Adsorption-diffusion control

Mixed adsorption diffusion control was considered by Lorenz and Mockel [18], and they derived the following equations for the frequency-normalized admittance ... 

Mixed adsorption-diffusion control was considered by Lorenz and Mockel... 

Alternative approaches treat the adsorbed layer as an ideal solution or in terms of a Polanyi potential model (see Refs. 12-14 and Section XVII-7) a related approach has been presented by Myers and Sircar [15]. Adsorption rates have been modeled as diffusion controlled [16,17]. 

The Langmuir-Hinshelwood picture is essentially that of Fig. XVIII-14. If the process is unimolecular, the species meanders around on the surface until it receives the activation energy to go over to product(s), which then desorb. If the process is bimolecular, two species diffuse around until a reactive encounter occurs. The reaction will be diffusion controlled if it occurs on every encounter (see Ref. 211) the theory of surface diffusional encounters has been treated (see Ref. 212) the subject may also be approached by means of Monte Carlo/molecular dynamics techniques [213]. In the case of activated bimolecular reactions, however, there will in general be many encounters before the reactive one, and the rate law for the surface reaction is generally written by analogy to the mass action law for solutions. That is, for a bimolecular process, the rate is taken to be proportional to the product of the two surface concentrations. It is interesting, however, that essentially the same rate law is obtained if the adsorption is strictly localized and species react only if they happen to adsorb on adjacent sites (note Ref. 214). (The apparent rate law, that is, the rate law in terms of gas pressures, depends on the form of the adsorption isotherm, as discussed in the next section.)... 

Rapid Adsorption-Desorption Cycles For rapid cycles with particle diffusion controlling, when the cycle time is much smaller than the time constant for intraparticle transport, the LDF approximation becomes inaccurate. The generalized expression... 

FIG. 16-16 Batch adsorption curves for solid diffusion control. The curve for A = 0 corresponds to an infinite fluid volume (adapted from Ruthven, gen. refs., with permission). 

For a linear isotherm tij = KjCj), this equation is identical to the con-seiwation equation for sohd diffusion, except that the solid diffusivity D,i is replaced by the equivalent diffusivity = pDj,i/ p + Ppi< ). Thus, Eqs. (16-96) and (16-99) can be used for pore diffusion control with infinite and finite fluid volumes simply by replacing D,j with D. When the adsorption isotherm is nonhnear, a numerical solution is... 

This equation has the same form of that obtained for solid diffusion control with D,j replaced by the equivalent concentration-dependent diffusivity = pDpj/[ pn]Ki l - /i,//i)) ]. Numerical results for the case of adsorption on an initially clean particle are given in Fig. 16-18 for different values of A = = 1 - R. The upt e curves become... 

FIG. 16-18 Constant separation factor batch adsorption curves for pore diffusion control with an infinite fluid volume. X is defined in the text. 

The major difference between diffusion controlled dispersion and that resulting from adsorption and desorption is that the transfer process is concentration controlled. Reiterating equation (7),... 

While it is possible that surface defects may be preferentially involved in initial product formation, this has not been experimentally verified for most systems of interest. Such zones of preferred reactivity would, however, be of limited significance as they would soon be covered with the coherent product layer developed by reaction proceeding at all reactant surfaces. The higher temperatures usually employed in kinetic studies of diffusion-controlled reactions do not usually permit the measurements of rates of the initial adsorption and nucleation steps. 

Comprehensive evaluation of the results shows that the adsorption and desorption of alkanesulfonates, and other surfactants too, is diffusion-controlled. The dilatation modulus increases with increasing number of carbons because of the enhanced intermolecular interaction. This information is particularly significant in, for instance, foams. 

Valette-Hamelin approach,67 and other similar methods 24,63,74,218,225 (2) mass transfer under diffusion control with an assumption of homogeneous current distribution73 226 (3) adsorption of radioactive organic compounds or of H, O, or metal monolayers73,142,227 231 (4) voltammetry232,233 and (5) microscopy [optical, electron, scanning tunneling microscopy (STM), and atomic force microscopy (AFM)]234"236 as well as a number of ex situ methods.237 246... 

