BET theory

By following the standard procedures of statistical mechanics, one arrives at an equation which can be converted into the BET equation (2.12) by the simple substitution 9i/9i , ki = c. Thus parameter c acquires a significance different from that in the BET theory in essence it now involves entropic terms as well as energetic terms. 

The external surface area of the filler can be estimated from a psd by summing the area of all of the equivalent spheres. This method does not take into account the morphology of the surface. It usually yields low results which provide Htde information on the actual area of the filler that induences physical and chemical processes in compounded systems. In practice, surface area is usually determined (5) from the measured quantity of nitrogen gas that adsorbs in a monolayer at the particle surface according to the BET theory. From this monolayer capacity value the specific surface area can be determined (6), which is an area per unit mass, usually expressed in m /g. 

Most surface area measurements are based on the interpretation of the low temperature equilibrium adsorption of nitrogen or of krypton on the solid using the BET theory [33,269,276—278]. There is an extensive literature devoted to area determinations from gas adsorption data. Estimates of surfaces may also be obtained from electron micrographs, X-ray diffraction line broadening [279] and changes in the catalytic activity of the solid phase [ 280]. 

Conventional bulk measurements of adsorption are performed by determining the amount of gas adsorbed at equilibrium as a function of pressure, at a constant temperature [23-25], These bulk adsorption isotherms are commonly analyzed using a kinetic theory for multilayer adsorption developed in 1938 by Brunauer, Emmett and Teller (the BET Theory) [23]. BET adsorption isotherms are a common material science technique for surface area analysis of porous solids, and also permit calculation of adsorption energy and fractional surface coverage. While more advanced analysis methods, such as Density Functional Theory, have been developed in recent years, BET remains a mainstay of material science, and is the recommended method for the experimental measurement of pore surface area. This is largely due to the clear physical meaning of its principal assumptions, and its ability to handle the primary effects of adsorbate-adsorbate and adsorbate-substrate interactions. 

The adsorption/desorption isotherms measured by NMR (equivalent to conventionally measured isotherms), extracted from two different regions of the imaging field of view corresponding to the two ceramics, are shown in Figure 3.5.9. Once these local isotherms are extracted, they are simply the local adsorption for that point in space contained within the material, measured non-invasively and non-destructively. Conventional analysis techniques for adsorption isotherms (such as BET theory) can therefore be applied to the data, to determine the microstructural properties corresponding to that isotherm curve. 

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 major advance in adsorption theory generalized the treatment of monolayer adsorption and incorporated the concept of multilayer adsorption. This is known as the BET theory after Brunauer, Emmett and Teller [32], The adsorption of a gas on a solid surface can be described by... 

Table 5.2 Comparison of specific surface areas calculated from the ascending branch of the water isotherms of selected cellulosic materials by the BET theory, the ct procedure and the Zimm and Lundberg (ZL) cluster theory.

In 1938, Brunauer, Emmett and Teller(12) and Emmett and de Witt(13) developed what is now known as the BET theory. As in the case in Langmuir s isotherm, the theory is based on the concept of an adsorbed molecule which is not free to move over the surface, and which exerts no lateral forces on adjacent molecules of adsorbate. The BET theory does, however, allow different numbers of adsorbed layers to build up on different parts of the surface, although it assumes that the net amount of surface which is empty or which is associated with a monolayer, bilayer and so on is constant for any particular equilibrium condition. Monolayers are created by adsorption on to empty surface and by desorption from bilayers. Monolayers are lost both through desorption and through the adsorption of additional layers. The rate of adsorption is proportional to the frequency with which molecules strike the surface and the area of that surface. From the kinetic theory of gases, the frequency is proportional to the pressure of the molecules and hence ... 

The BET theory assumes that the reasoning used for one or two layers of molecules may be extended to n layers. It argues that energies of activation after the first layer are all equal to the latent heat of condensation, so that ... 

Popper and Bariska (1972) studied the moisture sorption properties of wood chemically modified with acetic (or phthalic) anhydride and analysed the results using Brunauer-Emmett-Teller (BET) theory and the H-H model. Acetylation was found to reduce the number of sorption sites, whereas little effect was noted with phthaloylation. By dividing the sorption isotherm into a monolayer component and a multilayer component using the H-H model, it was shown that there was a large reduction in the... 

