Adsorption model, generalized

The distinctive properties of densely tethered chains were first noted by Alexander [7] in 1977. His theoretical analysis concerned the end-adsorption of terminally functionalized polymers on a flat surface. Further elaboration by de Gennes [8] and by Cantor [9] stressed the utility of tethered chains to the description of self-assembled block copolymers. The next important step was taken by Daoud and Cotton [10] in 1982 in a model for star polymers. This model generalizes the... 

The linear equilibrium isotherm adsorption relationship (Eq. 11) requires a constant rate of adsorption, and is most often not physically valid because the ability of clay solid particles to absorb pollutants decreases as the adsorbed amount of pollutant increases, contrary to expectations from the liner model. If the rate of adsorption decreases rapidly as the concentration in the pore fluid increases, the simple Freundlich type model (Eqs. 8 and 9) must be extended to properly portray the adsorption relationship. Few models can faithfully portray the adsorption relationship for multicomponent COM-pollutant systems where some of the components are adsorbed and others are desorbed. It is therefore necessary to perform initial tests with the natural system to choose the adsorption model specific to the problem at hand. From leaching-column experimental data, using field materials (soil solids and COMs solutions), and model calibration, the following general function can be successfully applied [155] ... 

The estimation of the surface area of finely divided solid particles from solution adsorption studies is subject to many of the same considerations as in the case of gas adsorption, but with the added complication that larger molecules are involved whose surface orientation and pore penetrability may be uncertain. A first condition is that a definite adsorption model is obeyed, which in practice means that area determination data are valid within the simple Langmuir Equation 5.23 relation. The constant rate is found, for example, from a plot of the data, according to Equation 5.23, and the specific surface area then follows from Equations 5.21 and 5.22. The surface area of the adsorbent is generally found easily in the literature. 

These two equations represent the generalized Szyszkowski-Langmuir adsorption model. 

A multi-component gas solubility model and a multi-component surface adsorption model are generally required to estimate the monomer concentration at active sites. If the latter equilibrium can be neglected then the gas-solubility in the suspending agent determines the monomer concentration near the active site, which changes significantly with temperature and pressure. 

Multilayer adsorption models have been used by Asada [147,148] to account for the zero-order desorption kinetics. The two layers are equilibrated. Desorption goes from the rarefied phase only. This model has been generalized [148] for an arbitrary number of layers. The filling of the upper layer was studied with allowance for the three neighboring molecules being located in the lower one. The desorption frequency factor (CM) was regarded as being independent of the layer number. The theory has been correlated with experiment for the Xe/CO/W system [149]. Analysis of the two-layer model has been continued in Ref. [150], to see how the ratios of the adspecies flows from the rarefied phases of the first and the second layers vary if the frequency factors for the adspecies of the individual layers differ from one another. In the thermodynamic equilibrium conditions these flows were found to be the same at different ratios of the above factors. 

Lattice models play a central role in the description of polymer solutions as well as adsorbed polymer layers. All of the adsorption models reviewed so far assume a one-to-one correspondence between lattice random-walks and polymer configurations. In particular, the general scheme was to postulate the train-loop or train-loop—tail architecture, formulate the partition function, and then calculate the equilibrium statistics, e.g., bound fraction, average loop... 

Ionization of the oxide/water interface and the resultant electrical double layer have been studied intensively by a variety of techniques within the last decade. Although many electrical double layer and adsorption models have been proposed, few are sufficiently general to consider surface equilibria in complex electrolyte solutions. Recently we proposed a comprehensive adsorption model for the oxide/water interface which can simultaneously estimate adsorption density, surface charge, and electro-kinetic potential in a self-consistent manner (jL, 2, 3). One advantage of the model was that it could be incorporated within the computer program, MINEQL ( ), by adding charge and mass-balance equations for the surface. 

Adsorption of Anions. The general nature of the adsorption model and computational method allow one to describe the uptake of anions also ( ). Similar to the approach for metal ions, we included a term in the mass-law expression to correct for the effect of potential on surface equilibria. Although adsorption of some anions (e.g. chloride, nitrate, syringic acid, thiosulfate) can be simulated by one surface reaction (24), formation of two surface complexes is probable for other anions, e.g. chromate, selenate (J ). Model calculations were more consistent with experimental adsorption data when the following surface reactions were considered, i.e. 

