Derivation of the BET Equation

The basic assumption of the BET theory is that the Langmuir isotherm can be applied to every adsorption layer. The theory postulates that the first layer of adsorbed molecules acts as a base for the adsorption of the second layer of molecules, which in turn acts as a base for the third layer, and so on, so that the concept of localization is maintained in all layers. Furthermore, the forces of interaction between the adsorbed molecules are also neglected, as in the case of Langmuir concept. 

Let us suppose 5q, 2... 5, represents the surface areas that are covered by 

i molecular layers of adsorbed molecules. Because at equilibrium the surface area is constant, the rate of condensation on the bare surface is equal to the rate of evaporation from the first layer, so that 

Extending the argument to the second and consecutive layers, the general equation of equilibrium between the (i -1) layer and ith layer can be written as 

The total surface area S of the adsorbent is equal to the surface area of all the layer and is, therefore, given by 

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The assumption of monolayer adsorption in the Langmuir isotherm model is unrealistic in most cases, and a modification to multilayer adsorption should be considered. In 1938, Brunauer, Emmett, and Teller modified the Langmuir approach of balancing the rates of adsorption and desorption for the various molecular layers [Brunauer et al., 1938], This approach is known as the BET method. The BET isotherm assumes that the adsorption of the first layer has a characteristic heat of adsorption A Ha and the adsorption and desorption on subsequent layers are controlled by the heat of condensation of the vapor, A Hc. The derivation of the BET equation is beyond the scope of this book however, a common form of the BET equation is given as... 

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 original derivation of the BET equation was an extension and generalization of Langmuir s treatment of monolayer adsorption. This derivation is based on kinetic considerations—in particular on the fact that at equilibrium the rate of condensation of... 

Based on the assumptions in the kinetic derivation of the BET equation, the constant C is related to the... 

It is an empirical fact that adsorption of vapor molecules may occur in two places within the carbon pores (1) the activated sites, and (2) the vapor molecules already adsorbed on the sites. The former is adsorption of a single layer of solvent the latter is adsorption of multiple layers. Derivation of the BET Equation described in Appendix A2 was based on this fact. 

Derivation of the BET equation is outside the scope of this book (Box A2.1). But it is shown as Equation A2-17. 

In general, the BET equation fits adsorption data quite well over the relative pressure range 0.05-0.35, but it predicts considerably more adsorption at higher relative pressures than is experimentally observed. This is consistent with an assumption built into the BET derivation that an infinite number of layers are adsorbed at a relative pressure of unity. Application of the BET equation to nonpolar gas adsorption results is carried out quite frequently to obtain estimates of the specific surface area of solid samples. By assuming a cross-sectional area for the adsorbate molecule, one can use Wm to calculate specific surface area by the following relationship ... 

The success of the BET equation in representing experimental data should not be regarded as a measure of the accuracy of the model on which it is based. Its capability of modelling the mobile multilayers of a Type IV isotherm is entirely fortuitous because, in the derivation of the equation, it is assumed that adsorbed molecules are immobile. 

This kinetic-theory-based view of the Langmuir result provides no new information, but it does draw attention to the common starting assumptions of the Langmuir derivation and the BET derivation (Section 9.5a). This kinetic derivation of the Langmuir equation is especially convenient for obtaining an isotherm for the adsorption of two gases. This is illustrated in Example 9.4. 

In principle, each adsorbed layer has a different set of values of a, b, and , but the derivation of the BET isotherm equation is dependent on two main assumptions ... 

Bulk Si/Al ratios were determined by AAS. Surface areas and pore volumes were determined by N2 absorption isotherms measured at liquid nitrogen temperature using a Micromeritics ASAP 2000M (Table 1). The zeolites were degassed under vacuum at 150°C for the as-s)mthesised and 450°C for the modified zeolites for at least 3 hours. The total surface area was derived using the BET equation [12], the micropore volume and the external surface area (ESA) were estimated by means of the t-plot method of Lippens et al [13] and the total and mesopore volumes were calculated by Barrett-Joyner-Halenda anaylsis of the desorption branch of the N2 isotherm [14]. 

