Langmuir and BET theories

The success of kinetic theories directed toward the measurements of surface areas depends upon their ability to predict the number of adsorbate molecules required to exactly cover the solid with a single molecular layer. Equally important is the cross-sectional area of each molecule or the effective area covered by each adsorbed molecule on the surface. The surface area then, is the product of the number of molecules in a completed monolayer and the effective cross-sectional area of an adsorbate molecule. The number of molecules required for the completion of a monolayer will be considered in this chapter and the adsorbate cross-sectional area will be discussed in Chapter 6. 

---

S. Brunaer, L. E. Copeland, and D. L. Kantro, The Langmuir and BET theories, in The Solid-Gas Interface (E. A. Flood, ed.), Vol. 1. Dekker, New York, 1967. Equation 1.7 is derived under the assumption that an infinite number of layers build up on the absorbing surface. If the number of layers is finite, a more general expression results, but it cannot be distinguished from Eq. 1.7 when plotted as in Fig. 1,12 unless the number of layers is fewer than three. 

The Langmuir and BET theories, originating from one common assumption, assume existence of the interface geometrical surface on which mono- or... 

Discussing and comparing the basic adsorption theories presented so far, it is difficult to say which one is fundamentally right. The Polanyi thermodynamic theory neither determines a definite adsorption isotherm equation nor gives a detailed mechanism of the process. Development of this theory for description of adsorption on microporous substances leads to the analytical DR equation but its character is semiempirical. The Langmuir and BET theories introduce the concept of localized mono- and multilayer formed on the energetically homogeneous solid surface. 

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 ... 

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. 

In Langmuir model, the maximal adsorption is that of a monolayer. Langmuir adsorption isotherms all saturate at high vapor pressures. This is unrealistic for many cases. In order to consider the adsorption of multilayers, Brunauer, Emmett, and Teller extended the Langmuir theory and derived the so-called BET adsorption isotherm [378], The basic idea in the BET theory was to assume a Langmuir adsorption for each of the layers (Fig. 9.8). 

The BET theory, developed by Brunauer, Emmett, and Teller [26], is based on the kinetic model of adsorption proposed by Langmuir [24,25] and was extended to describe the multilayer adsorption by the introduction of some assumptions listed below ... 

The appearance of Langmuir s comprehensive review of the nature of adsorption (1916, 1918) prompted several investigators to consider the possibility of using gas adsorption for surface area determination. Early attempts were made by Williams (1919) and Benton (1926), but these led to inconclusive findings. The first significant advances were made by Brunauer and Emmett (1935, 1937) and their work prepared the way for the development of the Brunauer-Emmett-Teller (BET) theory in 1938. 

In interfacial science the term monolayer is used in a number of different ways. Although this rarely leads to confusion one needs to be aware of them. In the chapters on adsorption on solids (chapters II. 1 and 2) the notion of monolayer was sometimes used to distinguish it from bilayer or multilayer, and implying that all adsorbed molecules are in contact with the adsorbent. Whether or not this layer is completely filled does not matter in that case. However, the packing did matter in other instances where the amount adsorbed in a completely filled layer was at issue, as for example in the plateau of the Langmuir adsorption isotherm (r(< )), or in the volume in the BET theory, corresponding to the volume of gas that would... 

These assumptions are justifiable as the heat of adsorption of the small inert sorbate (e.g., N2 or Ar) is rather low and, hence, differences between sorption sites at the surface will be very small. Similarly, the interaction between the first and the following layers will be close to the heat of condensation, as the effect of polarization by the surface will be small beyond the first layer (screening of the long-range van der Waals forces). From its conception, the BET theory extends the Langmuir model to multilayer adsorption. It postulates that under dynamic equilibrium conditions the rate of adsorption in each layer is equal to the rate of desorption from that layer. Molecules in the first layer are located on sites of constant interaction strength and the molecules in that layer serve as sorption sites for the second layer and so forth. The surface is, therefore, composed of stacks of sorbed molecules. Lateral interactions are assumed to be absent. With these simplifications one arrives at the BET equation... 

One of the most widely used isotherms is based on the BET theory, named from the initials of Brunauer, Emmett, and Teller. The BET theory is an extension of the Langmuir isotherm to include the adsorption of two or more molecular layers assuming, in addition to assumptions 1-3 listed above, that... 

With respect to the values of the apparent specific surfiice area, we have to note here that modem instmmentation permitting the measurement of nitrogen adsorption isotherms is usually provided with software permitting surface area calculation according to several mathematical models, including the BJH (corresponding to the names of Barrett, Joyner, Halenda) method, Langmuir adsorption isotherms, and the BET theory these methods may provide for the same sample of a porous material, values that may differ by as much as 30%. 

The BET theory accounts for monolayer and multilayer molecular adsorption according to the following hypotheses (a) gas molecules physically adsorb on a solid in layers infinitely (b) there is no interaction between each adsorption layer and (c) the Langmuir theory can be applied to each layer. The resulting BET equation is expressed by ... 

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. 

The adsorption on a solid surface, the types of adsorption, the energetics of adsorption, the theories of adsorption, and the adsorption isotherm equations (e.g., the Langmuir equation, BET equation, Dubinin equation, Temkin equation, and the Freundlich equation) are the subject matter of Chapter 2. The validity of each adsorption isotherm equation to the adsorption data has been examined. The theory of capillary condensation, the adsorption-desorption hysteresis, and the Dubinin theory of volume fllhng of micropores (TVFM) for microporous activated carbons are also discussed in this chapter. 

---