Localized adsorption model

Many adsorption systems can be shown to correspond either to a localized adsorption model or to a two-dimensional gas. However, there must also be a large number of adsorbate-adsorbent pairs for which the potential energy functions will have variations intermediate between the extremes required for the limiting models. The formal equations applicable to these systems are included in the theory presented here, and furthermore, it appears to be feasible to carry out specific computations of the isotherms and heats for such systems up to moderate coverages (11, 24). 

In the case of the localized adsorption model, we must take into account Eq. 5.3 and also accept an eligible standard value for the fractional coverage 6°. The standard coverage can obviously be interchanged with the ratio of the minimum possible molar area, Na/Ca, to the standard molar area ... 

Finally we pay attention to the ideal frontal TC (cf. Fig. 4.1). The high temperature front of the zone profile is obviously proportional to the adsorption isobar and so, at least for the localized adsorption model, to the adsorption constant. As such, it would obey Eq. 5.14. It holds for the activities which do not appreciably decay in the course of run. As for the shorter-lived nuclides, both the elution and the formally frontal TC result in non-ideal frontal chromatograms. Their shapes are close to what would arise from ideal processing during t . but they are smeared due to the random lifetimes of nuclei. Still the initial part of the thermochromatogram might be useful for evaluation of the required quantity, provided that the statistics of detected decay events is good. 

The above wells are never completely separated, which would be the rationale for the localized adsorption model. Actually, the potential barriers between adjacent minima are smaller than the local desorption energies, and this results in lateral diffusion of the adsorbate. 

Experimental results have elearly shown that during the formation of the monolayer, a change from nonlocalized to locahzed adsorption occurs. Several theoretical studies have been made of the so-called partially mobile or partially localized adsorption models [11,60,161,164,228,230]. These theories must explain the phase transitions in the adsorbed monolayers and may also be useful in describing surface transport phenomena [11]. 

The state of an adsorbate is often described as mobile or localized, usually in connection with adsorption models and analyses of adsorption entropies (see Section XVII-3C). A more direct criterion is, in analogy to that of the fluidity of a bulk phase, the degree of mobility as reflected by the surface diffusion coefficient. This may be estimated from the dielectric relaxation time Resing [115] gives values of the diffusion coefficient for adsorbed water ranging from near bulk liquids values (lO cm /sec) to as low as 10 cm /sec. 

On the other hand, as applied to the submonolayer region, the same comment can be made as for the localized model. That is, the two-dimensional non-ideal-gas equation of state is a perfectly acceptable concept, but one that, in practice, is remarkably difficult to distinguish from the localized adsorption picture. If there can be even a small amount of surface heterogeneity the distinction becomes virtually impossible (see Section XVll-14). Even the cases of phase change are susceptible to explanation on either basis. 

One may choose 6(Q,P,T) such that the integral equation can be inverted to give f Q) from the observed isotherm. Hobson [150] chose a local isotherm function that was essentially a stylized van der Waals form with a linear low-pressure region followed by a vertical step tod = 1. Sips [151] showed that Eq. XVII-127 could be converted to a standard transform if the Langmuir adsorption model was used. One writes... 

J. P. Badiali, L. Blum, M. L. Rosinberg. Localized adsorption at solid-liquid interface the sticky site hard wall model. Chem Phys Lett 729 149-154, 1986. 

The thermodynamic properties of clathrates can be derived from a simple model which corresponds to the three-dimensional generalization of ideal localized adsorption. In ref. 52 the deriva-... 

Contrary to the last two isotherms, which take into the account interactions between the neighboring molecities ortiy, the Kiselev model assumes the singlecomponent localized adsorption, with the specific lateral interactions among all the adsorbed molecules in the monolayer [4—6]. The equation of the Kiselev isotherm is given below ... 

Buckled surface models for the two other copper sulfur phases were proposed that rationalised the STM, LEED and SEXAFS data (Figure 10.5). Note that the local adsorption site of the sulfur adatoms in both the p(3 x 2) and p(5 x 2) structures is the four-fold hollow in agreement with the SEXAFS18 measurements but looking at the structure from above with the STM, the periodicity of the structure suggests a variety of different adsorption sites. 

Quantitative determination of the local adsorption structure of carbonate on Ag(llO) has been done by Kittel etal. [110]. They have found that the carbonate species is essentially planar and adsorbs almost parallel to the surface at the off-atop site with respect to the outermost layer Ag atom. The C—Ag layer spacing was 0.264 0.009 nm, with a well-defined azimuthal orientation. This geometry is understood best in terms of the added-row model proposed by Guo and Madix. This model assumes that additional Ag atoms lie adjacent to the carbonate, such that the... 

The two-column process (Figure 2) is the heatless adsorption process of Skarstrom (5 ) This process can perform both recovery and purification of a light component Local equilibrium models of this process have been presented by Shendalman and Mitchell (6), Chan et al (7), and Knaebel and Hill (8) In each case the method of characteristics was used. Models with finite mass transfer rates have been published by Kawazoe and Kawai (9 ), Mitchell and Shendalman (10), Chihara and Suzuki (11), and Richter et al (12). In these models the method of characteristics and direct integration of partial differential equations were employed ... 

Langmuir (1918) laid down a systematic and theoretically grounded adsorption isotherm, the well-known Langmuir isotherm, which has served as a basis for a number of future isotherms, such as the BET isotherm, described in 1.4.1.3. The basic assumptions of the Langmuir adsorption model include monolayer coverage, localized adsorption (i.e., the specific adsorption sites exist and the interactions are between a specific molecule and the site), and homogeneous materials. In addition, the heat of adsorption is independent of coverage. He proposed that the rate of adsorption is proportional to the dimensionless pressure of the adsorbate, p/po (= p ), where po is the saturated vapor pressure, and the... 

The two micromechanism of LMIE are (i) the dissolution-condensation model (DCM) and (ii) the adsorption-induced localized slip model. In the DCM mechanism the crack is filled with melt and grows by dissolution of atoms from the crack tip where the chemical potential is increased due to applied stress.1,2 In the second model,... 

Figure 7.92 Sketches of (a) dissolution-condensation and (b) adsorption-induced localized-slip Models of LMIE...

For example, the local equilibrium theory combined with an accurate adsorption model (fitted to experimental adsorption equilibrium data) was succesfully used to predict experimental desorption data [2,3],... 

The experimental desorption concentration profiles were obtained in the same way as the experimental adsorption profiles. However, using the local equilibrium model to describe the process, only qualitatively results can be obtained. Mass transfer resistance has stronger effect on desorption in comparison to adsorption so its importance cannot be neglected in the modeling. 

Adsorption models are only as accurate as the local free-energy density that determines the SCF. A limited number of comparisons (Figs. 18, 24, and 25) between results based on the Flory-Huggins SCF, and those generated from... 

Various attempts have been made to modify the Langmuir model. One of the best known is that of Fowler and Guggenheim (1939), which allowed for adsorbate-adsorbate interactions in a localized monolayer on a uniform surface. However, on an empirical basis the Fowler-Guggenheim equation turns out to be no more successful than the original Langmuir isotherm. The highly complex problem of localized adsorption on heterogeneous surfaces has been discussed by Rudzinski and Everett (1992). 

For polymer/penetrant combinations where strong interactions between specific functional groups occur and binding to specific sites predominates, a localized sorption model is more appropriate. Figure 5.15(b) represents such a model, which is equivalent to the Langmuir and Freundlich isotherm models presented in the context of adsorption in the previous section. This behavior has been ob-... 

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