Competitive Equilibrium Isotherms

4 Simple, Thermodynamically-consistent, Competitive Langmuir Isotherm. 165 

12 Competitive Isotherms Models for Other Modes of Chromatography..186 

2 Determination of a Multi-component Langmuir Isotherm by Frontal Analysis 

All cases of practical importance in liquid chromatography deal with the separation of multicomponent feed mixtures. As shown in Chapter 2, the combination of the mass balance equations for the components of the feed, their isotherm equations, and a chromatography model that accounts for the kinetics of mass transfer between the two phases of the system permits the calculation of the individual band profiles of these compounds. To address this problem, we need first to understand, measure, and model the equilibrium isotherms of multicomponent mixtures. These equilibria are more complex than single-component ones, due to the competition between the different components for interaction with the stationary phase, a phenomenon that is imderstood but not yet predictable. We observe that the adsorption isotherms of the different compounds that are simultaneously present in a solution are almost always neither linear nor independent. In a finite-concentration solution, the amount of a component adsorbed at equilib- 

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. 

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The theory of simple waves applies to large-volume injections, i.e., to the profiles obtained upon injection of rectangular profiles which are so wide that the injection plateau has not been entirely eroded when the band elutes. Then, simplifications of the solution occur because there is a constant state, the concentration plateau. This solution is not valid in overloaded elution chromatography when the injection volume is sufficiently small that the injection plateau has eroded and disappeared by the time the band elutes from the column. It is important to discuss this solution, however, because it takes a finite time for the profile of even a narrow rectangular injection to decay, and the band profile during that period is given by the simple wave solution. Also, this solution is the basis for a method of determination of competitive equilibrium isotherms (Chapter 4, Section 4.2.4). 

The theory of the hodograph transform and the relationship derived between the equations of the two lines given by this transform in the case of a binary mixture and those of the competitive equilibrium isotherms were briefly presented in Section 8.1.2. The theory is easily extended to multicomponent mixtures, although in this case we must represent the hodograph transform in an n-dimensional coordinate system, Ci, C2, , C , or in its planar projections. If the solution presents a constant state (Figure 8.1), it is a simple wave solution, and there is a relationship between the concentrations of the different components in the eluent at the column exit (Figure 8.2). This result is valid for any convex-upward isotherm. In the particular case in which the competitive Langmuir isotherm apphes, these relationships are linear. 

All these results demonstrate that the calculation of band profiles that are in very good agreement with experimental results is possible provided an accurate model of the competitive equilibrium isotherms is available. The main practical difficulty in the modeling of separations is in the accurate measurement and modeling of these competitive isotherms. Once such a model has been validated, it is possible to calculate the performance of a chromatographic unit and to optimize its performance [27]. 

Although for the sake of clarity the previous discussion was limited to the case of a binary mixture, these results are easily generalized to the study of an n-component mixture. Because of the coupling between the mobile phase components, the velocity eigenvalues are related to the slopes of the tangents to the n-dimensional isotherm surface, in the n composition path directions. These slopes can be calculated when the isotherm surface is known. Conversely, systematic measurement of the retention times of very small vacancy pulses for various compositions of the mobile phase may permit the determination of competitive equilibrium isotherms, but only if a proper isotherm model is available. Least-squares fitting of the set of slope data to the isotherm equations allows the calculation of the isotherm parameters. If an isotherm model, i.e., a set of competitive isotherm equations, is not available, the experimental data cannot be used to derive an empirical isotherm (see Chapter 4). 

Piqtkowski el al. measured the single-component and the competitive equilibrium isotherms of phenetole (ethoxy-benzene) and n-propyl benzoate on a 150 x 3.9 mm S3onmetry -Cig (endcapped) column (Waters), using a methanol/water (65 35, v/v) as the mobile phase [26]. The adsorption equilibrium data of the single-component systems were acquired by frontal analysis. For both compoimds. 

The competitive equilibrium isotherm model best fitting the FA experimental data for the R and S enantiomers of 1-phenyl-l-propanol on cellulose tiibenzoate was the Toth model. This model was used to calculate the elution profiles of samples of mixtures of the two enantiomers [29]. The General Rate model combined with the Generalized Maxwell-Stefan equation (GR-GMS) was used to model and describe surface diffusion (see Chapter 5). The mass transfer kinetics is slow and this model fits the experimental data well over a wide concentration range with one single set of numerical parameters to account for the band profiles in a wide range of concentrations, as shown in Figure 16.24. 

The accurate determination of the competitive equilibrium isotherms of the feed components of importance in the chromatographic system selected, and the measurement of the other parameters of importance (column efficiency as a function of the mobile phase velocity, viscosity of feed solutions in the mobile phase). 

Cm, ) functional dependence of the competitive equilibrium isotherm on the mobile phase concentrations of all components, 2 FeC feed cost (cost of product lost), 18... 

James, F., Sepulveda, M., Charton, F., Quinones, L, and Guiochon, G. (1999) Determination of binary competitive equilibrium isotherms from the individual chromatographic band profiles. Chem. Eng. Sci., 54,1677-1696. 

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