Model adsorption

Some models have been proposed interpreting the adsorption-desorption phenomena. The most important models are described by the isotherms introduced by Langmuir, Freundlich, and Temkin. 

Ziegler-Natta catalyst systems being mostly heterogeneous in nature, adsorption reactions are most likely to occur in such polymerizations and feature in their kinetic schemes (Erich and Mark, 1956). A number of kinetic schemes have thus been proposed based on the assumption that the polymerization centers are formed by the adsorption of metal alkyl species on to the surface of a crystalline transition metal halide and that chain propagation occurs between the adsorbed metal alkyl and monomer. In this regard the Rideal rate law and the Langmuir-Hinshelwood rate law for adsorption and reaction on solids assume importance see Problem 9.4). 

Problem 9.4 Considering reaction between A and B catalyzed by a solid there are two possible mechanisms by which this reaction could occur. The first is that one of them, say A, gets adsorbed on the solid surface and the adsorbed A then reacts chemically with the other component B, which is in the gas phase or in solution and is not adsorbed on the surface. The second mechanism is that both A and B are adsorbed, and the adsorbed 

Both the Rideal and Langmuir-Hinshelwood rate laws are based upon the Langmuir adsorption equation which is applicable for gas-solid as well as liquid-solid systems where diffusion of the sorbate to the solid surface is not rate limiting (generally true). The basic assumption of the Langmuir adsorption is that adsorption occurs at adsorption sites and all these sites are equivalent. For gas-solid systems, the rate of adsorption, r, of the gas A is proportional to the gas pressure, px, and the number of vacant sites, i.e., 

At equilibrium the rates of adsorption and desorption are equal. Equating Eqs. (P9.4.1) and (P9.4.2) then gives 

Defining the fraction of adsorption sites covered by A as 0a = a/ o and the equilibrium constant for the adsorption equilibrium as ZX = kjkd, the above equation reduces 

The present discussion is focused on models of primary surface charging, that is, of adsorption of protons in the presence of inert electrolytes. These models are elements of more general models, which describe adsorptions of all kinds of species. Basically, the models discussed in this section apply to metal oxides, but similar models have been used for other materials. For example, a model of proton and heavy metal binding by humic acid described in [700] is similar to models used for oxides. 

In the absence of surface-active species, the surface charge of metal oxides is due solely to the adsorption/desorption of protons. The ions of an inert electrolyte remain at some distance from the surface and affect the surface charging indirectly. In most models, proton adsorption and desorption is interpreted as the protonation and deprotonation of discrete surface sites. Direct spectroscopic observation of these surface sites is difficult, and it does not give clear and 

The surface site density obtained by means of the aforementioned independent methods can be used to interpret potentiometric titration data. Alternatively, the titration data can be used to calculate the best-fit surface site density (or densities of various types of sites) as parameter(s) of a certain model. In such a calculation, knowledge about the nature of the surface sites is not required. The site densities have also been calculated (e.g., in [723]) as parameters of adsorption isotherms of various adsorbates (usually small ions). The problem with such site densities is that protons behave differently from other adsorbates, and sites that are capable of binding protons are not necessarily capable of binding other species, and vice versa. 

As early as 1956 Erich and Mark [17] pointed out that since most Ziegler-Natta catalyst systems were heterogeneous in nature, it was most likely 

At equilibrium these rates are equal and one obtains l apjkd 

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Ruch and Bartell [84], studying the aqueous decylamine-platinum system, combined direct estimates of the adsorption at the platinum-solution interface with contact angle data and the Young equation to determine a solid-vapor interfacial energy change of up to 40 ergs/cm due to decylamine adsorption. Healy (85) discusses an adsorption model for the contact angle in surfactant solutions and these aspects are discussed further in Ref. 86. 

The acid monolayers adsorb via physical forces [30] however, the interactions between the head group and the surface are very strong [29]. While chemisorption controls the SAMs created from alkylthiols or silanes, it is often preceded by a physical adsorption step [42]. This has been shown quantitatively by FTIR for siloxane polymers chemisorbing to alumina illustrated in Fig. XI-2. The fact that irreversible chemisorption is preceded by physical adsorption explains the utility of equilibrium adsorption models for these processes. 

Stahlberg has presented models for ion-exchange chromatography combining the Gouy-Chapman theory for the electrical double layer (see Section V-2) with the Langmuir isotherm (. XI-4) [193] and with a specific adsorption model [194]. 

While a thermodynamic treatment can be developed entirely in terms of f(P,T), to apply adsorption models, it is highly desirable to know on a per square centimeter basis rather than a per gram basis or, alternatively, to know B, the fraction of surface covered. In both the physical chemistry and the applied chemistry of the solid-gas interface, the specific surface area is thus of extreme importance. 

