Role of Diffusion

Kinetic coefficients ki entering equations for the well-stirred systems like (8.1.4) to (8.1.6) are defined, in principle, by mutual diffusive approach of 

When deriving (8.1.7) and (8.1.8), it was assumed that the concentrations riE and nc of initial reactants remain to be homogeneously distributed in space. 

The extended Brusselator [2, 5], Oregonator [5, 10] and other similar systems [4, 7] demonstrate other autowave processes whose distinctive spatial and temporal properties are independent on initial concentrations, boundary conditions and often even on geometrical size of a system. As it was noted by Zhabotinsky [4], Vasiliev, Romanovsky and Yakhno [5], a number of well-documented results obtained in the theory of autowave processes is much less than a number of problems to be solved. In fact, mathematical methods for analytical solution of the autowave equations and for analysis of their stability are practically absent so far. 

The principal role of diffusion in these processes could be established considering rather simple examples [2]. If the kinetic equations for a well-stirred system are able to reproduce self-oscillations (the limit cycle), the extended system could be presented as a set of non-linear oscillators continuously distributed in space. Diffusion acts to conjunct these local oscillations and under certain conditions it can result in the synchronisation of oscillations. Thus, autowave solutions could be interpreted as a result of a weak coupling (conjunction) of local oscillators when they are not synchronised completely. The stationary spatial distributions in an initially homogeneous systems can also arise due to diffusion, which makes homogeneous solutions unstable. 

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Adrian F J 1971 Theory of anomalous electron spin resonance spectra of free radicals in solution. Role of diffusion-controlled separation and reencounter of radical pairs J. Chem. Rhys. 54 3918-23... 

Climb unlocks dislocations from the precipitates which pin them and further slip (or glide ) can then take place (Fig. 19.3). Similar behaviour takes place for pinning by solute, and by other dislocations. After a little glide, of course, the unlocked dislocations bump into the next obstacles, and the whole cycle repeats itself. This explains the progressive, continuous, nature of creep, and the role of diffusion, with diffusion coefficient... 

Satterfield, C.N. and T.K. Sherwood, 1963, The Role of Diffusion in Catalysis, Addison-Wesley. 

In one dimension the truncation of the equations of motion has been worked out in detail [59]. This has allowed an accurate examination of the role of diffusion in desorption, and implications for the Arrhenius analysis in nonequilibrium situations. The largest deviations from the desorption kinetics of a mobile adsorbate obviously occur for an immobile adsorbate... 

Whitmer, A.M., Ramenofsky, A.F., Thomas, J., Thibodeaux, L., Field, S.D. and Miller, B.J. 1989 Stability or instability. The role of diffusion in trace element studies. Archaeological Method and Theory 1 205-273. 

The theory of radiation-induced grafting has received extensive treatment [21,131,132]. The typical steps involved in free-radical polymerization are also applicable to graft polymerization including initiation, propagation, and chain transfer [133]. However, the complicating role of diffusion prevents any simple correlation of individual rate constants to the overall reaction rates. Changes in temperamre, for example, increase the rate of monomer diffusion and monomer... 

Shoup, D. and Szabo, A. (1982). Role of diffusion in ligand binding to macromolecules and cell-bound receptors, Biophys. J., 40, 33-39. 

Para-selectivity for a wide variety of ZSM-5 preparations of comparable activity are shown in Figure 12. These data include results for unmodified HZSM-5 s of varying crystal size as well as chemically modified HZSM-5 s. Since the activity of these catalysts is nearly identical, these data clearly establish the major role of diffusion in the para-xylene content of the xylenes produced in TDP. We have examined in more detail the effect of the concentration of one of these chemical modifiers, MgO. 

An important parameter for observing the role of diffusion is the electrode rotation rate. Figure 5.20 gives an experimental example of... 

Table 18.2 The Role of Diffusion in Pores for First-Order Reactions in Series...

The nature of oxygen transport within minerals in the presence of hydrothermal water and the role of diffusion. Chem. Geol. 53, 197-206. 

Trial A.F. and Spera F.J. (1988) Natural convection boundary layer flows in isothermal ternary systems role of diffusive coupling. Int. J. Heat Transfer 31, 941-955. 

Professor Prigogine showed us the wide variety of phenomena that may appear in a nonlinear reaction-diffusion system kept far from thermodynamic equilibrium. The role of diffusion in these systems is to connect the concentrations in different parts of space. When the process of diffusion is approximated by Fick s law, this coupling is linear in the concentration of the chemicals. 

Numerical simulations illustrating the role of diffusion in the onset of instabilities have been carried out by Hannusse.5 They confirm the modification of macroscopic behavior by local fluctuations that introduce such phenomena as delays or metastable states. 

If kinetic runs over the same catalyst in powder form are available, comparing them with tests over small monolith samples at the same conditions permits also a direct experimental evaluation of the role of diffusion processes in determining the catalytic performances. 

Temperature Dependence of the Activity and Selectivity of Xylene Isomerization over AP Catalyst. Based upon our analysis of the intracrystalline diffusional resistance in AP catalyst, we would expect that when the reaction temperature is increased, the selectivity would shift toward p-xylene since the diffusional effects are increased as the activity increases. A shift in selectivity toward p-xylene as the reaction temperature was increased was observed and is shown in Figure 6. The role of diffusion in changing the selectivity can be seen in the Arrhenius plot of Figure 7. The reaction rate constant for the o-xylene - p-xylene path, fc+3i, goes from an almost negligible value at 300°F to a substantial value at 600°F. Furthermore, the diffusional effects are also demonstrated by the changing... 

To treat the stochastic Lotka and Lotka-Volterra models, we have now to extend the formalism presented in Section 2.2.2, where collective variables-numbers of particles iVA and Vg were used to describe reactions. The point is that this approach neglects local density fluctuations in small element volumes. To incorporate both these fluctuations and their correlations due to diffusive conjunction, we are in position now to reformulate these models in terms of the diffusion-controlled processes - in contrast to the rather primitive birth-death formalism used in Section 2.2.2. It permits also to demonstrate in the non-trivial way a role of diffusion in the autowave processes. The main results of this Chapter are published in [21, 25]. 

Ward, A.F.H. and Tordai, L. 1946. Time dependence of boundary tensions of solutions. 1. The role of diffusion in time effects. J. Chem. Phys. 14 453-461. 

S. Kamat, E. J. Beckman, and A. J. Russell, Role of diffusion in nonaqueous enzymo-logy. 1. Theory, Enz. Microb. Technol. 1992b, 34, 265-271. 

Diffusional mass transfer processes can be essential in complex catalytic reactions. The role of diffusion inside a porous catalyst pellet, its effect on the observed reaction rate, activation energy, etc. (see, for example, ref. 123 and the fundamental work of Aris [124]) have been studied in detail, but so far several studies report only on models accounting for the diffusion of material on the catalyst surface and the surface-to-bulk material exchange. We will describe only some macroscopic models accounting for diffusion (without claiming a thorough analysis of every such model described in the available literature). 

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