Bi-Langmuir isotherm

A. Gentilini, C. Migliorini, M. Mazzotti and M. Morbidelli, Optimal operation of simulated moving-bed units for non-linear cliromatograpliic separ ations. II. Bi-Langmuir isotherm , 7. Chromatogr. 805 37-44 (1998). 

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 ... 

Gentilini A., Migliorini C., Mazzotti M., Morbidelli M. (1998) Optimal Operation of Simulated Moving-Bed Units for Non-Linear Chromatographie Separations. II. Bi-Langmuir Isotherm, J. Chromatogr. A 805 37-44. 

Fig. 5.12. The concentration dependence of k( obtained from frontal analysis for L- (solid lines and plus symbols) and D- (dashed lines and crosses) PA at 40°C using a bi-Langmuir isotherm (main figure) or a Freundlich isotherm (inset) to fit the data. Experimental data (symbols) were fitted (lines) to the function k( = kf oC". Mobile phase MeCN/potassium phosphate 0.05 M, pH 5.85 70/30 (v/v). The MIP was prepared using L-PA as template and dichloromethane as diluent following the procedure shown in Fig. 5.2. From Sajonz et al. [40].

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] ... 

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. 

Figure 18. Isotherms for (R) and (S)-2-phenylbutyric acid, experimentally acquired by frontal analysis and compared with their fits with bi-Langmuir adsorption isotherm parameters. The lines are calculated data using the best single bi-Langmuir isotherm parameters.

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). 

Modified multi-component Langmuir and multi-component Bi-Langmuir isotherms offer a maximum flexibility for adjustment to measured data if all coefficients are chosen individually. But in the same way as for multi-component Langmuir isotherms (Eq. 2.43) it is possible to use, for Eqs. 2.47 and 2.48, constant Langmuir terms (by = by, bjy = b]i , b2y = b2u) as well as constant adjustment terms (Xj = X) or equal saturation capacities (qsaUii = [Pg.37]

Gentilini, A., Migliorini, C., Mazzotti, M., Morbi-delli, M. Optimal operation of simulated moving-bed units for non-linear chromatographic separations, II. Bi-langmuir isotherm, J. Chromatogr. A, 1998, 805, 37-44. 

Figure 3.2 Linear, Langmuir, and bi-Langmuir isotherms. The Linear, Langmuir, and bi-Langmuir isotherms are used as models to fit the experimental adsorption data for N-Benzoyl-D-phenylalanine. Parameters Linear isotherm a = 13.1 Langmuir isotherm a = 13.1, b = 241 bi-Langmuir isotherm ai = 9.6, bi = 1920, fl2 = 7.1, 2 = 33-8- Data from S. Jacobson, S. Golshan-Shirazi and G. Guiochm, AIChE /., 37 (1991) 836. Reproduced by permission of the American Institute of Chemical Engineers. 1991 AIChE. All rights reserved.

Figure 3.13 Single-component equilibrium isotherms for (R)- and (S)- Propranolol on immobilized Cel-7A at increasing pH. Symbols experimental data, o R- and S-enantiomers. Lines best bi-Langmuir isotherms (dashed for R-, solid for S-enantiomer). Left Low concentration data (in iM). pH-values (1) 4.72, (2) 4.98, (3) 5.21, (4) 5.49, (5) 5.70, and (6) 5.93. Right High concentration data (in mM). Reproduced with permission from T. Fomstedt, G. Got-mar, M. Andersson, G. Guiochon, J. Am. Chem. Soc., 121 (1999) 1164 (Figs. 6a, 6c). 1999, American Chemical Society.

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... 

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. 

This model was used by Gritti and Guiochon to account for the behavior of propranolol on several Cig-bonded silica adsorbents, from methanol/water solutions. Similar results were obtained for Kromasil-Cig [101,102], XTerra-Cjs [84], Symmetry-Ci8 [103]. When the mobile phase contains a high concentration of a monovalent salt, the adsorption follows bi-Langmuir isotherm behavior [101]. In the absence of salt, at low concentrations of a monovalent salt or with di- (e.g., phthalate, succinate, naphthalene sulfonate) or tri-valent salts (e.g., citrate), the isotherm data are best modeled by a bi-Moreau isotherm [91,104]. 

Figure 3.31 Distribution of the equilibrium constants of adsorption calculated by means of the biToth model for 1-indanol on cellulose tribenzoate. The arrows indicate the equilibrium constants derived from the competitive bi-Langmuir isotherm. Reproduced with permission from A. Felinger, D. Zhou, G. Guiochon,. Chromatogr. A, 1005 (2003) 35.

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).

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. 

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