Nonlinear ideal chromatography

Figure 1.15. Isotherms for nonlinear ideal chromatography. Cg = cone, at surface or in stationary phase Cg = cone, in solution at equilibrium.

In ideal chromatography, we assume that the column efficiency is infinite, or in other words, that the axial dispersion is negligibly small and the rate of the mass transfer kinetics is infinite. In ideal chromatography, the surface inside the particles is constantly at equilibrium with the solution that percolates through the particle bed. Under such conditions, the band profiles are controlled only by the thermodynamics of phase equilibria. In linear, ideal chromatography, all the elution band profiles are identical to the injection profiles, with a time or volume delay that depends only on the retention factor, or slope of the linear isotherm, and on the mobile phase velocity. This situation is unrealistic, and is usually of little importance or practical interest (except in SMB, see Chapter 17). By contrast, nonlinear, ideal chromatography is an important model, because the profiles of high-concentration bands is essentially controlled by equilibrium thermodynamics and this model permits the detailed study of the influence of thermodynamics on these profiles, independently of the influence of the kinetics of mass transfer... 

Using these two sets of conditions, we can then describe four chromatographic systems (a) linear ideal chromatography, (b) linear nonideal chromatography, (c) nonlinear ideal chromatography, and (d) nonlinear nonideal chromatography. 

FIGURE 2.14 Nonlinear ideal chromatography to = start of separation (point of sample injection) Ja = retention time of component A ts = retention time of component B f = time of emergence of mobile phase from to-... 

The relative importance of the contributions of the nonlinear thermodynamics of phase equilibrium and of the finite mass transfer kinetics to the band profile is evidenced and quantified by the numerical value of e. When e = 0, Cg = cjsiCm), and Eqs. 10.59a and 10.59b reduce to Eq. 10.58 by summing them up, which gives the equation of ideal chromatography. This asymptotic result can be justified in a few cases. On the other hand, the slower the mass transfer kinetics, the larger will be the value of e. 

In ideal chromatography, frontal analysis results in the gradual extension of a band of adsorbate at constant concentration down the column, assuming constant column temperature and constant partial pressure of adsorbate in the gas entering the column. By equating total adsorbate introduced to the column to the amount present in the gas phase plus adsorbent phase in a band of length on the column, one arrives at f = (qL/P (CG + vA/P ) t, where t is total time of flow in minutes. In the case of a linear isotherm this is just = AF t, as one intuitively expects. ven if the isotherm is nonlinear, (v/P is constant everywhere in the band, for P constant. This... 

SAMs are generating attention for numerous potential uses ranging from chromatography [SO] to substrates for liquid crystal alignment [SI]. Most attention has been focused on future application as nonlinear optical devices [49] however, their use to control electron transfer at electrochemical surfaces has already been realized [S2], In addition, they provide ideal model surfaces for studies of protein adsorption [S3]. 

The ideal model of chromatography, which has great importance in nonlinear chromatography, has little interest in linear chromatography. Along an infinitely efficient column, with a linear isotherm, the injection profile travels unaltered and the elution profile is the same as the injection profile. We also note here that, because of the profound difference in the formulation of the two models, the solutions of the mass balance equation of chromatography for the ideal, nonlinear model and the nonideal, linear model rely on entirely different mathematical techniques. 

For all these reasons, the mathematical aspects of the theory become much more complex. The mathematics of nonlinear chromatography are so complex that even for a single solute, there is no analytical, closed-form solution available, except with two simplified models, the ideal model and the Thomas model [120]. The ideal model is based upon the assumption of an infinite column efficiency. Its solutions are discussed in detail in Chapters 7 to 9. The Thomas model is based upon the assumptions that there is a slow Langmuir adsorption-desorption kinetics and that there are no other nvass transfer resistances, nor any axial dispersion. The system of equations of this model has been solved by Goldstein [121], and this general solution has been simplified for pulse injection by Wade et al. [122]. In aU other cases, the problem must be solved numerically. The Thomas model is discussed with other kinetic models in Chapter 14 and 16. 

Furthermore, the theoretical analysis of the single-component problem in the ideal model provides some of the fimdamental concepts in nonlinear chromatography, such as the notions of the velocity associated with a concentration, of concentration shocks, and of diffuse bormdaries [1,2]. It also provides an understanding of the relationship between the thermod5mamics of phase equilibria, the shape of the isotherm (i.e., convex upward, linear, convex downward, or S-shaped) and the band profiles. Finally, it provides an explanation of the relative importance of the influences of the thermodynamics and the kinetics on the band profile. These concepts will provide a most useful framework for imderstanding the phenomena that occur in preparative chromatography. 

The application of the z-transform and of the coherence theory to the study of displacement chromatography were initially presented by Helfferich [35] and later described in detail by Helfferich and Klein [9]. These methods were used by Frenz and Horvath [14]. The coherence theory assumes local equilibrium between the mobile and the stationary phase gleets the influence of the mass transfer resistances and of axial dispersion (i.e., it uses the ideal model) and assumes also that the separation factors for all successive pairs of components of the system are constant. With these assumptions and using a nonlinear transform of the variables, the so-called li-transform, it is possible to derive a simple set of algebraic equations through which the displacement process can be described. In these critical publications, Helfferich [9,35] and Frenz and Horvath [14] used a convention that is opposite to ours regarding the definition of the elution order of the feed components. In this section as in the corresponding subsection of Chapter 4, we will assume with them that the most retained solute (i.e., the displacer) is component 1 and that component n is the least retained feed component, so that... 

For a number of nonlinear and competitive isotherm models analytical solutions of the mass balance equations can be provided for only one strongly simplified column model. This is the ideal model of chromatography, which considers just convection and neglects all mass transfer processes (Section 6.2.3). Using the method of characteristics within the elegant equilibrium theory, analytical expressions were derived capable to calculate single elution profiles for single components and mixtures (Helfferich and Klein, 1970 Helfferich and Carr 1993 Helfferich and Whitley 1996 Helfferich 1997 Rhee, Aris, and Amundson, 1970 ... 

By the derivation of Eq. (9.8) the ideal model is also useful for nonlinear chromatography as the shape and position of peaks can be reproduced satisfactorily, if a high number of theoretical plates can be assumed. A partly possible analytical solution of the model enables the determination of adsorption isotherms from experimental chromatograms. 

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