Specific surface areas (Sbet for investigated samples, were calculated from the linear form of the BET equation over the range of relative pressure between 0.05 and 0.4, taking the cross-sectional area of the nitrogen molecule to be 16.2 A. The pore size distributions were derived from the desorption isotherm using the BJH method [14]. The total pore volumes, V, were calculated from adsorption isotherms at p/Po 0-98 by assuming that complete pore filling by the condensate had occurred. 

X-ray powder diffraction (XRD) patterns were taken on a Spectrolab CPS Series 3000 120 diffractometer, using Ni filtered Cu Ka radiation. The nitrogen adsorption isotherms were determined at 77 K by means of a Micromeritics Gemini 2370 surface area analyser. Surface areas were derived from the BET equation in the relative pressure range 0.05-0.25, assuming a cross-sectional area of 0.162 nm" for the nitrogen molecule [ 18]. 

The Hiittig and BET models are the same except for a change in the kinetic mechanism of attaining equilibrium. Now the statistical derivation (48) of the BET equation is independent of any choice of kinetic mechanism, and since otherwise the two models are identical, one is forced to conclude that the Hiittig kinetic mechanism is impossible [see (3)]. 

Inasmuch as the Langmuir equation does not allow for nonuniform surfaces, interactions between neighboring adsorbed species, or multilayer adsorption, a variety of theoretical approaches that attempt to take one or more of these factors into account have been pursued by different investigators. The best known alternative is the BET isotherm, which derives its name from the initials of the three people responsible for its formulation Brunauer, Enunett, and Teller (8). It takes multilayer adsorption into account and is the basis of standard methods for determining specific surface areas of heterogeneous catalysts. The extended form of the BET equation (4) can be used to derive relations for all five types of isotherms as special cases. 

The second and further layers start to build up before the completion of the first one If the application of the BET equation was to be limited to the type of adsorbent assumed above (an energetically uniform surface and no pores), there would probably not be many people to remember it to-day. In reality, most interesting adsorbents are either heterogeneous from the viewpoint of adsorption energy, or porous, or both. Finally, assumptions 1/ and 2/ are exceptionally fulfilled, if ever, assumption 3/ does not hold for porous adsorbents, assumption 4/ is an acceptable approximation, assumption 5/ is incorrect, and, finally, only assumption 6/ is usually right. .. except for those ultra-micropores whose width cannot accommodate more than two molecules. Moreover, at the time of deriving a surface area from the monolayer content, three other assumptions are used ... 

Barrer et al. [SO] derived eighteen analogs of the BET equation by making various assumptions as to the evaporation-condensation properties of the molecules in each layer. 

Several attempts have been made to extend the scope of the BET equation. Hiittig [85] assumed that the evaporation of the ith layer molecule was unimpeded by the presence of molecules in the (/-l)th layer whereas in the BET derivation it is assumed that they are completely effective in preventing the evaporation of underlying molecules. Hiittig s final equation is ... 

A criterion as to the correctness of the BET equation is its agreement with other models such as that derived from the Kelvin equation for pressure lowering over a concave meniscus. This has been applied to a porous adsorbent as follows ... 

Because of their prevalence in physical adsorption studies on high-energy, powdered solids, type II isotherms are of considerable practical importance. Bmnauer, Emmett, and Teller (BET) [39] showed how to extent Langmuir s approach to multilayer adsorption, and their equation has come to be known as the BET equation. The derivation that follows is the traditional one, based on a detailed balancing of forward and reverse rates. 

Although the preceding derivation is the easier to follow, the BET equation also may be derived from statistical mechanics by a procedure similar to that described in the case of the Langmuir equation [41,42]. 

Detailed derivations of the isothemi can be found in many textbooks and exploit either statistical themio-dynaniic methods [1] or independently consider the kinetics of adsorption and desorption in each layer and set these equal to define the equilibrium coverage as a function of pressure [14]. The most conmion fomi of BET isothemi is written as a linear equation and given by ... 

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. 

When the value of c exceeds unity, the value of n can be derived from the slope and intercept of the BET plot in the usual way but because of deviations at low relative pressures, it is sometimes more convenient to locate the BET monolayer point , the relative pressure (p/p°) at which n/n = 1. First, the value of c is found by matching the experimental isotherm against a set of ideal BET isotherms, calculated by insertion of a succession of values of c (1, 2, 3, etc., including nonintegral values if necessary) into the BET equation in the form ... 

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