Langmuir adsorption isotherm kinetic theory

Langmuir adsorption isotherm A theoretical equation, derived from the kinetic theory of gases, which relates the amount of gas adsorbed at a plane solid surface to the pressure of gas in equilibrium with the surface. In the derivation it is assumed that the adsorption is restricted to a monolayer at the surface, which is considered to be energetically uniform. It is also assumed that there is no interaction between the adsorbed species. The equation shows that at a gas pressure, p, the fraction, 0, of the surface covered by the adsorbate is given by ... 

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 Freundlich equation proved to be applicable to the adsorption of liquids with only limited ranges of concentration. It was replaced by the Langmuir equation (see later on) and others which had a theoretical basis in the kinetic theory of gases. It is clear that neither the Freundlich nor the Langmuir equation can describe isotherms of the shape shown in Figure 10.5. 

Using a kinetic approach, Langmuir was able to describe the type I isotherm with the assumption that adsorption was limited to a monolayer. According to the kinetic theory of gases, the number of molecules striking each square centimeter of surface per second is given by... 

This kinetic-theory-based view of the Langmuir result provides no new information, but it does draw attention to the common starting assumptions of the Langmuir derivation and the BET derivation (Section 9.5a). This kinetic derivation of the Langmuir equation is especially convenient for obtaining an isotherm for the adsorption of two gases. This is illustrated in Example 9.4. 

EXAMPLE 9.4 Kinetic-Theory-Based Description of Binary Adsorption. Assume that two gases A and B individually follow the Langmuir isotherm in their adsorption on a particular solid. Use the logic that results in Equation (46) to derive an expression for the fraction of surface sites covered by one of the species when a mixture of the two gases is allowed to come to adsorption equilibrium with that solid. 

By introducing a number of simplifying assumptions, Brunauer, Emmett and Teller (1938) were able to extend the Langmuir mechanism to multilayer adsorption and obtain an isotherm equation (the BET equation), which has Type II character. The original BET treatment involved an extension of the Langmuir kinetic theory of monomolecular adsorption to the formation of an infinite number of adsorbed layers. 

Sorption in micropores can be described by the Dubinin-Radushkevic formalism that has been adapted by Stoeckli et al. This is a largely empirical approach and it should be emphasized that the use of a combination of Langmuir types isotherm leads to similar quantitative results. For evaluation of the distribution of micropores, one can either rely on high-resolution measurements of mostly nitrogen adsorption as suggested by Horvath and Kawazoe or use a combination of probe molecules of different minimum kinetic diameter. More recently, approaches based on density functional theory are put forward. 

The Langmuir isotherm equation can also be derived from the formal adsorption and desorption rate equations derived from chemical reaction kinetics. In Section 3.2.2, we see that the mass of molecules that strikes 1 m2 in one second can be calculated using Equation (186), by applying the kinetic theory of gases as [dmldt = P2 (MJ2nRT)m], where P2 is the vapor pressure of the gas in (Pa), Mw is the molecular mass in (kg mol ), T is the absolute temperature in Kelvin, R is the gas constant 8.3144 (nT3 Pa mol-K-1). If we consider the mass of a single molecule, mw (kg molecule-1), (m = Nmw), where N is the number of molecules, by considering the fact that (R = kNA), where k is the Boltzmann constant, and (Mw = NAmw), we can calculate the molecular collision rate per unit area (lm2) from Equation (186) so that... 

Physisorption arises from the van der Waals forces, and these forces also condense gas molecules into their liquid state. Thus, in principle, there is no reason to stop upon completion of a monolayer during physisorption. Indeed, the formation of multi-layers, which are basically liquid in nature, is very common in physisorption experiments. Brunauer, Emmett and Teller developed a theory in 1938 to describe physisorption, where the adsorbate thickness exceeds a monolayer, and this isotherm equation is known by the initials of the authors (B.E.T.). The original derivation of the B.E.T. equation is an extension of Langmuir s treatment of monolayer adsorption from kinetic arguments. Later, in 1946, Hill derived this equation from statistical mechanics. In the B.E.T. isotherm, it is assumed that ... 

As mentioned above, the adsorption kinetics for a kinetic-controlled mechanism is given by the balance of surfactant adsorption and desorption fluxes to and from the interface and for the Langmuir kinetics this balance has the form of Eq. (4.15). The rate constants kad and kdes are functions of the activation energies adsorption and desorption and can be specified on the basis of the molecular kinetic [9, 120] or transition state theory [121]. Eq. (4.15) was applied to adsorption kinetics data of surfactants at the water/air interface by many authors, for example in [24, 39, 83, 97, 122, 123, 124, 125, 126, 127]. In these works, it was shown that the values of kad and kdes are not constant hut depend on the surfactant bulk, the degree of adsorption layer saturation, or its lifetime. To obtain better correspondence with the experimental data, some authors had assumed that the adsorption and desorption activation energies depend on the degree of adsorption layer saturation. These rather complicated kinetic equation are more or less empirical, although they transforms into a valid adsorption isotherm at equilibrium... 

