Langmuir isotherm model

Sorption and desorption are usually modeled as one fully reversible process, although hystersis is sometimes observed. Four types of equations are commonly used to describe sorption/desorption processes Langmuir, Freundlich, overall and ion or cation exchange. The Langmuir isotherm model was developed for single layer adsorption and is based on the assumption that maximum adsorption corresponds to a saturated monolayer of solute molecules on the adsorbent surface, that the energy of adsorption is constant, and that there is no transmigration of adsorbate on the surface phase. 

It is interesting, however, to compare the different dissolution constants. K, obtained with the simple Langmuir isotherm model with experimental data of dissolution rates over a 24-hour period. Figure 15 gives concentrations obtained consecutively with fresh solvent every 24... 

Although the rotational anisotropy scans are informative, considerably more information can be obtained by separate determination of changes in the isotropic and anisotropic components of the surface susceptibility tensor as done by Koos et al. [122]. The experiments consist of monitoring the SH intensity at a fixed angle of 0 = 30° where / oc a 2 and Tj a b0) 2. The results shown in Fig. 5.16 are displayed in terms of thallium coverage. The data has been fitted to a simple linear Langmuir isotherm model of Heinz [79] where the adsorbate contribution to x(2) varies linearly with coverage such that... 

The assumption of monolayer adsorption in the Langmuir isotherm model is unrealistic in most cases, and a modification to multilayer adsorption should be considered. In 1938, Brunauer, Emmett, and Teller modified the Langmuir approach of balancing the rates of adsorption and desorption for the various molecular layers [Brunauer et al., 1938], This approach is known as the BET method. The BET isotherm assumes that the adsorption of the first layer has a characteristic heat of adsorption A Ha and the adsorption and desorption on subsequent layers are controlled by the heat of condensation of the vapor, A Hc. The derivation of the BET equation is beyond the scope of this book however, a common form of the BET equation is given as... 

Figure 6.3 shows the basis of the adsorption model, and the four rate constants used. For simplicity, we assume a Langmuir isotherm model for the occupation of the interface. If the fraction of interface occupied is then the adsorption from the octanol side is... 

The isotherm fields measured could be described with sufficient accuracy by using a modified Langmuir isotherm model, considering the dependence of adsorption enthalpy and adsorption entropy on ACC loading [4]. 

The Langmuir isotherm model can be extended to multi-component systems [109], When several components are simultaneously present in a solution, the amount of each component adsorbed at equilibrium is smaller than if that component were alone [13] because the different components compete to be adsorbed on the stationary phase. The adsorption isotherm for the / th component in a multicomponent system is written ... 

Since the Langmuir isotherm model is widely applicable, it is normal to take the column saturation capacity as a unit to measure the degree of column overload achieved, by reporting the actual amoimt injected as a fraction of the column saturation capacity. The loading factor, Ly, is defined for each component as the ratio of its amount in the sample to the column saturation capacity for that component ... 

Finally, it should be emphasized that the successful use of a bi-Langmuir isotherm model (as of any other combination of models that multiplies the number of model parameters) to account for a set of experimental adsorption data requires that these data are acquired in a wide concentration range [55]. This is even more... 

The StShlberg model is based on the Langmuir isotherm model, which describes the competition of the ions for available surface area but it assumes that ions are adsorbed as separate, individual ions that xmdergo electrostatic interactions. The intensity of these interactions is calculated using the Stem-Gouy-Chapman theory. The isotherm equation obtained [92] is... 

As a consequence, the competitive Langmuir isotherm model offers no possibility to account for a reversal in the elution order of two components with increasing concentration. On the contrary, experimental results show that such an inversion is possible, and that it is not even unusual when the column saturation capacities of the two single-component isotherms are different. For example, experimental adsorption data and chromatograms of mixtures of tmns- and a s-androsterone show an inversion of the elution order when the sample size increases (see later. Figure 4.8 and the related discussion below) [9-11]. 

