Equations current-potential

Unless there is a large excess of indifferent ions that assume the burden of carrying the current (as indeed was assumed above), the electron acceptors and donors do not move only by diffusion or convection they also move under the influence of the electric field. In fact, this is generally the case unless one has diminished the fraction of the current in the solution which reactants need for carrying, by adding an excess of ions of another kind that do not undergo electrodic reaction, e.g., the indifferent electrolyte. How must the current-potential equations be modified ... 

If Eq. 11 is inserted into the current-potential equation (Eq. 8), the so-called current-overpotential equation results (Eq. 12), where the overpotential q is defined as the potential relative to the equilibrium potential, q = E - eq. 

The dc current in such a case is described by the current-potential equation [17]... 

By solving the standard current-potential equation for an electrochemical reaction (see, for example, Bard and Faulkner [1980]) under the conditions of kinetic... 

Let us write the current-potential equation for reactions XIV and XV the anodic current will depend on the surface concentration of Red,... 

From the above qualitative considerations, it appears that the mass transport to microelectrodes, apart from a few cases, is complicated theoretically, and in the next sections, we describe the current-time and current-potential equations, which have been derived by using either analytical solutions or simulation procedures for MEs having the geometries depicted in the scheme of Fig. 15.5, which have been largely employed for practical applications. Detailed information on how MEs and NEs can be fabricated can be found in several reports and reviews. ... 

The expression for the mass-transport-limiting current density may be employed together with the Nemst equation to deduce the complete current-potential response in a solution containing only oxidized or reduced species... 

It must be emphasised that in evaluating the limiting cathode potential to be applied in the separation of two given metals, simple calculation of the equilbrium potentials from the Nernst Equation is insufficient due account must be taken of any overpotential effects. If we carry out, for each metal, the procedure described in Section 12.2 for determination of decomposition potentials, but include a reference electrode (calomel electrode) in the circuit, then we can ascertain the value of the cathode potential for each current setting and plot the current-potential curves. Schematic current-cathode potential... 

In deriving theoretical equations of the current-potential (or time) curves of ion-transfer voltammetry of a dibase we shall make the following assumptions ... 

In deriving theoretical equations of the current-potential (or time) curves of ion transfer of an acid we shall make essentially the same assumptions as the assumption 1-6 above. It is noted here that theoretical equations of the more general case, that is, of a dibasic acid, such as expressed by AH2 = AH + H, AH = A + H, can be derived [24], but are not included here, to save space. The formal formation constant, and formal dissociation constant,, in the a phase is defined by... 

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 ... 

The basic theory of mass transfer to a RHSE is similar to that of a RDE. In laminar flow, the limiting current densities on both electrodes are proportional to the square-root of rotational speed they differ only in the numerical values of a proportional constant in the mass transfer equations. Thus, the methods of application of a RHSE for electrochemical studies are identical to those of the RDE. The basic procedure involves a potential sweep measurement to determine a series of current density vs. electrode potential curves at various rotational speeds. The portion of the curves in the limiting current regime where the current is independent of the potential, may be used to determine the diffusivity or concentration of a diffusing ion in the electrolyte. The current-potential curves below the limiting current potentials are used for evaluating kinetic information of the electrode reaction. 

The two-step charge transfer [cf. Eqs. (7) and (8)] with formation of a significant amount of monovalent aluminum ion is indicated by experimental evidence. As early as 1857, Wholer and Buff discovered that aluminum dissolves with a current efficiency larger than 100% if calculated on the basis of three electrons per atom.22 The anomalous overall valency (between 1 and 3) is likely to result from some monovalent ions going away from the M/O interface, before they are further oxidized electrochemically, and reacting chemically with water further away in the oxide or at the O/S interface.23,24 If such a mechanism was operative with activation-controlled kinetics,25 the current-potential relationship should be given by the Butler-Volmer equation... 

Figure 5.2 Current-potential curves according to the Butler-Volmer equation.

