Mixing inhomogeneity

Conditions //en time- causes mixing Inhomogenous Homogenous... 

Analogous considerations apply to spatially distributed reacting media where diffusion is tire only mechanism for mixing chemical species. Under equilibrium conditions any inhomogeneity in tire system will be removed by diffusion and tire system will relax to a state where chemical concentrations are unifonn tliroughout tire medium. However, under non-equilibrium conditions chemical patterns can fonn. These patterns may be regular, stationary variations of high and low chemical concentrations in space or may take tire fonn of time-dependent stmctures where chemical concentrations vary in botli space and time witli complex or chaotic fonns. 

The oxidation methods described previously are heterogeneous in nature since they involve chemical reactions between substances located partly in an organic phase and partly in an aqueous phase. Such reactions are usually slow, suffer from mixing problems, and often result in inhomogeneous reaction mixtures. On the other hand, using polar, aprotic solvents to achieve homogeneous solutions increases both cost and procedural difficulties. Recently, a technique that is commonly referred to as phase-transfer catalysis has come into prominence. This technique provides a powerful alternative to the usual methods for conducting these kinds of reactions. 

The present work demonstrates that the mixed oxide catalyst with inhomogeneous nanocrystalline MosOu-type oxide with minor amount of M0O3- and Mo02-type material. Thermal treatment of the catalyst shows a better performance in the formation of the crystals and the catalytic activity. The structural analysis suggests that the catalytic performance of the MoVW- mixed oxide catalyst in the partial oxidation of methanol is related to the formation of the M05O14 t3 e mixed oxide. 

When the two phases separate the distribution of the solvent molecules is inhomogeneous at the interface this gives rise to an additional contribution to the free energy, which Henderson and Schmickler treated in the square gradient approximation [36]. Using simple trial functions, they calculated the density profiles at the interface for a number of system parameters. The results show the same qualitative behavior as those obtained by Monte Carlo simulations for the lattice gas the lower the interfacial tension, the wider is the interfacial region in which the two solvents mix (see Table 3). 

Illustration Short-time behavior in well mixed systems. Consider the initial evolution of the size distribution of an aggregation process for small deviations from monodisperse initial conditions. Assume, as well, that the system is well-mixed so that spatial inhomogeneities may be ignored. Of particular interest is the growth rate of the average cluster size and how the polydispersity scales with the average cluster size. 

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