The Mo adsorption process can be studied by using adsorption isotherms. Langmuir and Freundlich equations are the two major types of isotherms used to describe the Mo adsorption process. The Langmuir equation is based on the kinetic theory of gaseous adsorption onto solids, but is often used to model the adsorption of ions from solution (Ellis and Knezek, 1972). A common form of the Langmuir equation is... 

Fundamentals of sorption and sorption kinetics by zeohtes are described and analyzed in the first Chapter which was written by D. M. Ruthven. It includes the treatment of the sorption equilibrium in microporous sohds as described by basic laws as well as the discussion of appropriate models such as the Ideal Langmuir Model for mono- and multi-component systems, the Dual-Site Langmuir Model, the Unilan and Toth Model, and the Simphfied Statistical Model. Similarly, the Gibbs Adsorption Isotherm, the Dubinin-Polanyi Theory, and the Ideal Adsorbed Solution Theory are discussed. With respect to sorption kinetics, the cases of self-diffusion and transport diffusion are discriminated, their relationship is analyzed and, in this context, the Maxwell-Stefan Model discussed. Finally, basic aspects of measurements of micropore diffusion both under equilibrium and non-equilibrium conditions are elucidated. The important role of micropore diffusion in separation and catalytic processes is illustrated. 

The classical work of Clark (1937), based on the application of Langmuir s adsorption isotherm (see Section 8.1), assumed that the effect of a drug is proportional to the fraction of receptors occupied by drug molecules, and that a maximal effect is obtainable only when all receptors are occupied by the drug. As a statement of the action of inhibitors (so important for work in chemotherapy and agriculture) this simple occupation theory can hardly be bettered. However, it is inadequate to explain the kinetics of agents which elicit a positive response substances like the synaptic transmitter acetylcholine and innumerable artificial agonists. 

In the last chapter, we discussed the description of pure component adsorption equilibrium from the fundamental point of view, for example Langmuir isotherm equation derived from the kinetic approach, and Volmer equation from the Gibbs thermodynamic equation. Practical solids, due to their complex pore and surface structure, rarely conform to the fundamental description, that is very often than not fundamental adsorption isotherm equations such as the classical Langmuir equation do not describe the data well because the basic assumptions made in the Langmuir theory are not readily satisfied. To this end, many semi-empirical approaches have been proposed and the resulting adsorption equations are used with success in describing equilibrium data. This chapter will particularly deal with these approaches. We first present a number of commonly used empirical equations, and will discuss some of these equations in more detail in Chapter 6. 

We note here that there are other theories of adsorption/desorption kinetics that offer expressions for the adsorption/desorption rate that are different from the ART expression but also lead to the Langmuir isotherm when d0/dt = 0. It is rather strange that adsorption systems with different kinetics of the adsorption/ desorption processes have the same form at equilibrium. 

The Langmuir isotherm equation is the first theoretically developed adsorption isotherm. Many of the equations proposed later and which fit the experimental results over a wide range are either based on this equation, or these equations have been developed using the Langmuir concept. Thus, the Langmuir equation still retains an important position in physisorption as well as chemisorption theories. The equation has also been derived using thermodynamic and statistical approaches but we shall discuss the commonly used kinetic approach for its derivation. 

Langmuir s theory for deriving an isotherm is a kinetic one, assuming the adsorption system is in dynamic equilibrium, where therate of adsorption is equal to that of desorption. The Langmuir isotherm, described by the following equation, is still the most useful for data correlation. 

We wrote that both for the case when the sticking coefficient S is assumed to be constant and when it is assumed to be equal to 5 o(l — 9), Kreuzer s approach leads to the Langmuir isotherm at equilibrium, i.e., when = R - Ward and coworkers [17,18,62,63] present a different idea that applies the statistical rate theory to the kinetics of adsorption. 

Although the Langmuir theory depicts a highly idealized situation that many real systems do not follow, it is nevertheless of great value in developing the kinetic interpretation of the heterogeneous reactions on solid surfaces. The concepts of the Langmuir theory have been extended to multilayer physical adsorption by Brunauer et al. [55]. The BET isotherm takes the form... 

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