The Langmuir Model

The Langmuir Model Describes Adsorption of Gas Molecules on a Surface 

To get the translational entropy, apply the Boltzmann relation and Stirling s approximation, Equation (4.26), to Equation (27.3), 

Equation (27.4) has appeared before as the translational entropy of a lattice gas (Equation (7.9)) and as the entropy of mixing in a three-dimensional system (Equation (15.2)). Remarkably, this distributional entropy does not depend on whether a system is one-, two-, or three-dimensional. Nor does it depend on the arrangement of the binding sites. They could form a contiguous two-dimensional plane, or each binding site could be found on a different protein or polymer molecule. 

Now we compute the adsorption free energy. If N particles stick to the surface, each with energy w 0, then binding contributes an amount 

The degrees of freedom are (N, A, T). They correspond to (N, V, T) in a singlecomponent three-dimensional system of size V. To get the chemical potential Psurf of the adsorbate at the surface, take the derivative with respect to N, holding A and T constant  

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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. 

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. 

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

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... 

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 ... 

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 ... 

In the Langmuir model, a species sorbs and desorbs according to a reaction such as,... 

No value for dmq/dmp is needed to evaluate the and Freundlich models. For the Langmuir model and the ion exchange model under the Gaines-Thomas and Gapon conventions,... 

The Langmuir model was extended to include interaction between the adsorbed atoms/molecules by Fowler and Guggenheim [31], The model now becomes... 

The Langmuir model first assumes the adsorption sites are energetically identical. Actually, this assumption is not borne out when adsorption occurs predominantly by physisorption. The spread of A/Tads values between the various sites can be as high as 2 kJmol-1, which is often a significant fraction of the overall enthalpy of adsorption when physisorption is the sole mode of adsorption. By contrast, energetic discrepancies between sites can be ignored when adsorption occurs by chemisorption. 

Simple evidence about the information content of the dynamic properties can be obtained by considering the Langmuir model of adsorption of molecules from gas phase to a limited number of interacting sites on a sensor surface [3]. 

Several investigators [7,123] suggested the use of a Langmuir-type saturation model in addition to the electrostatic model to account for saturation effects. The Langmuir model implies that there are a finite number of localised sorption sites [15] ... 

The adsorption of a number of organic pollutants on various solid surfaces was found to fit the Langmuir-model isotherm [139,143-145]. 

Correspondence of experimental data to the Langmuir model does not mean that the stated assumptions are valid for the particular system being studied, because departure from the assumptions can have a canceling effect. An advantage of this model is that it can approach Henry s law at low concentrations. 

The Langmuir model for competitive adsorption can be used as a common model for predicting adsorption equilibria in multicomponent systems. This was first developed by Butler and Ockrent [77] and is based on the same assumptions as the Langmuir model for single adsorbates. It assumes, as in the case of the Langmuir model, that the rate of adsorption of a species at equilibrium is equal to its desorption rate. This is expressed by Eq. (18) ... 

Jain and Snoeyink [93] reported that if the Langmuir model for competitive adsorption satisfactorily predicts the extent of adsorption from a bisolute system when it is probably due to the competition for all available sites. They... 

The first term on the right side of Eq. (19) is the Langmuir expression for the number of moles of species 1 which adsorb without competition on the surface area proportional to (Q, - Q2). The second term represents the number of moles of species 1 adsorbed on the surface area proportional to Q2 under competition with species 2 and is based on the Langmuir model for competitive adsorption. The number of moles of species 2 adsorbed on the surface area proportional to Q2 and under competition with species 1 can be calculated from Eq. (20). 

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