Adsorption 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. 

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. 

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. 

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. 

Therefore, for the reorientation model the two methods lead to similar values of AG , because, on the one hand, this model transforms into the ideal (Langmuir - von Szyszkowski) model at high surface pressure, when only one of two possible adsorption states exists, namely that possessing the minimum area, cf. Eq. (3.7). On the other hand, this coincidence also indicates that the CMC and the adsorption characteristics (b and CO2) calculated from the fitting program, are reliable. 

The Langmuir model is based on the assumption of ideal localized adsorption without interaction on a set of identical sites as outlined in Section 2.4. The special case of sorption of CH4 or Ar in sodalite was noted in Section 3.5 as an example of a system for which the basic assumptions of the Langmuir model are in fact fulfilled and for which the isotherms conform, as expected, to the Langmuir equation (Eq. (2.28)]. Sorption of normal tri- or tetradecane in 5A zeolite is another example of a zeolitic system in which each cage can accommodate only one sorbate molecule. Approximate conformity... 

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... 

Note that the treatment above is based on the totally reversible electrochemical adsorphon and desorption of H+ (or an ideal Langmuir isotherm model with a monolayer adsorption) on a metal surface. 

In a Langmuir model for adsorption, it is assumed that the two-dimensional phase is an ideal solution, where water, oxygen or hydroxyl, and sulfur adsorb competitively on the same surface sites and there are no interactions between adsorbed species. 

Detailed Modeling Results. The results of a series of detailed calculations for an ideal isothermal plug-flow Langmuir system are summarized in Figure 15. The soHd lines show the form of the theoretical breakthrough curves for adsorption and desorption, calculated from the following set of model equations and expressed in terms of the dimensionless variables T, and P ... 

---