Diffusion of species

For phase-boundary controlled reactions, the situation differs somewhat. Diffusion of species is fast but the reaction is slow so that the dlfiusing species pile up. That is, the reaction to rearrange the structure is slow in relation to the arrival of the diffusing ions or atoms. TTius, a phaseboundary (difference in structure) focus exists which controls the overall rate of solid state reaction. This rate may be described by ... 

A discussion of the charge transfer reaction on the polymer-modified electrode should consider not only the interaction of the mediator with the electrode and a solution species (as with chemically modified electrodes), but also the transport processes across the film. Let us assume that a solution species S reacts with the mediator Red/Ox couple as depicted in Fig. 5.32. Besides the simple charge transfer reaction with the mediator at the interface film/solution, we have also to include diffusion of species S in the polymer film (the diffusion coefficient DSp, which is usually much lower than in solution), and also charge propagation via immobilized redox centres in the film. This can formally be described by a diffusion coefficient Dp which is dependent on the concentration of the redox sites and their mutual distance (cf. Eq. (2.6.33). 

Am pseudo-binary diffusivity of species A in a multicomponent gas mixture... 

External reflectance. As was seen above, in this approach the strong electrolyte absorption is minimised by employing a thin layer configuration and the possible severe restrictions imposed on the diffusion of species to and from the electrode in such an arrangement can lead to some difficulties. 

The thin-layer thickness calculated in this manner may well have some serious errors associated with it due to sample purity, errors in the weighing out of the solution, the diffusion of species near the thin layer into it within the timescale of the experiment, etc. 

This system was subsequently investigated by Christensen et at. (1990) also using in situ FTIR, who observed identical product features (see Figure 3.48). In order first to compare directly the IR spectrum of oxalate generated in situ, the authors took advantage of the surface reactivity of Pt and the poor diffusion of species to and from the thin layer. Thus, a solution of oxalic acid in the electrolyte was placed in the spectroelectrochemical cell, the potential of the platinum working electrode stepped to successively lower values and spectra taken at each step. The spectra were all normalised to the reference spectrum collected at the base potential of 0 V vs. SCE. As a result of the deprotonation of adventitious water ... 

One of the most important parts of the FTIR technique is the movable electrode in order to allow proper diffusion of species to and fro... 

Figure 1. Outline of the uptake model showing the spherical diffusion of species M through the medium towards two different sites where adsorption is followed by internalisation. The radius of the organism is taken as ro...

The dissolution rate and mechanism of dissolution for the soluble pigment particles. Related to that, the characteristics of the porous layer subsequently formed and its influence on the diffusion of species in and out of the paint. 

Polymers dynamics of polymer chains microviscosity free volume orientation of chains in stretched samples miscibility phase separation diffusion of species through polymer networks end-to-end macrocyclization dynamics monitoring of polymerization degradation... 

As implied in the previous section, the Russian investigators Zeldovich, Frank-Kamenetskii, and Semenov derived an expression for the laminar flame speed by an important extension of the very simplified Mallard-Le Chatelier approach. Their basic equation included diffusion of species as well as heat. Since their initial insight was that flame propagation was fundamentally a thermal mechanism, they were not concerned with the diffusion of radicals and its effect on the reaction rate. They were concerned with the energy transported by the diffusion of species. 

A flame is quenched in a tube when the two mechanisms that permit flame propagation—diffusion of species and of heat—are affected. Tube walls extract heat the smaller the tube, the greater is the surface area to volume ratio within the tube and hence the greater is the volumetric heat loss. Similarly, the smaller the tube, the greater the number of collisions of the active radical species that are destroyed. Since the condition and the material composition of the tube wall affect the rate of destruction of the active species [5], a specific analytical determination of the quenching distance is not feasible. 

Defining mA as the mass fraction of A, one obtains the following proper form for the diffusion of species A in terms of mass fraction ... 

Here q denotes density, m represents collision-dynamic mass (which may depart substantially from the molecular mass), Vf is the specific critical volume for diffusion of species i, and g is an overlap factor 0.5 g 1. The fractional free volumes fj... 

In the case of the mixture-averaged formulation, we desire to calculate Dkm (i.e., a diffusion coefficient for diffusion of species k into a mixture of other gases). This can result in savings in computational cost. At the same time the results in this section are approximations, although in some cases good ones. 

The self-diffusivity of species 1 in a chemically homogeneous solution of concentration ci, corresponding to D in Eq. 3.5, can be compared with the intrinsic diffusivity of the same species in a chemically inhomogeneous solution at the same concentration, corresponding to D in Eq. 3.12. Typically, in addition to the approximation of a concentration-independent average site volume (fi), it is reasonable to assume that the coupling (off-diagonal) terms, 12/02 and Lm/c- in Eqs. 3.5 and 3.12, are small compared with the direct term L fc. In this approximation,... 

The preceding analysis provides a powerful method for determining the diffusivities of species that produce an anelastic relaxation, such as the split-dumbbell interstitial point defects. A torsional pendulum can be used to find the frequency, u>p, corresponding to the Debye peak. The relaxation time is then calculated using the relation r = 1/ojp, and the diffusivity is obtained from the known relationships among the relaxation time, the jump frequency, and the diffusivity. For the split-dumbbell interstitials, the relaxation time is related to the jump frequency by Eq. 8.63, and the expression for the diffusivity (i.e., D = ra2/12), is derived in Exercise 8.6. Therefore, D = a2/18r. This method has been used to determine the diffusivities of a wide variety of interstitial species, particularly at low temperatures, where the jump frequency is low but still measurable through use of a torsion pendulum. A particularly important example is the determination of the diffusivity of C in b.c.c. Fe, which is taken up in Exercise 8.22. 

Such a high reaction rate strongly suggested the potential for mass transport problems. Indeed, a turn-over frequency on the order of 1 s 1 is considered appropriate for the purpose of mechanistic studies normally conducted in the gas phase (ref. 5). At higher rates, various complications including intraparticle diffusion problems, often arise. The situation is even more severe in the liquid phase where the bulk diffusivity of species is considerably reduced. A... 

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