Reaction-diffusion sequence

In like manner, if we substitute the complex p from Eq. (44) into Eq. (62), we obtain the circuit shown in Figure 1.1.1b, appropriate for a reaction-diffusion sequence without an intermediate adsorption stage. Here is given by Eq. (64) as before and... 

Reaction-diffusion systems provide a means to subdivide successively a domain at a sequence of critical parameter values due to size, shape, diffusion constants, or other parameters. The chemical patterns that arise are the eigenfunctions of the Laplacian operator on that geometry. The succession of eigenfunctions on geometries close to the wing, leg, haltere, and genital discs yield sequential nodal lines reasonably similar to the observed sequence and symmetries and geometries of the observed com-... 

From a theoretical viewpoint, predicting the sequence of layer occurrence at the A-B interface would present no difficulties if the values of all the chemical constants entering a system of differential equations like (3.27) were known. For any multiphase binary system A-B, these values are determined by the physical-chemical properties of the elements A and B and their compounds. With their dependence on those properties established, the sequence of formation of compound layers would readily be predicted from the system of equations (3.27) or similar. Unfortunately, the theory of reaction diffusion has not yet reached this stage of its development. 

At least with intermetallics, the effect of melting points and atomic radii on the sequence of occurrence of compound layers at the A-B interface seems to be more or less straightforward. On the contrary, the influence of the crystal structure of the compounds is rather obscure. Probably, those with less symmetrical and loosely packed structures may be expected to form first under highly non-equilibrium and stressed conditions usually encountered in reaction-diffusion experiments. 

While it is often possible to demonstrate that a surface process is rate limiting, identification of the step concerned is not always so readily achieved (as in heterogeneous catalysis which involve comparable mechanistic steps). Reaction rates are determined by reactant areas and are slow compared with the rate of diffusive transport of material to the appropriate boundaries. Surface limited reactions are also sensitive to the ease of removal of volatile products, which may be hampered by the presence of an inert gas. Readsorption may influence the effective concentrations of participating surface intermediates. As in catalytic heterogeneous reactions, the sequence of changes which precede product evolution may involve several interlinked steps, and the parameters which determine the overall progress of reaction are not always readily identified. 

Reduction metabolic transition time. This relates to the temporality of the lag phase during the transition of a metabolic process from one steady-state to another (see Figure 3). It is a function of metabolite diffusion, enzyme density, and kinetic parameters. Close spatial proximity of sequentially-acting enzymes in organized microcompartments can sharply reduce the temporality associated with reaction-diffusion events for sequences of enzyme reactions in dilute solution. Accordingly, the flux condition in metabolic pathways can rapidly switch in response to external stimuli. 

Fig. 7.8. Schematic diagram showing development of concentration waves in the Belousov reaction, (a) Sequence A is triggered in central domain diffusion flows are indicated, (b) At a later time sequence B dominates in central region rings of concentration variations propagate outward.

The sequence of events in a surface-catalyzed reaction comprises (1) diffusion of reactants to the surface (usually considered to be fast) (2) adsorption of the reactants on the surface (slow if activated) (3) surface diffusion of reactants to active sites (if the adsorption is mobile) (4) reaction of the adsorbed species (often rate-determining) (5) desorption of the reaction products (often slow) and (6) diffusion of the products away from the surface. Processes 1 and 6 may be rate-determining where one is dealing with a porous catalyst [197]. The situation is illustrated in Fig. XVIII-22 (see also Ref. 198 notice in the figure the variety of processes that may be present). 

The generalized transport equation, equation 17, can be dissected into terms describing bulk flow (term 2), turbulent diffusion (term 3) and other processes, eg, sources or chemical reactions (term 4), each having an impact on the time evolution of the transported property. In many systems, such as urban smog, the processes have very different time scales and can be viewed as being relatively independent over a short time period, allowing the equation to be "spht" into separate operators. This greatly shortens solution times (74). The solution sequence is... 

The automated method differs from the ICSH method chiefly in that oxidation and ligation of heme iron occur after the hemes have been released from globin. Therefore, ferricyanide and cyanide need not diffuse into the hemoglobin and methemoglobin, respectively. Because diffusion is rate-limiting in this reaction sequence, the overall reaction time is reduced from approximately three minutes for the manual method to 3 —15 seconds for the automated method. Reaction sequences in the Coulter S + II and the Technicon H 1 and H 2 are similar. Moreover, similar reactions are used in the other Coulter systems and in the TOA and Unipath instmments. 

The detailed mechanism of battery electrode reactions often involves a series of chemical and electrochemical or charge-transfer steps. Electrode reaction sequences can also include diffusion steps on the electrode surface. Because of the high activation energy required to transfer two electrons at one time, the charge-transfer reactions are beheved to occur by a series of one electron-transfer steps illustrated by the reactions of the 2inc electrode in strongly alkaline medium (41). 

Kinetic studies have shown that the enolate and phosphorus nucleophiles all react at about the same rate. This suggests that the only step directly involving the nucleophile (step 2 of the propagation sequence) occurs at essentially the diffusion-controlled rate so that there is little selectivity among the individual nucleophiles. The synthetic potential of the reaction lies in the fact that other substituents which activate the halide to substitution are not required in this reaction, in contrast to aromatic nucleophilic substitution which proceeds by an addition-elimination mechanism (see Seetion 10.5). 

Several authors " have suggested that in some systems voids, far from acting as diffusion barriers, may actually assist transport by permitting a dissociation-recombination mechanism. The presence of elements which could give rise to carrier molecules, e.g. carbon or hydrogen , and thus to the behaviour illustrated in Fig. 1.87, would particularly favour this mechanism. The oxidant side of the pore functions as a sink for vacancies diffusing from the oxide/gas interface by a reaction which yields gas of sufficiently high chemical potential to oxidise the metal side of the pore. The vacancies created by this reaction then travel to the metal/oxide interface where they are accommodated by plastic flow, or they may form additional voids by the mechanisms already discussed. The reaction sequence at the various interfaces (Fig. 1.87b) for the oxidation of iron (prior to the formation of Fe Oj) would be... 

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