For membranes with pore diameters smaller than 3.5 nm, the nitrogen adsorption/desorption method based on the widely used BET theory ean be employed. This measurement technique, however, is good only for pore diameters ranging from 1.5 nm to 100 nm ( = 0.1 micron). Typical data from this method are split into two portions adsorption and desorption. The nitrogen desorption curve is usually used to describe the pore size distribution and corresponds better to the mercury intrusion curve. Given in Figure... 

Models for Calculation of Surface Area and Pore Sizes BET Theory for the determination of surface area... 

The enthalpy involved in multilayers is related to the differences, and was defined by BET theory as... 

Brunauer, Emmett and Teller, in 1938, extended Langmuir s kinetic theory to multilayer adsorption. The BET theory assumes that the uppermost molecules in adsorbed stacks are in dynamic equilibrium with the vapor. This means that, where the surface is covered with only one layer of adsorbate, an equilibrium exists between that layer and the vapor, and where two layers are adsorbed, the upper layer is in equilibrium with the vapor, and so forth. Since the equilibrium is dynamic, the actual location of the surface sites covered by one, two or more layers may vary but the number of molecules in each layer will remain constant. 

The BET theory assumes that the terms v, E and A remain constant for the second and higher layers. This assumption is justifiable only on the grounds that the second and higher layers are all equivalent to the liquid... 

Rarely, if ever, does the BET theory exactly match an experimental isotherm over its entire range of relative pressures. In a qualitative sense, however, it does provide a theoretical foundation for the various isotherm shapes. Of equal significance is the fact that in the region of relative pressures near completed monolayers (0.05 P/Pq 0.35) the BET theory and experimental isotherms do agree very well, leading to a powerful and extremely useful method of surface area determination. 

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. 

Figure 4.3 illustrates how the lateral interactions and the surface interactions can sum to a nearly constant overall adsorption energy up to completion of the monolayer, an implicit assumption of the BET theory. This results in a constant C value from equation (4.22)... 

A further criticism of the BET theory is the assumption that the heat of adsorption of the second and higher layers is equal to the heat of liquefaction. It seems reasonable to expect that polarization forces would induce a higher heat of adsorption in the second layer than in the third, and so forth. Only after several layers are adsorbed should the heat of adsorption equal the heat of liquefaction. It is, therefore, difficult to resolve a model of molecules adsorbed in stacks while postulating that all layers above the first are thermodynamically a true liquid structure. The apparent validity of these criticisms contributes to the failure of the BET equation at high relative pressures (P/Pq > 0.35). However, in the range of relative pressure leading to coverage near W/ = 1, the BET C values... 

The BET theory requires that a plot of [W Pq/P — 1)] versus P/Pq be linear with a finite intercept [cf. equation (4.38) and Fig. 4.1]. By reducing the experimental requirement to only one data point, the single-point method offers the advantages of simplicity and speed often with little loss in accuracy. 

Because of its simplicity and straightforward applicability, the BET theory is almost universally employed for surface area measurements. However, other methods and theoretical models have been developed which are briefly outlined in this chapter. No attempt is made to completely derive and discuss these alternate methods but rather to present their essential features and to indicate how they may be used to calculate surface areas. 

An important contribution made by the Harkins and Jura absolute method, however, must not be overlooked. Their measurements of some specific surface areas give confirmation to the value of 16.2 for the cross-sectional area of nitrogen. This value, when employed with the BET theory, gave exactly the same specific surface area as the HJ absolute method. 

Several modifications of Poiseuille s equation have been attempted by various authors to describe permeability in the transitional region between viscous and diffusional flow. The assumptions underlying these modifications are often questionable and the results obtained offer little or no theoretical or experimental advantage over the BET theory for surface area measurements. Allen" " discusses these modifications as well as diffusional flow at low pressures. 

As stated earlier the C value in micropores will be large due to the overlapping wall potentials. Under these circumstances, the surface Will be covered well over 90% by stacks of adsorbate not in excess of two molecules in depth as shown by equation (4.45) and Table 4.1. Therefore, the close proximity of the walls offer no special condition which is not already allowed for by the BET theory. 

The most definitive surface area measurements are probably those made by nitrogen adsorption using the BET theory. Neither the Brunauer, Emmett and Teller (BET) theory nor equation (11.5), used to calculate surface area from mercury intrusion data makes any assumptions regarding pore shape for surface area determinations. When these two methods are compared there is often surprisingly good agreement. When... 

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