Pure Diffusion Control

Equation (96) can be solved analytically for two cases, as characterized by the kinetics of hydrogen transfer across the metal interface (i) pure diffusion control and (ii) diffusion control with a limited rate of entry (see Section III.1). As with the permeation methods, in both cases the layer of adsorbed hydrogen is assumed to adjust very rapidly to a potential step. 

For pure diffusion control, the initial and boundary conditions are written as 

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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... 

As an example, consider a simple reaction of the type (6.2) taking place under pure diffusion control. At all times the electrode potential, according to the Nemst equation, is determined by the reactant concentrations at the electrode surface. It was shown in Section 11.2.3 that periodic changes in the surface concentrations which can be described by Eq. (11.19) are produced by ac flow. We shall assume that the amplitude of these changes is small (i.e., that Ac electrode polarization. With this substitution and using Eq. (11.19), we obtain... 

We have seen that purely diffusion-controlled biouptake fluxes may require time spans of O(103) s to decay to their eventual steady-state values (see Section 2.3.6). In reality the situation of pure diffusion as the mode of mass transfer in... 

A very important factor in ensuring that full cell wall penetration has occurred is to allow sufficient time for the impregnant molecules to diffuse into the intracellular spaces. Many workers allow several days (weeks in some cases) for this to occur. It is important to emphasize that pressure treatment will aid penetration of larger wood samples, but will not in any way result in cell wall penetration, which is a purely diffusion-controlled process. 

The gradient of this graph therefore permits the determination of n, and the intercept allows k to be calculated. The advantage of using a Sharp-Hancock plot rather than a least squares fitting process with the Avrami equation is that if Avrami kinetics are not applicable, this can be seen in the former plot, and hence other kinetic models may be investigated. Purely diffusion controlled processes can be identified using a Sharp-Hancock plot n is foimd to be 0.5 in such cases. 

The homogeneous catalysis method is suitable to measure rate constants over a very wide range, up to the diffusion limit. The lower limit is determined by interferences, such as convection, which occur at very slow scan rates. It is our experience that, unless special precautions are taken, scan rates below lOOmV/s result in significant deviations from a purely diffusion-controlled voltammetric wave. For small values of rate constants (down to 10 s ), other potentiostatic techniques are best suited, such as chronoamperometry at a rotating disk electrode UV dip probe and stopped-flow UV-vis techniques. ... 

P. Delahay, New Instrumental Methods in Electrochemistry, Interscience, New York (1952). Contains much seminal work showing the evolution away from the pure diffusion control and the introduction of activation overpotential. i.e.. interfacial control. 

For the intermediate case, the shape of the wave is determined by both diffusion and kinetic parameters resulting in a peak-shaped voltam-metric wave with a smaller peak current and a broader peak width compared with a purely diffusion-controlled voltammetric peak. 

The flow modulation technique, in general, appears therefore very well suited for this specific purpose of quantitative diffusivity measurement. However, it also reveals any deviation from a purely diffusion controlled kinetics more clearly than do steady state measurements when, for example, a slow series process is concealed in an apparent diffusion plateau. 

In this chapter we consider systems under conditions in which the kinetics of the electrode reaction is sufficiently fast that the control of the electrode process is totally by mass transport. This situation can, in principle, always be achieved if the applied potential is sufficiently positive (oxidation) or negative (reduction). First we consider the case of pure diffusion control, and secondly systems where there is a convection component. 

In the next two sections we use the Laplace transform to solve Fick s second law for two important cases under conditions of pure diffusion control ... 

Equation (187) is only strictly applicable to crystals in a stationary liquid. Particles less than 5 fim in size (the value depending on the difference in the densities of the solid and solution) will tend to be carried with the solution during stirring and so grow by a purely diffusion-controlled growth rate. However, for larger particles, the transport of ions to the surface will depend upon the solution movement around the particles and so both convection and diffusion have to be taken into account. In this case, the assumption that r2 P r( that led to eqn. (184) is no longer valid. Instead, if (r2 - r4) = 3 and r, r2, then eqn. (184) becomes... 

When the solute is spherical, or close to be so, its radius is easily obtained otherwise, estimations can be made on the basis of the geometry and arrangement of the constituting atoms or ions. For solutes having a complex stucture (e.g., micelles), a distinction should be made between the hydrodynamic radius (which appears in the Stokes-Einstein equation of the diffusion coefficient) and the reaction radius [98]. For Ps, RPs should represent the bubble radius. However, as shown in Table 4.4, the experimental data are systematically very well recovered by using the free Ps radius, RPs = 0.053 nm using the bubble radius results in a calculated value of kD (noted kDb) that is too small by a factor of 2 or 3. Table 4.4 does not include such cases where k kD, as these do not correspond to purely diffusion-controlled reactions. 

