Electrode processes steady-state mass transfer

For many electrode processes of interest, the rates of electron transfer, and of any coupled chemical reactions, are high compared with that of steady state mass transport. Therefore during any steady state experiment, Nernstian equilibrium is maintained at the electrode and no kinetic or mechanistic information may be obtained from current or potential measurements. Apart from in a few areas of study, most notably in the field of corrosion, steady state measurements are not therefore widely used by electrochemists. For the majority of electrode processes it is only possible to determine kinetic parameters if the Nernstian equilibrium is disturbed by increasing the rate of mass transport. In this way the process is forced into a mixed control region where the rates of mass transport and of the electrode reaction are comparable. The current, or potential, is then measured as a function of the rate of mass transport, and the data are, then either extrapolated or curve fitted to obtain the desired kinetic parameters. There are basically three different ways in which the rate of mass transport may be enhanced, and these are now discussed. 

Study of the charge transfer processes (step 3 above), free from the effects of mass transport, is possible by the use of transient techniques. In the transient techniques the interface at equilibrium is changed from an equilibrium state to a steady state characterized by a new potential difference A. The analysis of the time dependence of this transition is a basis of transient electrochemical techniques. We will discuss galvanostatic and potentiostatic transient techniques. For other techniques [e.g., alternating current (ac) and rotating electrodes], the reader is referred to references in the Further Reading list. 

In these electrode processes, the use of macroelectrodes is recommended when the homogeneous kinetics is slow in order to achieve a commitment between the diffusive and chemical rates. When the chemical kinetics is very fast with respect to the mass transport and macroelectrodes are employed, the electrochemical response is insensitive to the homogeneous kinetics of the chemical reactions—except for first-order catalytic reactions and irreversible chemical reactions follow up the electron transfer—because the reaction layer becomes negligible compared with the diffusion layer. Under the above conditions, the equilibria behave as fully labile and it can be supposed that they are maintained at any point in the solution at any time and at any applied potential pulse. This means an independent of time (stationary) response cannot be obtained at planar electrodes except in the case of a first-order catalytic mechanism. Under these conditions, the use of microelectrodes is recommended to determine large rate constants. However, there is a range of microelectrode radii with which a kinetic-dependent stationary response is obtained beyond the upper limit, a transient response is recorded, whereas beyond the lower limit, the steady-state response is insensitive to the chemical kinetics because the kinetic contribution is masked by the diffusion mass transport. In the case of spherical microelectrodes, the lower limit corresponds to the situation where the reaction layer thickness does not exceed 80 % of the diffusion layer thickness. 

The very fast metal-metal ion electrode processes, for which the exchange current density is very high. At steady state the overall rates of those electrode processes are controlled by the rates of mass transfer of the electroactive components to and from the electrode-melt interface. 

In this form of voltammetry, the concentration distributions of each species in the electrode reaction mechanism are temporally invariant at each applied potential. This condition applies to a good approximation despite various processes still occurring such as mass transport (e.g. diffusion), heterogeneous electron transfer and homogeneous chemical processes. Theoretically it takes an infinite time to reach the steady state. Thus, in a practical sense steady-state voltammetric experiments are conducted under conditions that approach sufficiently close to the true steady state that the experimental uncertainty of the steady-state value of the parameter being probed (e.g. electrode current) is greater than that associated with not fully reaching the steady state. The... 

A reversible one-electron transfer process (19) is initially examined. For all forms of hydrodynamic electrode, material reaches the electrode via diffusion and convection. In the cases of the RDE and ChE under steady-state conditions, solutions to the mass transport equations are combined with the Nernst equation to obtain the reversible response shown in Fig. 26. A sigmoidal-shaped voltammogram is obtained, in contrast to the peak-shaped voltammetric response obtained in cyclic voltammetry. 

A second important property of Eq. (149) is that it provides an estimate of the rate, in terms of a characteristic time 6, associated with mass transfer. Indeed, this is the time 9 needed for a molecule to reach the electrode, that is, to cover the space interval in which the molecular concentration differs from that in the bulk. In transient methods this time is identical to that elapsed since the beginning of the experiment, provided that it is lower than tmax = conv/2D. For steady-state methods, the length to be covered is (Sconv and thus from Eq. (149) it follows that 9 = 5conv/2D. The rate of mass transfer can be defined as 1 /9, since it is obviously equivalent to a first-order process (see Chapter 3 for a demonstration of this point). Yet in light of the previous discussion, it is preferable to think in terms of a characteristic time 9 associated with a given electrochemical method rather than in terms of mass transfer rate, although this intuitive latter notion was extremely worthwhile up to this point. ... 

