Diffusivity, bulk combined

Fig. 2a-c. Kinetic zone diagram for the catalysis at redox modified electrodes a. The kinetic zones are characterized by capital letters R control by rate of mediation reaction, S control by rate of subtrate diffusion, E control by electron diffusion rate, combinations are mixed and borderline cases b. The kinetic parameters on the axes are given in the form of characteristic currents i, current due to exchange reaction, ig current due to electron diffusion, iji current due to substrate diffusion c. The signpost on the left indicates how a position in the diagram will move on changing experimental parameters c% bulk concentration of substrate c, Cq catalyst concentration in the film Dj, Dg diffusion coefficients of substrate and electrons k, rate constant of exchange reaction k distribution coefficient of substrate between film and solution d> film thickness (from ref. 

Mass transport in an adsorbent particle can be viewed as a combination of several mechanisms, as shown in Figure 14.8. Macropore and micropore diffnsion are shown and conld be considered as examples of intraparticle diffusion. Bulk flow or conveyance through the particle (e.g., via connected pores) is shown in that figure but, to be significant, requires high porosity or a large... 

In extreme cases, the shrinkage and drop in diffusivity may combine to give a skin, practically impervious to moisture, which encloses the bulk of the material so the interior moisture cannot be removed. This is called casehardening. 

Pulsed field gradient NMR (PFG-NMR) is primarily used for measuring molecular difiusion coefficients in solution. In ILs, diffusion is the process of charge transport via cations or anions activated by internal kinetic energy. Self-diffusion in combination with other bulk properties, such as viscosity, density, and electrical conductivity, provides a thorough understanding of molecular transport in ILs. 

A third approach is suggested by Hugo s formulation of material balances at the limit of bulk diffusion control, described in Section 11.3. Hugo found expressions for the fluxes by combining the stoichiometric conditions and the Stefan-Maxvell relations, and this led to no inconsistencies since there are only n - 1 independent Stefan-Maxwell relations for the n fluxes. An analogous procedure can be followed when the diffusion is of intermediate type, using the dusty gas model equations in the form (5.10) and (5.11). Equations (5.11), which have the following scalar form ... 

The assumption that k values are constant over the entire duration of the reaction breaks down for termination reactions in bulk polymerizations. Here, as in Sec. 5.2, we can consider the termination process—whether by combination or disproportionation to depend on the rates at which polymer molecules can diffuse into (characterized by kj) or out of (characterized by k ) the same solvent cage and the rate at which chemical reaction between them (characterized by kj.) occurs in that cage. In Chap. 5 we saw that two limiting cases of Eq. (5.8) could be readily identified ... 

An interesting finding is a slow increase (f 1/2 0.5 ms), observed in PBS in tetrahydro-furan solution, after the faster decrease (t1/2 30/rs). This indicates that some of the fragments diffuse while still carrying reactive sites and can combine in the bulk. 

Many theoretical embellishments have been made to the basic model of pore diffusion as presented here. Effectiveness factors have been derived for reaction orders other than first and for Hougen and Watson kinetics. These require a numerical solution of Equation (10.3). Shape and tortuosity factors have been introduced to treat pores that have geometries other than the idealized cylinders considered here. The Knudsen diffusivity or a combination of Knudsen and bulk diffusivities has been used for very small pores. While these studies have theoretical importance and may help explain some observations, they are not yet developed well enough for predictive use. Our knowledge of the internal structure of a porous catalyst is still rather rudimentary and imposes a basic limitation on theoretical predictions. We will give a brief account of Knudsen diffusion. 

The physical meaning of the g (ion) potential depends on the accepted model of an ionic double layer. The proposed models correspond to the Gouy-Chapman diffuse layer, with or without allowance for the Stem modification and/or the penetration of small counter-ions above the plane of the ionic heads of the adsorbed large ions. " The experimental data obtained for the adsorption of dodecyl trimethylammonium bromide and sodium dodecyl sulfate strongly support the Haydon and Taylor mode According to this model, there is a considerable space between the ionic heads and the surface boundary between, for instance, water and heptane. The presence in this space of small inorganic ions forms an additional diffuse layer that partly compensates for the diffuse layer potential between the ionic heads and the bulk solution. Thus, the Eq. (31) may be considered as a linear combination of two linear functions, one of which [A% - g (dip)] crosses the zero point of the coordinates (A% and 1/A are equal to zero), and the other has an intercept on the potential axis. This, of course, implies that the orientation of the apparent dipole moments of the long-chain ions is independent of A. 

Gartner s equation can be derived by calculating that part of the photocurrent which comes from the bulk. The concentration p(x) of holes obeys the following equation, which combines the familiar diffusion equation with a source and a loss term ... 

As depicted in Figure 2.8, mass transport of substrate from the bulk water phase takes place through a fluid boundary layer (liquid film) and into a biofilm followed by a combined diffusion and utilization of the substrate in the biofilm. 

The commonest multiple step control mechanism in use is that of diffusion to the surface of the catalyst combined with one of the adsorption or surface reaction steps. Mass transfer by diffusion is proportional to the difference between partial pressures in the bulk of the gas and at the catalyst surface,... 

In the particular case dealt with now (fully labile complexation), due to the linearity of a combined diffusion equation for DmCm + DmlL ml, the flux in equation (65) can still be seen as the sum of the independent diffusional fluxes of metal and complex, each contribution depending on the difference between the surface and bulk concentration value of each species. But equation (66) warns against using just a rescaling factor for the total metal or for the free metal alone. In general, if the diffusion is coupled with some nonlinear process, the resulting flux is not proportional to bulk-to-surface differences, and this complicates the use of mass transfer coefficients (see ref. [II] or Chapter 3 in this volume). 

The degradation profile can be detected by measuring a property, such as microhardness, as a function of depth after ageing so that the magnitude of any effect from the limitation of oxygen diffusion could be measured for any temperature and material combination. The effect of a degradation profile on a bulk property will depend on the particular... 

Hoftyzer and van Krevelen [100] investigated the combination of mass transfer together with chemical reactions in polycondensation, and deduced the ratedetermining factors from the description of gas absorption processes. They proposed three possible cases for poly condensation reactions, i.e. (1) the polycondensation takes place in the bulk of the polymer melt and the volatile compound produced has to be removed by a physical desorption process, (2) the polycondensation takes place exclusively in the vicinity of the interface at a rate determined by both reaction and diffusion, and (3) the reaction zone is located close to the interface and mass transport of the reactants to this zone is the rate-determining step. 

B , while for an n-type semiconductor the reverse is true. An analog to the SCR in the semiconductor is an extended layer of ions in the bulk of the electrolyte, which is present especially in the case of electrolytes of low concentration (typically below 0.1 rnolh1). This diffuse double layer is described by the Gouy-Chap-man model. The Stern model, a combination of the Helmholtz and the Gouy-Chapman models, was developed in order to find a realistic description of the electrolytic interface layer. 

When chemisorption is involved, or when some additional surface chemical reaction occurs, the process is more complicated. The most common combinations of surface mechanisms have been expressed in the Langmuir-Hinshelwood relationships 36). Since the adsorption process results in the net transfer of molecules from the gas to the adsorbed phase, it is accompanied by a bulk flow of fluid which keeps the total pressure constant. The effect is small and usually neglected. As adsorption proceeds, diffusing molecules may be denied access to parts of the internal surface because the pore system becomes blocked at critical points with condensate. Complex as the situation may be in theory,... 

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