Bi-Langmuir model

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... 

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]. 

Thermodynamic Characteristics of the Adsorption Sites According to the Bi-Langmuir Model... 

Very similar characteristics (bi-Langmuir model) and overall trends have been also found for 3CPP on the tBuCQD-CSP in the NPLC mode, yet the saturation capacities at the enantioselective sites were much larger than in case of TFAE [54]. 

A reexamination of the chiral discrimination of PPA herbicides was done on a TE CSP using the perturbation method to calculate the solute distribution isotherms [86]. The effect of temperature was well described by the bi-Langmuir model and enabled confirmation of the previous results by the same authors [85]. 

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. 

Fig. 5.8. Frontal analysis of the binding of D- and L-PA to an MIP imprinted with L-PA. The MIP was prepared using dichloromethane as diluent following the procedure shown in Fig. 5.2. Fitting (lines) of the experimental isotherm data (symbols) for L-PA (A) and D-PA (B) to the bi-Langmuir model (main figure), the Langmuir model (left inset) and the Freundlich model (right inset). For the runs at 40°C solid lines and plus symbols, at 50°C long dashed lines and crosses, at 60°C short dashed lines and stars, at 70°C dotted lines and squares. Mobile phase MeCN/potassium phosphate buffer 0.05 M, pH 5.85 70/30 (v/ v). From Sajonz et al. [40].

All processes are modeled as series of countercurrent equilibrium cells. Parameters were determined experimentally (section 3). A liquid-phase reaction is accounted for by Da = (rate constant)x(cell volume)/(solid flow rate). Adsorption is described by the bi-Langmuir model. All equations were implemented in the simulation environment DIVA [3] details on the implementation of a largely analogous model can be found in [1,4]. The following set of performance parameters were used to evaluate each process ... 

The isotherm data acquired from frontal analysis over a broad concentration range fitted well to the bi-Langmuir model, see Figure 18, demonstrating that the adsorption on Kromasil CHI-TBB is heterogeneous with two types of sites. The saturation capacity of site II obtained from the bi-Langmuir isotherm parameters were qs>n = 130 mM for (R)-(-)-2-phenylbutyric acid and qs,n= 123 mM for (S)-(+)-2-phenylbutyric. 

Similar adsorption data have also been reported for the adsorption isotherms of many compounds in various systems. For example, the adsorption data of several jS-blockers, particularly those of propranolol acquired in a 1 to 7000 relative concentration range, on an immobilized cellulase. Cel 7A, fit very well to the bi-Langmuir model, as illustrated in Figure 3.13 [47,55]). The enantioselective site was identified as a pair of amino acid residues in the tunnel formed by the main chain of the protein. The parameters of the isotherm depend on the pH as illustrated in Figure 3.14. A bi-Langmuir model was also found to accoimt well for the separation of pairs of enantiomers on polymers molecularly imprinted with one of the enantiomers [56]. Note, however, that there are also many systems in which the adsorption isotherms of enantiomers are not accounted for by a bi-Langmuir model showing that enantioselectivity is often achieved by a complex... 

Toth and the bi-Langmuir models give excellent fits however, the bi-Langmuir isotherm is quantitatively better. The main figure illustrates all the data and the insert shows only the data at low concentrations. 

Lisec et al. [23] measured by frontal analysis the ternary isotherm data for phenol, 2-phenylethanol and 3-phenyl-l-propanol on Kromasd-Cig, with a water/methanol (1/1) solution. The data were fitted to the model equations of the competitive Langmuir and competitive bi-Langmuir models and to the LAS and RAS models derived from the Langmuir model. No substantial improvements were observed with the more complex models. Satisfactory agreement was observed between experimental band profiles and the profiles calculated from the ternary Langmuir isotherm. 

This isotherm model has been used successfully to accoimt for the adsorption behavior of numerous compounds, particularly (but not only) pairs of enantiomers on different chiral stationary phases. For example, Zhou et ah [28] foimd that the competitive isotherms of two homologous peptides, kallidin and bradyki-nine are well described by the bi-Langmuir model (see Figure 4.3). However, most examples of applications of the bi-Langmuir isotherm are found with enantiomers. lire N-benzoyl derivatives of several amino acids were separated on bovine serum albumin immobilized on silica [26]. Figure 4.25c compares the competitive isotherms measured by frontal analysis with the racemic (1 1) mixture of N-benzoyl-D and L-alanine, and with the single-component isotherms of these compounds determined by ECP [29]. Charton et al. foimd that the competitive adsorption isotherms of the enantiomers of ketoprofen on cellulose tris-(4-methyl benzoate) are well accounted for by a bi-Langmuir isotherm [30]. Fornstedt et al. obtained the same results for several jS-blockers (amino-alcohols) on immobilized Cel-7A, a protein [31,32]. 

Similar results were obtained with the enantiomers of methyl mandelate separated on 4-methylcellulose tribenzoate immobilized on silica [30]. Figure 4.4a shows the experimental adsorption data for the two pure enantiomers (symbols), the best bi-Langmuir isotherms (solid lines) and the best LeVan-Vermeulen isotherms [33]. The data (symbols) were obtained by ECP. Figures 4.4b-d compare the competitive isotherm data measured with three mixtures of different composition and the isotherms calculated from the single component isotherms (Figure 4.4a) using the competitive bi-Langmuir model (Eq. 4.10). Results obtained... 

Recently, Lindholm et al. [24] measured by the perturbation method the quaternary isotherms of the two enantiomers of methyl- and ethyl- mandelate on a chiral stationary phase. The method developed earlier by Forss6n et al. [47] consists in injecting perturbation pulses that contains concentrations of the different components that are alternately higher and lower than the plateau concentrations of the different components (see Eq. 4.104). This method ensures that the different perturbations (four in the present case) are easily detectable and that their retention times can be measured accurately. The isotherm data were modeled using a quaternary bi-Langmuir model and excellent agreement was reported between experimental and calculated band profiles. 

Figure 4.33 Top. Simultaneous fitting to a competitive bi-Langmuir model of the chromatograms obtained with a large (50.7 mg) and a moderate (10.14 mg) sample of the racemic mixture of 1-indanol. Bottom. Comparison of the FA adsorption data points (s)rmboIs) and the best competitive bi-Langmuir isotherms obtained by the inverse method (lines) for the racemic mixture. Reproduced with permission from A. Felinger, D. Zhou, G. Guiochon, f. Chromatogr. A, 1005 (2003) 35 (Figures 7 and 8).

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