LeVan-Vermeulen isotherm

On the other hand, the quantitative prediction of competitive isotherm behavior for the components of binary mixtmes is not possible using the competitive Langmuir isotherm model when the difference between the column satmation capacities for the two components exceeds 5 to 10%. For example, the adsorption isotherms of pure cis- and trans-androsterone on sihca are well accoimted for by the Langmuir model [9]. However, the two column saturation capacities differ by 30%, due to the nearly flat structure of the trans isomer compared to the folded structure of the cis isomer. As a consequence, the competitive Langmuir model accounts poorly for the competitive adsorption data [9,10]. Much improved results are obtained with the more complex LeVan-Vermeulen isotherm (Section 4.1.5). Another approach could use the random adsorption site model, with different exclusion siuface areas for the competing molecules [12],... 

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... 

Figure 4.4 Competitive isotherms of the 0 and enantiomers of methyl mandelate on 4-methylcellulose tribenzoate, with hexane/2-propanol (90 10) as the mobile phase, (a) Single-component isotherms at 30°C solid lines, competitive Langmuir model dotted lines, LeVan-Vermeulen isotherm, (b) Experimental (symbols) and calculated (lines) competitive isotherms ratio C(+)/C(-) = 1.05. (c) Same as (b), but ratio C(+)/C(-) = 2.43. (d) Same as (b), but ratio C(+)/C(-) = 0.32. Reproduced from F. Charton and G. Guiochon, ]. Chro-matogr., 630 (1993) 21 (Figs. 2 and 3).

The three-term expansion of the LeVan-Vermeulen isotherm is... 

The LeVan-Vermeulen isotherm has been used by Golshan-Shirazi et al. [10] to accoimt for experimental results obtained with cis- and frans-androsterone, two... 

Figure 4.8 Comparison of the two-term and three-term LeVan-Vermeulen isotherms and the Langmuir competitive isotherm. Solid line two-term LeVan-Venneulen isotherm dotted line three-term LeVan-Vermeulen isotherm dashed Une Langmuir competitive isotherm. F = 0.25, qg,l = 1 mmol/ml, 2 = 2 mmol/ml kg =3 = 2/ 01 ...

Figure 4.10 Experimental competitive adsorption isotherm data of ds- and fraws-andro-sterone. Same phase system as in Figure 4.9. Comparison of the Langmuir competitive model (bottom) and the two-term expansion of the LeVan-Vermeulen isotherm (top). In both cases, the best-fit parameters are used to calculate the Unes. Experimental data A ds-androsterone o trafjs-androsterone. Theory czs-androsterone (dotted lines) trans-androsterone (solid Hnes). a and d 2 1 mixture b and e 1 1 mixture and c and f 1 2 mixture. Reproduced with permission from S. Golshan-Shirazi, J.-X. Huang and G. Guiochon, Anal. Chem., 63 (1991) 1147 (Figs. 1 and 2), ( )1991 American Chemical Society.

Extension of the Series Solution of the LeVan-Vermeulen Isotherm to multi-component systems Frey and Rodrigues [56] have extended the binary LeVan-Vermeulen isotherm to the case of multicomponent systems. They showed that, if we assume that the adsorption isotherm of each single component follows Langmuir isotherm behavior, the multicomponent isotherm is given by [56]... 

The FA method gives isotherm data. To be useful in preparative chromatography, these data must be fitted to an isotherm model. There are presently no numerical procedures available to smooth the data from multidimensional plots, similar to the 2-D splines or French curves and obtain purely empirical isotherms. Therefore, the major difficulty is the selection of adequate models. The Langmuir isotherm is too simplistic in most cases, and the LeVan-Vermeulen isotherm is complicated and difficult to use as a fitting fimction. Several methods have been described to extract the "best" set of Langmuir parameters which could accormt for a set of competitive adsorption data [108]. These methods have been compared. The most suitable method seems to depend on the aim of the determination and on the deviation of the system from true Langmuir behavior [108]. 

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.

LeVan-Vermeulen isotherm Isotherm model which often applies well when the single-component isotherms both follow the Langmuir model, but with different values of the column saturation capacity. 

To satisfy the Gibbs adsorption isotherm for unequal monolayer capacities, exphcit isotherms can be obtained in the form of a series expansion [LeVan and Vermeulen,/. Phy.s. Chem., 85, 3247 (1981)]. A two-term form is... 

The rectangular isotherm has received special attention. For this, many of the constant patterns are developed fuUy at the bed inlet, as shown for external mass transfer [Klotz, Chem. Rev.s., 39, 241 (1946)], pore diffusion [Vermeulen, Adv. Chem. Eng., 2, 147 (1958) Hall et al., Jnd. Eng. Chem. Fundam., 5, 212 (1966)], the linear driving force approximation [Cooper, Jnd. Eng. Chem. Fundam., 4, 308 (1965)], reaction kinetics [Hiester and Vermeulen, Chem. Eng. Progre.s.s, 48, 505 (1952) Bohart and Adams, J. Amei Chem. Soc., 42, 523 (1920)], and axial dispersion [Coppola and LeVan, Chem. Eng. ScL, 38, 991 (1983)]. 

Using the IAS theory, LeVan and Vermeulen derived a competitive binary isotherm equation that accounts for differences in the column saturation capacities for the two components, when the single-component adsorption isotherms follow a Langmuir or a Freundlich isotherm model [33]. 

When (Ji g = ij2,s = = this equation reduces to the Langmuir isotherm. For two components, this equation reduces to the two-term Taylor series expansion derived by LeVan and Vermeulen [33]. 

The last three chapters deal with the fundamental and empirical approaches of adsorption isotherm for pure components. They provide the foundation for the investigation of adsorption systems. Most, if not all, adsorption systems usually involve more than one component, and therefore adsorption equilibria involving competition between molecules of different type is needed for the understanding of the system as well as for the design purposes. In this chapter, we will discuss adsorption equilibria for multicomponent system, and we start with the simplest theory for describing multicomponent equilibria, the extended Langmuir isotherm equation. This is then followed by a very popularly used IAS theory. Since this theory is based on the solution thermodynamics, it is independent of the actual model of adsorption. Various versions of the IAS theory are presented, starting with the Myers and Prausnitz theory, followed by the LeVan and Vermeulen approach for binary systems, and then other versions, such as the Fast IAS theory which is developed to speed up the computation. Other multicomponent equilibria theories, such as the Real Adsorption Solution Theory (RAST), the Nitta et al. s theory, the potential theory, etc. are also discussed in this chapter. 

The approach of IAS of Myers and Prausnitz presented in Sections 5.3 and 5.4 is widely used to calculate the multicomponent adsorption isotherm for systems not deviated too far from ideality. For binary systems, the treatment of LeVan and Vermeulen presented below provides a useful solution for the adsorbed phase compositions when the pure component isotherms follow either Langmuir equation or Freundlich equation. These expressions are in the form of series, which converges rapidly. These arise as a result of the analytical expression of the spreading pressure in terms of the gaseous partial pressures and the application of the Gibbs isotherm equation. 

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