Simple diffusion, criteria

Madon and Boudart propose a simple experimental criterion for the absence of artifacts in the measurement of rates of heterogeneous catalytic reactions [R. J. Madon and M. Boudart, Ind. Eng. Chem. Fundam., 21 (1982) 438]. The experiment involves making rate measurements on catalysts in which the concentration of active material has been purposely changed. In the absence of artifacts from transport limitations, the reaction rate is directly proportional to the concentration of active material. In other words, the intrinsic turnover frequency should be independent of the concentration of active material in a catalyst. One way of varying the concentration of active material in a catalyst pellet is to mix inert particles together with active catalyst particles and then pelletize the mixture. Of course, the diffusional characteristics of the inert particles must be the same as the catalyst particles, and the initial particles in the mixture must be much smaller than the final pellet size. If the diluted catalyst pellets contain 50 percent inert powder, then the observed reaction rate should be 50 percent of the rate observed over the undiluted pellets. An intriguing aspect of this experiment is that measurement of the number of active catalytic sites is not involved with this test. However, care should be exercised when the dilution method is used with catalysts having a bimodal pore size distribution. Internal diffusion in the micropores may be important for both the diluted and undiluted catalysts. 

For many laboratoiy studies, a suitable reactor is a cell with independent agitation of each phase and an undisturbed interface of known area, like the item shown in Fig. 23-29d, Whether a rate process is controlled by a mass-transfer rate or a chemical reaction rate sometimes can be identified by simple parameters. When agitation is sufficient to produce a homogeneous dispersion and the rate varies with further increases of agitation, mass-transfer rates are likely to be significant. The effect of change in temperature is a major criterion-, a rise of 10°C (18°F) normally raises the rate of a chemical reaction by a factor of 2 to 3, but the mass-transfer rate by much less. There may be instances, however, where the combined effect on chemical equilibrium, diffusivity, viscosity, and surface tension also may give a comparable enhancement. 

The Merrill and Hamrin criterion was derived for a first-order reaction. It should apply reasonably well to other simple reactions, but reactions exist that are quite sensitive to diffusion. Examples include the decomposition of free-radical initiators where a few initial events can cause a large number of propagation reactions, and coupling or cross-linking reactions where a few events can have a large effect on product properties. 

In assessing whether a reactor is influenced by intraparticle mass transfer effects WeiSZ and Prater 24 developed a criterion for isothermal reactions based upon the observation that the effectiveness factor approaches unity when the generalised Thiele modulus is of the order of unity. It has been showneffectiveness factor for all catalyst geometries and reaction orders (except zero order) tends to unity when the generalised Thiele modulus falls below a value of one. Since tj is about unity when 0 < ll for zero-order reactions, a quite general criterion for diffusion control of simple isothermal reactions not affected by product inhibition is < 1. Since the Thiele modulus (see equation 3.19) contains the specific rate constant for chemical reaction, which is often unknown, a more useful criterion is obtained by substituting l v/CAm (for a first-order reaction) for k to give ... 

Potential at half-height — (in voltammetry) This is a diagnostic criterion in -> linear scan voltammetry. The potential at half-height Ep/2 is the potential at which the current is equal to one-half of the peak current fp Ep/2 = h (/=/p/2)- I he first of two potentials at half-height, the one that precedes the peak potential (Ep) is considered only. If a simple electrode reaction is reversible (- reversibility) and controlled by the planar, semi-infinite - diffusion, the absolute value of the difference between Ep/2 and Ep is equal to 56.6/n mV and independent of the - scan rate. If the -> electrode reaction of dissolved reactant is totally irreversible (-> reversibility), the difference Ep/2 - Ep is equal to 47.7/an mV for the cathodic process and -47.7/(l - a)n mV for the anodic process. 

The main difference between the cases discussed in this section on selectivity and the previously discussed case of a simple polystep reaction resides in the fact that here the single component which generates the species that becomes the intermediate in the polyfunctional composite can itself generate a distinct product species with appreciable yield. Since the coupling between the Y- and the X-system occurs in any event through mass-transport of intermediates between X-sites and T-sites, the diffusion criteria already discussed must apply or the kinetic schemes which accomplish interception or selectivity control will not be physically and effectively accomplished. The criterion, of formula (15) should be satisfied. ... 

We introduce a simple model to investigate and calculate a diffusion coefficient as a basic quantity describing transport in Section II, and then we visualize resonances to detect the structure of the Arnold web and overlapped resonances in Section III. With the aid of this representation, to clarify the relevance of Arnold diffusion and diffusion induced by resonance overlap to global transport in the phase space, we compute transition diagrams in the frequency space in Section IV. In Section V, we extend the resonance overlap criterion to multidimensional systems to identify the pathway for fast transport, and in Section VI we revisit the diffusion coefficient to ensure fast transport affecting the global diffusion. A brief summary is given in Section VII. 

A simple yet valuable criterion for candidate membrane material selection is the characteristic membrane thickness [41 ]. In the theory, the transport equations for diffusion in the solid, and for the surface exchange are linearized. It should therefore strictly be used when small Po -gradients are imposed across the membrane. In the next section, methods for measuring are briefly discussed. 

The results of numerous investigations on the kinetics of sorption of pure substances in zeolites have since then appeared in the literature and the field has been reviewed recently by Walker et al. 42). The total uptake or loss of sorbate in a large number of crystallites is commonly observed, and it is generally assumed that the rate of these processes is controlled by diffusion in the solid. Variable diffusion coefficients were sometimes observed by this method, and it appears possible that other processes than diffusion in the solid had some influence on the rate in these cases. The apparent diffusivity will depend only on concentration (besides temperature) if the migration of sorbate particles in the solid is rate controlling. A simple criterion whether this condition exists can be obtained by measuring sorption or desorption rates repeatedly for various initial concentrations and boundary conditions, as described by Diinwald and Wagner 43). 

A simple criterion whether the observed reaction rate in solid catalysts is not limited by mass transport has been derived by Weisz 61). If this criterion is not fulfilled, one can obtain the dimensiordess moduli 0 and 99 from the observed rate and known or estimated values of the respective diffusion coefficient, as described in the literature (57, 59, 60). In the case of a pelleted zeolite, these methods have first to be applied to the bulk pellet and subsequently to the individual crystals. The procedure requires that the concentration (ca)o at the phase boundary of the zeolite crystals and the binary diffusivity at steady state are known must be estimated from transient measurements in the same... 

For the practical use a criterion is needed to decide whether it is justified that such a kinetic equation is applied. This criterion must ensure that the kinetic constants are independent of the parameters of the adsorption process, mainly the surfactant concentration and the monolayer coverage. Experimental data for various surfactants show that for surface lifetimes shorter than 20 ms the reduced desorption rate constant k , = kj /T is nearly constant and of the order of 100 s [16]. This important result allows to define a simple criterion for a non-diffusional adsorption mechanism by comparing the characteristic times of diffusion and adsorption kinetics according to the model of Eq. (4.15). The condition for mixed or kinetic controlled... 

A simple criterion for the effectiveness of the interfacial film area was proposed by Pohorecki [49]. The criterion is based on the characteristic diffusion time of the species in the film and the film contact time 0 For physical... 

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