Adsorption model multilayer

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... 

While Eq. (2) models submonolayer order-disorder transitions and Eq. (60) model multilayer adsorption, it is of course possible to formulate a combined model which considers the competition between order-disorder phenomena in the first layer and adsorption of further layers . Then instead of Eqs. (2), (60) we write, for the simple cubic lattice,... 

In Langmuir model, the maximal adsorption is that of a monolayer. Langmuir adsorption isotherms all saturate at high vapor pressures. This is unrealistic for many cases. In order to consider the adsorption of multilayers, Brunauer, Emmett, and Teller extended the Langmuir theory and derived the so-called BET adsorption isotherm [378], The basic idea in the BET theory was to assume a Langmuir adsorption for each of the layers (Fig. 9.8). 

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. 

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. 

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. 

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,... 

A. Dabrowskl, M. Jaroniec, Adv. Colloid Interface Set 27 (1987) 211, Theoretical Foundations of Physical Adsorption from Binary Non-electrolytic Liquid Mixtures on Solid Surfaces Present and Future (emphasis on adsorption models covering heterogeneity, adsorptlves of different sizes and multilayers). 

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-... 

The BET model (8) assumes that layers of molecules are adsorbed on top of previously adsorbed molecules. Each layer adsorbs according to the Langmuir adsorption model. The four basic assumptions for the BET multilayer adsorption model are (a) adsorbed molecules do not migrate on the surface (b) the enthalpy of adsorption is constant for all molecules in a given layer (c) all molecules in layers beyond the first have equal energies of adsorption and (d) layers need not be completed for adsorption before the next one starts. The BET equation has the following formula for adsorption from a liquid solution ... 

Inclusion of multilayer effects in Langmuir s original model leads to the BET adsorption model, which can be written in the following form [103] ... 

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]. 

Figure 3.8 Multilayer surface adsorption model for the BET isotherm.

Fig. 2 Coordination binding of Cd + to carboxylate-terminated SAMs, (a) The number of electrons in the counterion overlayers per carboxylic acid group at the SAM surface held in 1 mM Cd + solutions at different pH values, obtained by fitting X-ray reflectivity curves. The filled circles represent measurements using synchrotron radiation the open circles were obtained using a rotating anode X-ray source. The line is a fit calculated using a modified BET multilayer adsorption model, (b) Schematic presentation of the pH-dependent equilibrium for binding of Cd + ions to a MHA SAM (adapted from Ref 28).

FIGURE 4.23 Schematic drawing of the ideal multilayer adsorption, as assumed in the BET model multilayer adsorption takes place, but in this example, the adsorbate-surface interaction is stronger than adsorbate-adsorbate interaction, thus the first layer forms preferentially to multilayer growth. The adsorbate molecules are assumed to form stacks independent of each other, so that every molecule has only up and down interactions, not lateral ones. 

The widely used empirical Freundlich model expresses adsorption at multilayer and on energetically heterogeneous surface miflti-site adsorption isotherm for hetero-... 

An adsorption process can be described by isotherms, i.e. by the functional relationship between the adsorbed quantities of a species vs. its activity. A direct consequence of the two possible interactions of a protic electrolyte (e.g. phosphoric, sulfuric or perchloric acid) to a polymer chain with basic groups is a multilayer-like adsorption process. Therefore, the use of an adsorption isotherm as described by the BET model (Brunauer-Emmett-Teller) is convenient. The BET model is originally derived for gas adsorption on surfaces [62, 63]. To derive a multilayer-like adsorption model for a basic ionogen polymer in analogy to the original BET model, we attribute the basic groups of the polymer chains, which can be protonated by the protic electrolyte, as adsorption sites. In case of PBI-type polymers the basic groups are the imidazole centres. 

The multilayer adsorption model, as shown in Fig. 2.11, was proposed by Brunauer, Emmet and Teller (BET) in 1938 to modify Langmuir s monolayer one. BET theory developed from the multilayer model can be applied to explain all types of isotherms. Based on the BET theory, a standard method for determining the specific surface area of solid catalysts was developed, which brought catalysis study into a new stage. 

The assumptions and main points of multilayer adsorption model are as follows ... 

Some authors have extended the above adsorption model by including lateral, attractive interactions between admolecules, the multilayer nature of the surface film, and adsorption mobility (see Refs. 5,9,167,168 and references therein). 

