Polarization curve equation

This represents two opposite elementaiy reactions a reduction toward the left, and an oxidation toward the right. 

The total current density will therefore be the algebraic sum of two oxidation and reduction current densities  

By proceeding as we did and introducing the activation equilibrium constants, we obtain  

We now consider the energy paths of these two opposing steps. In each of these reactions, the electron crosses a potential barrier AO, which is the absolute potential of the electrode. We can therefore calculate the Gibbs free energy of activation  

Substituting these expressions into equation [11.79], the current density will be  

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The cathodic polarization curve equation for flat or large spherical electrode is given by... 

It is necessary to note that (44) is an approximation, because the value of y is lower than unity. This approximation is widely used in qualitative discussions, because it permits the simple mathematical treatment of electrochemical processes with relatively small errors and with clear physical meaning. If y 1 is included in the derivation of the general polarization curve equation, simple analytical solutions are not available and numerical solutions are required. 

Equation (44) is the polarization curve equation for a modified inert electrode for y = 1. It is valid for inert substrates modified by active microparticles or nanoparticles as well as by 2D and 3D islands of active metal. 

Equation (7) is valid for the complete active electrode surface. On the other hand, if the inert substrate is partially covered with the same active material, the polarization curve equation is given by (44).7... 

The initiation of dendritic growth is followed by an increase of the deposition current density, and the overall current density will be larger than the limiting diffusion current on a flat active electrode. Based on the above discussion, the polarization curve equation in the Ohmic-controlled electrodeposition of metals can be determined now by 9... 

Since both the exchange current density and the limiting diffusion current density are included in the general cathodic polarization curve equation given by (7), it is necessary to standardize them in the same way, hence, to the apparent surface area. In that case, the exchange current density has some effective value, y o,eff, given by (68)... 

A completely new approach to the analysis of experimental data is introduced by the use of the complete polarization curve equation and by the method of digital simulation. It was possible in this way to elucidate the polarization behavior of the partially covered inert... 

As already given in Chap. 1, the most frequently used form of the cathodic polarization curve equation for flat or large spherical electrode of massive metal is given by ... 

Zivkovic PM, Grgur BN, Popov KI (2008) The validity of the general polarization curve equation approximation for the metal deposition process. J Serb Chem Soc 73 227-231... 

In corrosion and in electrochemistry, the potential sweep technique is commonly used to measure polarization curves, and the result is referred to as potentiodynamic polarization curves. Equation (5.98) offers a criterion for the selection of the maximum sweep rate while still working under steady state conditions with respect to mass transport. As a rule of thumb, for a value of > 20 the error in the measured steady-state limiting current is less than 1% [7]. Equation (5.98) shows that to attain a steady state, the sweep rate must be the slower the larger the diffusion layer thickness, in other words, the weaker the convection. If, for example, D = 10 m s, = 2 and 5= 10 pm, the sweep rate must not exceed 15 mV s ... 

The Polarization Curve Equation for Partially Covered Inert Electrode... 

Different forms of polarization curve equation were discussed in detail [25] and this form was chosen for digital simulation. The use of any other form of the polarization curve equation will give some similar results. 

The analytical CCL polarization curve Equation 4.189 is compared to the exact numerical solution of the system (4.53) and (4.54) in Eigure 4.21. A reference value ofDrt = 1.37 10 cm s is taken from measurements (Shen et al., 2011). The curves in Eigure 4.21 correspond to the indicated ratios D/Dre/- Clearly, as this ratio tends to infinity, the analytical and numerical results tend to the diffusion-free polarization curve Equation 4.141. Note that as b decreases, the overpotential due to oxygen transport increases, and the accuracy of the model drops. Nonetheless, in the region of currents Jo < 1, the model works well for D/Dref as small as 0.1 (Figure 4.21). 

Based on the data from entries 2 and 3, the desired function of the current density i and electrode potential is found (the polarization curve equation). Sometimes other relationships are investigated. For example, in the case of potentiostatic electrolysis one determines the rate as a function of time (chronoamperometry), and the potential does not change. Under galvanostatic conditions (i = const), kinetics of the process is described by a potential-time relationship (chronopo-tentiometry). 

Function [11.91] is the polarization curve equation. This curve admits one inflexion point defined by ... 

In this case, the polarization curve equation becomes ... 

