Polarization equation

The practical polarization-resistance technique involves one more assumption than those listed in Section 11.1. It is assumed that the polarization equation can be approximated by a linear relation between the measuring current density and the polarization for small polarization values. Equation (7) can be obtained by expanding the exponentials in Eq. (1) into Taylor series and ignoring all terms higher than first power. There has been a certain amount of controversy about the criteria of validity for this approach, both for the corrosion case and for the electrode kinetics case (see Ref. 116 and references therein). By now, this controversy seems to have been cleared up with the following conclusions. While the polarization curve is never absolutely linear at finite polarization values, the assumption of linearity often introduces only a small error. Furthermore, this error approaches zero, by mathematical definition, as A approaches zero, but the rate of approach depends strongly on the values of and b. The error 

Under certain conditions, the even-order terms in the Taylor series expansion of the exponentials cancel consequently, Eq. (7) can be a better approximation than indicated by Eq. (36). For electrode kinetic measurements, this has been shown to be the case when the polarization resistance is measured using two identical working electrodes, that is, when one electrode is polarized negatively and the other positively to the same extent. A similar, two-specimen approach was also proposed for corrosion measurements, but the measurements can also be carried out with one specimen using two successive polarization measurements one anodic and one cathodic.When the two polarization values used are A and — A , with the corresponding two current densities and the corrosion current density can be expressed 

It has been suggested that the linearization error can be even further reduced by making four measurements two sets of positive-negative polarizations at two different polarization values. However, the use of the three-point technique may be more advantageous as long as multiple measurements are carried out. 

The error is slightly different if two identical specimen are used and polarized by 2 E with respect to each other, without the use of a reference electrode. In this case, the current densities are equal and opposite in sign, while the polarizations are only approximately equal to E. Under these conditions, the error cannot be expressed analytically, but the error still can be shown to be less than that of the single specimen polarization-resistance technique. 

Equation (1) is based on a single anodic and a single cathodic reaction. The polarization curves are distorted if more than one anodic process takes place (alloy dissolution) or more than one cathodic reactant is present in the solution. This effect was discussed by Stern and Skorchelletti, but a quantitative error analysis was not reported. 

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The activity coefficient (y) based corrector is calculated using any applicable activity correlating equation such as the van Laar (slightly polar) or Wilson (more polar) equations. The average absolute error is 20 percent. 

The polarization equation describes polarization as a fnnction of current density. In the case of concentration polarization, the form of the polarization eqnation is nnre-lated to the natnre of reaction or electrode. In the case of activation polarization, the parameters of the polarization eqnations depend decisively on the natnre of the reaction. At identical values of current density and otherwise identical conditions, the values of polarization for different reactions will vary within wide limits, from less than 1 mV to more than 2 or 3 V. However, these equations still have common features. A relatively simple set of equations is obtained for simple redox reactions of the type... 

Here we consider the polarization equations for simple redox reactions and for reactions that are similar to them. The special features of more complex reactions are discussed in Part 11 of this book. 

Equations (6.9) and (6.10), which contain the rate constants, the electrode potential, and the concentrations, are equivalent to Eqs. (6.12) and (6.13), which contain the exchange CD and the electrode s polarization. But in the second set of equations the concentrations do not appear explicitly they enter the equations through the values of exchange CD and equilibrium potential. By convention, equations of the former type will be called kinetic equations, and those of the latter type will be called polarization equations. 

Polarization equations are convenient when (1) the measurements are made in solutions of a particular constant composition, and (2) the equilibrium potential is established at the electrode, and the polarization curve can be measured both at high and low values of polarization. The kinetic equations are more appropriate in other cases, when the equilibrium potential is not established (e.g., for noninvertible reactions, or when the concentration of one of the components is zero), and also when the influence of component concentrations on reaction kinetics is of interest. 

In the intermediate region of moderate polarization (between 10 and 80 mV) we must use the polarization equation (6.13) in its general form. 

The kinetic and polarization equations described in Sections 6.1 and 6.2 have been derived under the assumption that the component concentrations do not change during the reaction. Therefore, the current density appearing in these equations is the kinetic current density 4. Similarly, the current density appearing in the equations of Section 6.3 is the diffusion current density When the two types of polarization are effective simultaneously, the real current density i (Fig. 6.6, curve 3) will be smaller than current densities and ij (Fig. 6.6, curves 1 and 2) for a given value of polarization. 

In the case considered in this section of a joint action of concentration and activation polarization, in the polarization equation (6.10) we must take into account the concentration changes of the rectants near the electrode surface ... 

For an analysis of the polarization curves at low values of polarization (low overpotentials), we shall use the general polarization equation... 

So the popular polarization equations of the type (6.5) for electrochemical reactions thus acquire some physical basis. However, according to current concepts the nature of the activated state is different, and quantum-mechanical approaches must be used for a theoretical calculation of the values and These concepts are discussed in more detail in Chapter 34. 

With Eq. (14.32) for the reaction rate and Eq. (14.34) for polarization, we obtain the following general form of the polarization equation ... 

After complete formation of each successive monolayer of atoms, the next layer should start to form. This requires two-dimensional nucleation by the union of several adatoms in a position 1. Like three-dimensional nucleation, two-dimensional nucleation requires some excess energy (i.e., elevated electrode polarization). Introducing the concept of excess linear energy p of the one-dimensional face (of length L) of the nucleus, we can derive an expression for the work of formation of such a nucleus (analogous to that used in Section 14.2.2). When the step of two-dimensional nucleation is rate determining, the polarization equation becomes, instead of (14.39),... 

Polarization equations of the type (14.35) or (14.38) contain the mean values of true current density. However, the rate-determining step is more often concentrated at just a few segments of the electrode the true working area changes continuously and an exact determination of this area is practically impossible. This gives rise to difficulties in an interpretation of polarization data. 

Polarization equations are based on a planar description of light, which is reasonable when considering the excitation source. Polarization, however, fails to account for the symmetrical distribution of fluorophore emission/31 The anisotropy, r, more accurately describes the emission by... 

If the effect of potential in diffusion layer is considered, then the electrochemical polarization equation is ... 

A more sophisticated and more common treatment of the catalyst layers still models them as interfaces but incorporates kinetic expressions at the interfaces. Hence, it differs from the above approach in not using an overall polarization equation with the results, but using kinetic expressions directly in the simulations at the membrane/diffusion medium interfaces. This allows for the models to account for multidimensional effects, where the current density or potential changes 16,24,46-48,51,52,54,56,60-62,66,80,82,87,107,125 although... 

ORR rate constant as defined by eq 61, 1/s ORR rate constant in Figure 11, cm/s thermal conductivity of phase k, J/cm K relative hydraulic permeability saturated hydraulic permeability, cm electrokinetic permeability, cm catalyst layer thickness, cm parameter in the polarization equation (eq 20) loading of platinum, g/cm molecular weight of species i, g/mol symbol for the chemical formula of species i in phase k having charge Zi... 

MOs can be different, and this permits spin polarization. Equations (6.8) and (6.9) define umestricted Hartree-Fock (UHF) theory. 

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