Diffusion-controlled case

Although there are differences in the approach curves with the constant-composition model, it would be extremely difficult to distinguish between any of the K cases practically, unless K was below 10. Even for K = 10, an uncertainty in the tip position from the interface of 0. d/a would not allow the experimental behavior for this rate constant to be distinguished from the diffusion-controlled case. For a typical value of Z)Red, = 10 cm s and electrode radius, a= 12.5/rm, this corresponds to an effective first-order heterogeneous rate constant of just 0.08 cm s. Assuming K,. > 20 is necessary... 

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. 

This section will only cover reactions in aqueous solutions. Water molecules acting as either a proton acceptor or a proton donor will thus be in close contact with an acid or a base undergoing excited-state deprotonation or protonation, respectively. Therefore, these processes will not be diffusion-controlled (Case A in Section 4.2.1). 

This situation, as discussed in the last section, closely resembles that of the droplet diffusion flame, in which the oxygen concentration approaches zero at the flame front. Now, however, the flame front is at the particle surface and there is no fuel volatility. Of course, the droplet flame discussed earlier had a specified spherical geometry and was in a quiescent atmosphere. Thus, hD must contain the transfer number term because the surface regresses and the carbon oxide formed will diffuse away from the surface. For the diffusion-controlled case, however, one need not proceed through the conductance hD, as the system developed earlier is superior. 

The quasi-steady-state hopping recombination rate K(oo) = Kq is related to the coefficient i eff via equation (4.2.14) as in the diffusion-controlled case. As in equation (4.2.15), this Ifu is defined by the asymptotics of the solution, Y(r,oo) = y(r), as r —> oo. It is important, however, that R ff cannot generally be treated as the effective recombination radius. It holds provided that the hop length is much smaller than the distinctive scale ro of tunnelling recombination... 

Figure 50. Snapshots of oxygen incorporation experiments in Fe-doped SrTi03, recorded by in situ time and space resolved optical absorption spectroscopy.256 Rhs column refers to the corresponding oxygen concentration profiles, in a normalized representation. Top row refers a predominantly diffusion controlled case (single crystal), center row to a predominandy surface reaction controlled case (single crystal), bottom row to transport across depletion layers at a bicrystal interface.257,258 For more details on temperature, partial pressure, doping content, structure see Part I and Ref.257-259 Reprinted from J. Maier, Solid State Ionics, 135 (2000) 575-588. Copyright 2000 with permission from Elsevier.

For combustion reactions Levenspiel (4) gives the constant temperature integration for reaction and gas or ash diffusion controlled cases. The integration of the pyrolysis kinetics will be demonstrated in the following section. 

This equation looks similar to Eq. (7.33) which was derived for the diffusion-controlled case. Eq. (7.42) differs from Eq. (7.33), however, insofar as here the half-wave potential depends on Ldift and therefore on the rotation speed of the electrode. On the other hand, if the current is only diffusion-controlled, Eq. (7.33) determines the current-potential curve. In this case f/i/2 is independent of Ljjff and therefore also independent of the rotation speed (Eq. 7.32). 

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.

Although this is a very complex equation, it allows to take into consideration any function of R(t), and consequently A(t), resulting from experiments with growing drops or bubbles. In combination with an adsorption isotherm (diffusion-controlled case) or a transfer mechanism (mixed diffusion-kinetic-controlled model) it describes the adsorption process at a growing or even receding drop. Eq. (4.48) can be applied in its present form only via numerical calculations and an algorithm is given by MacLeod Radke (1994). 

Steady State Solutions the Diffusion Controlled Case. 140... 

Steady State Solutions The Diffusion Controlled Case... 

The influences on the IMPS response of the illumination geometry, absorption coefficient, and electron recombination lifetime predicted for the diffusion controlled case are illustrated in Fig. 46 and Fig. 47. Two cases are considered corresponding to interband excitation and to dye sensitisation. In the first the penetration depth of the light is chosen to be much smaller than the film thickness. The calculated IMPS responses for illumination from the electrolyte and from the substrate side are contrasted in Fig. 46a and 46b respectively. The first thing to note is that the imaginary component of the IMPS response is negative under most conditions. This contrasts... 

For simple reversal chronopotentiometry, the ratio of reversal transition time T2 to the forward time t is 1/3, just as in the diffusion-controlled case, independent of the rate constants. However, for cyclic chronopotentiometry the transition times for the third (73) and subsequent reversals differ from those of the diffusion-controlled case (31). 

One of the key features of the voltammogram is the peak height. For a planar diffusion-controlled case, the peak current is given by ... 

We note that when greater than diffusional resistance), the system reduces to the chemical-reaction-controlled case and when is large, to the diffusion-controlled case. 

For comparison with the diffusion controlled case, expressing Eq. (2.26) in the form of Eq. (2.21) gives... 

When ash diffusion controls (case E), the mass transfer coefficient depends on the thickness of the ash layer. The usual assumption is that this coefficient is... 

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