Current-overpotential equation linearization

The potential of the working electrode is ramped at a scan rate of v. The resultant trace of current against potential is termed a voltamnu ram. In linear-sweep voltammetry (LSV), the potential of the working electrode is ramped from an initial potential Ei to a final potential Ef (cf. Figure 6.2). Figure 6.12 shows a linear-sweep voltammogram for the reduction of a solution-phase analyte, depicted as a function of scan rate. Note that the jc-axis is drawn as a function of overpotential (equation (6.1)), and that the peak occurs just after = 0. 

Steady-state Current Overpotential Behaviour - For a simple single charge-transfer process equation (2.28) describes the closed-circuit behaviour. At low overpotentials, the current and overpotential are linearly related and the exchange current density can be evaluated from the gradient (see equation... 

The aforementioned models include three governing equations (i) mass transport equation for oxygen, (ii) proton current conservation equation with the Tafel rate of electrochemical reaction on the right side and (iii) Ohm s law, which relates proton current to the gradient of overpotential. Due to the exponential dependence of the rate of ORR on overpotential this system is strongly non-linear. 

Equations 21.15 and 21.16 indicate that net current density varies linearly with overpotential. At high overpotential, if the forward reaction rate is mueh higher than that of the backward reaction, i.e., 4, Equation 21.14 ean be simplified... 

The distribution of current (local rate of reaction) on an electrode surface is important in many appHcations. When surface overpotentials can also be neglected, the resulting current distribution is called primary. Primary current distributions depend on geometry only and are often highly nonuniform. If electrode kinetics is also considered, Laplace s equation stiU appHes but is subject to different boundary conditions. The resulting current distribution is called a secondary current distribution. Here, for linear kinetics the current distribution is characterized by the Wagner number, Wa, a dimensionless ratio of kinetic to ohmic resistance. 

These equations cannot be used at higher overpotentials 77 > kT/e0. If the reaction is not too fast, a simple extrapolation by eye can be used. The potential transient then shows a steeply rising portion dominated by double-layer charging followed by a linear region where practically all the current is due to the reaction (see Fig. 13.2). Extrapolation of the linear part to t = 0 gives a good estimate for the corresponding overpotential. 

Equation 1.7 for the reduction of protons at a mercury surface in dilute sulphuric add is followed with a high degree of accuracy over the range -9 Tafel plot i.s shown in Figure 1.5. At large values of the overpotential, one reaction dominates and the polarization curve shows linear behaviour. At low values of the overpotential, both the forward and back reactions are important in determining the overall current density and the polarization curve is no longer linear. 

Two limiting forms of the Butler-Volmer equation of experimental interest are concerned with the current response of the system at both small and large overpotentials. For small overpotentials (rj < 8 mV/n), the exponential terms may be linearized (remember that e x = 1 - x for small x), so that... 

The solution of Equation (2.59) at a constant Nemst voltage shows a linear dependency of the cell voltage from the overpotential. The Ohmic law is similar to the term, where the voltage drop is proportional to the current density represented by the quotient of the utilised part of the maximum current and the cell area. 

Since the a.c. perturbation is small, the linearized relation between current and overpotential, rj (equation (6.50)), considering ara = arc = 0.5, may be used, that is... 

Whereas the charge-injection method is a small-amplitude perturbation method in which measurement is conducted during open-circuit decay, we now discuss a different open-circuit measurement, in which the initial overpotential is high, in the linear Tafel region. The equations we need to solve are similar to Eqs. 9K and lOK, except that the value of the current in Eq. lOK is that corresponding to the linear Tafel region, namely... 

This is the well-known Tafel equation, expressing overpotential as a linear function of the logarithm of the current density. An equation of this form has long been used to describe hydrogen and oxygen evolution at various electrodes. If the linear logarithmic plot of (14-21) is extrapolated back to zero overpotential, the cathodic component approaches the exchange current density y o. Thus log jo = —a/b. 

It is shown in Chapter 3 that a simple kinetic model of half-cell reactions leads to Tafel equations in which the overpotentials (r ) or polarizations of the oxidation and reduction components of a half-cell reaction are linearly dependent on the logarithm of the oxidation and reduction currents (Iox and Ired), respectively, or... 

In Fig. 26, plots of log vs ij are given for three values of (up to one tenth of the limiting current ij, the one-dimensional treatment is in complete agreement with the two-dimensional one between and the solutions of the two-dimensional equation were used), which show clearly that electrocatalytic effects of the substrate are dominant at all overpotentials, d log I d log has values of 1 and at low and high overpotentials, respectively. Similar equations set up for the thin film model show that, in this case, a linear dependence of 7j on % occurs over an even more extended region of tj values. 

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