On the other hand, the quantitative prediction of competitive isotherm behavior for the components of binary mixtmes is not possible using the competitive Langmuir isotherm model when the difference between the column satmation capacities for the two components exceeds 5 to 10%. For example, the adsorption isotherms of pure cis- and trans-androsterone on sihca are well accoimted for by the Langmuir model [9]. However, the two column saturation capacities differ by 30%, due to the nearly flat structure of the trans isomer compared to the folded structure of the cis isomer. As a consequence, the competitive Langmuir model accounts poorly for the competitive adsorption data [9,10]. Much improved results are obtained with the more complex LeVan-Vermeulen isotherm (Section 4.1.5). Another approach could use the random adsorption site model, with different exclusion siuface areas for the competing molecules [12],... 

Finally, it can be shown that the multicomponent competitive Langmuir isotherm (Eq. 4.5) does not satisfy the Gibbs-Duhem equation if the column saturation capacities are different for the components involved [13]. This profound inconsistency may explain in part why this model does not accoimt well for experimental results. There are two very different alternative approaches to the problem of competitive Langmuir isotherms when the saturation capacities for the two pure compounds are different. Before discussing this important problem and an interesting extension of the competitive Langmuir isotherm, we must first present the competitive bi-Langmuir isotherm model. 

Figure 4.5 Single-component adsorption data for L- and D-methyl mandelate determined by FA and best isotherm model. Symbols experimental data LM (A) and DM ( ). Lines best bi-Langmuir isotherm models, LM (soHd) and DM (dashed). The main figure shows the medium-concentration range (up to 0.25 mM), the top left inset the low-concentration range (below 12.5 M), and the bottom right inset the high- concentration range (up to 5.0 mM). Reproduced with permission from ]. Lindholm, P. Porssen, T. Fomstedt, Anal. Chem., 76 (2004) 4856 (Fig. 5). ( )2004, American Chemical Society.

Because of unsatisfactory results obtained with the competitive Langmuir isotherm model, several empirical equations have been suggested, based on hybrids of common models. One of the most popular of these isotherm models addresses the problem of strong adsorption observed at low concentrations and its rapid subsidence at increasingly large concentrations. This equation combines conven-... 

This method is ctunbersome compared to MCV because it requires the measurement of series of values of the mezzanine concentrations, which is time consuming. In addition, if values of C, are measured, it is much better to use the MMB method which gives the actual isotherm graph, a graph that is independent of the model while the MMC is valid only for the Langmuir isotherm model. 

Figure 6.19 Comparison of the band profiles calculated with a one-site and a two-site model, using the Langmuir isotherm model. Dimensionless plot of versus f(j. Solid line one-site Langmuir isotherm model q = 24C/(1 + 6C). Dotted line two-site bi-Langmuir isotherm q = 24C/(1- -6C) + 2.4C/(1 + 600C). In both cases, N = 5000. Reprinted hy permission of Kluwer Academic Publishing, from S. Golshan-Shirazi and G. Guiochon, NATO ASl Series C, vol 383, 61 (Fig. 8), with kind permission of Springer Science and Business Media.

In Chapter 7 (Section 7.3.2), we have shown that the solution of the ideal model in the case of a single component following the Langmuir isotherm model can be plotted in a dimensionless graph such that the band profile depends on the single value of the loading factor, Lf (Eq.7.26) [30]. In this plot, the reduced concentration... 

Like Helfferich and Klein [9], Rhee et al. [10] studied the separation of multicomponent mixtures by displacement chromatography using the restrictive assumption of the validity of the Langmuir isotherm model and the ideal model. They used a different approach, based on the method of characteristics, and studied the interactions between concentration shocks and centered simple waves [15]. This approach is more directly suited to adsorption chromatography than the... 

All profiles were calculated with a Langmuir isotherm model. The dashed line profile was calculated with a rectangular injection profile. 

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