The Adsorption Model

FIGURE 9.8. At a solid-liquid interface two adsorption situations may be encountered (a) in the case of a pure liquid, the molecular distribution will be approximately uniform, although some molecules may be adsorbed to produce a small surface excess (black circles) (b) for a solution of surface-active solute (black ovals), extensive adsorption will occur, producing a significant interfacial region of excess solute concentration. 

We have already seen (and will continue to see) the Gibbs adsorption equation which can be conveniently written as 

As will be seen in later sections concerned with Uquid-fluid systems, this equation normally employed to determine the amount of adsorbed material at an interface as a function of interfacial tension, r, in the case of solid surfaces, it is difficult or impossible to determine a directly. It is, however, relatively easy to determine the amount of adsorbed material directly and use that information to calculate a value of the interfacial tension. Such exercises are of great theoretical importance in understanding why and how molecules are adsorbed at an interface, and of even greater practical importance for understanding how such adsorption affects the characteristics of the interface and its interaction with its surroundings, especially in the context of colloidal stability and wetting phenomena. 

When the adsorption of a molecule from solution onto a solid surface is considered, there are several quantitative and qualitative points that are of interest, including (1) the amount of material adsorbed per unit mass or area of solid, (2) the solute concentration required to produce a given surface coverage or degree of adsorption, (3) the solute concentration at which surface saturation occurs, (4) the orientation of the adsorbed molecules relative to the surface and solution, and (5) the effect of adsorption on the properties of the solid relative to the rest of the system. In all the above, it is assumed that such factors as temperature and pressure are held constant. 

The conventional method for determining the above quantities in a given system is by way of the adsorption isotherm, using the basic concepts aheady introduced. The basic quantitative equation describing the adsorption of one component of a binary solution onto a solid substrate can be written as 

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A pulse of a racemic mixture (5 g each enantiomer) was carried out to check the adsorption model and to predict the mass transfer coefficient. The other model parameters used in simulation were = 0.4 and Pe = 1000. The mass transfer coefficient used to fit experimental and model predictions in the pulse experiment was k = 0.4 s k Model and experimental results are compared in Figs. 9-16 and 9-17. 

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] ... 

This definition erases some of the differences between the interpretation ul experimental data by different workers. Fur instance, I lie conclusions of Scott and Kucera (2/0), which apparently support a partitioning mechanism, are also consistent with the adsorption model in the view of the above definition. 

The low-temperature physisorption (type I isotherm) of hydrogen in zeolites is in good agreement with the adsorption model mentioned above for nanostructured carbon. The desorption isotherm followed the same path as the adsorption, which indicates that no pore condensation occurred. The hydrogen adsorption in zeolites depends linearly on the specific surface areas of the materials and is in very good agreement with the results on carbon nanostructures [24]. 

The deviations from the Szyszkowski-Langmuir adsorption theory have led to the proposal of a munber of models for the equihbrium adsorption of surfactants at the gas-Uquid interface. The aim of this paper is to critically analyze the theories and assess their applicabihty to the adsorption of both ionic and nonionic surfactants at the gas-hquid interface. The thermodynamic approach of Butler [14] and the Lucassen-Reynders dividing surface [15] will be used to describe the adsorption layer state and adsorption isotherm as a function of partial molecular area for adsorbed nonionic surfactants. The traditional approach with the Gibbs dividing surface and Gibbs adsorption isotherm, and the Gouy-Chapman electrical double layer electrostatics will be used to describe the adsorption of ionic surfactants and ionic-nonionic surfactant mixtures. The fimdamental modeling of the adsorption processes and the molecular interactions in the adsorption layers will be developed to predict the parameters of the proposed models and improve the adsorption models for ionic surfactants. Finally, experimental data for surface tension will be used to validate the proposed adsorption models. 

Furthermore, as will be analyzed in practical applications, the adsorption models can also be used as a fust approximation for ion-exchange systems, i.e. in the exchange of ions of different valences. 

Fig. 6.112. Comparison of coverage vs. potential curves for (o) experimental and ( ) theoretical [Eq. (6.270)] results predicted by the adsorption model for the adsorption of (a) 10-5 M phenol and (b) 10""4 M n-valeric acid on platinum electrodes. Under these experimental conditions, the pzc was found to be at about 0.35 V. (Reprinted from J. O M. Bockris and K. T. Jeng, J. Electroanal. Chem. 330 541, Copyright 1992, Figs. 18,19, and 20, with permission of Elsevier Science.)...

Considering the method of preparation of these ZnO samples, these results correspond to what one would expect on the basis of the adsorption model. The samples were sintered or evaporated at some high temperature, and then cooled to room temperature in air. As discussed in Section IV, 1, adsorption will occur until the rate of electrons crossing the surface barrier is pinched off to zero at room temperature. If the temperature is now lowered below room temperature, no electron transfer will be possible between the surface level and the bulk of the solid due to this high surface barrier. Thus the surface levels will be isolated and unable to affect the conductivity, which will therefore reflect bulk properties of the zinc oxide. 

The above analyses show the qualitative agreement of the adsorption model with recent conductivity experiments, and indicate how the trapping energy in adsorption may be indirectly obtained by conductivity measurements in the temperature range where slow desorption occurs. The energy can be obtained accurately, however, only with more careful examination of the theory and experiment than is presented here. 

