Contracting sphere model

The mathematical treatment that describes the nucleation mechanism of reduction can be found in Ref. [2]. The final integral equation rate, Eq.5.13), represents the rising part of the a-vs-t curve before the infiection point (Fig. 5.8 left) and shows that the rate of nucleation is proportional to (time)  

In fhe cafalysf preparation, not only the choice of the active phase precursor is cra-cial, the method of catalyst preparation is decisive, too, for obtaining good dispersion of the active phase. Active phase can be deposited on supports by impregnation, ion-exchange, adsorption, etc. Once selected the nature of support and active phase, the observed differences in dispersion should only be due to the method of preparation. Dispersed iron oxide catalysts (FeOx) have received much attention because their potentiality for many applications in environmental catalysis (N2O decomposition and reduction) and in fine chemical industry (Friedel-Crafts, isomerisations, etc.). For most applications, high dispersion of the metal centres is desirable to enhance the activity-selectivity pattern of the catalysts. 

Iron oxide is a well reducible phase, the comparison of reduction profiles of supported Fe-catalysts may give information on the dispersion state of iron. In Fig. 5.10, two TPR spectra of two dispersed FeOx catalysts on zirconia prepared by conventional impregnation and adsorption methods are shown [17]. Reduction of pure hematite (Fe203) by hydrogen is a complex event that can proceed in different steps 

A very different and complex situation occurs when supported iron phases are concerned. The TPR profile of the Fe0x/Zr02 catalysts prepared by impregnation presents a convoluted curve (Fig. 5.10, left). The complexity of the TPR profile reveals 

In alternative, AOS can be calculated from the balanced redox reaction here below written for a trivalent metal species, M +, which can be reduced to bivalent or univalent or zerovalent metal species  

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If the initiation step, the activation of H2, is fast, as may be the case on noble metal oxides or highly defective oxide surfaces, the shrinking core or contracting sphere model applies (see Figure 2.3). The essence of this model is that nuclei of reduced metal atoms form rapidly over the entire surface of the particle and grow into a shell of reduced metal. Further reduction is limited by the transport of lattice oxygen out of the particle. The extent of reduction increases rapidly initially, but slows down as the metal shell grows. 

Figure 2.3 Left, reduction models. In the shrinking core or contracting sphere model the rate of reduction is initially fast and decreases progressively due to diffusion limitations. The nucleation model applies when the initial reaction of the oxide with molecular hydrogen is difficult. Once metal nuclei are available for the dissociation of hydrogen, reduction proceeds at a higher rate until the system comes into the shrinking core regime. Right the reduction rate depends on the concentration of unreduced sample (1-a) as f(a) see Expressions (2-5) and (2-6).

The function /(1 -a) depends on the model that describes the reduction process. The simplest choices would be/fl-ar) = (1 -a) or (1 -af, as in the review of Hurst et al. [1]. More realistic expressions are those for the nucleation and the contracting sphere models of Fig. 2.3 ... 

The rate of decomposition has been measured as a function of C02 pressure by Centnerszwer and Brusz14 and Prodan and Pavlyuchenko.15 The decomposition yields CdO without the formation of intermediates. In vacuum the activation energy is 151 kJ. The rate of decomposition depends on the method of preparation. The kinetics follows the contracting sphere model. 

Since the volume of A1203 is much less than that of the sulfate, it is not surprising that the product is porous and that the kinetics follows the contracting sphere model.6... 

Thermodynamic calculations (Tables 4.18 to 4.21) indicate that BeS04 is unstable above 1000 K. Kinetics of the decomposition have been studied by Johnson and Gallagher.11 The authors use their data to study various kinetic models and find the contracting sphere model most satisfactory. 

A contracting sphere model applies, meaning that the reaction occurs at the oxide-sulfate boundary. Similar results were obtained by Tagawa and Saijo.30 The specific gravity is 3.543.27... 

The kinetics was found to follow the contracting sphere model.56 Two groups827 have reported different densities at 298 K 3.74 and 3.546. 

The kinetics consist of rapid nucleation, followed by a contracting sphere model.17... 

If the rate of reaction with diffusion control is also proportional to the amount of reactant present based on the contracting sphere model, then ... 

In addition, spherical copper particles allow the detailed determination of the kinetics of the reduction process with respect to the nucleation and contracting sphere models of the reduction of transition metals as described in literature [9]. 

Apparently, a significant decrease in the neodymium chloride particle size in the first 4-5 h of the synthesis is not due to the occurrence of the complexation on the particle surface (contracting sphere model), because the yield of the complex in this period is low. Most probably, a decrease in the particle size of the initial NdCl3 is associated with disintegration of the solid phase under the action of mechanical stirring, with the wedging effect produced by solvent molecules solvating the particle surface. This... 

The aim of the analysis of the TPR/TPO data is to derive kinetic parameters relating to the reduction/oxidation process. It is common to observe the reduced/oxidized fraction (o ) as a function of time for various temperatures and pressures/concentrations of reducing/oxidant gas. The reduction/oxidation of a sample with redox property or of part of it (in the case of supported phase system) is a bulk phenomenon and its degree of reduction/oxidation is interpreted in terms of mechanism by which the reaction occurs. Nucleation model or contracting sphere model are the most successful utilized kinetic models [2]. 

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