The BET model

To obtain the monolayer capacity from the isotherm, it is necessary to interpret the (Type II) isotherm in quantitative terms. A number of theories have been advanced for this purpose from time to time, none with complete success. The best known of them, and perhaps the most useful in relation to surface area determination, is that of Brunauer, Emmett and Teller. Though based on a model which is admittedly over-simplified and open to criticism on a number of grounds, the theory leads to an expression—the BET equation —which, when applied with discrimination, has proved remarkably successful in evaluating the specific surface from a Type II isotherm. 

The BET treatment is based on a kinetic model of the adsorption process put forward more than sixty years ago by Langmuir, in which the surface of the solid was regarded as an array of adsorption sites. A state of dynamic equilibrium was postulated in which the rate at which molecules arriving from the gas phrase and condensing on to bare sites is equal to the rate at which molecules evaporate from occupied sites. 

If the fraction of sites occupied is 0, and the fraction of bare sites is 0q (so that 00 + 1 = 0 then the rate of condensation on unit area of surface is OikOo where p is the pressure and k is a constant given by the kinetic theory of gases (k = jL/(MRT) ) a, is the condensation coefficient, i.e. the fraction of incident molecules which actually condense on a surface. The evaporation of an adsorbed molecule from the surface is essentially an activated process in which the energy of activation may be equated to the isosteric heat of adsorption 4,. The rate of evaporation from unit area of surface is therefore equal to 

If rt (in moles) is the amount adsorbed on 1 g of adsorbent, then 6i = n/n , where rt is the monolayer capacity. Insertion into (2.5) leads to 

Equation (2.6) is the familiar Langmuir equation for the case when adsorption is confined to a monolayer. In practice B is an empirical constant and cannot be evaluated from the relationship in Equation (2.7). The question as to how well the Langmuir equation reproduces experimental isotherms will be dealt with in Chapter 4. 

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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. 

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. 

An alternative way of deriving the BET equation is to express the problem in statistical-mechanical rather than kinetic terms. Adsorption is explicitly assumed to be localized the surface is regarded as an array of identical adsorption sites, and each of these sites is assumed to form the base of a stack of sites extending out from the surface each stack is treated as a separate system, i.e. the occupancy of any site is independent of the occupancy of sites in neighbouring stacks—a condition which corresponds to the neglect of lateral interactions in the BET model. The further postulate that in any stack the site in the ith layer can be occupied only if all the underlying sites are already occupied, corresponds to the BET picture in which condensation of molecules to form the ith layer can only take place on to molecules which are present in the (i — l)th layer. 

From the earliest days, the BET model has been subject to a number of criticisms. The model assumes all the adsorption sites on the surface to be energetically identical, but as was indicated in Section 1.5 (p. 18) homogeneous surfaces of this kind are the exception and energetically heterogeneous surfaces are the rule. Experimental evidence—e.g. in curves of the heat of adsorption as a function of the amount adsorbed (cf. Fig. 2.14)—demonstrates that the degree of heterogeneity can be very considerable. Indeed, Brunauer, Emmett and Teller adduced this nonuniformity as the reason for the failure of their equation to reproduce experimental data in the low-pressure region. 

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). 

A number of the assumptions used in the BET theory have been questioned for real samples [6]. One assumption states that all adsorption sites are energetically equivalent, which is not the case for normal samples. The BET model ignores lateral adsorbate interactions on the surface, and it also assumes that the heat of adsorption for the second layer and above is equal to the heat of liquefaction. This assumption is not valid at high pressures and is the reason for using adsorbate pressures less than 0.35. In spite of these concerns, the BET method has proven to be an accurate representation of surface area for the majority of samples [9,10]. 

A number of models have been developed for the analysis of the adsorption data, including the most common Langmuir [49] and BET (Brunauer, Emmet, and Teller) [50] equations, and others such as t-plot [51], H-K (Horvath-Kawazoe) [52], and BJH (Barrett, Joyner, and Halenda) [53] methods. The BET model is often the method of choice, and is usually used for the measurement of total surface areas. In contrast, t-plots and the BJH method are best employed to calculate total micropore and mesopore volume, respectively [46], A combination of isothermal adsorption measurements can provide a fairly complete picture of the pore size distribution in sohd catalysts. Mary surface area analyzers and software based on this methodology are commercially available nowadays. 

Brunauer-Emmett-Teller (BET) adsorption describes multi-layer Langmuir adsorption. Multi-layer adsorption occurs in physical or van der Waals bonding of gases or vapors to solid phases. The BET model, originally used to describe this adsorption, has been applied to the description of adsorption from solid solutions. The adsorption of molecules to the surface of particles forms a new surface layer to which additional molecules can adsorb. If it is assumed that the energy of adsorption on all successive layers is equal, the BET adsorption model [36] is expressed as Eq. (6) ... 

When measured adsorption data are plotted against the concentration value of the adsorbate at equilibrium, the resulting graph is called an adsorption isotherm. The mathematical description of isotherms invariably involves adsorption models described by Langmuir, Freundlich, or Brauner, Emmet and Teller (known as the BET-model). Discussion of these models is given in Part 111, as conditions relevant to chemical-subsurface interactions are examined. 

In spite of the success of the BET theory, some of the assumptions upon which it is founded are not above criticism. One questionable assumption is that of an energetically homogeneous surface, that is, all the adsorption sites are energetically identical. Further, the BET model ignores the influence of lateral adsorbate interactions. 

Usually adsorption is more realistically described by the BET model. BET theory accounts for multilayer adsorption. The adsorption isotherm goes to infinity at relative partial pressures close to one, which corresponds to condensation. 