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. 

Thus from an adsorption isotherm and its temperature variation, one can calculate either the differential or the integral entropy of adsorption as a function of surface coverage. The former probably has the greater direct physical meaning, but the latter is the quantity usually first obtained in a statistical thermodynamic adsorption model. 

The data on heats and entropies of adsorption do allow a more discriminating test of an adsorption model, although even so only some rather qualitative conclusions can be reached. The discussion of these follows. 

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... 

Because of the relatively strong adsorption bond supposed to be present in chemisorption, the fundamental adsorption model has been that of Langmuir (as opposed to that of a two-dimensional nonideal gas). The Langmuir model is therefore basic to the present discussion, but for economy in presentation, the reader is referred to Section XVII-3 as prerequisite material. However, the Langmuir equation (Eq. XVlI-5) as such,... 

Figure Al.7.8. Sticking probability as a fimction of surface coverage for tliree different adsorption models.

Surface Area and Permeability or Porosity. Gas or solute adsorption is typicaUy used to evaluate surface area (74,75), and mercury porosimetry is used, ia coajuactioa with at least oae other particle-size analysis, eg, electron microscopy, to assess permeabUity (76). Experimental techniques and theoretical models have been developed to elucidate the nature and quantity of pores (74,77). These iaclude the kinetic approach to gas adsorptioa of Bmaauer, Emmett, and TeUer (78), known as the BET method and which is based on Langmuir s adsorption model (79), the potential theory of Polanyi (25,80) for gas adsorption, the experimental aspects of solute adsorption (25,81), and the principles of mercury porosimetry, based on the Young-Duprn expression (24,25). 

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. 

Adsorption beds of activated carbon for the purification of citric acid, and adsorption of organic chemicals by charcoal or porous polymers, are good examples of ion-exchange adsorption systems. Synthetic resins such as styrene, divinylbenzene, acrylamide polymers activated carbon are porous media with total surface area of 450-1800 m2-g h There are a few well-known adsorption systems such as isothermal adsorption systems. The best known adsorption model is Langmuir isotherm adsorption. 

FHH (Frenkel-Halsey-Hill) theory is valid for multi molecules adsorption model of the flat surfrtce material. When this model is applied for the surface fractal in the range of capillary condensation, in other words, in the state of interface which was controlled by the surface tension between liquid and gas, the modified FHH equation can be expressed as Eq. (3). 

The S-S adsorption model assumes a linear relationship given by Equation 4.30 ... 

The application of the Diffusion-Adsorption model to dating bone (by AP) was funded by a NERC grant to Robert Hedges at the Research Laboratory for Archaeology, University of Oxford. The U-series date profiles shown here were measured at the NERC U-series dating facility at Open University, and the laser ablation U-series profile was measured at the Research School for Earth Sciences, Australian National University, Canberra in collaboration with Steve Eggins and Rainer Griin. 

Non-compliance with the simple Langmuir adsorption model is indicative of violation under experimental conditions of certain assumptions used to derive the model. Therefore, while developing the theoretical models adequately describing experimental data one usually resorts to one of two approaches either introduces the notion of a inhomogeneous surface [36, 37] or accounts for various types of interaction developing between the particles absorbed [4, 38]. 

As noted above, adsorption isotherms are largely derived empirically and give no information on the types of adsorption that may be involved. Scrivner and colleagues39 have developed an adsorption model for montmorillonite clay that can predict the exchange of binary and ternary ions in solution (two and three ions in the chemical system). This model would be more relevant for modeling the behavior of heavy metals that actively participate in ion-exchange reactions than for organics, in which physical adsorption is more important. 

These assumptions are akin to those taken in account in the mixed adsorption model of Trogus (12). The difference between the two models lies in the relationship linking CMCs of single and mixed surfactants and monomer molar fractions Trogus used the empirical equation proposed by Mysels and Otter (13) in our model, the application of RST leads to an equation of the same type. 

Adsorption and Film Formation. Inhibition of HC1 corrosion by organic compounds is a complicated multi-step process. Nevertheless, the effect of an inhibitor on corrosion of a metal is often treated mathematically with an equilibrium adsorption model for displacement of water (19,20) ... 

The lamellae grown in these Langevin dynamics simulations are very small in comparison with experimentally investigated lamellae. In view of this, we have developed the coarse-grained anisotropic adsorption model described... 

Fig. 4.6 Protein adsorption model on aluminum-substituted mesoporous silica. Adapted from [37], A. Vinu et al.,J. Nanosci. Nanotechnol. 2006, 6, 1510.

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