Figure 63. The kinetics in Lao. Sro.iCoOj.x> under the conditions given, is strongly influenced by the surface reaction. For pure diffusion control the normalized surface concentration would be unity.207 (Reprinted from R. A. De Souza, J. A. Kilner, Oxygen transport in La,.xSrxMni.yCoy03is. , Solid State Ionics, 106, 175-187. Copyright 1998 wih permission from Elsevier.)...

Equation 9D is strictly applicable only in unstirred solutions, but for short transients the situation is better. As long as the value of 8, as calculated from Eq. 9D for a purely diffusion-controlled process, is small compared to the diffusion layer thickness set up by stirring,... 

When adsorption takes place much faster than according to [2.8.2] the process cannot be purely diffusion-controlled. For instance, when the molecules and the adsorbent are oppositely charged, conduction also plays a role. In that case the material flux is given by the Nemst-Planck equation, [1.6.7.1 or 3]. 

However, the straight lines do not intersect with the zero point for /w - 0, thus indicating that the measured limiting cathodic current density values are not diffusion-controlled limiting current densities according to the Levich-equation [138]. The deviation of pure diffusion control may be an indication of additional electrochemical or chemical hindrance [18, 95]. A steeper inclination of the slope indicates stronger diffusion control. 

Both m and t in the Ilkovic equation [Eq. (74)] depend on the height of the mercury reservoir h, and since m is equal to c h and t is equal to c"/h, where c and c" are constants, we have the relationship between id and h given by Eq. (82). Thus the magnitude of a purely diffusion-controlled polarographic wave is proportional to the square root of the height of the mercury reservoir. 

An additional benefit to be derived from direct measurements of k2 involves its relation to rates of exciton migration using the theory of diffusion controlled processes. The rate constant for a purely diffusion controlled reaction may be written in the form (10)... 

Figure 9.22A illustrates the purely diffusion-controlled process, in which the effects of boundary layers and interfacial reaction rates are negligible. In this case, the concentrations of the complex at the interfaces are the equilibrium concentrations. Figure 9.22B illustrates the partially boundary-layer-controlled case. Here, prior to steady state, the permeant diffuses across the membrane faster in the feed-side boundary layer and accumulation of permeant in the product-side boundary layer. The consequence of this concentration polarization is a reduction in the net concentration gradient across the membrane, and a reduced flux compared with the diffusion-controlled case. The last case is that of partially reaction-rate-controlled flux, illustrated by the concentration profile in Figure 9.22C. Here, either the permeant initially diffuses away from the feed interface faster than it can be replenished by the interfacial reaction, or the dissociation reaction is not fast enough to prevent accumulation of the complex at the product interface. Again, the net result is a decrease in the concentration gradient compared with that in the purely diffusion-controlled case. In all three cases, the flux is proportional to the slope of the concentration profile across the liquid membrane.

In experiments on nonionic surfactants, namely Triton X-405 Geeraerts at al. (1993) performed simultaneously dynamic surface tension and potential measurements in order to discuss peculiarities of nonionic surfactants containing oxethylene chains of different lengths as hydrophilic part. Deviations from a diffusion controlled adsorption were explained by dipole relaxations. In recent papers by Fainerman et al. (1994b, c, d) and Fainerman Miller (1994a, b) developed a new model to explain the adsorption kinetics of a series of Triton X molecules with 4 to 40 oxethylene groups. This model assumes two different orientations of the nonionic molecule and explains the observed deviations of the experimental data from a pure diffusion controlled adsorption very well. Measurements in a wide temperature interval and in presence of salts known as structure breaker were performed which supported the new idea of different molecular interfacial orientations. At small concentration and short adsorption times the kinetics can be described by a usual diffusion model. Experiments of Liggieri et al. (1994) on Triton X-100 at the hexane/water interface show the same results. 

The transformed current data can be used directly, by (6.7.2), to obtain Cq(0, t). Under conditions where Cq(0, 0 = 0 (i e under purely diffusion-controlled conditions), 7(0 reaches its limiting or maximum value, 7/ [or, in semi-integral notation, m(0max] where... 

If the dnjdt versus t 1/2 plots do not extrapolate to zero at infinite time, as required by a purely diffusion controlled process, this indicates the presence of an activation barrier to the desorption. This behavior was rationalized by MacRitchie [16]. As shown in Fig. 2, most of the segments are located at the interface at low surface pressure. The probability that the transition state configuration could oc-... 

Diffusion plays a role in the process since the reaction rate decreases with increasing particle size although the illustrating data are not given. The effect of particle diameter on rate is, however, not as great as would be predicted from a purely diffusion-controlled mechanism. 

Figure 4 shows data obtained by Sehested and Christensen [18] for this reaction (R6 in Table 1). In this case the data are accurately represented by equation (9) with p = 0.25, taking Dh = I x 10 m s at 25 °C [19] and rn = 0.19 nm [14]. Evidently this reaction is purely diffusion controlled, as would be expected for such a simple isotropic reactant, and p= 025 is consistent with > 13 ps [20]. 

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