Figure C2.8.2. (a) Schematic polarization curves for tire anodic and catliodic reaction of an Fe/Fe electrode. The anodic and catliodic branches of tire curve correspond to equations (C2.8.12 ) and (C2.8.13 ) respectively. The equilibrium potential of tliis electrode will adjust so tliat j = j tire corresponding potential is of Fe. (b)...

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. 

Thus, in the region of very high anodic or cathodic polarization, the RDS is always the first step in the reaction path. The transfer coefficient of the full reaction which is equal to that of this step is always smaller than unity (for a one-electron RDS), while slope i in the Tafel equation is always larger than 0.06 V. When the potential is outside the region of low polarization, a section will appear in the polarization curve at intermediate values of anodic or cathodic polarization where the transfer coefficient is larger than unity and b is smaller than 0.06 V. This indicates that in this region the step that is second in the reaction path is rate determining. 

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

The form of the kinetic equation depends on the way in which the surface potential X varies with electrode potential E. When the surface potential is practically constant, the first factor in Eq. (14.24) will also be constant, and the potential dependence of the reaction rate is governed by the second factor alone. The slope b of the polarization curve will be RT/ F (i.e., has the same value as that found when the same reaction occurs at a metal electrode). When in another case a change in electrode potential E produces an equally large change in surface potential (i.e., E = x + const), while there is practically no change in interfacial potential. Then Eq. (14.24) changes into... 

The shape of polarization curves for metals with low polarizability depends primarily on concentration polarization. In the case of highly polarizable metals, where activation polarization can be measured sufficiently accurately, the polarization curve can usually be described by an equation of the type (6.3) (i.e., by a Tafel equation). For metals forming polyvalent ions, slope b in this equation often has values between 30 and 60 mV. 

Whereas is relatively easy to determine from the calculated binding energies, it is not easy to measure experimentally, since the measured potentials are always related to a specific current. Therefore, in order to compare directly with experiment, we have to calculate polarization curves, i.e., the current. The link between Gqrr and the current is the Tafel equation. 

Applying the Tafel equation with Uq, we obtain the polarization curves for Pt and PtsNi (Fig. 3.10). The experimental polarization curves fall off at the transport limiting current since the model only deals with the surface catalysis, this part of the polarization curve is not included in the theoretical curves. Looking at the low current limit, the model actually predicts the relative activity semiquantitatively. We call it semiquantitative since the absolute value for the prefactor on Pt is really a fitting parameter. 

Plotting the overpotential against the decadic logarithm of the absolute value of the current density yields the Tafel plot (see Fig. 5.3). Both branches of the resultant curve approach the asymptotes for r RT/F. When this condition is fulfilled, either the first or second exponential term on the right-hand side of Eq. (5.2.28) can be neglected. The electrode reaction then becomes irreversible (cf. page 257) and the polarization curve is given by the Tafel equation... 

This equation describes the cathodic current-potential curve (polarization curve or voltammogram) at steady state when the rate of the process is simultaneously controlled by the rate of the transport and of the electrode reaction. This equation leads to the following conclusions ... 

This is the equation of a reversible polarization curve. The anodic polarization curve of the reduced form obeys an identical equation... 

Irreversible polarization curve (voltammogram). If the value of is so small that the first term on the right-hand side of Eq. (5.4.22) is much smaller than the second term, even when j approaches d, then the equation assumes the form... 

This equation is analogous to Eq. (5.4.18) or (5.4.19) for the steady-state current density, although the instantaneous current depends on time. Thus, the results for a stationary polarization curve (Eqs (5.4.18) to (5.4.32)) can also be used as a satisfactory approximation even for electrolysis with the dropping mercury electrode, where the mean current must be considered... 

The anodic evolution of oxygen takes place at platinum and other noble metal electrodes at high overpotentials. The polarization curve obeys the Tafel equation in the potential range from 1.2 to 2.0 V with a b value between 0.10 and 0.13. Under these conditions, the rate-controlling process is probably the oxidation of hydroxide ions or water molecules on the surface of the electrode covered with surface oxide ... 

The plot of ( versus q which results from Eqn. 9-9 is a polarization curve this polarization crirve is usually divided into two ranges of polarization as shown in Fig. 9-3 one is a range of polarization where a linear rate equation holds near the equilibrium potential (t) - 0) the other is a range of polarization (the Tafel range) where an exponential rate equation applies at potentials away from the equilibrium potential (ti 0). 

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