Factors influencing diffusion constants

Diffusion is a physical process that involves the random motion of molecules as they collide with other molecules (Brownian motion) and, on a macroscopic scale, move from one part of a system to another. The average distance that molecules move per unit time is described by a physical constant called the diffusion coefficient, D (in units of mm2/s). In pure water, molecules diffuse at a rate of approximately 3xl0"3 mm2 s 1 at 37°C. The factors influencing diffusion in a solution (or self-diffusion in a pure liquid) are molecular weight, intermolecular... 

In this exercise we shall estimate the influence of transport limitations when testing an ammonia catalyst such as that described in Exercise 5.1 by estimating the effectiveness factor e. We are aware that the radius of the catalyst particles is essential so the fused and reduced catalyst is crushed into small particles. A fraction with a narrow distribution of = 0.2 mm is used for the experiment. We shall assume that the particles are ideally spherical. The effective diffusion constant is not easily accessible but we assume that it is approximately a factor of 100 lower than the free diffusion, which is in the proximity of 0.4 cm s . A test is then made with a stoichiometric mixture of N2/H2 at 4 bar under the assumption that the process is far from equilibrium and first order in nitrogen. The reaction is planned to run at 600 K, and from fundamental studies on a single crystal the TOP is roughly 0.05 per iron atom in the surface. From Exercise 5.1 we utilize that 1 g of reduced catalyst has a volume of 0.2 cm g , that the pore volume constitutes 0.1 cm g and that the total surface area, which we will assume is the pore area, is 29 m g , and that of this is the 18 m g- is the pure iron Fe(lOO) surface. Note that there is some dispute as to which are the active sites on iron (a dispute that we disregard here). 

An interesting aspect of Eq. (8) was discussed by Jumars et al. (1993). The only factor directly influenced by temperature is the diffusion constant D, which is a function of water viscosity. The temperature dependence of this physical process is less than the Q10 of 2.7-3 normally found for biological processes. In our expression for BCD [Eq. (2)], however, such an argument would apply to both aB in the numerator and aA in the denominator the effect on affinity constants would therefore be expected to cancel out. From the preceding arguments, clearance rates and affinity constants... 

The proportionality factor D characterising the medium is called the diffusion constant. It is assumed that each particle, i.e. each parcel of the fluid, is moving imder the influence of the average field from the rest of the fluid. Pick s law is then combined with the continuity equation similar to (3.51)... 

There are several variants of the DCR theory differing from one another by the way in which account is taken of the physical factors influenced by the diffusion control on the description of elementary reactions rate. As a rule, the main factor influenced is the bimolecular chain termination process. The constant rate of chain termination is considered as a fnnction of the macroradical s mobility, their length [9-14], free volume [12,15-17] or characteristic viscosity of monomer-polymeric system. However, with the aim of explaining the auto deacceleration stage, the efficiency of initiation and constants of rate chain propagation are also considered to be functions of the macroradical s mobility [12,15,18]. 

Rouse-like behavior is not in fact observed in dilute solutions, for which it is necessary to take into account the influence of the chain on the motion of the solvent, and deviations from Gaussian statistics arising from polymer-solvent interactions [17, 18]. These factors are incorporated in the Zimm model, which predicts the diffusion constant to be proportional to N, for example, where v depends on the solvent quality, in better agreement with experimental data [4,14]. Indeed, although it was first proposed for isolated chains, the Rouse model turns out to be more appropriate to polymer melts, where flexible linear chain conformations are approximately Gaussian and hydrodynamic interactions are relatively unimportant [4, 14-16]. 

In the study of dielectric relaxation, temperature is an important variable, and it is observed that relaxation times decrease as the temperature increases. In Debye s model for the rotational diffusion of dipoles, the temperature dependence of the relaxation is determined by the diffusion constant or microscopic viscosity. For liquid crystals the nematic ordering potential contributes to rotational relaxation, and the temperature dependence of the order parameter influences the retardation factors. If rotational diffusion is an activated process, then it is appropriate to use an Arrhenius equation for the relaxation times ... 

Figure 10 shows that Tj is a unique function of the Thiele modulus. When the modulus ( ) is small (- SdSl), the effectiveness factor is unity, which means that there is no effect of mass transport on the rate of the catalytic reaction. When ( ) is greater than about 1, the effectiveness factor is less than unity and the reaction rate is influenced by mass transport in the pores. When the modulus is large (- 10), the effectiveness factor is inversely proportional to the modulus, and the reaction rate (eq. 19) is proportional to k ( ), which, from the definition of ( ), implies that the rate and the observed reaction rate constant are proportional to (1 /R)(f9This result shows that both the rate constant, ie, a measure of the intrinsic activity of the catalyst, and the effective diffusion coefficient, ie, a measure of the resistance to transport of the reactant offered by the pore stmcture, influence the rate. It is not appropriate to say that the reaction is diffusion controlled it depends on both the diffusion and the chemical kinetics. In contrast, as shown by equation 3, a reaction in solution can be diffusion controlled, depending on D but not on k. 

The constant 607 is a combination of natural constants, including the Faraday constant it is slightly temperature-dependent and the value 607 is for 25 °C. The IlkoviC equation is important because it accounts quantitatively for the many factors which influence the diffusion current in particular, the linear dependence of the diffusion current upon n and C. Thus, with all the other factors remaining constant, the diffusion current is directly proportional to the concentration of the electro-active material — this is of great importance in quantitative polarographic analysis. 

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