Determination of Single-Component Isotherms

The primary use of isotherm data measurements carried out on single-component elution profiles or breakthrough curves is the determination of the single-component adsorption isotherms. This could also be done directly, by conventional static methods. However, these methods are slow and less accurate than chromatographic methods, which, for these reasons, have become very popular. Five direct chromatographic methods are available for this purpose frontal analysis (FA) [132,133], frontal analysis by characteristic point (FACP) [134], elution by characteristic point (ECP) [134,135], pulse methods e.g., elution on a plateau or step and pulse method) [136], and the retention time method (RTM) [137]. 

The inverse problem consists in determining the isotherm and the rate constants, knowing one or several solutions of the system of equations ie., band profiles acquired under known experimental conditions, i.e. with known initial and boimdary conditions). There is a paucity of mathematical results to guide this last quest. However, the selection of proper experimental conditions i.e., of simple bormdary conditions) can profoimdly simplify the solution of the inverse problem. 

When the column efficiency is high, an acceptable approximation is given by  

Frontal analysis is a very popular method of isotherm determination, and rightly so. It has been apphed to the determination of a great number of equi-Ubrlum isotherms, in many modes of chromatography. Among others, it has been used for the measiu ment of the isotherms of peptides [38] and proteins [8,41,65]. It is the method most often used in our group. 

The advantages and drawbacks of FA relative to those of the other methods of isotherm determination are discussed later, in Section 3.6.4. 

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Gritti, F. et al. Determination of single component isotherms and affinity energy distribution by chromatography J. Chromatogr. A. 2003, 988, 185-203. 

In most of the cases investigated so far, as in those already discussed and as explained in Chapter 3 (Figures 3.6 to 3.8), the experimental data obtained in the determination of single-component isotherms can be fitted correctly to the Langmuir equation q — (aC) /(1-F bC)) [77]. Occasionally, however, the isotherm data are better fitted when using other isotherm models. The bi-Langmuir isotherm... 

Determination of Single Component Adsorption Isotherm Parameters - Characterization of a New CSP (Paper III)... 

One of the most important applications of frontal chromatography is the determination of equilibrium adsorption isotherms. It was introduced for this purpose by Shay and Szekely and by James and Phillips.The simplicity as well as the accuracy and precision of this method are reasons why the method is so popular today and why it is often preferred over other chromatographic methods, for example, elution by characteristic points (ECP) or frontal analysis by characteristic points (FACP). Frontal chromatography as a tool for the determination of single-component adsorption isotherms will be discussed in the following section. 

A novel and simple method for determination of micropore network connectivity of activated carbon using liquid phase adsorption is presented in this paper. The method is applied to three different commercial carbons with eight different liquid phase adsorptives as probes. The effect of the pore network connectivity on the prediction of multicomponent adsorption equilibria was also studied. For this purpose, the Ideal Adsorbed Solution Theory (lAST) was used in conjuction with the modified DR single component isotherm. The results of comparison with experimental data show that incorporation of the connectivity, and consideration of percolation processes associated with the different molecular sizes of the adsorptives in the mixture, can improve the performance of the lAST in predicting multicomponent adsorption equilibria. 

Single-component isotherm parameters cannot always predict elution profiles with satisfied accuracy [122, 123], Therefore, to be able to predict accurate overloaded multi-component elution profiles where competition occurs competitive adsorption isotherm parameters are often necessary. Measurement of isotherms from a mixture is also often necessary because the pure enantiomers are not always accessible in large quantities. However, there exist only a small number of reports on the determination of multi-component adsorption isotherm parameters. FA can be used to determine binary isotherm data but it is time-consuming. The PP method is an alternative method to determine isotherm parameters from binary mixtures. It has been reported that the PP method works well up to weakly non-linear conditions [118, 119],... 

The most dramatic difference between analytical and preparative chromatography is the extension of the working range of the adsorption isotherm into its nonlinear region. The behavior of single components as well as their mixtures over the complete range of the adsorption isotherm has, therefore, to be determined with great accuracy. 

In contrast to the well-developed thermodynamic methods for determining gas/ liquid equilibriums the theoretical determination of adsorption isotherms is not yet feasible. Only approaches to determining multi-component isotherms from experimentally determined single-component isotherms are known. Such approaches are explained in more detail in Section 2.5.2.3. Careful experimental determination of the adsorption isotherm is therefore absolutely necessary. The different approaches for isotherm determination are discussed in Chapter 6.5.7. 

