Adsorption model monolayer

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. 

The number of gas molecules can be measured either directly with a balance (gravimetric method) or calculated from the pressure difference of the gas in a fixed volume upon adsorption (manometric method). The most frequently apphed method to derive the monolayer capacity is a method developed by Brunauer, Emmett, and Teller (BET) [1], Starting from the Langmuir equation (monolayer adsorption) they developed a multilayer adsorption model that allows the calculation of the specific surface area of a sohd. The BET equation is typically expressed in its linear form as... 

This equation assumes that only one layer of gas molecule adsorbs. A monolayer of gas adsorbs in the case in which there are only a given number of adsorption sites for a monolayer. This is the most simple adsorption model. The amount adsorbed, Ns, is related to the monolayer coverage, Nsm, as follows (Appendix C) ... 

Although the BET equation is open to a great deal of criticism, because of the simplified adsorption model upon which it is based, it nevertheless fits many experimental multilayer adsorption isotherms particularly well at pressures between about 0.05 p0 and 0.35 pQ (within which range the monolayer capacity is usually reached). However, with porous solids (for which adsorption hysteresis is characteristic), or when point B on the isotherm (Figure 5.5) is not very well defined, the validity of values of Vm calculated using the BET equation is doubtful. 

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... 

A frequently used adsorption model that allows for adsorption in multilayers has been introduced by Brunauer, Emmett and Teller [10] and is known as the BET equation. With the exception of the assumption that the adsorption process terminates at monolayer coverage, these authors have retained all the other assumptions made in deriving the Langmuir adsorption isotherm. Hence all objections to the application of the Langmuir equation apply here, too. 

Figure 7.7. Schematic representation of the monolayer adsorption model in the Ni-Si/SiC system. YNi and XNi are the molar fractions of Ni in the interfacial layer and bulk liquid respectively.

The Langmuir-adsorption model predicts an asymptotic approach to monolayer surface coverage as adsorbate partial pressure approaches saturation this is the Type-I isotherm of Figure 5.7. The Langmuir model, though proven for many ultraclean, well-ordered surfaces interacting with small-molecule adsorbates, is oversimplified for many real-world systems. Nonetheless, it is the foundation upon which much of adsorption theory is built and as such provides a useful con-... 

The Brunauer-Emmett-Teller (BET) adsorption model was developed to account for multilayer adsorption. The BET model can be thought of as the sum of two terms the Langmuir model is used to account for coverage from zero to the completion of the first monolayer, while the second and all subsequent layers (not treated by the Langmuir model) are assumed to have a heat of adsorption equal to the heat of vaporization of the bulk liquid phase of the adsorbing species. The heat of adsorption of the first monolayer usually exceeds the liquid s heat of vaporization. Although it might seem a erode oversimplification, the BET model works well for many systems that involve physisorption of simple molecules,... 

The problem is to relate v (z) to the surface potential - v (0) or the surface charge density a° = a(O)) and the volume fraction profiles of the components. Early versions t-2) of a polyelectrolyte adsorption model neglected the volume of the small Ions and solved (numerically) the Poisson-Boltzmann equation 13.5.6). A more sophisticated, yet simpler, approach was proposed by Bflhmer et al. who accounted for the Ion volume by adopting a multilayer Stem model, see fig. 5.17. This Is a straightforward extension of the monolayer Stern model discussed in sec. 3.6c. The charges of the ions and the segments are assumed to be located on planes in the centres of the lattice layers. The lattice is thus con-... 

In the limit of monolayer adsorption model Eq. (16) reduces to the Langmuir-Freund-lich isotherm which, nowadays, seems to be the most widely used expression to correlate experimental adsorption isotherms in the submonolayer coverage region. 

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]. 

Some criticism can be made of the assumptions of the B.E.T. adsorption model. If the second and other layers are assumed to be in the liquid state, how can localized adsorption take place on these layers Also, the assumption that the stacks of molecules do not interact energetically seems to be unrealistic. In spite of these theoretical weaknesses, the B.E.T. adsorption expression is very useful for qualitative application to type II and III isotherms, the B.E.T equation is very widely used in the estimation of specific surface areas of solids. The surface area of the adsorbent is estimated from the value of Vm. The most commonly used adsorbate in this method for area determination is nitrogen at 77 K. The knee in the type II isotherm is assumed to correspond to the completion of a monolayer. In the most strict sense, the cross-sectional area of an adsorption site, rather than that of the adsorbate molecule, ought to be used, but the former is an unknown quantity however, this fact does not prevent the B.E.T. expression from being useful for the evaluation of surface areas of adsorbents. 

The Langmuir theory describes the quantitatively formation a monolayer adsorbate on the outer surface of the adsorbent without further adsorption taken place [3]. This model assumes an adsorption homogeneity, which includes the equally available adsorption sites, monolayer surface coverage and no mutual interactions. The equation of Langmuir adsorption isotherm is expressed in Equation (11.1) ... 

Apart from diffusion in continuum phase, the transport of surface-adsorbed molecules and capillary condensate takes place in meso- and macroporous media. In order to model transport of adsorbable vapor at elevated pressure, it is necessary to consider the type of adsorption occurring monolayer adsorption, multilayer adsorption, or capUlary condensation [38]. Models for surface diffusion have been proposed... 

Langmuir and Freundlich adsorption models can be used to describe the adsorption behavior of dispersants on an abrasive surface. The Langmuir adsorption isotherm model assumes that the particle surface is homogeneously covered by the monolayer dispersant. It is expressed as follows ... 

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