BET equation for multilayer adsorption

The most important step in the study of adsorption came with a derivation by Brunauer, Emmett and Teller for the multilayer adsorption of gases on solid surfaces [22]. The multilayer adsorption theory, known generally as the BET equation, has occupied a central position in gas adsorption studies and surface area measurement ever since. 

On the assumption that the forces that produce condensation are chiefly responsible for the binding energy of multilayer adsorption, ey proceeded to derive an isotherm equation for multilayer adsorption by a method that was a generalization of Langmuir s treatment of the unimolecular layer. The generalization of the ideal localized monolayer treatment is effected by assuming that each first layer adsorbed molecule serves as a site for the adsorption of a molecule into the second layer and so on. Hence, the concept of localization prevails at all layers and forces of mutual interaction are neglected. 

Si represent the fractional surface covered with 0. 1, 2,1 layers of adsorbate molecules. At equilibrium, the rate of condensation on Sq equals the rate of evaporation from Sj giving  

An essentially similar equation had been arrived at earlier by Baly [23], who could proceed further only by empirical means. 

Brunauer et al. [24] proceeded to solve this summation using two simplifying assumptions, that  

---

The most common way of analyzing such data is by using the so-called BET equation. For multilayer adsorption, this equation can be set out in the form... 

Particle Surface Area Determination Methods From the standard definitions of particle surface area, it can be seen that various determination methods are used for surface area measurement, such as adsorption (including Langmuir s equation for monolayer adsorption and the BET equation for multilayer adsorption), particle size distribution, and permeability methods. The different methods are rarely in agreement because the value obtained depends upon the procedures used and also on the assumptions made in the theory relating the surface area to the phenomena measured. The most common methods used for measuring particle surface area are described below. 

Surface area determination by gas adsorption 47 2.4 BET equation for multilayer adsorption... 

Perhaps the most important uses of gas adsorption are the estimation of the surface area of materials, the definition of the type of porosity, the computation of pore volumes, and the calculation of pore-size disfribution. For these purposes, an equation for multilayer adsorption will clearly be most usefirl. As is well known, the generalization of Langmuir s equation to the multilayer case is the so-called BET (Brunauer-Emmett-Teller) equation or BET model [164],... 

For microporous materials the 5bet values obtained are usually much higher than the real surface area, because in the region where the BET equation is applied (this equation assumes multilayer adsorption but not condensation) conden.sation already takes place. 

The BET equation has been derived for multilayer adsorption data. 

The theory of Brunauer, Emmett and Teller167 is an extension of the Langmuir treatment to allow for multilayer adsorption on non-porous solid surfaces. The BET equation is derived by balancing the rates of evaporation and condensation for the various adsorbed molecular layers, and is based on the simplifying assumption that a characteristic heat of adsorption A Hi applies to the first monolayer, while the heat of liquefaction, AHL, of the vapour in question applies to adsorption in the second and subsequent molecular layers. The equation is usually written in the form... 

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... 

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... 

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°. 

For multilayer adsorption, i.e., the Type II isotherm in Fig. 4.3, Langmuir equation can be used after a modification with several assumptions [38]. The equation is usually known as the BET equation, which is given by ... 

This is not the total capadty of the activated carbon to hold solvent because the BET Equation allows for multilayer adsorption As used in the Langmuir Equation, this parameter can take on values no greater than s ax (as solvent Is present only in a monolayer)... 

In the first articles eonneeted with the eonsidered extensions of the BET isotherm, diserete energy distributions were used [90,91]. The later studies, based on the integral equation (10), have a more universal eharacter [6]. For multilayer adsorption, this equation can be easily solved in two cases [67] (1) when the loeal adsorption isotherm is a product of the monolayer eoverage and a function describing the formation of higher adsorbed layers, namely... 

Various boundary conditions limit each of the theories, hence a range of equations have been developed to cover the various phenomena equation developed by Brunauer, Emmett and Teller commonly known as the BET equation. This equation is for multilayer adsorption, but is based upon the Langmuir equation where adsorption is restricted to a monolayer. Both of these equations are developed below, although the application of the Langmuir equation to gas adsorption is restricted to adsorption in micropores where adsorption is limited to a monolayer due to pore geometry. Langmuir adsorption isotherms are common in adsoiption of solute from solution. 

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. 

BET method. The most commonly used method for determining the specific surface area is the so-called BET method, which obtained its name from three Nobel prize winners Brunauer, Emmett and Teller (1938). It is a modification of the Langmuir theory, which, besides monolayer adsorption, also considers multilayer adsorption. The equation allows easy calculation of the surface area, commonly referred to as the BET surface area ( bet). From the isotherms also pore-radii and pore-volumes can be calculated (from classical equation for condensation in the pores). 

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 individuals responsible for its formulation, Brunauer, Em-... 

The BET equation describes the phenomenon of multilayer adsorption, which is characteristic of physical or van der Waals interactions. In the case of gas adsorption, for example, multilayer adsorption merges directly into capillary condensation... 

Until now, we have focused our attention on those adsorption isotherms that show a saturation limit, an effect usually associated with monolayer coverage. We have seen two ways of arriving at equations that describe such adsorption from the two-dimensional equation of state via the Gibbs equation or from the partition function via statistical thermodynamics. Before we turn our attention to multilayer adsorption, we introduce a third method for the derivation of isotherms, a kinetic approach, since this is the approach adopted in the derivation of the multilayer, BET adsorption isotherm discussed in Section 9.5. We introduce this approach using the Langmuir isotherm as this would be useful in appreciating the common features of (and the differences between) the Langmuir and BET isotherms. 

As noted above, the range of pressures over which gas adsorption studies are conducted extends from zero to the normal vapor pressure of the adsorbed species p0. An adsorbed layer on a small particle may readily be seen as a potential nucleation center for phase separation at p0. Thus at the upper limit of the pressure range, adsorption and liquefaction appear to converge. At very low pressures it is plausible to restrict the adsorbed molecules to a mono-layer. At the upper limit, however, the imminence of liquefaction suggests that the adsorbed molecules may be more than one layer thick. There is a good deal of evidence supporting the idea that multilayer adsorption is a very common form of physical adsorption on nonporous solids. In this section we are primarily concerned with an adsorption isotherm derived by Brunauer, Emmett, and Teller in 1938 the theory and final equation are invariably known by the initials of the authors BET. 

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. 

---