Diffusion shape influence

Thiele(I4>, who predicted how in-pore diffusion would influence chemical reaction rates, employed a geometric model with isotropic properties. Both the effective diffusivity and the effective thermal conductivity are independent of position for such a model. Although idealised geometric shapes are used to depict the situation within a particle such models, as we shall see later, are quite good approximations to practical catalyst pellets. 

Figure 5. Effectiveness factor rj as a function of the Thiele modulus for different pellet shapes. Influence of intraparticle diffusion on the effective reaction rate (isothermal, first order, irreversible reaction).

If the diffusing bound particle is composed of many molecules, as, for example, in the case of a rigid macroscopic particle, rotational diffusion may influence the shape of the Mossbauer or neutron scattering spectra. Ofer et al. (1984) treat the case in which the particle participates in both bound translational and free rotational diffusion. The Mossbauer spectrum in this case can again be represented by a sum of Lorentzian lines. In the... 

Magnitudes of n have been empirically established for those kinetic expressions which have found most extensive application e.g. values of n for diffusion-limited equations are usually between 0.53 and 0.58, for the contracting area and volume relations are 1.08 and 1.04, respectively and for the Avrami—Erofe ev equation [eqn. (6)] are 2.00, 3.00 etc. The most significant problem in the use of this approach is in making an accurate allowance for any error in the measured induction period since variations in t [i.e. (f + f0)] can introduce large influences upon the initial shape of the plot. Care is needed in estimating the time required for the sample to reach reaction temperature, particularly in deceleratory reactions, and in considering the influences of an induction period and/or an initial preliminary reaction. 

The retarding influence of the product barrier in many solid—solid interactions is a rate-controlling factor that is not usually apparent in the decompositions of single solids. However, even where diffusion control operates, this is often in addition to, and in conjunction with, geometric factors (i.e. changes in reaction interfacial area with a) and kinetic equations based on contributions from both sources are discussed in Chap. 3, Sect. 3.3. As in the decompositions of single solids, reaction rate coefficients (and the shapes of a—time curves) for solid + solid reactions are sensitive to sizes, shapes and, here, also on the relative dispositions of the components of the reactant mixture. Inevitably as the number of different crystalline components present initially is increased, the number of variables requiring specification to define the reactant completely rises the parameters concerned are mentioned in Table 17. 

Empedocles, S. A. and Bawendi, M. G. (1999) Influence of spectral diffusion on the line shapes of single CdSe nanocrystallite quantum dots. J. Phys. Chem. B, 103, 1826-1830. 

Principles and Characteristics Supercritical fluid extraction uses the principles of traditional LSE. Recently SFE has become a much studied means of analytical sample preparation, particularly for the removal of analytes of interest from solid matrices prior to chromatography. SFE has also been evaluated for its potential for extraction of in-polymer additives. In SFE three interrelated factors, solubility, diffusion and matrix, influence recovery. For successful extraction, the solute must be sufficiently soluble in the SCF. The timescale for diffusion/transport depends on the shape and dimensions of the matrix particles. Mass transfer from the polymer surface to the SCF extractant is very fast because of the high diffusivity in SCFs and the layer of stagnant SCF around the solid particles is very thin. Therefore, the rate-limiting step in SFE is either... 

A high diffusion coefficient increases the rate of diffusion, all else being the same. The diffusion coefficient is determined in part by molecular size and shape. Small molecules tend to have high diffusion coefficients, which is one reason why formaldehyde penetrates faster than glutaraldehyde. In addition, interactions between the chemical and its environment will influence the diffusion coefficient. Thus, if the chemical hydrogen bonds to the water around it, the diffusion coefficient will be lower and the rate of diffusion will be reduced. 

The coefficients are defined for infinitely dilute solution of solute in the solvent L. However, they are assumed to be valid even for concentrations of solute of 5 to 10 mol.%. The relationships are available for pure solvent, and could be used for mixture of solvents composed of molecules of close size and shape. They all refer to the solvent viscosity which can be estimated or measured. Pressure has a negligible influence on liquid viscosity, which decreases with temperature. As a consequence, pressure has a weak influence on liquid diffusion coefficient conversely, diffusivity increases significantly with temperature (Table 45.4). For mixtures of liquids, an averaged value for the viscosity should be employed. 

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