In case of multilayer adsorbate (P> 2) certmn hypotheses on the capture mechanism and interaction potential of colliding particle with the adsorbate are necessary for the calculation of the adsorption probabilities. Particularly, in some cases the expressions (7.1.6) and (7.1.7) are valid. Omitting the question of the calculation of detailed adsoi-ption-desorption probabilities let us formulate a model based on ideas of the two-layer adsorption model and BET-approximation for the adsorption isotherms of multilayer fihns (Flood 1967). The main assumptions of this generalized kinetic BET-model are ... 

The first term on the right is the common inverse cube law, the second is taken to be the empirically more important form for moderate film thickness (and also conforms to the polarization model, Section XVII-7C), and the last term allows for structural perturbation in the adsorbed film relative to bulk liquid adsorbate. In effect, the vapor pressure of a thin multilayer film is taken to be P and to relax toward P as the film thickens. The equation has been useful in relating adsorption isotherms to contact angle behavior (see Section X-7). Roy and Halsey [73] have used a similar equation earlier, Halsey [74] allowed for surface heterogeneity by assuming a distribution of Uq values in Eq. XVII-79. Dubinin s equation (Eq. XVII-75) has been mentioned another variant has been used by Bonnetain and co-workers [7S]. 

Brunauer (see Refs. 136-138) defended these defects as deliberate approximations needed to obtain a practical two-constant equation. The assumption of a constant heat of adsorption in the first layer represents a balance between the effects of surface heterogeneity and of lateral interaction, and the assumption of a constant instead of a decreasing heat of adsorption for the succeeding layers balances the overestimate of the entropy of adsorption. These comments do help to explain why the model works as well as it does. However, since these approximations are inherent in the treatment, one can see why the BET model does not lend itself readily to any detailed insight into the real physical nature of multilayers. In summary, the BET equation will undoubtedly maintain its usefulness in surface area determinations, and it does provide some physical information about the nature of the adsorbed film, but only at the level of approximation inherent in the model. Mainly, the c value provides an estimate of the first layer heat of adsorption, averaged over the region of fit. 

Returning to multilayer adsorption, the potential model appears to be fundamentally correct. It accounts for the empirical fact that systems at the same value of / rin P/F ) are in essentially corresponding states, and that the multilayer approaches bulk liquid in properties as P approaches F. However, the specific treatments must be regarded as still somewhat primitive. The various proposed functions for U r) can only be rather approximate. Even the general-appearing Eq. XVn-79 cannot be correct, since it does not allow for structural perturbations that make the film different from bulk liquid. Such perturbations should in general be present and must be present in the case of liquids that do not spread on the adsorbent (Section X-7). The last term of Eq. XVII-80, while reasonable, represents at best a semiempirical attempt to take structural perturbation into account. 

In considering isotherm models for chemisorption, it is important to remember the types of systems that are involved. As pointed out, conditions are generally such that physical adsorption is not important, nor is multilayer adsorption, in determining the equilibrium state, although the former especially can play a role in the kinetics of chemisorption. 

Another limitation of tire Langmuir model is that it does not account for multilayer adsorption. The Braunauer, Ennnett and Teller (BET) model is a refinement of Langmuir adsorption in which multiple layers of adsorbates are allowed [29, 31]. In the BET model, the particles in each layer act as the adsorption sites for the subsequent layers. There are many refinements to this approach, in which parameters such as sticking coefficient, activation energy, etc, are considered to be different for each layer. 

Surface areas are deterrnined routinely and exactiy from measurements of the amount of physically adsorbed, physisorbed, nitrogen. Physical adsorption is a process akin to condensation the adsorbed molecules interact weakly with the surface and multilayers form. The standard interpretation of nitrogen adsorption data is based on the BET model (45), which accounts for multilayer adsorption. From a measured adsorption isotherm and the known area of an adsorbed N2 molecule, taken to be 0.162 nm, the surface area of the soHd is calculated (see Adsorption). 

In this chapter, we are going to show that using the one- and the two-component multilayer adsorption isotherm models or the models taking into the account lateral interactions among the molecules in the monolayer (discussed in Section 2.1), the overload peak profiles presented in Section 2.4 can be qualitatively modeled. 

The most spectacular peak profiles, which suggest self-associative interactions, were obtained for 5-phenyl-1-pentanol on the Whatman No. 1 and No. 3 chromatographic papers (see Figure 2.15 and Figure 2.16). Very similar band profiles can be obtained using the mass-transfer model (Eqnation 2.21), coupled with the Fowler-Guggenheim isotherm of adsorption (Equation 2.4), or with the multilayer isotherm (Equation 2.7). 

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