BET Isotherm model

Besides the heterogeneity of the adsorbent surface, the second major reason for the adsorption of a compound to deviate from Langmuir isotherm behavior is that the adsorbed molecules interact. In this category, we find the Fowler isotherm, the anti-Langmuirian isotherm, and several S-shaped isotherm models, including the quadratic isotherm, the extended BET isotherm models, and the Moreau model. 

The extended liquid-solid BET isotherm describes well the adsorption behavior corresponding to types II or III isotherms of the van der Waals classification of isotherms (see Figure 3.1). Its expression parallels that of the BET isotherm model which is often applied in gas-solid equiUbtia [3]. It assiunes the same molecular description the solute molecules can adsorb from the solution onto either the bare surface of the adsorbent or a layer of solute already adsorbed. The equation of the model is derived from kinetic adsorption-desorption relationships, assuming first order kinetics [10,85]. The expression obtained after a rather lengthy derivation is... 

Total catalyst area is generally determined using a physically adsorbed species, such as N2. The procedure was developed in the 1930s by Brunauer, Emmett and and Teller [7], and the isotherm they developed is referred to as the BET isotherm. In the physisorption process multi-layers are allowed to form. The BET isotherm model treats the first layer differently from all subsequent layers and leads to the following expression... 

W. Piatkowski, D. Antos, F. Gritti and G. Guiochon, Study of the competitive isotherm model and mass transfer kinetics for a BET binary system. J. Chromatogr.A 1003 (2003) 73-89. 

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... 

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.)... 

Semiernpirical Isotherm Models. Some of these models have been shown to have some thermodynamic inconsistencies and should be used with due care. They include models based on the Polanyi adsorption potential (Dubinin-Radushkevich, Dubinin-Astakhov, Radke-Prausnitz, Toth, UNI LAN. and BET). 

For most practical purposes, the isotherm can be modeled by an empirical (Brunauer, Emmett and Teller BET) or theoretical (Guggenheim, Anderson, and DeBoer GAB) equation (see below) however, none of the isotherm models in the literature is valid over the entire aw range of 0 to 1. The GAB model is one of the most widely accepted models for foods over a wide range of aw (from 0.10 to 0.90). The details of the different isotherm models with their parameters have been compiled by Rahman (1995). The BET (Eq. A2.3.4) and GAB (Eq. A2.3.5) equations are given as follows ... 

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... 

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... 

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. 

To determine the surface area of dry powders, it is only necessary to record the first part of the adsorption branch, reducing the experimental time significantly (to less than 0.5 h). When increasing the partial pressure of the adsorbate over the sample, a monolayer of adsorptive builds up, while with increasing relative pressure, multilayer adsorption occurs. Brunauer et al. derived a relation from gas-kinetic and statistical models on how this monolayer coverage can be determined from the mentioned experiment, which nowadays is often called BET-isotherm ... 

Adsorption isotherms at 77 K were simulated for each model material. These data are shown in Figure 2. These isotherms all show standard Type IV behavior characteristic of mesoporous materials. They are described well by BET-type models at low pressures, with capillary rises at high pressures and pore filling at pre.ssures near saturation. The (a) and (b) models have considerably higher maximum adsorption than the (c) and (d) models due to their higher porosity. The (a) and (c) models both have capillary upswings at relative pressures around 0.6, while the (b) and (d) models, which have larger pores, show sharper capillary rises between relative pressures of 0.7 and 0.8. 

BET models. The Freundlich isotherm is an empirical relationship, while the Langmuir and BET isotherms are based on theoretical developments. The Langmuir isotherm is based on the concept of monolayer adsorption, while the BET isotherm assumes multilayer adsorption. 

Series of overloaded band profiles for single-component samples were acquired, the experimental conditions being varied systematically. These profiles were compared to those calculated using the GR or the TD models and the BET isotherm. The lumped mass transfer rate coefficient, kf ,i was found to be practically propor-... 

Since the definition of surface area can sometimes be ambiguous, it is important to recognize that the macroscopic determination of surface area generally involves the measurement of a certain property of the solid (e.g., equilibrium adsorption capacity) which can be a qualitative measure of the development of the surface. This property can then be related to the actual surface area through an appropriate theory (e.g., BET isotherm). Therefore, one should not be surprised that the results from different models may give different values of the surface area. 

The difficulty of gas adsorption methods lies in the fact that purely monomolecular layers are never formed. Already before the adsorbent is completely covered, multiple layers build up locally. Brunauer, Emmett and Teller derived a relation between gas pressure and the amount of gas adsorbed at the surface which is known as the BET isothermal line. They used both a gas kinetic and a statistical model. 

The Brunauer-Emmett-Teller (BET) and Guggenheim-Anderson-deBoer (GAB) sorption isotherm models were used to obtain experimental steady-state moisture contents in dry basis by linear regression analysis according to Kouassi and Roos (2002). These equations provide the value of monolayer water content, which is an important parameter in food deterioration studies. 

Figure 3.8 Multilayer surface adsorption model for the BET isotherm.

Statistical models of the sorption behavior of foods have been applied for predicting the sorption behavior of foods. One of the most well-known and used model is the equation proposed by Brunauer, Emmett, and Teller known as the BET sorption isotherm model. This model is used extensively in food research. Another model extensively used for foods proposed by Guggenheim, Anderson, and de Boer is known as the GAB model or equation. The predicted values by either... 

The use of the Brunauer-Emmett-Teller (BET) sorption model by TMT is problematic due to the faet that the BET is fit to sorption data at 30°C and does not consider the temperature dependence of sorption behaviour. One way the model of TMT eould be improved is by using more recent models for sorption isotherms, e.g. that of Choi and Datta [29], or by using conductivity data measured as a funetion of water content. 

One of the most conamon uses of the BET isotherm is for determining the surface area of finely divided solids by physical adsorption. Such information can be of great importance in a number of areas including heterogeneous catalysis and various sorption applications. While the BET model for multilayer adsorption contains several potential sources of error due to the assumptions of the absence of lateral interactions between adsorbed molecules, the... 

The parameter n can be correlated with the number of multilayers formed. If n is unity the isotherm converts to the Langmuir model, and if n is infinity it becomes the standard BET isotherm. 

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