The Langmuir and Brumnauer, Emmett, and Teller (BET) models also have been used to describe nonlinear sorption behavior for environmental solids, particularly for mineral dominated sorption (Ruthven, 1984 Weber et al., 1992). The Langmuir model assumes that maximum adsorption corresponds to a saturated monolayer of solute molecule on the absorbent surface, that there is no migration of the solute on the surface phase, and that the energy of adsorption is constant. The BET model is an extension of the Langmuir model that postulates multilayer sorption. It assumes that the first layer is attracted most strongly to the surface, while the second and subsequent layers are more weakly held. 

The BET model can also be applied to a situation which might be applicable to porous solids. If adsorption is limited to n molecular layers (where n is related to the pore size), the equation... 

There is substantial theoretical objections to the BET model. Nevertheless, experimental data often conform to Equation (2.11) admirably (Figure 2.2), and the BET model is used extensively to determine areas of microscopically complicated surfaces. 

In spite of some grounded criticisms11,12,13,14,15 16 (the BET model assumes energetically identical adsorption sites, neglects lateral interactions and assumes identical behaviour in all layers of the multilayer adsorption), and the proposal of alternative models,17,18,19,20 the BET equation has retained its utility. It is a relatively easy approach and the method is applicable to a great variety of adsorption isotherms. 

Both the BET model and pore size distribution models have a rather low degree of accuracy. A divergence of at least 10% from the actual area is not exceptional.36 A way of checking the validity of the different models consists in comparing the cumulative pore area, as calculated by the BJH model, with the BET surface. Some results are presented in table 2.2. 

Besides specific surface area, silicas are also characterised by their porosity. Most of the silica s are made out of dense spherical amorphous particles linked together in a three dimensional network, this crosslinked network building up the porosity of the silica. Where the reactivity of diborane towards the silica surface has been profoundly investigated, little attention has been paid to the effect of those reactions on the pore structure. However different methods are developed to define the porosity and physisorption measurements to characterise the porosity parameters are well established. Adsorption isotherms give the specific surface area using the BET model, while the analysis desorption hysteresis yields the pore size distribution. 

The BET isotherm, like the isotherm developed by Langmuir (the first person to develop a rigorous model for gas adsorption), assumes that the adsorbing surface is energetically uniform, and that only one molecule could adsorb at each surface site. The BET isotherm is a generalized form of the Langmuir equation to account for multilayer adsorption, and assumes that after the adsorption of the first layer, the heat of condensation is equal to the heat of evaporation, and that the rates of adsorption for the second adsorbed layer and beyond are the same.29-31 From a practical perspective, variables in the equation must have specific values for the BET model to be valid, namely the y-intercept and BET constant, C, must be positive. Several excellent reviews of surface area measurement and gas adsorption can be found in References.6,32 34... 

Model 1. Inert adsorbent-hard sphere adsorbate—the BET model and its early modifications. 

An alternative and elegant derivation of the BET equation is by a statistical mechanical treatment (Hill, 1946 Steele, 1974). The adsorbed phase is pictured as a lattice gas that is molecules are located at specific sites in all layers. The first layer is localized and these molecules act as sites for molecules in the second layer, which in turn act as sites for molecules in the third layer, and so on for the higher layers. As the surface is assumed to be planar and uniform, it follows that all surface sites are identical. It is also assumed that the occupation probability of a site is independent of the occupancy of neighbouring sites. This is equivalent to the assumption that there are no lateral interactions between adsorbed molecules. In accordance with the BET model, the probability for site occupation is zero unless all its underlying sites are occupied. Furthermore, it is assumed that it is only the molecular partition function for the first layer which differs from that for molecules in the liquid state. 

The BET model appears to be unrealistic in a number of respects. For example, in addition to the Langmuir concept of an ideal localized monolayer adsorption, it is assumed that all the adsorption sites for multilayer adsorption are energetically identical and that all layers after the first have liquid-like properties. It is now generally recognized that the significance of the parameter C is oversimplified and that Equation (4.33) cannot provide a reliable evaluation of... 

A recent molecular simulation study (Seri-Levy and Avnir, 1993) has also revealed the artificial nature of the BET model and has illustrated the effect of taking adsorbate-adsorbate interactions into account. Thus, the addition of lateral interactions appears to flatten the BET stacks into more realistically shaped islands. 

An extension to the BET model was put forward by Brunauer, Deming, Deming and Teller (BDDT) in 1940. The BDDT equation contains four adjustable parameters and was designed to fit the isotherm Types I-V. From a theoretical standpoint, the BDDT treatment appears to offer very little more than the original BET theory and the cumbersome equation has very rarely been applied to experimental data. 

As was pointed out in Chapter 4, the BET model does not provide a realistic description of any known physisorption system. Indeed, it has been implied by some, critics that the BET plot is little more than a mathematical device for locating Point B. This opinion may be too harsh, but it must be acknowledged that if it were removed from the context of surface area determination, the BET theory would be unlikely to continue to attract much interest. This situation reinforces the need to, examine the limitations of the BET method and in particular to attempt to define the conditions which govern its application. 

Various other aspects of fractal analysis have been discussed by Van Damme and Fripiat and their co-workers. For example, by extending the BET model to fractal surfaces, Fripiat et al. (1986) were able to show that the apparent fractal dimension is reduced by the progressive smoothing of a molecularly rough surface. Alternatively, the effect of a micropore filling contribution is to enhance the fractal dimension. 

The BET model is strictly incompatible with the energetic heterogeneity exhibited by most solid surfaces. The range of linearity of the BET plot is always restricted to a limited part of a Type II isotherm, which rarely extends above p/p° 0.35 and in some cases no higher than pjp° 0.1. In fact, a more useful empirical relation for multilayer adsorption is the FHH equation, which is generally applicable over a wide range of pjp°. 

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