Adsorption isotherms 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 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 4.12 Competitive isotherms of the (-)- and (+)- enantiomers of Troger base on microcrystalline cellulose triacetate, with ethanol as mobile phase, (a) Single-component adsorption isotherm of (+)-TB (squares) and (-)-TB (triangles) at 40° C. Experimental data and best fit to a Langmuir, (+)-TB, and a quadratic, (-)-TB, isotherm model, (b) Competitive isotherms of (-)-TB in enantiomeric mixtures for increasing concentrations (0, 0.5,1,1.5,2, 2.5, 3 g/L) of (+)-TB, calculated with IAS theory, (c) Competitive isotherms of (+)-TB in enantiomeric mixtures for increasing concentrations (0, 0.5, 1, 1.5, 2, 2.5, 3 g/L) of (-)-TB calculated with IAS theory. Reproduced from A. Seidel-Morgenstem and G. Guiockon, Chem. Eng. Sci., 48 (1993) 2787 (Figs. 4, 6, and7).

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.

Up to this point the treatment is strictly thermodynamic. However, the next steps necessarily demand the use of extra-thermodynamic assumptions. The first of them is the choice of an adsorption isotherm, which in general may be written as Pa = f( , 0). The most commonly adopted isotherm is the Frumkin isotherm, Eq. (1), assuming arbitrarily a linear or a quadratic dependence of upon E. The next necessary assumption concerns the dependence of p upon E, for which the following expression is usually adopted P = Pn,axexp -5( - moU, where b is a constant. Thus if a certain adsorption isotherm px = f( , 0) is selected, the partial derivative of In x with respect to E at constant 0 can be calculated and therefore Eq. (4) after integration yields the relationship = g( ). From this relationship the dependence of y upon E is obtained by integration and the differential capacity C is calculated fromC = dCT /d . 

Recently, Ilic, Elockerzi, and Seidel-Morgenstern (2010) derived a useful new explicit solution of the IAS theory equations for the calculation of adsorbed phase concentrations of binary systems whose individual component behavior can be represented by flexible second-order (quadratic) adsorption isotherms (Equation 2.56). 

Besides the switching time, the feed concentration is the most important parameter for enhancing productivity of an SMB process. The shape and the location of the region of complete separation in a (m, m3)-diagram are mainly influenced by the feed concentration. Therefore, the feed concentration for the simulation was gradually increased. Quadratic Hill-adsorption isotherms dependent on pressure were determined for the phytol-isomers. The mixture interactions were taken into account by the ideal adsorbed solution theory (lAST) [58). 

The improvement of the HILDA algorithm in comparison to the above method is connected with the following features evaluation of the monolayer capacity by systematic normalization of /(e)) application of an odd-ordered quadratic smoothing routine for the adsorption isotherm and distribution function, and evaluation of x(e) in a finite region of adsorption energies corresponding to the measured region of the adsorption isotherm [127]. 

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