The BET Isotherm

a plot of x/[n(l - x)] versus x gives a straight line and c and Mq can be obtained from the slope and the intercept of the line. 

The BET equation has been used as the general method for the determination of surface area from adsorption data because experimentally it is relatively easy to apply. This will be discussed in detail in Section 3.3.2. 

Equation (3.12) can fit well in the usual region of the BET equation and has been used to estimate surface area with a single point [11]. This one point method has been incorporated into commercial equipment for rapid surface area determinations. 

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The classical isotherm for multilayer adsorption on a homogeneous, flat surface is the BET isotherm [Brunauer, Emmett, and Teller, J. Am. Chem. Soc., 60, 309 (1938)]... 

Figure 5.17. To derive the BET isotherm the surface is divided into regions covered by / adsorbate layers each region is characterized by a fractional coverage Oj...

The BET method has its limitations and several improvements exist, but these are beyond the scope of our treatment. We note that the BET isotherm is valid under the following assumptions ... 

Give the essential assumptions made in deriving the BET isotherm. Ho v does the BET isotherm differ from the Langmuir isotherm ... 

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-... 

For a given value of n, a plot of straight line from the slope of which V may be calculated. The simpler, infinite, form of the BET isotherm is given by equation 17.15. The appropriate plot gives a straight line, from the slope and intercept of which V may be calculated. Equation 17.15 is most likely to apply at low relative pressures. 

The results of Na versus Pa are analyzed by an equation first derived by Brunauer, Emmet, and Teller, and the resultant isotherm is called the BET isotherm. Typically one measures the amount of N2 adsorbed for a particular pressure at 78 K (the boiling point of N2 at a pressure of 1 atm) as sketched in Figure 7-24. There are several regimes of an adsorption isotherm. At low densities the density increases linearly with pressure. When the density approaches one monolayer, the surface saturates. As the pressure approaches the saturation pressure of the gas, bulk condensation of liquid OCCUrs. This condensation can occur preferentially in pores of the solid due to capillary condensation, and the amount of gas and pressure where this occurs can be used to determine the pore volume of the catalyst. 

As we see in the course of the chapter, these two approaches frequently clash. The adsorption isotherm of Brunauer, Emmett, and Teller (BET), which is discussed in Section 9.5, is an excellent example of this. The model on which the BET isotherm is based has been criticized by many theoreticians. At the same time, the isotherm itself has become virtually... 

The BET isotherm has subsequently been derived by a variety of methods, but, as mentioned in the previous section, we follow the approach of the original derivation, namely, a... 

How is the BET isotherm used to determine the surface area of powders ... 

At low pressures, the coverage of adsorbed species increases linearly with. However, as the overpressure of A gets large, the amount of adsorbed A(s) begins to saturate. That is, the coverage begins to approach 6a = 1, which is a monolayer, and (in this model) further adsorption cannot take place. (The BET isotherm, discussed in Section 11.4.6, describes multilayer adsorption.)... 

Usually specific surface areas are determined from adsorption experiments. To illustrate this let us assume that adsorption of a specific sample is adequately described by the Langmuir Eq. (9.22). From fitting experimental results we obtain Tmon in units of mol/g. Then we assume a reasonable value for the cross-section area of a gas molecule a a, and obtain the specific surface from J2 = rmon ANA- In most practical applications the BET adsorption isotherm is used instead of the Langmuir Eq. (9.22) because it fits better. From a fit with the BET isotherm we get Tmon or nmon. Some cross-sectional areas for suitable gases in A2 are N2 16.2 02 14.1 Ar 13.8 n-C4Hi0 18.1. 

In these equations, a, n, a, and fi are constants. Adsorption of noninteracting molecules on top of each other to form multiple layers leads to the BET isotherm (Brunnauer, Emmett, and Teller). 

Langmuir (1918) laid down a systematic and theoretically grounded adsorption isotherm, the well-known Langmuir isotherm, which has served as a basis for a number of future isotherms, such as the BET isotherm, described in 1.4.1.3. The basic assumptions of the Langmuir adsorption model include monolayer coverage, localized adsorption (i.e., the specific adsorption sites exist and the interactions are between a specific molecule and the site), and homogeneous materials. In addition, the heat of adsorption is independent of coverage. He proposed that the rate of adsorption is proportional to the dimensionless pressure of the adsorbate, p/po (= p ), where po is the saturated vapor pressure, and the... 

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... 

