Current-potential curves equilibrium

Fig. 2. Current-potential curves in Evans diagram [29] format for reduction of Cu2+ ions and oxidation of H2CO. and are the equilibrium, or open circuit, potentials for the Cu2+ reduction and H2CO oxidation reactions, respectively. Assuming negligible interfering reactions, the vertical dashed lines indicate the exchange current densities for the two half reactions, and the deposition current for the complete electroless solution. Adapted from ref. 23.

Contrarily to the CE mechanism, in this case the current-potential curve is more affected by the chemical reaction the higher the rate and the equilibrium constants / and Keq = 1 /K. 

It is of interest at this point to compare the study of Multipulse Chronoamperometry and Staircase Voltammetry with those corresponding to Single Pulse Chronoamperometry and Normal Pulse Voltammetry (NPV) developed in Chaps. 2 and 3 in order to understand how the same perturbation (i.e., a staircase potential) leads to a sigmoidal or a peak-shaped current-potential response as the equilibrium between two consecutive potential pulses is restored, or not. This different behavior is due to the fact that in SCV the current corresponding to a given potential pulse depends on the previous potential pulses, i.e., its history. In contrast, in NPV, since the equilibrium is restored, for a reversible process the current-potential curve is similar to a stationary one, because in this last technique the current corresponding to any potential pulse is independent of its history [8]. 

The value of the constant is nearly independent of the electrolyte used and the type and value of the conductivity of the electrode. Furthermore, all Ge samples showed identical current-potential curves in the same electrolyte in the region of small currents, where no deviations from the equilibrium n(x) p(x) nf are to be expected. Consequently different values of exist at the same electrode potential for differently doped samples, There-... 

Current-potential curves calculated from Eq. 35E are shown in Fig. 2E, for three values of p. The plots are linear near the equilibrium potential, and the current density increases exponentially at large overpotentials. 

If the cathodic current - / is plotted as ordinate and the cathodic polarization — t) as abscissa, then the quantity iotiFIRT is the slope (dildrj)i=o of the current-potential curve at the point of zero current (the equilibrium potential). Its reciprocal RTInFi has the dimensions of resistance (ohms) and is often called the polarization resistance. It is the effective resistance imposed at the electrode surface by the finite rate of the electron-transfer process (Section 12-9). 

Equation (78) assumes that the deposition reactions are activation controlled, rj = rjct (see Sect. 3.1.3). Figure 28 shows current-potential curves for the case when the deposition of metal Mi is limited by transport (iy = i 11 diffusion control). It can be seen that in this case, at the cathode potential E(i), metal M2 with a more negative equilibrium potential ( eq,2 < h eq/i) is deposited to a greater extent than metal Mi, i2 > i. ... 

Thin-film ideal or Nemstian behavior is the starting point to explain the voltammetric behavior of polyelectrolyte-modified electrodes. This condition is fulfilled when (i) the timescale of the experiment is slower than the characteristic timescale for charge transport (12/Dapp, with (the film thickness) in the film, that is all redox within the film are in electrochemical equilibrium at any time, (ii) the activity of redox sites is equal to their concentration and (iii) all couples have the same redox potential. For these conditions, anodic and cathodic current-potential waves are mirror images (zero peak splitting) and current is proportional to the scan rate [121]. Under this regime, there exists an analytical expression for the current-potential curve ... 

Figure 3 shows current - potential curves (b-f) for 5 solutions with 2 mM KAu(CN)2 + 1 M KOH with different concentrations of KCN, hence, with different equilibrium potentials for the Au(CN)2 / Au redox couple. The scan rate was 10 mV s 1 in all cases. The KCN concentration was varied from 2 M (curve b), corresponding to a redox potential of -1.01 V, to 0.02 M (curve e) which corresponds to a redox potential of -0.78 V. Curve a corresponds to the current - potential characteristics in 0.04 M KCN at pH 14. It can be seen that the gold deposition peak shifts to more negative potential with... 