The goal is to obtain the unknown parameters for a selected isotherm equation. Special parameters of nearly all types of isotherms are the Henry coefficient as well as the saturation capacities for large concentrations. It is advisable to check the validity of the single-component isotherm equation before determining the component interaction parameters. In general the decision on a certain isotherm equation should be made on the basis of the ability to predict the experimental overloaded concentration profiles rather than fitting the experimental isotherm data. In any case, consistency with the Henry coefficient determined from initial pulse experiments with very low sample amounts must be fulfilled. 

Determination of component interactions from single component isotherms ... 

This isotherm model has been used successfully to accoimt for the adsorption behavior of numerous compounds, particularly (but not only) pairs of enantiomers on different chiral stationary phases. For example, Zhou et ah [28] foimd that the competitive isotherms of two homologous peptides, kallidin and bradyki-nine are well described by the bi-Langmuir model (see Figure 4.3). However, most examples of applications of the bi-Langmuir isotherm are found with enantiomers. lire N-benzoyl derivatives of several amino acids were separated on bovine serum albumin immobilized on silica [26]. Figure 4.25c compares the competitive isotherms measured by frontal analysis with the racemic (1 1) mixture of N-benzoyl-D and L-alanine, and with the single-component isotherms of these compounds determined by ECP [29]. Charton et al. foimd that the competitive adsorption isotherms of the enantiomers of ketoprofen on cellulose tris-(4-methyl benzoate) are well accounted for by a bi-Langmuir isotherm [30]. Fornstedt et al. obtained the same results for several jS-blockers (amino-alcohols) on immobilized Cel-7A, a protein [31,32]. 

Step 1. Determine, analytically or numerically, the ftmetions JT(pP) and Ff(IT) for each component of the system, using the known single-component isotherm. 

Although this procedure involves the fitting of the function Pf (17) to a Pad6 approximation, it is generally simpler to fit the experimentally determined single component isotherm q Pf) to the approximate form. If Eq. 4.23 applies, Eq. 4.22 yields the following single-component isotherm for each component ... 

If we apply one of these equations to single-component isotherm data, we see that Eqs. 4.54 and 4.55 can be applied to the competitive adsorption data for a binary mixtiue only if Eq. 3.31 applies to the single-component data for each component. Then the six parameters can be derived from the single-component isotherms and only the coefficient b has to be measured with the mixture. Using more complicated models, Lin et al. [70] and Moreau et al. [71] have derived similar isotherms. Attempts at reducing the number of independent parameters as well as at determining these parameters from sets of experimental data have had limited success so far. 0onsiderable attention is required to clarify this issue. 

Figure 9.3 Illustration of the operating line and determination of the plateau concentration of the individual bands, (a) Single-component isotherms with Q > Q cnt (Eq. 9.7). (b) Profiles of the zones in an ideal isotachic train.

In all these figures, we used the competitive Langmuir isotherm model to calculate the band profiles. However, the coefficients of the isotherms used for Figures 11.21 are the coefficients of the single-component isotherms determined by frontal analysis, while the coefficients of the isotherms used to calculate the profiles in Figure 11.22 are measured by the simple wave method (Chapter 4, Section 4.2.4). These latter coefficients are certainly empirical coefficients, and their use would not permit an accurate prediction of single-component bands. However, they permit the calculation of band profiles that are in much better agreement... 

Single component isotherm of T and P were measmed using the ECP method and fitted to single component Langmuir isotherms in order to determine the column saturation capacity of each solute, qs,T arid qg,p. The binary Langmuir isotherms were written as ... 

A number of experimental techniques have been described for the determination of isotherms based on frontal analysis, frontal analysis by characteristic point, elution by characteristic point, and perturbation methods [12,21,27,169,176-179]. Most authors report single-component isotherm results. Multiple-component isotherm data are more complicated because all components are simultaneously in competition for the sorption sites on the stationary phase. The retention time and peak shapes of any solute is dependent on the concentration and properties of all other solutes in the mixture [12,170,180]. For multicomponent mobile phases in liquid and supercritical fluid chromatography this includes each component of the mobile phase. 

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