Most commercial instruments using gas adsorption for surface area and porosity determination are based on the BET isotherm. In Eq. (1.45), the monolayer capacity Vm can be used to calculate the surface area on the basis of the area occupied by each adsorbed gas molecule. According to Eq. (1.45), a plot of p/[Va(p0 - p)] versus p/po is linear. From the slope and the intercept, Vm can be obtained. Thus, the specific surface area S can be obtained as... 

Consider the physical assumptions used to derive the BET isotherm. List three ways in which these assumptions might be in error. 

The Langmuir isotherm describes chemisorption, whereas the BET isotherm describes physisorption. Why is more appropriate to estimate surface areas using the latter isotherm ... 

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... 

In most applications, measurement of specific surface area is synonymous with the use of nitrogen and the BET isotherm. For most commercial surface area instruments, at least 1 m2 of surface area must be present in the sample cell for the measurement to be repeatable. This requires that a significant mass of material (> 1 g) must be present in the sample cell for low surface-area materials. When these conditions cannot be met due to material availability, an instrument that uses krypton as the adsorbate should be utilized. Krypton allows specific surface area measurement with as little as 0.1 m2 present in the sample. 

Inverse gas chromatography (IGC) is another technique that can be used to measure the specific surface area of a particulate material, as well as to measure a number of surface thermodynamic properties of powders. Such instrumentation operates on a different principle than traditional nitrogen/krypton adsorption using the BET isotherm. 

The deduction of the BET isotherm equation can be carried out by means of the grand canonical ensemble approach applying a methodology developed by Hill [12], which is exposed elsewhere [2], The BET isotherm obtained is expressed as follows [2,4,5]... 

In order to apply real adsorption data to the BET isotherm equation, it is customary to use Equation 6.28a in the linear form... 

The fitting process of the BET isotherm equation, as was persistently stated before, can be also carried out with the help of a nonlinear regression method [54],... 

The Langmuir isotherm is based on the formation of a saturated unimolecular layer of adsorbate molecules on the surface of the adsorbent at equilibrium. The BET isotherm (Bruauser, Emmett, and Teller) assumed that a multimolecular layer of adsorbate molecules covers the surface of the adsorbent and that each layer behaves as the Langmuir isotherm. It is not necessary to cover the entire bare surface with the adsorbate to form the multilayers. When using the Langmuir treatment, take into account the multilayer adsorption in which the sequential layers have equal adsorption energies. The BET isotherm is then written in the form of ... 

FIGURE 4.5 The BET isotherm of water vapor on highly hydroscopic herbal extracts (a) regular form (b) linear form. [Graph reconstructed from data by Chu and Chow, Pharm. Res., 17, 1133 (2000).]... 

The BET isotherm (Adamson, 1982) addresses some of the shortcomings of the Langmuir isotherm most important, the restriction of one adsorbed layer. The resulting isotherm is... 

In the pressure range 0.05 < P/P < 0.35, the quantity [V(P /P) — 1] 1 increases linearly with P/P from the slope s" and intercept "i", one obtains bm0n0 = 1 /(s + i) and c = 1 + s/i. Thus, the BET isotherm measures the effective specific surface of a particulate solid support substances like silica gel, zeolite, and so on, can have a specific surface useful for heterogeneous catalysis, provided that it exceeds 200 m2 g... 

Fig. 12.3. The adsorption of nitrogen on nonporous samples of silica and alumina, the surface areas ranging from 2.6-11.5 m2 g 1 for silica and from 58-153 m2 g I for alumina. n/nm is plotted against p/p°. n is the number of moles adsorbed per gram of adsorbent and nm is the monolayer capacity, expressed in the same units. Line A = silica + = alumina. Line B the BET isotherm...

A form of isotherm widely used for foods is the one described by Brunauer et al. (1938) and known as the BET isotherm or equation. A form of the BET equation given by Labuza(1968)is... 

A plot of a/(l - a)V versus a gives a straight line, as indicated in Figure 1-10. The mono-layer coverage value can be calculated from the slope and the intercept of the line. The BET isotherm is only applicable for values of a from 0.1 to 0.5. In addition to monolayer coverage, the water surface area can be calculated by means of the following equation ... 

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 ... 

X-Ray powder diffraction patterns of these materials were obtained on a Philips APD 3720 Automated Powder X-Ray diffractometer using CuK alpha radiation. Surface areas were measured with a Micromeritics Flowsorb II Model 2300 Analyzer with P/Po =0.3 utilizing the BET isotherm. 

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