Models of linear chromatography

The second consequence of the assmnption of a linear isotherm is to make simple the mathematics of describing the migration of these independent, individual bands and of calculating their retention times and profiles. As we show later in this chapter, an analytical solution or, at least, a closed-form solution in the Laplace domain can be obtained with any model of linear chromatography. This is certainly not the case in nonlinear chromatography. 

The two Eqs. 6.57a and 6.57b are classical relationships of the most critical importance in linear chromatography. Combined, they constitute the famous Van Deemter equation, which shows that the effects of the axial dispersion and of the mass transfer resistances are additive. This is the basic tenet of the equilibrium-dispersive model of linear chromatography. We will assume that this rule of additivity and Eqs. 6.57a remain valid when we apply the equilibrium-dispersive model to nonlinear chromatography. In this case, however, it is only an approximation because the retention factor, k = dq/dC, is concentration dependent. These equations have been derived from the lumped kinetic model. Thus, they show that the kinetic model and the equilibrium-dispersive model are equivalent as long as the rate of the equilibrium kinetics in the chromatographic system is not very slow. 

A "microscopic probabilistic" method can be used for the modeling of linear chromatography. In this case, the probability density function at I and t of a single molecule of solute is derived. The "random walk" approach [29] is the simplest method of that type. It has been used to calculate the profile of the chromatographic band in a simple way, and to study the mechanism of band broadening. 

When the loading factor tends toward zero and the condition of infinite dilution is approached, the solution of the Thomas model becomes identical to the Giddings-Eyring model of linear chromatography ... 

Description of models of linear chromatography with an incompressible mobile phase... 

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. 

Long jumps cannot be described in the model of linear gas chromatography. 

The importance of linear chromatography comes from the fact that almost all analytical applications of chromatography are carried out xmder such experimental conditions that the sample size is small, the mobile phase concentrations low, and thus, the equilibrixim isotherm linear. The development in the late 1960s and early 1970s of highly sensitive, on-line detectors, with detection limits in the low ppb range or lower, permits the use of very small samples in most analyses. In such cases the concentrations of the sample components are very low, the equilibrium isotherms are practically linear, the band profiles are symmetrical (phenomena other than nonlinear equilibrium behavior may take place see Section 6.6), and the bands of the different sample components are independent of each other. Qualitative and quantitative analyses are based on this linear model. We must note, however, that the assumption of a linear isotherm is nearly always approximate. It may often be a reasonable approximation, but the cases in which the isotherm is truly linear remain exceptional. Most often, when the sample size is small, the effects of a nonlinear isotherm (e.g., the dependence of the retention time on the sample size, the peak asymmetry) are only smaller than what the precision of the experiments permits us to detect, or simply smaller than what we are ready to tolerate in order to benefit from entertaining a simple model. 

Villemiaux, J. (1987) Chemical engineering approach to dynamic modeling of linear cliromatography, J. Chromatography, 406, 11-26. 

The solutions of the equilibrium-di.spersive model exhibit the same features as those of the ideal model at high conceniraiions. where thermodynamic effects are dominant and dispersion due to finite column efficiency merely smoothes the edges. When concentrations decrease, the solutions tend toward the Gaussian profiles of linear chromatography. 

Possible long jumps cannot be described in the model of linear gas chromatography. Hence separation factors are overestimated, especially in open columns. 

This is the principle of superposition. Since equation (7.1.18c) or equation (7.1.95a) are linear equations due to constant k,i, the superposition of two simpler solutions by simple addition leads to the solution of the problem under consideration. This is the advantage of linear chromatography models. 

The competitive adsorption isotherms were determined experimentally for the separation of chiral epoxide enantiomers at 25 °C by the adsorption-desorption method [37]. A mass balance allows the knowledge of the concentration of each component retained in the particle, q, in equilibrium with the feed concentration, < In fact includes both the adsorbed phase concentration and the concentration in the fluid inside pores. This overall retained concentration is used to be consistent with the models presented for the SMB simulations based on homogeneous particles. The bed porosity was taken as = 0.4 since the total porosity was measured as Ej = 0.67 and the particle porosity of microcrystalline cellulose triacetate is p = 0.45 [38]. This procedure provides one point of the adsorption isotherm for each component (Cp q. The determination of the complete isotherm will require a set of experiments using different feed concentrations. To support the measured isotherms, a dynamic method of frontal chromatography is implemented based on the analysis of the response curves to a step change in feed concentration (adsorption) followed by the desorption of the column with pure eluent. It is well known that often the selectivity factor decreases with the increase of the concentration of chiral species and therefore the linear -i- Langmuir competitive isotherm was used ... 

Zhong G., Guioehon G. (1996) Analytieal Solution for the Linear Ideal Model of Simulated Moving Bed Chromatography, Chem. Eng. Sci. 51 4307-4319. 

Trone, M. D., Khaledi, M. G. Statistical evaluation of linear solvation energy relationship models used to characterize chemical selectivity in micellar electrokinetic chromatography. J. Chromatogr. A 2000, 886, 245-257. 

T. Baczek and R. Kaliszan, Combination of linear solvent strength model and quantitative structure-retention relationships as a comprehensive procedure of approximate prediction of retention in gradient liquid chromatography. J. Chromatogr.A 962 (2002) 41-55. 

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. 

Ishihara T, Kadoya T, Yamamoto S. Application of a chromatography model with linear gradient elution experimental data to the rapid sacle-up in ion-exchange process chromatography of proteins. Journal of Chromatography A 2007 1162 34-40. 

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