Determination of Competitive Isotherms

There is a dearth of competitive adsorption data, in a large part because they are difficult to measme, but also because little interest has been devoted to them, as, until recently, there were few problems of importance whose solution depended on their understanding. Besides the static methods, which are extremely long and tedious and require a large amoimt of material, the main methods of measurement of competitive isotherms use column chromatography. Frontal analysis can be extended to competitive binary isotherms [14,73,93-99], as well as pulse techniques [100-104]. The hodograph transform is a powerful method that permits an approach similar to FACP for competitive binary isotherms [105,106]. 

Frontal analysis can easily be extended to binary mixtures. The shape of the breakthrough profiles and the effect of axial dispersion on these shapes have been studied theoretically [93,94] and experimentally [14,73,95-99]. These profiles are characterized by the successive elution of two steep fronts (shock layers) for a binary mixture. The use of these profiles for the determination of the competitive isotherms of two components has been developed by Jacobson et al. [14]. 

The experiment is repeated with a series of solutions corresponding to other ratios of the concentrations, in order to determine the entire isotherm surface [14]. Rather than making successive step changes in the feed concentration, it is also 

The major drawbacks of the frontal analysis method are the important number of measurements to be made, the considerable amount of time that it takes to determine a set of competitive isotherms and the large amount of sample required. The competitive isotherms are sets of n surfaces in an n -b 1 space where n is the number of components. For a binary mixture, we have two surfaces, /i(Ci, C2) and /2(Ci, C2). These surfaces depend minimally on four parameters, often on more, depending on the isotherm model selected. 

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]. 

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The use of these profiles for the determination of competitive isotherms in the binary case has been developed by Jacobsen et al. [8]. 

In this chapter, we discuss first a number of models that have been used to accoimt for competitive isotherm data. Although the multi-component extension of many of these models is straightforward, most of them have been used almost exclusively with binary mixtures. In the second part of this chapter, we describe the methods of determination of competitive isotherms. Finally, we discuss the methods of data acquisition for multi-component adsorption and we present a few examples. 

Figure 4.23 Schematic of an equipment designed for the experimental determination of competitive isotherms by binary frontal analysis.

Figure 4.31 Experimental determination of competitive isotherms by the method of the hodograph transform, (a) Individual elution profiles of the two components of a binary mixture, (b) Hodograph transform of the profiles in (a). Reproduced with permission from Z. Mfl and G. Guiochon,. Chromatogr. 603 (1992) 13 (Fig. 3).

The method has been used for the determination of competitive isotherms in cases in which the deviation from the Langmuir model is moderate [115]. An HPLC chromatograph configured so as to permit injecting into the column a wide rectangular pulse, e.g., by pumping into it either the pure mobile phase or, for a known time, a solution of the compound of interest, was used to make the measurements described [115]. Excellent agreement was observed with other experimental data and with the experimental band profiles recorded in overloaded elution for binary samples of various compositions [48]. 

This value is in agreement with the one derived from band profiles calculated with the equilibrium-dispersive model [9]. The time given by Eq. 16.20 provides useful information regarding the specifications for the experimental conditions under which staircase binary frontal analysis must be carried out to give correct results in the determination of competitive isotherms. The concentration of the intermediate plateau is needed to calculate the integral mass balances of the two components, a critical step in the application of the method (Chapter 4). This does not apply to single-pulse frontal analysis in which series of wide rectangular pulses are injected into the column which is washed of solute between successive pulses. 

Competitive frontal analysis Frontal analysis carried out with multicomponent mixtures, for the determination of competitive isotherms or the enrichment of certain feed components. 

It is possible to extend the frontal chromatography method for the measurement of binary and multicomponent isotherms. In this case, the profiles are characterized by successive elution of several steep fronts. The use of these profiles for the determination of competitive isotherms in the binary case has been developed by Jacobsen et al. ... 

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