Models Langmuir

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 

FIGURE 4.1. Langmuir constanis calculated according to Eq. (3.78) for Kr and Xe in 5.A zeolite. (Temperatures are in kelvin. Saturation capacities Kr,, = 11 molecules/cage Xc, 8.0 molecules/cage). (From ref. 3 reprinted by permission of the National Research Council of Canada from the Canadian Journal of Chemistry, Volume 53, 1975.) 

HGURE 4.2. Langmuir constants j(r ) and h) and heals of adsorption (c) plotted against fractional coverage (0) for I, on 5A and 13X zeolites, fleats of adsorption data for L on chabazilc as well as the heat of condensation of molecular iodine ( -A//,.) are also shown. (From ref. 4, with permission.) 

The Langmuir model is the basic model for chemisorption equihbria. It rehes on a number of assumptions  

Given the assumption that species x and G - 5 are two components of the same phase , the apphcation of the law of mass action to balance the previous reaction is written as follows  

Let Sq be the total number of sites per unit area before adsorption and S the number of sites unfilled at a given time. The amount ofG-s will then he Sq-S and the fractional coverage will be defined by  

At the adsorption equilibrium, the fractional coverage is and equilibrium equation [3.52] becomes  

Since is an equihbrium constant, it obeys van t Hoffs law with temperature  

The first quantitative theoretical model of gas adsorption in solids was proposed by Langmuir in 1916. Langmnir assumed the following hypotheses in this model  

The solid surface has a finite nnmber of adsorption sites. 

All sites have equivalent energies or equal adsorption enthalpies. 

The adsorption is independent of neighboring adsorbed species or the enthalpy is independent of the surface coverage. 

At equilibrium and constant temperature and pressure, the number of adsorbed molecules at the surface is ha- The fraction of sites occupied by A molecules is Oa  

Consider the adsorption of a mixture of two gases A and B on the same surface. Then, two cases can be distinguished  

on applying the law of mass actions to both eqirilibriums, the isotherms become  

Within the ideal solution of the Langmuir assumption framework, we can obtain  

It is noted that the ratio of the fractions of covering is proportional to the ratio of the partial pressures. 

These results spread easily to the adsorption of more complex mixtures, j represents all of the gases present in the nuxture and which are adsorbed on the same sites as gas A. For gas A belonging to a mixture, we have  

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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. 

Thus the thermodynamic description of the Langmuir model is that the energy of adsorption Q is constant and that the entropy of adsorption varies with 6 according to Eq. XVII-37. 

A variety of experimental data has been found to fit the Langmuir equation reasonably well. Data are generally plotted according to the linear form, Eq. XVn-9, to obtain the constants b and n from the best fitting straight line. The specific surface area, E, can then be obtained from Eq. XVII-10. A widely used practice is to take to be the molecular area of the adsorbate, estimated from liquid or solid adsorbate densities. On the other hand, the Langmuir model is cast around the concept of adsorption sites, whose spacing one would suppose to be characteristic of the adsorbent. See Section XVII-5B for an additional discussion of the problem. 

Because of the relatively strong adsorption bond supposed to be present in chemisorption, the fundamental adsorption model has been that of Langmuir (as opposed to that of a two-dimensional nonideal gas). The Langmuir model is therefore basic to the present discussion, but for economy in presentation, the reader is referred to Section XVII-3 as prerequisite material. However, the Langmuir equation (Eq. XVlI-5) as such,... 

This means that desorption activation energies can be much larger than those for adsorption and very dependent on 6 since the variation of Q with 6 now contributes directly. The rate of desorption may be written, following the kinetic treatment of the Langmuir model. 

Another limitation of tire Langmuir model is that it does not account for multilayer adsorption. The Braunauer, Ennnett and Teller (BET) model is a refinement of Langmuir adsorption in which multiple layers of adsorbates are allowed [29, 31]. In the BET model, the particles in each layer act as the adsorption sites for the subsequent layers. There are many refinements to this approach, in which parameters such as sticking coefficient, activation energy, etc, are considered to be different for each layer. 