Current-potential curves, particularly those obtained under steady-state conditions, are sometimes called polarization curves. We have seen that an ideal polarized electrode (Section 1.2.1) shows a very large change in potential upon the passage of an infinitesimal current thus ideal polarizability is characterized by a horizontal region of an i-E curve (Figure 1.3.5a). A substance that tends to cause the potential of an electrode to be nearer to its equilibrium value by virtue of being oxidized or reduced is called a depolarizer An... 

The current-potential curves discussed so far can be used to measure concentrations, mass-transfer coefficients, and standard potentials. Under conditions where the electron-transfer rate at the interface is rate-determining, they can be employed to measure heterogeneous kinetic parameters as well (see Chapters 3 and 9). Often, however, one is interested in using electrochemical methods to find equilibrium constants and rate constants of homogeneous reactions that are coupled to the electron-transfer step. This section provides a brief introduction to these applications. 

For slow electron-transfer (irreversible) processes, the eventual extent of the electrode process will be governed by equilibrium considerations and the Nemst equation, but the rate of electrolysis will be small at the potentials predicted in the previous sections and long-duration electrolyses would result. For these processes, reduction must be carried out at somewhat more negative potentials the actual potential is usually selected on the basis of experimental current-potential curves taken under conditions near those for the intended bulk electrolysis. Processes that are controlled by the rate of a homogeneous reaction, such as... 

Here, we assume that electron transfer only occurs via the CB and not via surface states. As in a Schottky diode, j generally increases exponentially with (decreasing) potential (Fig. 3a). The form of the dark current-potential curve, however, depends on the mechanism and kinetics of the charge-transfer reaction. At high overpotential, corresponding to a large deviation from equilibrium, the reaction expressed by Eq. (4) may become limited by mass transport in solution, that is, the cathodic current becomes potential-independent (this is not shown in Fig. 3). 

In this section, the behavior of a redox system at the equilibrium potential has been discussed. It should, however, be noted that impedance spectroscopy of irreversible systems can also yield useful information. For example, the charge-transfer resistance determined at the corrosion potential corresponds to the slope of the current-potential curve (/ ct = dV(t)/dI (t) at that potential and allows calculation of the rate of corrosion [1]. 

Another important point, as far as the steady-state current-potential curves are concerned, is the position on the potential axis of the branch plotted, which in the example above is anodic. As previously underlined, no equilibrium potential can be spotted because the open-circuit potential cannot be defined by the Nernst law. However, it is quite useful to have the thermodynamic constant associated with the redox couple in question, which in this case is the standard potentiali °(Cu VCu), because this enables one to position the current-potential curve, i.e., the potential zone where the current undergoes sharp variations (see figures 2.18 above and 2.21 in section 2.B.2.2) Our aim here is to plot and use the shapes of the curves rather than to estimate their precise mathematical expression or any potential values to the mV. 

On the other hand, the shape of the current-potential curve is different in the oxidation branch because metallic copper is not concerned by any mass transport phenomenon, since copper is always present at the interface. The zone where the current undergoes large variations is close to the open-circuit potential, which is, in this case, equal to the equilibrium potential of the system. The latter, which can be calculated using the Nernst law, is shifted slightly from the standard potential. To give an order of magnitude for a concentration in Cu ions equal to 10" mol L , there is the following ... 

When establishing a method for determining redox kinetics, one can consider using the value of the slope of the current-potential curve for the open-circuit potential. However, it is impossible to assign a specific slope value to a fast couple because this slope is highly dependent on the limiting anodic and cathodic currents. To approach this issue from a quantitative point of view, one can start from the expression of the polarisation resistance around the equilibrium potential (see section 4.3.3.4). 

The first example discussed here involves predicting reactions occurring in a system with no current. Remember that such a system may or may not be in equilibrium. When describing an entire electrochemical system at open circuit, one usually considers what is occurring at each of the two interfaces as two separate entities. Here we will therefore focus on only one current-potential curve, in other words, one single interface. 

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