Additional information on adsorption mechanisms and models is in Stollenwerk (2003), 93-99 and Prasad (1994). Foster (2003) also discusses in considerable detail how As(III) and As(V) may adsorb and coordinate on the surfaces of various iron, aluminum, and manganese (oxy)(hydr)oxides. In adsorption studies, relevant laboratory parameters include arsenic and adsorbent concentrations, adsorbent chemistry and surface area, surface site densities, and the equilibrium constants of the relevant reactions (Stollenwerk, 2003), 95. Once laboratory data are available, MINTEQA2 (Allison, Brown and Novo-Gradac, 1991), PHREEQC (Parkhurst and Appelo, 1999), and other geochemical computer programs may be used to derive the adsorption models. 

Assumptions underlying the adsorption models are not often discussed in the literature, since the exact nature of the relevant surface complexes or phases is difficult to identify. In particular, lateral interactions between adsorbed ions, site heterogeneity as well as phenomena involving the oxide dissolution or rchy-dration are not taken into account systematically. The latter phenomena are discussed in section D.d. Lateral interactions between adsorbed ions (ion coadsorption) have been reported [27, 28] and make questionable the use of mass action equations at interfaces. The effect of surface structure, site heterogeneity and surface composition, in particular on the ZPC value, were also pointed out [29, 30]. 

Figure 6.3 shows the basis of the adsorption model, and the four rate constants used. For simplicity, we assume a Langmuir isotherm model for the occupation of the interface. If the fraction of interface occupied is then the adsorption from the octanol side is... 

Therefore, the Nyquist plot can be simulated as shown in Figure 4.146. The deformation of the adsorption model caused by the mixing of the parameters is plotted in Appendix D (Model D13). 

This model is a result of modifying the adsorption model by taking into account the irregularities of Cad. Replacing Cad with CPE represents the non-uniform behaviour of the adsorbed species. The equivalent circuit is shown in Figure 4.17a. 

Figure 4.176 shows the simulated Nyquist plot of a modified adsorption model with an extra CPE in the Randles circuit. This modification of the adsorption model strongly influences the low-fiequency shape of the complex plane impedance diagram. More examples with variations on the CPE exponent are presented in Appendix D (Model D16). 

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... 

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. 

In this work, a comparative study of the textural properties developed by the intercalation and pillaring of a saponite with various aluminium oligomers is presented. The adsorption models above summarized have been applied to the nitrogen adsorption at 77 K results in order to evaluate the effect of the oligomer nature and the calcination temperature on the MPSD of the prepared solids. 

Fig. 16. Left A one-dimensional intensity profile along the line AB of the image shown in Fig. 15 (b). Lower part of the figure is the result of the curve-fitting using Gaussians. Right The adsorption model illustrating the bidentate and bridging forms of the acetate. [159].

In most gas-solid heterogeneous catalyst systems, the effect of pressure often is correlated with an adsorption model of the Langmuir-Hinshelwood type. The over-all rate constant for the first order reaction is related to the adsorption model constants by... 

According to the adsorption-site theory, the model constants should follow an Arrhenius-type temperature relationship. An Arrhenius-type plot of the adsorption model constants is shown in Figure 3. The rate constant, ko, increases with an increase in temperature, and the adsorption constants decrease with an increase in temperature. These opposing effects are in agreement with a physically realistic model. The activation energies found from these data are 29.3 kcal/mole for reaction, —28.9 kcal/mole for hydrocarbon adsorption, and —35.4 kcal/mole for hydrogen adsorption. 

FIG. 14 Schematic representation of the adsorption model for silver cyanide on activated carbons. (From Ref. 264.)... 

The fit of experimental data to a Langmuir (or another) adsorption isotherm does not constitute evidence that adsorption satisfies the criteria of the adsorption model. Frequently, adsorption to a surface is followed by additional interactions at the surface for example, a surfactant undergoes two-dimensional association subsequent to becoming adsorbed or charged ions tend to repel each other within the adsorbed layer. 

It is by now clear that several mechanisms and phases are present in the adsorption process in micropores, due to the interplay between gas-gas and gas-solid interactions, depending on their geometry and size. For this reason all those methods assuming a particular pore-filling mechanism, or adsorption model, should show shortcomings in some regions of the relevant parameters and their predictions should be compared with those based on more fundamental formulations of the adsorption process, like Density Functional Theory (DFT) [13] or Monte Carlo simulation [13,15]. Then, one question that arises is how the adsorption model affects the determination of the MSD ... 

We have presented a simulation method to obtain the MSD of activated carbons, and have analyzed the influence of the adsorption model, pore geometry and surface energetic heterogeneity on this determination, in particular for two series of activated carbons... 

Wheeler and Robell [18] were the first to make use of the Bohart-Adams model which, as we have said, was developed for a fixed-bed adsorption problem. They used the adsorption model to establish the sluq)e and motion of the poison proffles (c.f Figure 1) within the bed and then essentially solved a reaction/deactivation problem based on these profiles (in their case a simple first order system). However, it was also necessary to specify how the reaction rate constant, k, varied with the concentration of poison on the catdyst. In the simplest case we can choose a linear relationship (nonselective poisoning) as they did and write... 

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