Pure diffusion limit

Often, we need only a qualitative estimate that is, we want to know whether the limiting current is raised or lowered by migration relative to the purely diffusion-limited current, or whether a, is larger or smaller than unity. It is evident that a, will be larger than unity when migration and diffusion are in the same direction. This is found in four cases for cations that are reactants in a cathodic reaction (as in the example above) or products in an anodic reaction, and for anions that are reactants in an anodic reaction or products in a cathodic reaction. In the other four cases (for cations that are reactants in an anodic or products in a cathodic reaction, and for anions that are reactants in a cathodic or products in an anodic reaction), we have a, < 1, a typical example being the cathodic deposition of metals from complex anions. 

Returning to the survival probability, in Fig. 57, the kinetic theory and diffusion equation [cf. eqn. (132)] predictions are compared. Three values of the activation rate coefficient are used, being 0.5, 1.0 and 2.0 times the Smoluehowski rate coefficient for a purely diffusion-limited homogeneous reaction, 4ttoabD. With a diffusion coefficient of 5x 10 9 m2 s1 and encounter distance of 0.5 nm, significant differences are noted between the kinetic theory and diffusion equation approaches [286]. In all cases, the diffusion equation leads to a faster rate of reaction. In their measurements of the recombination rate of iodine atoms in hydrocarbon solvents, Langhoff et al. [293] have noted that the diffusion equation analysis consistently predicts a faster rate of iodine atom recombination than is actually measured. Thus there is already some experimental support for the value of the kinetic theory approach compared with the diffusion equation analysis. Further developments cannot fail to be exciting. 

In spite of these theoretical results, dendritic-type forms for solid-state precipitation processes are the exception rather than the rule. This may happen because the theory is for pure diffusion-limited growth. Interface-limited growth tends to be stabilizing because the composition gradient close to a growing precipitate is less steep when the reaction is partly interface-limited. Thus, G is smaller, and Eq. 20.73 shows that this is a stabilizing effect. This and several other possible explanations for the paucity of observations for unstable growth forms in solid-state... 

Deciding on how many terms to keep is sometimes a difficult problem. If we keep only the first term (namely unity) in the expansion for each of the three exponentials, then we regain the pure diffusion limit given by eqn. (83). Keeping, in addition, the next term in the first two exponentials leads to... 

There is an extreme case that of very strong adsorption (b is large), leading to the approximate condition cq 0 (all t). This is just like the electrochemical purely diffusion limited potential step case, for which we have the solution G(T), (2.44) and (2.26). G can now be inserted into (10.1), and simple integration then gives ... 

Some elaborations are given In fig. 3.22. Figure 3.22a Illustrates, in line with [3.6.25], that for two capacitances In series the lower one plays the greater role in determining the overall value. For C o the purely diffuse limit is recovered, identical to the 10 M curve of fig. 3.5. Integration gives the [Pg.309]

Thus, y essentially quantifies the extent to which receptor/ligand association is rate-limited by the reaction step. As y approaches 0, association is severely reaction-limited, while as y nears 1, binding is almost purely diffusion-limited. 

To obtain the overall uncoupling rate constant, ku, we can again make use of the capture probability y [see Eq. (35)], such that kc = yk+. Recall that y quantifies the extent to which receptor/ligand association is rate-limited by the reaction step. As y approaches 0, association is severely reaction-limited, while as y nears 1, binding is almost purely diffusion-limited. ku is thus given by... 

Reaction (l) accounts for 0 release from the oxygen evolving system (OES), assuming a first order reaction with a time constant of 1 ms, or 10 ms. Reaction (2) was assumed to be diffusion-limited, taking a distance of 0.5 pm between the OES and the oxidase. A first order reaction with a time constant of 2 ms was taken for (3). This treatment is obviously a crude approximation, for a number of reasons such as the assumption of a fixed OES-oxidase distance, of a purely diffusion-limited reaction (2), the disregard for other redox centers (CuA and CuB) and further transfer steps (second reduction of a3, turnover of Cyt c, etc..). Nevertheless, this model accounts qualitatively for several features of the experimental data, such as the absence of a detectable lag in the oxidation, or the presence of such a lag for Cyt c. 

