Langmuir model equation

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... 

Adsorption Isotherm Measurements and Site-Selective Thermodynamics. For heterogeneous surfaces like CSPs, the adsorption isotherms are usually composite isotherms and often a Bi-Langmuir model (Equation 1.15) describes reasonably well the adsorption behavior [54]. 

The bi-Langmuir model (equation (4)) or tri-Langmuir model, the sum of two or three Langmuir isotherms, correspond to models that assume the adsorbent surface to be heterogeneous and composed of two or three different site classes. Finally the Freundlich isotherm model (equation (5)) assumes no saturation capacity, but instead, a continuous distribution of sites of different binding energies. 

In this particular case, the first (1st) and second (2nd) ran isotherms are virtually coincident indicating that CO adsorption was entirely reversible upon evacuation of the CO equilibrium pressure. For the experimental and samples details vide infra Sect. 1.4. It is here only recalled that the 2nd ran isotherms were performed after the overnight outgassing of the reversible adsorbed phase. The isotherms experimental points were interpolated by the Langmuir model equation (vide infra). 

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. 

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,... 

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. 

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 ... 

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

Markham andBenton. This model (34) is known as the extended Langmuir isotherm equation for two components, i and j. 

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... 

The most widely used model of adsorption is Langmuir s equation for reversible molecular adsorption [163]. However, this is inappropriate for charged latex particles, because... 

Another widely used sorption model is the Langmuir equation. It was developed by Irving Langmuir [140] to describe the adsorption of gas molecules on a planar surface. It was first applied to soils by Fried and Shapiro [ 141 ] and Olsen and Watanabe [142] to describe phosphate sorption on soils. Since that time, it has been heavily employed in many environmental fields to describe sorption on various solid surfaces [19,65]. The general Langmuir model is... 

The double-reciprocal Langmuir model has been extensively used in site assessment projects for elemental adsorption data. The double-reciprocal Langmuir is an adaptation of the traditional equation for elemental sorption of solid phases exhibiting two primary adsorbing surface sites. The double-reciprocal Langmuir model is as follows ... 

The Freundlich form is often employed along with the Langmuir form and is referred to as the Langmuir—FreundUch equation. That model is the basis of a great many useful modifications including many empirical forms that serve to describe adsorption data quite accurately. 

Figure 13.6 Step feed and shut-off of 700 ppm NH3 in He -I- 700 ppm NO -I- l%v/v Oj over VjO., WO,/liO, model catalyst at 220 C. Dashed lines, inlet NH, concentration solid lines, model fit with Temkin-type coverage dependence and modified Langmuir kinetics, Equation (13.10). (Adapted from ref. [52]).

Selecting the isotherm model and initial estimations for the values of its parameters. For instance, in the case of the Langmuir isotherm (Equation 10.39), an estimation for the a parameter can be achieved by an injection made under linear conditions (through Equation 10.50). At this point, the least-squares estimation of b—starting from an initial guess— does not present any difficulty. 

Equations 21 and 22 present the useful extension of the Szyszkowski-Langmuir model to the adsorption with two orientational states at the interface. If the molecular interactions are considered, a similar simphfied model with P = 2 = P and b = b2 = b can be obtained from Eqs. 10 and 11, giving... 

Equations 27 and 28 present the extension of the Szyszkowski-Langmuir model to the adsorption of one-surfactant systems with aggregation at the interface. For the formation of dimmers on the surface, n = 2 and Eqs. 27 and 28 can be expanded to obtain the Frumkin equation of adsorption state. In general, the surface aggregation model described by Eqs. 27 and 28 contains four free parameters, including coi, n, b and Fc, which can be obtained by regression analysis of the data for surface tension versus surfactant concentration in the solution. 

To resolve the problem of negative /3 values obtained with the Frumkin theory, the improved Szyszkowski-Langmuir models which consider surfactant orientational states and aggregation at the interface have been considered [17]. For one-surfactant system with two orientational states at the interface, we have two balances, i.e., Ft = Fi + F2 and Ftco = Ficoi + F2C02, which can be used in conjunction with Eq. 24 to derive two important equations for determining the total surface excess and averaged molecular area required in the calculation of surface tension, i.e.,... 

Microporous materials exhibit type I isotherms which are described by equation (4.12), the Langmuir equation. The Langmuir model was developed on the assumption that adsorption was limited to at most one monolayer. However, any factor which can limit the quantity adsorbed to a few monolayers will also produce a type I isotherm. In the case of very narrow pores, the close proximity of the walls will prevent multilayer adsorption and limit the amount adsorbed. 

Model equations give only the relation of Xi versus 0T - Xp and when the direct relationship between X and 0T - XT is required, it is desirable to represent X1 explicitly by X. For the Langmuir isotherm,... 

Although the assumptions of the Langmuir model are generally not fulfilled for molecular sieve adsorbents, this equation has been found to provide a useful empirical representation of the isotherms (2). The parameters b and qB must, however, be regarded simply as empirical constants. 

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