Mixed Kinetic and Diffusion Control

Irreversible Follow-Up Reactions, Mixed Diffusion and Kinetic Control (CV and DCV)... 

Activation control of an overall dissolution rate can, of course, reside in the reduction process, in the oxidation process, in a mixture of both, or in a mixture including some transport control. The reduction process is usually more influential in determining the overall rate. Thus, in the absence of transport control, the kinetics of the electrode process for reduction of hydrated protons, or water molecules, or dissolved molecular oxygen plays the major role in metal dissolution kinetics. Indeed the literature confirms the conclusion that many of the systems seen in experiment or in practice are diffusion controlled that most of the rest are under mixed diffusion and activation control and that those with some activation control... 

Fig. 18b.6. (a) Shape of the voltage pulses for diffusion control, mixed diffusion-kinetic control, and kinetic control, (b) concentration gradient of O showing expansion of the diffusion layer with time for complete diffusion controlled reaction, and (c) current transients show diffusion controlled, mixed kinetics and diffusion control, and complete kinetics controlled reactions corresponding to voltage pulses shown in (a). Note that the equations are derived only for the diffusion controlled case. 

XPS study by Buckley and Woods (1984b) showed that freshly fractured chalcopyrite surfaces exposed to air formed a ferric oxyhydroxide overlayer with an iron-deficient region composed of CuSi. Acid-treated surfaces of fractured chalcopyrite showed an increase in the thickness of the CuS2 layer and the presence of elemental sulfur. Hackl et al. (1995) suggested that dissolution of chalcopyrite is passivated by a thin (< 1 pm) copper-rich surface layer that forms as a result of solid-state changes. The passivating surface layer consists of copper polysulhde, CuS , where n > 2. Hackl et al. (1995) described the dissolution kinetics as a mixed diffusion and chemical reaction whose rate is controlled by the rate at which the copper polysulhde is leached. The oxidation of chalcopyrite in the presence of ferric ions under acidic conditions can be expressed as... 

Depending on the relative magnitudes of D and ks, we move from a situation in which diffusion control predominates (small D, large ks, reversible case, see Sect. 2.1.2.1) through a mixed-control regime (both diffusion and kinetics are important quasi-reversible case [9, 27, 28]) to a situation in which the rate of electron transfer controls the overall reaction (large D, small ks, irreversible case [12]). This continuum of conditions is characterized in LSV or CV by the dimensionless quantity... 

Microreactors proved to be much more eflicient for the phase transfer reactions (23). The two-phase reactions proceed on the phase boundary. As a result of mass transfer coefficient estimation, it can be ascertained that the application of microtechnology for the two-phase liquid reactions promotes instantaneous mixing and intensifies the interfusion of reagents, which is not to be assumed in standard reactors. By slow reactions due to increase in interfacial area, the reaction can be shifted from diffusion to kinetic control. Thus, Dan C 1, which means that there is no mass transfer limitation and the plug flow reactor model can be used to describe such a reaction (see Section 12.2). 

When concentration changes affect the operation of an electrode while activation polarization is not present (Section 6.3), the electrode is said to operate in the diffusion mode (nnder diffusion control), and the cnrrent is called a diffusion current i. When activation polarization is operative while marked concentration changes are absent (Section 6.2), the electrode is said to operate in the kinetic mode (under kinetic control), and the current is called a reaction or kinetic current i,. When both types of polarization are operative (Section 6.4), the electrode is said to operate in the mixed mode (nnder mixed control). 

It follows from the figures and also from an analysis of Eq. (6.40) that in the particular case being discussed, electrode operation is almost purely diffusion controlled at all potentials when flij>5. By convention, reactions of this type are called reversible (reactions thermodynamically in equilibrium). When this ratio is decreased, a region of mixed control arises at low current densities. When the ratio falls below 0.05, we are in a region of almost purely kinetic control. In the case of reactions for which the ratio has values of less than 0.02, the kinetic region is not restricted to low values of polarization but extends partly to high values of polarization. By convention, such reactions are called irreversible. We must remember... 

Figure 14 shows a schematic representation of a mixed potential diagram for the electroless deposition reaction. Oxidation of the reductant, in this case hypophos-phite, is considered to be under 100% kinetic control. A mixed kinetic-diffusion curve is shown for the reduction of the metal ion, in our case Co2+, in the region close to the mixed potential, Em. Thus, since Co deposition occurs under a condition of mixed kinetic and diffusion control, features small relative to the diffusion layer thickness for Co2+ will experience a higher concentration of the metal ion, and hence... 