Strictly speaking, the current-potential curve depends on the value of the starting potential, ), relative to the standard potential, E°. However, this dependence vanishes as ) becomes more and more positive. This is the reason that the lower limit of the integral in equation (1.5) is taken as... 

The peak current, potential, and width are consequently given (using equation 1.4) by... 

In cyclic voltammetry, simple relationships similar to equations (1.15) may also be derived from the current-potential curves thanks to convolutive manipulations of the raw data using the function 1 /s/nt, which is characteristic of transient linear and semi-infinite diffusion.24,25 Indeed, as... 

In more quantitative terms, the analysis developed in Section 5.3.2 may be applied here. It is, however, necessary to take into account inhibition by the substrate as depicted in Scheme 5.3. At low substrate concentration, however, inhibition can be neglected. When complete control by substrate diffusion prevails, the current-potential response obeys the conditions of total catalysis, being given by equation (5.25) (dotted line in Figure 5.24), as discussed in Section 5.3.2, introducing a stoichiometric factor of 2, while the peak potential is given by equation (5.26). 

In this section we establish the equation of the forward scan current potential curve in dimensionless form (equation 1.3), justify the construction of the reverse trace depicted in Figure 1.4, and derive the charge-potential forward and reverse curves, also in dimensionless form. Linear and semi-infinite diffusion is described by means of the one-dimensional first and second Fick s laws applied to the reactant concentrations. This does not imply necessarily that their activity coefficients are unity but merely that they are constant within the diffusion layer. In this case, the activity coefficient is integrated in the diffusion coefficient. The latter is assumed to be the same for A and B (D). 

The pure kinetic conditions, which are achieved for large values of A, implies that bo = t/ q/x/A —> 0 and thus, from equation (6.40), i//2 = (//,. It follows that the current is exactly the double of the irreversible EC current obtained under pure kinetic conditions along the entire current-potential curve. 

Thus, the dimensionless current-potential curves depend on the dimensionless parameters 1, A, A , oq, and a2. Simulating the dimensionless cyclic voltammograms then consists of finite difference resolutions of equations (6.57) and (6.58), taking into account all initial and boundary conditions. Examples of such responses are given in Section 2.5.2 (Figure 2.35). 

We are thus back to equation (6.64), showing that the current-potential responses are the same as in the concerted case. 

In the total catalysis zone (Figure 2.17), the current-potential response splits into two waves. One is the mediator reversible wave. The other is an irreversible wave arising in a much more positive potential region. The characteristics of the latter may be derived from the integral equation above, noting that since the wave is located at a very positive potential, 1 /[1 + exp(— )] is small compared to y and 1 + exp(— ) exp(— ). Thus,... 

The data shown in Figure 2.36 were gathered at constant current with a value of the current density that brought the electrode potential at the foot of the current-potential characteristic of the system. The concentration of substrate may thus be considered as constant. As discussed in Section 2.5, we consider only the case where the second electron transfer in the radical-substrate coupling pathway occurs at the electrode (ECE). The following equations and conditions apply. 

After the electrode reaction starts at a potential close to E°, the concentrations of both O and R in a thin layer of solution next to the electrode become different from those in the bulk, cQ and cR. This layer is known as the diffusion layer. Beyond the diffusion layer, the solution is maintained uniform by natural or forced convection. When the reaction continues, the diffusion layer s thickness, /, increases with time until it reaches a steady-state value. This behaviour is also known as the relaxation process and accounts for many features of a voltammogram. Besides the electrode potential, equations (A.3) and (A.4) show that the electrode current output is proportional to the concentration gradient dcourfa /dx or dcRrface/dx. If the concentration distribution in the diffusion layer is almost linear, which is true under a steady state, these gradients can be qualitatively approximated by equation (A.5). 

This current-potential relationship, also known as the Butler-Volmer equation, governs all the (fast and single step) heterogeneous electron transfers. 

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