From the a-values, it is clear that the transport regime strongly depends on the solvent. The transport varies from purely diffusion limited to mainly kinetically limited. Both transport parameters and k have been correlated to the viscosity r and polarity e, (Kirkwood function) of the solvent, respectively. 

Tracer Diffusivity Tracer diffusivity, denoted by D g is related to both mutual and self-diffusivity. It is evaluated in the presence of a second component B, again using a tagged isotope of the first component. In the dilute range, tagging A merely provides a convenient method for indirect composition analysis. As concentration varies, tracer diffusivities approach mutual diffusivities at the dilute limit, and they approach selr-diffusivities at the pure component limit. That is, at the limit of dilute A in B, D g D°g and... 

These results are plausible since according to Sand a two-fold concentration of a component yields a four-fold transition time. Now, these features show, in contrast to the net separation and pure additivity of polarographic waves and their diffusion-limited currents as concentration functions, that in chrono-potentiometry the transition times of components in mixtures are considerably increased by the preceding transition times of any other more reactive component, which complicates considerably the concentration evaluation of chronopotentiograms. 

Fig. 7. Limiting-current curves recorded for various current application rates in pure diffusion at a horizontal cathode facing downward. [From Hickman (H3).]...

An exothermic first-order reaction A—h B is conducted in an FBCR, operating adiabatically and isobarically. The bed has a radius of 1.25 m and is 4 m long. The feed contains pure A at a concentration of 2.0 mol m-3, and flowing at q = 39.3 m3 s 1. The reaction may be diffusion limited assume that the relationship between r) and is 7] = (tanh The diffusivity is proportional to Tia, and Le for the particles is 0.50 mm. Determine the fractional conversion of A and the temperature at the bed outlet. How would your answer change, if (a) diffusion limitations were ignored, and (b) a constant effectiveness factor, based on inlet conditions, was assumed. 

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. ... 

It is seen from Eq. (7237) that the current density i is always greater than in the case of pure diffusion [Eq. (7.202)], in which case tA = 0 and Eq. (7.237) reduces to (7.202). Similarly, the limiting current density must be greater for migration plus diffusion than for pure diffusion [Eq. (7.206)] and is given by... 

Over long times, this displays the limiting form t 3/2 characteristic of the diffusive recombination of radicals. The various forms of h(t) which have been developed are shown in Fig. 40. There are significant differences between these forms and, in particular, the form of h(t) at short times must be 0(t+") where n > — 1. The partially reflecting form [eqn. (194)] is satisfactory as its limiting short-time dependence is U1/2, so too is the Noyes random flights form of h(t), though its theoretical justification is limited. The purely diffusive form of h(f), eqn. (193), is an unnecessary contrivance. 

Both Felderhof and Deutch [25] and Pagistas and Kapral [37] find a correction to the rate coefficient (and kernel) which has a leading term of the form (ft )3 (3c)1/2 (ft + An 0ABD) 3 in the limit of slow back reactions. The higher-order correction terms differ. For the case of a purely diffusive reaction, ft - ft -+ 0, Pagistas and Kapral s expression reduces to a rate coefficient... 

In general, for each value of 2, there must be a different value of the eigenvalue T. Figure 5.11 illustrates this relationship for the radially inward flow. The two limiting situations (i.e., purely diffusive flow and purely convective flow) can be seen in the functional form of T ( 2). At high 2, the value of T may be fit from the solutions asT = —0.062(— 2)0 911. At low values, T = -1.9. 

This is of the same form as Equation 30, but involves the mixed diffusion coefficient, Jci9, instead of the thermal conductivity of the mixture. However, as seen from the kinetic theory of gases, the thermal conductivity is proportional to the diffusion coefficient. Therefore agreement of experimental results with either Equation 30 or 53a is not an adequate criterion for distinguishing between first explosion limits in branching chain reactions and purely thermal limits. It has been reported (52), that, empirically,... 

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