Competitive Langmuir model

The Langmuir isotherm (or Langmuir model) provides an improvement over the K( and Freundlich approaches by maintaining a mass balance on the sorbing sites (Stumm and Morgan, 1996). The model, for this reason, does not predict that species can sorb indefinitely, since the number of sites available is limited. When the calculation carries reactions for the sorption of more than one aqueous species, furthermore, it accounts for competition such a calculation is known as a competitive Langmuir model. 

Equation (51) represents a competitive Langmuir model for cation adsorption to the mineral surface. A similar competitive Langmuir model has been proposed to describe the effect of magnesium on forsterite dissolution (Pokrovsky and Schott, 2000b) ... 

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],... 

In spite of these major fundamental objections, the competitive Langmuir model has been foimd to be useful in several practical cases. Most of these cases were foimd in normal phase HPLC [14,15]. Similar results were also reported in RPLC for solutes of closely similar sizes and structures. 

Figure 4.4 Competitive isotherms of the 0 and enantiomers of methyl mandelate on 4-methylcellulose tribenzoate, with hexane/2-propanol (90 10) as the mobile phase, (a) Single-component isotherms at 30°C solid lines, competitive Langmuir model dotted lines, LeVan-Vermeulen isotherm, (b) Experimental (symbols) and calculated (lines) competitive isotherms ratio C(+)/C(-) = 1.05. (c) Same as (b), but ratio C(+)/C(-) = 2.43. (d) Same as (b), but ratio C(+)/C(-) = 0.32. Reproduced from F. Charton and G. Guiochon, ]. Chro-matogr., 630 (1993) 21 (Figs. 2 and 3).

The agreement that was observed between the experimental results and the prediction of a competitive Langmuir model based on the use of single-component Langmuir isotherms in the case of the adsorption of enantiomeric derivatives of amino acids on immobilized serum albumin [26] is imusual. It demonstrates the validity of the competitive Langmuir model based on the use of the parameters of the single-component Langmuir model. However, as explained before, the experimental conditions are exceptionally favorable since the column saturation capacities for the two enantiomers are equal. Nevertheless, Zhou et ah have shown that it is possible, in certain favorable cases, to derive the equilibrium isotherms of the pure enantiomers and to calculate isotherm equilibrium data for any mixture of... 

Adsorption on the sites accessible to both compounds. These sites are noted b. The classical competitive Langmuir model applies. 

Figure 11.23 Qualitative prediction of the elution order reversal of cis- and frans-androsterone. (Left) Anal5dical and preparative sample sizes. High concentration band profiles calculated with the LeVan-Vermeulen isotherm model. Inset experimental data [23], Stationary phase silica modified with a pH 6.8 phosphate buffer. Mobile phase (9 1) acetonitrile-dichloromethane, 0.98 mL/min. Samples, cis-androsterone 0.026 and 5.2 mg, trans-androsterone 0.15 and 1.8 mg. (Right) Band profiles calculated with the competitive Langmuir model. Reproduced with permission from S. Golshan-Shirazi, J.-X. Huang and G. Guiochon, Anal. Chem., 63, (1991) 1147 (Fig. 8). ( )1991, American Chemical Society.

It is possible to calculate from Eq. 13.16 the position, the sign, and the relative importance of the system peaks associated with the sample components if the competitive Langmuir model is assumed for the equilibrium isotherms of all the compormds involved, sample components and additives (Eq. 13.3). Since this isotherm model assumes that the column saturation capacity is the same for all the compounds i.e., qs = a /h = = Uilhi = = Uglbs), the partial differentials of the additive can be written as follows ... 

Competition between adsorbing species for the same site on a sorbent has been modeled with the so-called competitive Langmuir model. In this model an individual isotherm equation with its own constants is written for each species sorbed by a given sorbent (see Table 10.7). 

Preparative sepmations are rarely carried out between only two components. In practice one has many components. If one is to carry out computer simulations for such samples it is strictly necessary to have a competitive isotherm which describes the concentrations of all components. This can become a tall order, especially if all solutes do not fit the competitive Langmuir model which is one of the few available which can be used for more than two compounds. To a first approximation, one generally chooses the impurity peaks closest to the component of interest. It is not always a good assumption that the other components do not interact, especially at high loads when component bands overlap at high concentrations for much of their residence times in the column. Often experiments are performed to observe the peak shapes and position of impurities in order to assess their behaviour and to infer the type of isotherm which exists without necessarily determining it. Such experiments... 

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. 

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). 

The second model results from a bimolecular surface reaction, A + B — products, with competitive Langmuir-Hinshelwood kinetics, which occurs in a heterogeneous differential reactor with perfectly mixed gas phase. The reaction is first order in both adsorbed A and B, and two vacant sites are required in the reaction mechanism. If the reaction products desorb immediately, the... 

The third type of bimolecular surface reaction is also represented by eqns. (XI) and (26) but in this case the two reactants adsorb non-competitively on different kinds of surface site. In terms of the Langmuir model... 

The fact that the isotherm data for L-PA were better fitted to the Freundlich isotherm at lower concentrations agrees with previous studies on other template systems, where the isotherm data were often best fitted to tri-Langmuir models [25]. Competitive assays using radiolabelled substrates allowed the identification of a class of sites with a very low surface coverage ca. 1 nmol/g) and high binding constants (up to 1 x 10 /M) (Table 5.3). 

Figures 8a and 8b show the simulated elution profiles of two compounds using the simple Langmuir model (eq. 1 below), after injection of 135 pi of a solution (solute 1 = 20 mM, solute 2 = 60 mM). Figure 8a shows the situation when there is no competition between the two solutes. Since competition always exists, such a situation can only be achieved experimentally by overlaying the chromatograms resulting from separate injections. The second...

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. 

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