Ideal Langmuir Model

As a first-order deviation from the Langmuir model one may consider ideal adsorption on a set of localized sites with weak interaction between adsorbed molecules on neighboring sites. Such a model has been investigated theoretically by Lacher and by Fowler and Guggenheim. If the interaction is sufficiently weak that the random distribution of the adsorbed molecules is not significantly affected the resulting expression for the isotherm is 

Sorption on a set of independent pairs of identical sites with interaction energy 2w when both sites of the pair are occupied yields Eq. (3.86) which, expressed in terms of the Henry constant becomes 

FIGURE 43. a) Equilibrium i.sotherms and (/)) variation of Langmuir constant with coverage for n-heplane in 5A zeolite. (Frorn ref, 5 reproduced by permission of the National Research Council of Canada from the Canadian Journal of Chemhtrv. Volume 52, 1974.) 

The cage of 5A zeolite is large enough to contain just two molecules of n heplane at ordinary pressures so the models represented by both Eqs. (4.2) and (4.3) appear physically reasonable. Both equations contain two parameters (K, w and K[,K - and both fit the experimental data well over the entire concentration range. Since Eq. (4.3) may be written in the form 

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Eig. 4. The Bmnaner classification of isotherms (I V). Langmuir Isotherm. Type I isotherms are commonly represented by the ideal Langmuir model ... 

According to the ideal Langmuir model [156] the heats of adsorption should be independent of coverage, but this requirement is seldom fulfilled in real systems because the effects of surface heterogeneity and sorbate-sorbate interactions are generally significant. 

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. 

Various functional forms for / have been proposed either as a result of empirical observation or in terms of specific models. A particularly important example of the latter is that known as the Langmuir adsorption equation [2]. By analogy with the derivation for gas adsorption (see Section XVII-3), the Langmuir model assumes the surface to consist of adsorption sites, each having an area a. All adsorbed species interact only with a site and not with each other, and adsorption is thus limited to a monolayer. Related lattice models reduce to the Langmuir model under these assumptions [3,4]. In the case of adsorption from solution, however, it seems more plausible to consider an alternative phrasing of the model. Adsorption is still limited to a monolayer, but this layer is now regarded as an ideal two-dimensional solution of equal-size solute and solvent molecules of area a. Thus lateral interactions, absent in the site picture, cancel out in the ideal solution however, in the first version is a properly of the solid lattice, while in the second it is a properly of the adsorbed species. Both models attribute differences in adsorption behavior entirely to differences in adsorbate-solid interactions. Both present adsorption as a competition between solute and solvent. 

Many systems give linear plots of pjn against p over a limited ranges of pressure, but such linearity does not by itself imply conformity with the Langmuir model. As already indicated, a second condition is that the energy of adsorption should be independent of surface coverage. Thirdly, the differential entropy of adsorption should vary in accordance with the ideal localized model (Everett, 1950). That no real system has been found to satisfy all these requirements is not surprising in view of the complexities noted here and in subsequent chapters. 

Although many kinetic models assume that the catalyst is an ideal Langmuir surface (all sites have identical thermodynamic properties and there are no interactions among surface species), modern surface science has proven that ideality is often not the case. 

While assuming the Langmuir model for adsorption of atoms, and the ideal gas expression for p fiom eq. (2), we obtain. 

As has become clear adsorption phenomena play an important, if not, decisive role in this behaviour, and good data and modelling of adsorption are mandatory, too, to serve as the input parameters for the permeation description. This should not be l ted to the T.angmnir model, but other theories like the IAS (ideal adsorbed solution) and NIAS (non-ideal) should be considered, since they sometines work well for binary systems where the Langmuir model fails. 

A nonlinear local isotherm model is clearly required for description of sorption reactions between the TCB and the shale isolate. A variety of conceptual and empirical models for representing nonlinear sorption equilibria, exists (2). The Langmuir model is one of the ideal limiting-condition-type models cited earlier. It is predicated on a uniform surface affinity for the solute and prescribes a nonlinear asymptotic approach to some maximum sorption capacity. 

The major advantage of the Langmuir isotherm (Eq. 3.13) is that it can be inverted and solved for q in closed-form. This cannot be done with the Volmer isotherm (see next section), or with many other important isotherms, such as the Fowler or the virial isotherms (see below. Sections 3.1.6 and 3.2.3.1). Such isotherms are often called implicit isotherms. The other conditions of validity of the Langmuir model are the ideal behavior of the gas phase, the absence of adsorbate-adsorbate interactions, and the localized character of adsorption. 

As we have explained in the previous sections, the Langmuir model has been established on firm theoretical groimd for gas-solid adsorption, a case where there is no competition between the adsorbate and the mobile gas phase. On the contrary, in liquid-solid adsorption, there is competition for adsorption between the molecules of any component and those of the solvent. Although we can choose a convention canceling the apparent effect of this competition on the isotherm [30,36], the conditions of validity of Eq. 3.47 are not met. These conditions are (i) the solution is ideal (ii) the solute gives monolayer coverage (iii) the adsorption layer is ideal (iv) there are no solute-solute interactions in the monolayer (v) there are no solvent-solute interactions. These conditions cannot be valid in liquid-solid adsorption, especially at high concentrations. 

Figure 3.47 Dependence of the isotherm determined by ECP (or FACP) on the column efficiency. The ECP method is based on the ideal model profile (cf Eq. 7.4). A Langmuir isotherm (solid line, b) is used to calculate the band profiles obtained with columns of different efficiencies ( L/ = 10%). The profiles (a) are used to derive the isotherm following the ECP method. The isotherms differ from the initial Langmuir isotherm. The best fit of the data to a Langmuir model generates significant model errors, with deviations of the order of 1% between the initial and the measured coefficients of the isotherm for N = 5000 plates, larger at lower efficiencies and loading factors.

The most conspicuous difference between the experimental and calculated band profiles is the lower maximum concentration peak (center figures) and the lower plateau concentration (far right figures) measured for resorcinol and catechol than calculated. Note, however, that the isotherm was determined in a concentration range which does not extend to the displacer concentration used. Hence, there is a possible error in the exact position of the operating line. In this concentration range, deviations of the adsorption behavior from the Langmuir model is probable, as this model assumes the solution to be ideal. 

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