The bi-Langmuir Isotherm

Here q, is the solid phase concentration, C, is the mobile phase concentration and ati an,i btl bu, are isotherm parameters for the / th component. The column saturation capacity of the chiral site is defined as  

In papers III-VI the adsorption data were fitted to both the Langmuir and the bi-Langmuir equation. 

Also for the bi-Langmuir isotherm, that describes a surface which is covered with two different kinds of sites, we can account for the competitive behavior of a mixed sample by using the bi-Langmuir competitive isotherm [109, 112]  

Peak profiles can be calculated with a proper column model, the differential mass balance equation of the compound(s), the adsorption isotherm, the mass transfer kinetics of the compound(s) and the boundary and initial conditions [13], When a suitable column model has been chosen, the proper parameters (isotherm and mass transfer parameters and experimental conditions) are entered into the calculations. The results from these calculations can have great predictive value [13, 114], The most important of the column models are the ideal model , the equilibrium-dispersive (ED) model , the 

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For nonlinear systems, however, the evaluation of the flow rates is not straightforward. Morbidelli and co-workers developed a complete design of the binary separation by SMB chromatography in the frame of Equilibrium Theory for various adsorption equilibrium isotherms the constant selectivity stoichiometric model [21, 22], the constant selectivity Langmuir adsorption isotherm [23], the variable selectivity modified Langmuir isotherm [24], and the bi-Langmuir isotherm [25]. The region for complete separation was defined in terms of the flow rate ratios in the four sections of the equivalent TMB unit ... 

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. 

An even more flexible equation is given by the Bi-Langmuir isotherm (Graham, 1953) (Eq. 2.41) ... 

The Bi-Langmuir isotherm can be extended in the same way to give the multi-component Bi-Langmuir isotherm. (Eq. 2.48) (Guiochon, 1994). 

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. 

Figure 3.40 Illustration of the method of isotherm measurements by computation of elution profiles. R-l-indanol on cellulose tribenzoate chiral stationary phase. Mobile phase, n-hexane and 2-propanol (92.5 7.5, v/v). (Left) Calculated (using the bi-Langmuir isotherm) and experimental chromatograms recorded for 46.25 (main figure) and 9.251 mg (insert) of R-l-indanol. The isotherm was determined from the band profile obtained for 46.25 mg. (Right) Bi-Langmuir isotherms obtained by the inverse method (lines) and by frontal analysis (symbols) for the R- and S-l-indanol enantiomers. Cmax indicates the maximum elution concentration. Reproduced with permission from A. Felinger, D. Zhou, G. Guiochon, /. Chromatogr. A, 35 (2003) 1005 (Figs. 2 and 3).

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]. 

Figure 7.6 Comparison of the solutions of the ideal model for a Langmuir and a bi-Langmuir isotherm. Band profiles for different sample sizes (n) in mg on each profile, (a) Plots of a Langmuir and a bi-Langmuir isotherm. The Langmuir isotherm is used to calculate the profiles in (b), the bi-Langmuir isotherm for those in (c). Coefficients of the Langmuir isotherm a = 13.1, b = 0.803. Coefficients of the bi-Langmuir isotherm Ui = 9.6, 6i = 6.4, = 7.1, f>2 = O-ll- Inset enlargement of the low-concentration part of the...

The effect of the isotherm parameters on the band profiles is instructive. Figures 7.8a and 7.8b illustrate the influence on the isotherm (Figure 7.8a) and on the diffuse rear of the band profiles (Figure 7.8b) of the term 2 of the bi-Langmuir isotherm at constant column saturation capacity (i.e., constant fli/bi -I- a /hi). This term corresponds to the contribution of "active" sites. The increased tailing is illustrated in the inset. Since is equal to F(ai + 2)/ the retention time increases... 

In most of the cases investigated so far, as in those already discussed and as explained in Chapter 3 (Figures 3.6 to 3.8), the experimental data obtained in the determination of single-component isotherms can be fitted correctly to the Langmuir equation q — (aC) /(1-F bC)) [77]. Occasionally, however, the isotherm data are better fitted when using other isotherm models. The bi-Langmuir isotherm... 

Fornstedt et al. investigated the thermod5mamics and the mass transfer kinetics of the retention of the enantiomers of R- and S-Propranolol on a system using an acetic acid buffer solution as the mobile phase and the protein CBH I immobilized on silica as the stationary phase. The single-component isotherm of each isomer fitted well to the bi-Langmuir isotherm model... 

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