Langmuir adsorption isotherm approach assumptions

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. 

The local isotherm was assumed to follow the Langmuir equation, and by assuming a patchwise topography the overall isotherm can be readily obtained by averaging the local isotherm over the distribution of the Henry constant. One should note here that since there is no interaction between the adsorbed molecules (assumption of the Langmuir adsorption mechanism), the surface topography is irrelevant. This approach of Sircar yields an isotherm which has a finite slope at... 

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... 

Assuming that the second process is rapid, we obtain the following standard picture of adsorption on a uniform surface the equilibrium concentration q, which depends on the pressure of the gas, is determined by the Langmuir isotherm. The only difference from the standard picture is that the statistical sum for all states of the adsorbed molecule in a potential hole must be replaced by a combination of two statistical sums for all states of the adsorbed molecule and for all possible states of the surface element. This, of course, has no effect on the form of the Langmuir equation. Under very simple assumptions the kinetics of establishment of equilibrium will also not differ from those on a uniform surface. Thus, the initial velocity is proportional to the pressure and approaches equilibrium exponentially. 

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... 

In geological surfaces, the solid-gas and solid-liquid interfaces are important, so the correct thermodynamic adsorption equation (Gibbs isotherm) cannot be used. Instead, other adsorption equations are applied, some of them containing thermodynamic approaches, and others being empirical or semiempirical. One of the most widespread isotherms is the Langmuir equation, which was derived for the adsorption of gas molecules on planar surfaces (Langmuir 1918). It has four basic assumptions for adsorption (Fowler 1935) ... 

For type a curves the Intercept n° may be interpreted as n°, that is the real value of n° In a complete layer from which all 1 Is expelled (n = 0). This is the plateau value attained by the individual Isotherm of 2, Indicated by the dashed curve In fig. 2.23. When an assumption Is made about the molecular cross-section from n° the specific surface area can be obtained. In principle, this method Is not different from finding from, say the plateau In a Langmuir Isotherm, the only difference being that horizontal Langmuir plateaus (for adsorption from dilute solution) are replaced by linear upper parts, approaching zero at 1 (for excess adsorption from binary mixtures). 

Like Helfferich and Klein [9], Rhee et al. [10] studied the separation of multicomponent mixtures by displacement chromatography using the restrictive assumption of the validity of the Langmuir isotherm model and the ideal model. They used a different approach, based on the method of characteristics, and studied the interactions between concentration shocks and centered simple waves [15]. This approach is more directly suited to adsorption chromatography than the... 

The simplifications and assumptions of the retardation factor approach in treating chemical reactions can be illustrated by comparing the one-site Langmuir isotherm and surface complexation formulation for adsorption of a metal onto iron oxide with one type of surface sites. 

Because of its simplicity and wide utility, the Langmuir isotherm has found wide applicability in a number of useful situations. Like many such classic approaches, it has its fundamental weaknesses, but its utility generally outweighs its shortcomings. The Langmuir isotherm model is based on the assumptions that adsorption is restricted to monolayer coverage, that adsorption is localized (i.e., that specific adsorption sites exist and interactions are between the site and a specific molecule), and that the heat of adsorption is independent of the amount of material adsorbed. The Langmuir approach is based on a molecular kinetic model of the adsorption-desorption process in which the rate of adsorption (rate constant /ca) is assumed to be proportional to the partial pressure of the adsorbate (p) and the number of unoccupied adsorption sites (N - n), where N is the total number of adsorption sites on the surface and n is the number of occupied sites, and the rate of desorption (rate constant d) is proportional to n. 

We generalize the adsorption model of section 2.8. Most of the assumptions of the simple Langmuir model are still retained, except for two the sites are not localized, and they are solvated by the solvent (the solvent may be any mixture of liquids, but for simplicity we assume that it is a one-component liquid, denoted by w). The simplest way to obtain the modified binding isotherm is to use the thermodynamic approach of section 2.9.1 with the appropriate modification introduced into the chemical equilibrium condition in solution, as in section 8.3. 

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