As compared to the Nemstian case, the plateau is the same but the wave is shifted toward more negative potentials, the more so the slower the electrode electron transfer. An illustration is given in Figure 4.13 for a value of the kinetic parameter where the catalytic plateau is under mixed kinetic control, in between catalytic reaction and substrate diffusion control. For the kjet(E) function, rather than the classical Butler-Volmer law [equation (1.26)], we have chosen the nonlinear MHL law [equation (1.37)]. 

In chemically reacting flow systems, the overall reaction rate may be limited by the mixing rate of the reactants or by the rate of the chemical reaction upon mixing. If mixing is slow compared to chemical reaction, the system is diffusion or mixing controlled, while fast mixing and slow reaction results in a kinetically controlled system (Fig. 13.1). The assumption of fast chemistry is valid if the system is mixing controlled. 

This technique was proposed by Bruckenstein and co-workers [280, 281] and is useful in that the current due to the modulation of the fluid flow is essentially free of any electrode surface-controlled contributions in most cases. Thus, it can be used as an analytical tool to increase sensitivity [282]. Step changes were originally considered but this was later extended to sinusoidal hydrodynamic modulation (SHM) in the limiting current region and then to the region of mixed convective-diffusion/kinetic control [283—287]. If the modulation frequency is o, then the modulation, which is small, can be described by... 

Chemical reactions are classified usually as diffusion-controlled, whose rate is limited by a reactant spatial approach to each other, and reaction-controlled (kinetic stage), whose rate is limited by a reaction elementary event. For systems with ideal reactant mixing considered in Section 2.1.1, there is no mechanism of reactant mutual approach. On the other hand, the kinetic equations (2.1.40) distinguish between reaction in physically infinitesimal volumes and the distant reactant motion in a whole reaction volume. In the absence of reaction particle diffusion is described by equation... 

The field of predominantly kinetic influence (base of the voltammogram, BV relation valid) and the held of a mixed influence of kinetics and transport are suitable to determine parameters such as rate constant, reaction order and transfer coefficients. The held controlled by transport (Equation 1.51 valid, in practice usually with c0" or cR =0) can lead to the diffusion coefficient. 

Apparent rate laws include both chemical kinetics and transport-controlled processes. One can ascertain rate laws and rate constants using the previous techniques. However, one does not need to prove that only elementary reactions are being studied (Skopp, 1986). Apparent rate laws indicate that diffusion or other microscopic transport phenomena affect the rate law (Fokin and Chistova, 1967). Soil structure, stirring, mixing, and flow rate all affect the kinetic behavior when apparent rate laws are operational. 

Three different ways have been developed to produce nanoparticle of PE-surfs. The most simple one is the mixing of polyelectrolytes and surfactants in non-stoichiometric quantities. An example for this is the complexation of poly(ethylene imine) with dodecanoic acid (PEI-C12). It forms a solid-state complex that is water-insoluble when the number of complexable amino functions is equal to the number of carboxylic acid groups [128]. Its structure is smectic A-like. The same complex forms nanoparticles when the polymer is used in an excess of 50% [129]. The particles exhibit hydrodynamic diameters in the range of 80-150 nm, which depend on the preparation conditions, i.e., the particle formation is kinetically controlled. Each particle consists of a relatively compact core surrounded by a diffuse corona. PEI-C12 forms the core, while non-complexed PEI acts as a cationic-active dispersing agent. It was found that the nanoparticles show high zeta potentials (approximate to +40 mV) and are stable in NaCl solutions at concentrations of up to 0.3 mol l-1. The stabilization of the nanoparticles results from a combination of ionic and steric contributions. A variation of the pH value was used to activate the dissolution of the particles. 

FIGURE 1.21 An example of a complex-plane impedance plot (Nyquist plane) for an electrochemical system under mixed kinetic/diffusion control, with the mass transfer and kinetics (charge transfer) control regions, for a finite thickness 8N of the diffusion layer. Assumption was made that Kf Kh at the bias potential of the measurement, and D0I = Dmd = D, leading to RB = RCT (krb8N/ >). 

Different forms of the impedance plots can be obtained for an electrochemical system described by a mixed kinetic/diffusion control process, depending on the parameters of diffusion and charge transfer. An example of a Nyquist plot is presented in Figure 1.21. 

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