Any interpretation of the Type I isotherm must account for the fact that the uptake does not increase continuously as in the Type II isotherm, but comes to a limiting value manifested in the plateau BC (Fig. 4.1). According to the earlier, classical view, this limit exists because the pores are so narrow that they cannot accommodate more than a single molecular layer on their walls the plateau thus corresponds to the completion of the monolayer. The shape of the isotherm was explained in terms of the Langmuir model, even though this had initially been set up for an open surface, i.e. a non-porous solid. The Type I isotherm was therefore assumed to conform to the Langmuir equation already referred to, viz. 

According to the classical Langmuir model, n is actually equal to the monolayer capacity, and can be converted into the specific surface A of the solid by the standard relation A = n a L (cf. Equation (2.1)). A number of lines of argument would suggest, however, that this interpretation is invalid, and that the value of A arrived at does not represent a true specific surface. 

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

The Langmuir model is discussed in reference 19 the Volmer in reference 20 and the van der Waals and virial equations in reference 8. 

The bi-Langmuir model (Equation (2)) or tri-Langmuir model, the sum of two or three Langmuir isotherms, correspond to models that assume the adsorbent surface... 

The isotherms for the two enantiomers of phenylalanine anilide were measured at 40, 50, 60 and 70 C, and the data fitted to each of the models given in Equations (1-3) [42]. The isotherms obtained by fitting the data to the Langmuir equation were of a quality inferior to the other two. Fittings of the data to the Freundlich and to the bi-Langmuir equations were both good. A comparison of the residuals revealed that the different isotherms of d-PA were best fitted to a bi-Langmuir model, while the... 

Due to the inherent uncertainty of the Langmuir model and difficulties in solving the transcendental equation (41), probably the most accurate treatment in the near-equilibrium cases is a numerical or graphical integration of the expression... 

For a Langmuir model where a diatomic gas dlssoclatlvely adsorbs, the relationships between a j, 3j and the kinetic parameters are given by ... 

One of the simplest nonlinear isotherm models is the Langmuir model. Its basic assumption is that adsorbate deposits on the adsorbent surface in the form of the monomolecular layer, owing to the delocalized interactions with the adsorbent snrface. The Langmuir isotherm can be given by the following relationship ... 

The first STM evidence for the facile transport of metal atoms during chemisorption was for oxygen chemisorption at a Cu(110) surface at room temperature 10 the conventional Langmuir model is that the surface substrate atoms are immobile. The reconstruction involved the removal of copper atoms from steps [eqn (1)], resulting in an added row structure and the development of a (2 x 1)0 overlayer [eqn (2)]. The steps present at the Cu(llO) surface are... 

Chemisorption of oxygen at Pt(lll) has been studied in detail by Ertl s group25 and the STM evidence is for complex structural features present in the temperature range 54M60K (Figure 4.14). The limitations of the Langmuir model, frequently invoked for reactions at platinum surfaces, is obvious from... 

Spirodela intermedia, L. minor, and P. stratiotes were able to remove Pb(II), Cd(II), Ni(II), Cu(II), and Zn(II), although the two former ions were removed more efficiently. Data fitted the Langmuir model only for Ni and Cd, but the Freundlich isotherm for all metals tested. The adsorption capacity values (K ) showed that Pb was the metal more efficiently removed from water solution (166.49 and 447.95 mg/g for S. intermedia and L. minor, respectively). The adsorption process for the three species studied followed first-order kinetics. The mechanism involved in biosorption resulted in an ion-exchange process between monovalent metals as counterions present in the macrophytes biomass and heavy metal ions and protons taken up from water.112... 

The Langmuir model has been used to describe adsorption behavior of some organic compounds at near-surface conditions.137 However, three important assumptions must be made ... 

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