Morel, F. M. M., Yeasted, J. G., and WestaU, J. C., 1981, Adsorption Models A Mathematical Analysis in the Framework of General Equilibrium Calculations in Anderson, M. A., and Rubin, A. J., eds.. Adsorption of Inorganics at Solid-Liquid Interfaces Ann Arbor, Michigan, Ann Arbor Science, p. 263-294. 

The proposed competitive adsorption model was implemented and used to calculate the band profiles of cyclopentanone with mobile phases of different compositions. One more adjustable parameter, an equilibrium constant for additional interactions, was introduced in order to match calculated and experimental retention of cyclopentanone. Figure 15.3 compares some calculated and experimental band profiles of cyclopentanone for mobile phases containing different concentrations of methanol in the mobile phase. In general, the agreement observed with either methanol or acetonitrile is good. However, like as any other complicated model, there are many parameters which must be determined by fitting the experimental data to the model and stiU, at the end, the calculated retention times of the solute(s) must be adjusted using a last empirical parameter in order to match the experimental retention times. There are no independent ways to verify that these parameters are correct. Therefore, in practice, the use of a more simplified model remains preferable. 

In order to use sink models, important parameters must be available. For example, the Langmuir adsorption model requires information on the rates of adsorption and desorption. Diffusion models require information on the diffusion coefficients. These parameters are dependent upon the characteristics of both the VOC (or SVOC) and the sink material, and fundamental data are generally not available. Thus, experimental studies are required to determine the values of the important parameters of the sink models. 

Dubinin adsorption models have been used to calculate carbon micropore distributions from experimental isotherm measurements of a number of adsorbates, including nitrogen [120-122], carbon dioxide [122,123], methane [123], and several other organic molecules [124]. It is has generally been... 

Examples of correlations between stability constants of surface complexes (calculated using different adsorption models, cf. Chapter 5) on the one hand, and the first hydrolysis constant and other constants characterizing the stability of solution complexes on the other are more munerous than the studies of correlations involving directly measured quantities. It should be emphasized that there is no generally accepted model of adsorption of ions from solution, and stability constants of surface complexes are defined differently in particular models, thus, the numerical values of these constants depend on the choice of the model. Moreover, some publications reporting the correlations fail to define precisely the model. 

This model generalizes an early proposal made by Frennet and Lienard (55-57) for methane adsorption. These authors studied the surface composition of carbon and hydrogen atoms during the chemisorption of methane on metal films and found that the results were best explained by assuming that methane was adsorbed according to a bimolecular reaction between methane and adsorbed hydrogen ... 

Many different equations have been used to interpret monolayer—multilayer isotherms [7, 11, 18, 21, 22] (e.g., the equations associated with the names Langmuir, Vohner, HiU-de Boer, Fowler-Guggenheim, Brunauer-Emmett-Teller, and Frenkel-Halsey-Hill). Although these relations were originally based on adsorption models, they are generally applied to the experimental data in an empirical manner and they all have Hmitations of one sort or another [7, 10, 11]. 

Morel, F.M.M., J.G. Yeasted, and J.C. Westall. 1981. Adsorption models A mathematical analysis in the framework of general equilibrium calculations, p. 263-294. In M.A. Anderson and A.J. Rubin (ed.) Adsorption of inorganics at solid-liquid interfaces. Ann Arbor Sci., Ann Arbor, MI. 

For the study of adsorption to general oxide surfaces, the ideal ionic force-field would be consistent with a molecular mechanics model for dissociable water. Such a forcefield has indeed been developed and has been applied to the study of silicate (Rustad Hay, 1995) and iron(III) (Rustad et al., 1995) hydrolysis in solution, to bulk iron oxyhydroxide structures (Rustad et al., 1996a) and to the protonation of goethite surfaces (Rustad et al., 1996b). 

The adjustment of adsorption parameters with respect to changing temperature is generally not addressed in most modeling exercises, owing to the uncertainty of the adsorption modeling... 

Eq. (2.36) in its general form and Eqs (4.31) to (4.34) as particular cases have to be modified essentially in order to use them for the description of adsorption kinetics processes. This modification is the replacement of the bulk concentration c by the sublayer concentration c(0,t), which was first suggested by Baret (1969), which leads to Eq. (4.35) used in Chapter 4 as the basis for the so-called kinetic-controlled adsorption model. 

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