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Chapter 1. Introduction 13 frequency, for example when measuring across two different electrodes with different reaction rates as in the case of a full cell. Other phenomena and rate-limiting mechanisms besides charge-transfer, such as adsorption/desorption and diffusion of species, can also appear as semicircles or resemble semicircles. Gas concentration effects (not related to the electrochemical reactions) can also shown up as a R-C type process [2] [3] [4]. Another serial process would result in another semicircle adjacent to – or overlapping if the frequencies are close enough – the first semicircle. In this way multiple semicircles can be present, and it is useful to define the total resistance (at the low-frequency intercept with the Zreal axis) minus RS as the total polarization resistance, RP. In actual measurements, such a perfect semicircle is seldom encountered. Often the semicircle is depressed (Figure 1-5b). The reasons for this are dependent on the system and are a matter of debate, but most likely are related to a non-homogeneous electrode surface and/or interface in which reactions rates and capacitances have a wider distribution, with slightly different magnitudes and relaxation times. The impedance Z of a depressed semicircle has been modeled directly as a system with a distribution of time constants [5]. An equivalent circuit element called the constant phase element, , was introduced to describe such depressed arcs. Whereasthetimeconstant ofanR-Ccircuitis ,foranR-Qcircuititisdefinedas . The impedance Z of an R-Q circuit is therefore (1.5) where is the resistance of the parallel resistor ( in Figure 1-5b), is the constant phase element parameter, is the imaginary unit, and is the angular frequency ( ). Note that for , the equation corresponds to that of a capacitor. To calculate the full impedance of the R(RQ) circuit shown in Figure 1-5b, the serial resistor is simply added to equation (1.5). The ―true‖ or quasi-equivalent capacitance of a constant phase element in an R-Q circuit can be derived simply from the definitions of the time constants for the R-C and R-Q given above: . Since and , this can also be expressed as where is the summit frequency of the depressed arc. in reference to the in equation (1.4). A number of other equivalent circuit elements that do not exist in electrical circuits have also been introduced to describe various phenomena observed in electrochemical impedance data, such as diffusion limitations. For more information, the textbooks listed below are referred to. Equivalent circuits are also sometimes used as a ―measurement model‖ [6] before knowing enough to try to model the system, as a means to condense the impedance data and try to identify some characteristics such as the number of processes. The data can be fit to anPDF Image | Electrolysis of CO2 and H2O
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