Linear chromatography

3 From the Lumped Kinetic Model back to the Equilibrium-Dispersive Model. . 300 

2 Analytical Solution in the Case of Periodic Rectangular Injections.305 

3 Analytical Solution for Chromatography with a Packing Material having Nonuniform Particles. 308 

2 Plate Height Equation in the General Rate Model.313 

7 Evaluation of Transport Parameters from Chromatographic Peaks. 326 

---

Gentilini A., Migliorini C., Mazzotti M., Morbidelli M. (1998) Optimal Operation of Simulated Moving-Bed Units for Non-Linear Chromatographie Separations. II. Bi-Langmuir Isotherm, J. Chromatogr. A 805 37-44. 

In practice, experimental peaks can be affected by extracolumn retention and dispersion factors associated with the injector, connections, and any detector. For linear chromatography conditions, the apparent response parameters are related to their corresponding true column value by... 

Thermodynamic Studies by Linear Chromatography. The first one is under linear chromatography conditions at variable temperature making use of the van t Hoff equation (Equations 1.13 and 1.14) [48,50,52,53,89],... 

In Equation 1.15, q represents the adsorbed amount of solute, ns and qs are the saturation capacities (number of accessible binding sites) for site 1 (nonstereoselect-ive, subscript ns) and site 2 (stereoselective, subscript s), and fens and bs are the equilibrium constants for adsorption at the respective sites [54]. It is obvious that only the second term in this equation is supposed to be different for two enantiomers. Expressed in terms of linear chromatography conditions (under infinite dilution where the retention factor is independent of the loaded amount of solute) it follows that the retention factor k is composed of at least two distinct major binding increments corresponding to nonstereoselective and stereoselective sites according to the following... 

Gritti, F. and Guiochon, G, Effect of the surface heterogeneity of the stationary-phase on the range of concentrations for linear chromatography, Anal. Chem., 77, 1020, 2005. 

Guiochon, G., Goshen, S. and Katti, A., Fundamentals of Preparative and Non-linear Chromatography, Academic Press, Boston, 1994. 

G. Guiochon, S. Golshan-Shirazi and A. Katti, Fundamentals of Preparative and Non-Linear Chromatography, Aca.demic Press, New York, 1994. 

In analytical chromatography, the sample we analyze is nsnally rather dilnte and allows the development of a rather straightforward method. Due to the minute concentrations we deal with in analytical chromatography, we face a linear behavior. The retention time of the analytes and the selectivity of a given separation can be forecast by simple rules that tremendously help us to develop efficient and fast separations. However, when we increase the sample size and a finite amount of sample is introduced in a chromatographic column, we leave the shelter of linear chromatography and have to cope with more complex peak shapes and phenomena. 

The apparent dispersion coefficient in Equation 10.8 describes the zone spreading observed in linear chromatography. This phenomenon is mainly governed by axial dispersion in the mobile phase and by nonequilibrium effects (i.e., the consequence of a finite rate of mass transfer kinetics). The band spreading observed in preparative chromatography is far more extensive than it is in linear chromatography. It is predominantly caused by the consequences of the nonlinear thermodynamics, i.e., the concentration dependence of the velocity associated to each concentration. When the mass transfer kinetics is fast, the influence of the apparent axial dispersion is small or moderate and results in a mere correction to the band profile predicted by thermodynamics alone. 

A number of kinetic models of various degree of complexity have been used in chromatography. In linear chromatography, all these models have an analytical solution in the Laplace domain. The Laplace-domain solution makes rather simple the calculation of the moments of chromatographic peaks thus, the retention time, the peak width, its number of theoretical plates, the peak asymmetry, and other chromatographic parameters of interest can be calculated using algebraic expressions. The direct, analytical inverse Laplace transform of the solution of these models usually can only be calculated after substantial simplifications. Numerically, however, the peak profile can simply be calculated from the analytical solution in the Laplace domain. 

The assumption of linear chromatography reflects a condition in which there is no competition between analytes for the adsorption on the stationary phase. Molecules behave independently of each other as the free surface available for adsorption is much larger compared to molecular... 

The assumption of linear chromatography fails in most preparative applications. At high concentrations, the molecules of the various components of the feed and the mobile phase compete for the adsorption on an adsorbent surface with finite capacity. The problem of relating the stationary phase concentration of a component to the mobile phase concentration of the entire component in mobile phase is complex. In most cases, however, it suffices to take in consideration only a few other species to calculate the concentration of one of the components in the stationary phase at equilibrium. In order to model nonlinear chromatography, one needs physically realistic model isotherm equations for the adsorption from dilute solutions. 

The main conclusion from Fig. 1 is that the well-known ideas of linear chromatography cannot be used when the isotherm is not linear. For example, the position of the peak maximum should not be used for the determination of a retention factor k. As Fig. 1 shows, such a k value would depend on the sample concentration and thus it might not be used for characterizing the retention of the analyte independently from its concentration. The width of the peak also has little to do with the number of plates. It will also be shown later in this chapter that the separation of two analytes cannot be simply characterized by a selectivity factor a, which is calculated as the ratio of the corresponding k values. 

In the case of HPLC methods, selectivity is usually characterized by the ratio of the respective retention factors ( values) of two compounds. This a value has useful properties in linear chromatography, i.e., when the adsorption isotherms of both substances are linear, with slopes K and K2, these being the corresponding distribution ratios between the stationary phase and the eluent. In linear chromatography... 

It is worth mentioning here that comparisons between the efficiency of different MIP separation systems like two HPLC systems or two CEC systems, or an HPLC system with a CEC system, are quite difficult when the adsorption isotherms are nonlinear. One of the typical difficulties is that the phase ratios in the two systems may be different. The effect of phase ratio on the separation and particularly on the achievable optimum separation is a complex question even in linear chromatography. In nonlinear chromatography this is really difficult and also burdened by the differences between the isotherms of the two compounds to be separated. The complexity of this matter has been mostly overlooked in the MIP literature and the visual comparison of two separations in rather different systems, operated under very different conditions, has frequently lead to statements declaring one technique better than the other. 

Once thermodynamic parameters are known, R values can be obtained under a wide range of conditions. Knowledge of R values leads directly to all the important quantities related to the retardation or retention of zones in linear chromatography, as emphasized above. 

The limit of the split-peak fraction for zero amount injected is fo = e ". Combining with the value of the mass transfer unit term for a small amount injected [r = I with C, = 0 in Eq. (15)] gives the expression of the split-peak effect in linear chromatography [21] ... 

Fornstedt, T., Zhong, G., and Guiochon, G. Peak tailing and mass transfer kinetics in linear chromatography. J. Chromatogr. A. 1996, 741, 1-12. 

In linear chromatography, each analyte travels through the column with constant velocity u. Using this velocity, we can express the analyte retention time as... 

---