Quadratic isotherm

Besides the heterogeneity of the adsorbent surface, the second major reason for the adsorption of a compound to deviate from Langmuir isotherm behavior is that the adsorbed molecules interact. In this category, we find the Fowler isotherm, the anti-Langmuirian isotherm, and several S-shaped isotherm models, including the quadratic isotherm, the extended BET isotherm models, and the Moreau model. 

A Langmuir-t5rpe isotherm equation with a negative value of b could accormt qualitatively for the shape of convex downwards isotherms at low concentrations. This equation, however, accormts for an isotherm of class III and has a vertical asymptote of equation C = 1/b, whereas in liquid-solid equilibria the isotherm should rather exhibit a horizontal as3unptote and a finite saturation capacity. The quadratic isotherm, with a proper set of numerical coefficients, can accoimt for isotherms having up to three inflection points (Figure 3.1, isotherm of class IV). 

The Langmuir isotherm is the first-order isotherm predicted by statistical thermodynamics. The second-order isotherm, obtained with n = 2 in Eq. 3.61, is called the quadratic isotherm [76]. We see in Eq. 3.61 that the limit of 6 for C infinite is nqs- This results from the model, which considers that each site on the surface can accommodate n molecules and that there are qs such sites on the surface. 

Figure 3.25 Experimental isotherms of Troger s base enantiomers on microcrystalline cellulose triacetate. Experimental data by frontal analysis (symbols) and best quadratic isotherm (solid line). Experimental conditions column length, 25 cm column efficiency, N = 106 plates phase ratio, F = 0.515 flow velocity 0.076 cm/s, pure ethanol. Column (250 x4.6 mm) packed with cellulose microcrystalUne triacetate (CTA, 15-25ftm), previously boiled in ethanol for 30 min. (a) Isotherm data. Top line, (+)-TB, bottom line, (-)-TB. (b) Plot of q/C versus C. Reproduced with permission from A. Seidel-Morgenstem and G. Guiochon, Chem. Eng. Scl, 48 (1993) 2787 (Figs. 4 and 5).

The numerator of the second differential is a third-degree polynomial which may have 1 or 3 real roots and, thus may have 0,1,2, or 3 positive roots and as many inflection points. The quadratic isotherm is the first equation to which the experimental data should be fitted when an inflection point is observed. 

A quadratic isotherm has been used by Guiochon et al. [77] to calculate the band profiles obtained in the case of an S-shaped equilibrium isotherm. The same isotherm has been used by Svoboda [78]. An example of an isotherm with one inflection point, accormted for by the quadratic model is shown in Figure 3.25 [79]. It corresponds to the adsorption of the (+) isomer of Troger s base on microcrystalline cellulose triacetate, while the (-) isomer follows a Langmuir behavior in... 

Figure 12.29 Comparison of theoretical and experimental displacement separations of resorcinol and catechol by phenol. Calculations using the equilibrium-dispersive model, the LeVan- Vermeulen isotherm model, and single-component adsorption data. Experimental results on a 4.6x250 CIS Nucleosil 5 fim column, F = 0.4 carrier, water, Fj, = 0.2 mL/min, T = 20°C 1 1 mixture, = 0.5 mL displacer, 80 g/L phenol in water = 30%, Lf = 16.5%. (a) Calculation with LeVan-Vermeulen isotherm, (b) Calculation with quadratic isotherm, three floating parameters, (c) Calculation with competitive Langmuir isotherm, single-component isotherm parameters, (d) Calculation with Langmuir isotherm, best adjusted parameters. Reproduced with permission from. C. Bellot and J.S. Condoret, J. Chromatogr., 657 (1994) (Figs. 3c, 4c, 6c, 8c) 305.

This efficient approach was successfully validated experimentally by Kaspereit and Sainio (2011) for two cycloketones with bi-Langmuir isotherms, as well as for an enantioseparations for a pharmaceutically relevant compound with complex quadratic isotherms (von Langermarm et al., 2012). 

From Fig. 9.15 the influence of the quadratic isotherm can easily be seen for a feed concentration of lOmg/ml. In the figure, the boundaries of the region of complete separation for the linear isotherm are plotted as lines in order to show the difference between the linear and nonlinear separation regions. Point w, which has the maximal feed flow rate, moves to smaller m 2 and m. The size of the region of complete separation decreases. From Fig. 9.16 it can be seen that a further increase of feed concentration to 20mg/ml leads to a region of complete separations of completely different size and shape. 

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