Ideal Isothermal Chromatography

The first approach to understanding the regularities of gas-solid IC and TC was based on the laws of molecular kinetics. Within this framework, the time which a molecule spends in the adsorbed state when migrating down the column is seen as a sum of the random elementary adsorption sojourn times which accompany each adsorption event. The number of adsorption events is individual of major interest is, 

Let tp,c be the mean time necessary to travel the distance za down an IC column with the temperature Tc. It is the mean desorption time rd from Eq. 2.23 multiplied by the number of adsorption-desorption events on the way the latter is Z- calculated from Eq. 2.9. Then  

If za is substituted by the IC column length /c, the equation yields the net retention time in the column. We again stress that we discuss the internal chromatograms rather than the elution curves, unless stated otherwise. It means that now the total retention time equals the duration of the experiment. The latter is preset by the experimenter, thus characterizing experimental conditions rather than the results. Therefore, we need to know how za depends on rather than vice versa, and a more logical form of Eq. 4.1 would be  

On the other hand, relationships of the form of Eq. 4.1 have been used for a long time in publications on gas-phase radiochemistry, while the corresponding formulae for za might look uncommon. It especially concerns the most important equations for TC, which are discussed in the present and next chapters these, unlike Eq. 4.2, usually do not explicitly involve za- Therefore, it seems reasonable to adhere to the traditional presentation of the equations. 

Here and below, we assume no pressure drop along the column and a total pressure of one bar. It means that we account only for the temperature dependence of the appropriate quantities. Expanding Eq. 4.1 we obtain 

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On-line isothermal chromatography is ideally suited to rapidly and continuously separate short-lived radionuclides in the form of volatile species from less volatile ones. Since volatile species rapidly emerge at the exit of the column, they can be condensed and assayed with nuclear spectroscopic methods. Less volatile species are retained much longer and the radionuclides eventually decay inside the column. 

Golshan-Shirazi, S. and Guiochon, G., Analytical solution for the ideal model of chromatography in the case of a Langmuir isotherm, Anal. Chem., 60 , 2364, 1988. 

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... 

Figure 4.6 illustrates the use of the IAS model to account for the competitive isotherm data of a ternary mixture of benzyl alcohol (BA), 2-phenylethanol (PE) and 2-methyl benzyl alcohol (MBA) in reversed phase liquid chromatography. The RAS model accounts for the nonideal behaviors in the mobile and the stationary phases through the variation of the activity coefficients with the concentrations. Figures 4.6d and 4.6e illustrate the variations of the activity coefficients in the stationary and the mobile phases, respectively. The solutes exhibit positive deviations from ideal behavior in the adsorbed phase and negative deviations from ideal behavior in the mobile phase. 

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. 

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. 

Like Helfferich and Klein [9], Rhee et al. [10] studied the separation of multicomponent mixtures by displacement chromatography using the restrictive assumption of the validity of the Langmuir isotherm model and the ideal model. They used a different approach, based on the method of characteristics, and studied the interactions between concentration shocks and centered simple waves [15]. This approach is more directly suited to adsorption chromatography than the... 

The apparent plate munber can be calculated from the experimental profiles [27]. However, this number depends on the fractional height at which the bandwidth is measured. The value of Nth is calculated from the profiles predicted, under the same experimental conditions, by the ideal model. Finally, Nion is derived from the band profiles recorded in linear chromatography, e.g., with a very small sample size, using the relationships valid for Gaussian profiles. From Eqs. 7.24 and 7.26, we can derive the band width at half height, Wi/2, and the retention time of the band profile, ty, obtained with an infinitely efficient column. In the case of a Langmuir isotherm, we obtain [31]... 

In Chapters 3 and 4, we discussed the numerical analysis procedure suggested by James et al. [35] and applied by Felinger et al. [36] to calculate solutions of the inverse problem of ideal chromatography and, more specifically, to derive the best possible estimates of the numerical coefficients of an isotherm model together with a figure of merit for any isotherm model selected. The main drawback of this approach is that it is based on the use of the equilibrium-dispersive model since... 

A general mathematical treatment of system peaks and of the closely related method of vacancy chromatography was given by Helfferich and Klein [8]. This work includes a detailed analysis of the phenomena that take place upon injection of a sample into a chromatographic column. It is based on the use of the solution of the ideal model of chromatography for multicomponent systems, with competitive Langmuir isotherms (see Chapters 8 and 9), and of the ft-transform. 

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