To calculate the transfer processes in the I EG, the boundary-layer approximation is used. According to it, the current density in the bulk IEG is calculated on the basis of the height-average mass-, momentum-, and energy-transfer equations and those in the near-electrode layers. The transfer in the diffusion layers is calculated similarly to the case of quasi-steady-state approximation. 

In the type of linear-sweep voltammetry discussed thus far, the potential is changed slowly enough and mass transfer is rapid enough that a steady state is reached at the electrode surface. Hence, the mass transport rate of analyte A to the electrode just balances its reduction rate at the electrode. Likewise, the mass transport of P away from the electrode is just equal to its production rate at the electrode surface. There is another type of linear-sweep voltammetry in which fast scan rates (1 V/s or greater) are used with unstirred solutions. In this type of voltammetry, a peak-shaped current-time signal is obtained because of depletion of the analyte in the solution near the electrode. Cyclic voltammetry (see Section 23D) is an example of a process in which forward and reverse linear scans are applied. With cyclic voltammetry, products formed on the forward scan can be detected on the reverse scan if they have not moved away from the electrode or been altered by a chemical reaction. 

A mathematical model can be derived under the assumption that the electrochemical process on the microelectrodes inside the diffusion layer of a partially covered inert macroelectrode is under activation control, despite the overall rate being controlled by the diffusion layer of the macroelectrode. The process on the microelectrodes decreases the concentration of the electrochemically active ions on the surfaces of the microelectrodes inside the diffusion layer of the macroelectrode, and the zones of decreased concentration around them overlap, giving way to linear mass transfer to an effectively planar surface.15 Assuming that the surface concentration is the same on the total area of the electrode surface, under steady-state conditions, the current density on the whole electrode surface, j, is given by ... 

Figure 16a and b shows the effect of L on the radial dependence of the steady-state concentration and flux at the substrate/solution interface for a first-order dissolution process characterized by Ki = 10 and L = 0.1, 0.32, and 1.0. As the tip-substrate separation decreases, the effective rate of diffusion between the probe and the surface increases, forcing the crystal/so-lution interface to become more undersaturated. Conversely, as the UME is retracted from the substrate, the interfacial undersaturation approaches the saturated value, since the solution mass transfer coefficient decreases compared to the first-order dissolution rate constant. Movement of the tip electrode away from the substrate also has the effect of promoting radial diffusion, and consequently the area of the substrate probed by the UME increases. 

It is instructive to consider a more general derivation of (9.3.39), since the RDE can be used to study the kinetics of processes other than electron transfer at modified electrode surfaces (see Section 14.4). When mass transport and another process occur in series, the rates of both processes must be the same at steady state. Thus, for the case where electron transfer at the electrode surface is the rate-limiting process. 

The formation of 2D ordered, steady state adlayers at electrified interfaces is the result of several complex, nonequilibrium processes. In the simplest case, this sequence involves mass transfer of the molecules/ions from the bulk electrolyte towards the surface, adsorption and/or charge transfer, and association at the electrode surface [15,16] (Fig. 1). 

Usual conditions for LSV or CV experiments require a quiet solution in order to allow undisturbed development of the diffusion layer at the electrode. Some groups, however, have purposely used the interplay between diffusion and convection in electrolytes flowing in a channel or similar devices [23]. In these experiments (see also Chapter 2.4), mass transport to the electrode surface is dramatically enhanced. A steady state develops [54] with a diffusion layer of constant thickness. Thus, such conditions are in some way similar to the use of ultramicroelectrodes. Hydro-dynamic voltammetry is advantageous in studying processes (heterogeneous electron transfer, homogeneous kinetics) that are faster than mass transport under usual CV or LSV conditions. A recent review provides several examples [22]. 

Under steady-state conditions, the rate of charge-transfer process at the disk electrode is equal to the rate of removal of R from the disk surface (According to the definition of N, there is no further reaction of R at the disk electrode and chemical decomposition of R in the electrolyte solution). Therefore, the mass balance on disk electrode surface (x = 0) can be expressed as ... 

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