Abstract
Fundamentals of electrode thermodynamics and kinetics are given, aiming at furnishing a reference to the reader of the following chapters. No previous knowledge of electrochemistry is required; only the basic principles of chemical equilibria are supposed to be known. The suggested readings are given to deepen what is reported here, not constituting a premise in any way.
Effort has been made in order to couple to a rigorous, though simple mathematical treatment, intuitive elements that help the reader pick up the phenomenological aspects of what accounted for by the mathematical expressions. To similar purposes, basic theoretical and experimental aspects of the most frequently used amperometric techniques and of coulometry are also dealt with. Practical considerations are often made throughout the whole chapter.
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Notes
- 1.
as it will be clear in the following, the extent of overvoltage depends on the current density, rather than on the overall current. Just two examples to account for this fact: in the case of “charge transfer overvoltage” the electrode kinetics is accounted for by kinetic constants and by the concentrations at the electrode, that in no way depend on the electrode area. Similarly, as to the “concentration overvoltage,” once more the finite values of the concentration gradient and, consequently, of the concentration flux, are the sources of the relevant overvoltage, rather than the concentration flow rate, i.e., at the whole electrode surface.
- 2.
The knowledge of the operative electrode mechanism is of fundamental importance in a variety of situations, from analytical to industrial applications of electrochemical methods. In electroanalysis the current at a given time or potential is taken as an estimate of the concentration of the electroactive species in solution. A linear relationship between current intensity and concentration represents the case of choice. However, not always linearity is induced by the underlying electrode mechanism, nor the relationship should be forced to linearity, once repeatability and reproducibility are carefully verified.
- 3.
In the spectroscopic absorption measurements, the interaction of the radiation with matter causes alteration of the probe, namely of the intensity of the exiting radiation. However, the “flux of photons” that immediately afterwards crosses the samples is not affected by what happened to the previous photons: the probe is unaltered and in the case of most, though not all spectroscopies, also the sample does not undergo any changes as a consequence of the measurement. In voltammetric measurements the probe, viz. the electrode, also interacts with the sample, which is essentially unaltered; it is however possible that the probe is “modified” by the occurrence of the interaction: the effect of the modification may persist, inducing changes in the behavior of the probe afterwards. From these drawbacks the “history” of the electrode becomes one of the most meaningful limits of the voltammetric techniques. Polarography at dropping mercury electrode minimizes the history of the probe, since the electrode is periodically a new one, at a frequency that can minimize the effects of poisoning adsorption or other events altering the electroactive surface. Subsequent drops may be figured as the flux of photons, even if it is evident that the total absence of “history” proper of radiations is anyway far from being achieved, owing to the finite length of the life time of each drop. In solid electrodes, the history is the cause of eventual poor repeatability, or even of poor reliability, of the responses.
- 4.
For the sake of simplicity, along the whole chapter, unless otherwise specified, this simple process will be considered. Different mechanisms are often operative in electrode reactions of analytical interest; however, the treatment of these cases requires too much room for a book devoted to the issue of electroanalysis for environmental studies. On the other hand, the simplest mechanism constitutes the basis for the more complex ones. References to specific literature are found in the books of general interest that are listed at the end of the chapter. Noteworthy, a reduction reaction is considered, but negative sign is given in the following to the corresponding cathodic current [see from Eq. (10.5) onwards]. Such a choice is opposite to the “polarographic convention.” Electroanalysis was born thanks to the diffusion of amperometric techniques at Hg electrodes, at which reductions are for the very most part studied. It was then spontaneous to assign positive values to the most often encountered currents, so that this habit still survives, despite the subsequent diffusion of electrodes at which oxidations, i.e., flow of anodic currents, are induced.
- 5.
Bare electrodes, i.e., electrodes in which the interface with the solution consists of a metal such as Pt or Au, or of C, such as glassy carbon, are considered in this chapter. The diffusion of modified electrodes, which will be dealt with in different chapters of the book, offers a variety of solutions and of situations. In principle, everything becomes more versatile, more powerful, and more flexible and, as it often happens in similar cases, also more complex.
- 6.
In the very well-known relationship a = γ C, γ represents the activity coefficient. Throughout the whole chapter we make reference either to the activity, e.g., in the frame of rigorous thermodynamic issues, or to the concentration, e.g., when dealing with transfer of mass.
- 7.
The electrochemical potential of electrons in a given phase, \( {\overline{\mu}}_e^{\alpha } \), is the Fermi level or Fermi energy. The Fermi level represents the average energy of available electrons in phase α, related, similarly to any charged species, to the chemical potential of electrons in that phase, μ 0,α e (=μ 0,α e ), and the inner potential of α. In a solution phase, it may be computed from the electrochemical potentials of the oxidized and reduced species. For example, for a solution containing Fe(III) and Fe(II): \( {\overline{\mu}}_e^{\alpha }={\overline{\mu}}_{Fe(III)}^{\alpha }-{\overline{\mu}}_{Fe(II)}^{\alpha } \).
- 8.
The term “reaction coordinate,” reported qualitatively in the plots, assumes however quantitative meaning—with a precise quantity and relevant units—once the reaction path is followed through a quantity suitable to describe its progress. This quantity may be related, for instance, to (1) the bond length if a bond breaks or forms, e.g., in the reduction of iodine to iodide ions or oxidation in the opposite direction, respectively; (2) the angle formed by two atoms of one or two ligands and the metal in a complex, when passing, for instance, from a (regular) square planar coordination (90°) to a (regular) tetrahedral (109°) coordination by changing the oxidation state of the metal/complex; (3) the shortening of the bond length between the metal and one atom of the ligand set once the metal or complex are oxidized, and so on.
- 9.
Distinction should be made between the value of the geometrical and the electrochemical (active) areas of an electrode. The meaning of geometric area is obvious. The electrochemical area should be computed on the basis of the response to a benchmark species in one of the techniques discussed in the following. Once the diffusion coefficient of the species chosen, typically one partner of a reversible redox couple, such as the hexacyanoferrate anions in water or bis(cyclopentadienyl)iron(II)—ferrocene—in organic solvent, is known, the ratio between the measured current and the expected current density constitutes a reliable estimate of the electrochemical area. The dependence of this area value on the exact nature of the electroactive species may be discarded as a first approximation, once poisoning of the electrode and the occurrence of unknown complex electrode mechanisms can be excluded.
- 10.
The reason for this choice lies in the approximation to 1 of the values of the error function, erf(x), that accounts for the concentration profile of a species undergoing pure diffusion, with 0 concentration at the electrode; in particular, if C O (0,t) = 0 for t > 0, C O (x,t) = C O b erf[x/(2(D o t) 1/2 )]. It follows that \( {\mathrm{C}}_O\left(6\sqrt{D_Ot},\mathrm{t}\right)=0.99998\ {C_O}^b \)
- 11.
It is evident from the foregoing that a contribution to diffusion parallel to the planar electrode becomes more and more significant at decreasing the radius of the disk; the diffusional process tends asymptotically to a pure radial one, when the radius tends to zero. Such a kind of diffusion will be treated in detail in this book in the frame of microelectrodes (Chap. 15).
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Moretto, L.M., Seeber, R. (2014). Controlled Potential Techniques in Amperometric Sensing. In: Moretto, L., Kalcher, K. (eds) Environmental Analysis by Electrochemical Sensors and Biosensors. Nanostructure Science and Technology. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0676-5_10
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