a model of photosynthesis22-23 and in an electrochemical system for acid concentration.24 ... 

Figure 8.2 shows an electrochemical system - a model of a catalase-biomimetic sensor, consisting of the reference electrode (Ag/AlCl/Cl ) and biomimetic electrode. In this system, the electrochemical potential changed as a result of mimetic electrode interaction with... 

The equivalent circuit should be as simple as possible to represent the electrochemical system and it should give the best possible match between the model s impedance and the measured impedance of the system, whose equivalent circuit contains at least an electrolyte resistance, a double-layer capacity, and the impedance of the Faradaic or non-Faradaic process. Some common equivalent circuit elements for an electrochemical system are listed in Table 2.1. A detailed description of these elements will be introduced in Section 4.1. 

At first glance, it may not be obvious that such an approach should work. It is well known, for example, that the impedance spectrum associated with an electrochemical reaction limited by the rate of diffusion through a stagnant layer (either the Warburg or the finite-layer diffusion impedance) can be approximated by an infinite number of RC circuits in series (the Voigt model). In theory, then, a measurement model based on the Voigt circuit should require an infinite number of parameters to adequately describe the impedance response of any electrochemical system influenced by mass transfer. 

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