It has been seen from the above simple examples that the concentration of the substrate has a profound effect on the rate of the electrode process. It must be remembered, however, that the process may show different reaction orders in the different potential regions of the i-E curve. Thus, electron transfer is commonly the slow step in the Tafel region and diffusion control in the plateau region and these processes may have different reaction orders. Even at one potential the reaction order may vary with the substrate concentration as, for example, in the case discussed above where the electrode reaction requires adsorption of the starting material. 

An increase in pressure will also affect the rate of the diffusion of molecules to and from the electrode surface it will cause an increase in the viscosity of the medium and hence a decrease in diffusion controlled currents. The consequences of increased pressure on the electrode double layer and for the adsorption of molecules at the electrode surface are unclear and must await investigation. 

Figure 26 shows the redox potential of 40 monolayers of cytochrome P450scc on ITO glass plate in 0.1 KCl containing 10 mM phosphate buffer. It can be seen that when the cholesterol dissolved in X-triton 100 was added 50 pi at a time, the redox peaks were well distinguishable, and the cathodic peak at -90 mV was developed in addition to the anodic peak at 16 mV. When the potential was scanned from 400 to 400 mV, there could have been reaction of cholesterol. It is possible that the electrochemical process donated electrons to the cytochrome P450scc that reacted with the cholesterol. The kinetics of adsorption and the reduction process could have been the ion-diffusion-controlled process. 

Also, under continuous CO oxidation conditions, alkaline media exhibit a much higher activity than acidic media. Markovic and co-workers observed a shift of about 150 mV of the main oxidation wave, and a pre-wave corresponding to CO oxidation at potentials as low as 0.2-0.3 V [Markovic et al., 2002]. Remarkably, the hysteresis that is so prominently observed in the diffusion-controlled CO oxidation wave in acidic media (see Fig. 6.9), is no longer present in alkaline media. Markovic and co-workers also attribute the high activity of alkaline media to a pH-dependent adsorption of OH ds at defect/step sites. 

Again returning to the diffusion-controlled limiting current, we often meet a considerable influence on its height by catalysis, adsorption or other surface phenomena, so that we have to deal with irreversible electrode processes. For instance, when to a polarographic system with a diffusion-controlled limiting... 

In an interesting analysis of the effects of reduction of dimensionality on rates of adsorption/desorption reactions (26), the bimolecular rate of 10 M- s- has been reported as the lower limit of diffusion control. Based on this value, the rates given in Table III indicate the desorption step is chemical-reaction-controlled, likely controlled by the chemical activation energy of breaking the surface complex bond. On the other hand, the coupled adsorption step is probably diffusion controlled. 

Garg, D. R. and Ruthven, D. M. A.I.Ch.E. Jl 21 (1975) 200. Linear driving force approximation for diffusion controlled adsorption in molecular sieve columns. 

As already discussed, it indicates that for a diffusion controlled process (i.e. one which does not involve adsorption processes) taking place at a planar electrode, the current (which is proportional to... 

Understanding the adsorption, diffusivities and transport limitations of hydrocarbons inside zeolites is important for tailoring zeolites for desired applications. Knowledge about diffusion coefficients of hydrocarbons inside the micropores of zeolites is important in discriminating whether the transport process is micropore or macropore controlled. For example, if the diffusion rate is slow inside zeolite micropores, one can modify the post-synthesis treatment of zeolites such as calcination, steaming or acid leaching to create mesopores to enhance intracrystalline diffusion rates [223]. The connectivity of micro- and mesopores then becomes an... 

Cooper, R.S. and Liberman, D.A. (1970) Fixed-bed adsorption kinetics with pore diffusion control. Ind. Chem. Eng. Fund., 9, 620. 

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