Diffusivities rate-limiting step

Why adsorption, ion exchange and heterogeneous catalysis in one book The basic similarity between these phenomena is that they all are heterogeneous fluid-solid operations. Second, they are all driven by diffusion in the solid phase. Thus, mass transfer and solid-phase diffusion, rate-limiting steps, and other related phenomena are common. Third, the many aspects of the operations design of some reactors are essentially the same or at least similar, for example, the hydraulic analysis and scale-up. Furthermore, they all have important environmental applications, and more specifically they are all applied in gas and/or water treatment. 

Hiester and Vermeulin [39] extended the model of Thomas by deriving a breakthrough curve for a slow diffusion rate-limiting step with a nonlinear Langmuir equilibrium isotherm. The model was further adapted by Arnold and Blanch [40] to describe affinity chromatographic experiments ... 

It was shown I8 that the binding process is the rate-limiting step, with an adsorption rate constant of kd = 4.8 x 104 dm mol s. The calculated mass transfer coefficient for the diffusion into the pores of the support contributes 11% to the overall adsorption process. The value of ka is an order of magnitude lower than that reported in Table 2. The binding properties of the polyclonal immunoadsorbents used in these two studies may differ because of the different methods employed for protein immobilization. Another possible explanation may be an underestimation of the contribution for the diffusion rate-limiting step as the polyclonal anti-HSA antibody was attached to a silica matrix of large pores [18]. 

The rate-limiting step typically occurs at the air—Hquid interface and, for biological species without diffusion limitations, the overall relationship can be simply written at steady state as... 

The rate of hydrolysis of sarin on Dowex-50 cation exchange resin is insensitive to the stirring rate. However, with a more active catalyst (Amberlite-IRA 400), the rate constant at 20°C was 5.3, 7.5, and 8.5 h at 60,800 and 1000 revolutions/min , respectively, suggesting that film diffusion was the rate-limiting. step. Thus, the mechanism of the rate-limiting step depends on the nature of the catalyst [34]. 

The reaction of Si02 with SiC [1229] approximately obeyed the zero-order rate equation with E = 548—405 kJ mole 1 between 1543 and 1703 K. The proposed mechanism involved volatilized SiO and CO and the rate-limiting step was identified as product desorption from the SiC surface. The interaction of U02 + SiC above 1650 K [1230] obeyed the contracting area rate equation [eqn. (7), n = 2] with E = 525 and 350 kJ mole 1 for the evolution of CO and SiO, respectively. Kinetic control is identified as gas phase diffusion from the reaction site but E values were largely determined by equilibrium thermodynamics rather than by diffusion coefficients. 

Theoretical models available in the literature consider the electron loss, the counter-ion diffusion, or the nucleation process as the rate-limiting steps they follow traditional electrochemical models and avoid any structural treatment of the electrode. Our approach relies on the electro-chemically stimulated conformational relaxation control of the process. Although these conformational movements179 are present at any moment of the oxidation process (as proved by the experimental determination of the volume change or the continuous movements of artificial muscles), in order to be able to quantify them, we need to isolate them from either the electrons transfers, the counter-ion diffusion, or the solvent interchange we need electrochemical experiments in which the kinetics are under conformational relaxation control. Once the electrochemistry of these structural effects is quantified, we can again include the other components of the electrochemical reaction to obtain a complete description of electrochemical oxidation. 

Equations 4.31 and 4.32 also suggest another important fact regarding NEMCA on noble metal surfaces The rate limiting step for the backspillover of ions from the solid electrolyte over the entire gas exposed catalyst surface is not their surface diffusion, in which case the surfacediffusivity Ds would appear in Eq. 4.32, but rather their creation at the three-phase-boundaries (tpb). Since the surface diffusion length, L, in typical NEMCA catalyst-electrode film is of the order of 2 pm and the observed NEMCA time constants x are typically of the order of 1000 s, this suggests surface diffusivity values, Ds, of at least L2/t, i.e. of at least 4 10 11 cm2/s. Such values are reasonable, in view of the surface science literature for O on Pt(l 11).1314 For example this is exactly the value computed for the surface diffusivity of O on Pt(lll) and Pt(100) at 400°C from the experimental results of Lewis and Gomer14 which they described by the equation ... 

Figure 2.8. Rate-limiting steps in a CVD reaction (a) surface reaction kinetics control, (b) diffusion control.

To summarize, the surface kinetics (or near surface kinetics) is the limiting step at lower temperature and diffusion is the rate limiting step at higher temperature. It is possible to switch from one rate-limiting step to the other by changing the temperature. This is illustrated in Fig. 2.9, where the Arrhenius plot (logarithm of the deposition rate vs. the reciprocal temperature) is shown for several reactions leading to the deposition of silicon,... 

When diffusion becomes the rate-limiting step, the time required for falling-rate drying can be estimated using an approximate equation ... 

Upon formation of a metal chelate or complex, the next rate-limiting step in delivering iron to the cell is the diffusion of iron complexes through the. soil in response to diffusion gradients. In the vicinity of plant roots, metal chelates and complexes may also move by bulk flow in the transpiration stream as water moves from the soil into the plant. However, depending on their charge characteristics and hydrophobicity, metal chelators and complexes can become adsorbed to clay and organic matter, which may then decrease their mobility and bioavail-... 

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... 

As for polymers the rate-limiting step in SFE is generally diffusion, conditions should be selected that... 

If q is negligibly small a stagnant diffusion layer in aqueous phase is a rate-limiting step. This case is often the most useful from an analytical point of view. The well-known equation for a reversible polarographic wave can be obtained as ... 

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