Model from chromatographic theory

From chromatographic theory [2] it is clear that the R value should result in simple models. For this reason it is preferred over, the k or the Rj. These latter response values can be calculated from predicted R values. It is more difficult to determine the error structure of the R . It is believed however that logarithmic transformation of the k values should result in homoscedastical error structures [3]. 

It is clear from the above, that model equations for the description of retention surfaces have to meet high demands. Preferably, equations should be used that relate to reliable chromatographic theory, such as the one used to describe the retention behaviour as a function of pH in RPLC in the window diagram approach described in section 5.5.1. The use of such a chromatographic equation was clearly better in that case than a statistical approach using (for example) polynomial equations. 

Note that the physicochemical mechanisms that enables us to perform the chromatographic bioseparations are not always adsorption-like but can involve ion exchange, ion exclusion, or size exclusion. Even if it is generally possible to fit experimental data with a mathematical function derived from the adsorption theory, it is strongly advisable to refer to the proper physicochemical process before modeling the separation. For instance, ion exchange can be modeled with selectivity coefficients (derived from the mass action law) that can be constant or not,18,19 ion-exclusion can be modeled thanks to theories based on the Donnan exclusion, etc. 

These theories are based on the interaction of the solute ion with the charged surface layer established by the adsorbed counterion and by adsorbed competing ions. The nonstochiometric models apply the Poisson-Boltzmann equation to estimate retention from an electrostatic point of view. The electrical double-layer model applied uses different approaches such as liquid partition , surface adsorption, diffuse layer ion-exchange , and sru face adsorption doublelayer models. It is not possible to draw conclusions about the ion pair process from chromatographic retention data, but each model and theory may find use in describing experimental results under the particular conditions studied. 

Theoretical plate a concept borrowed from distillation theory and countercurrent extraction a chromatographic column is modeled as a series of discrete plates in each of which local equilibrium of analyte partitioning between stationary and mobile phases is established. The Plate Theory accounts for retention of analytes, i.e., retention times, but not the peak shapes (widths), for isocratic elution. 

Fritz and Scott (23) derived simple statistical expressions for calculating the mean and variance of chromatographic peaks that are still on a column (called position peaks) and these same peaks as they emerge from the column (called exit peaks). The classical plate theory is derived by use of simple concepts from probability theory and statistics. In this treatment, each sample chemical substance molecule is examined separately, whereas its movement through the colunm is described as a stochastic process. Equations are given for both discrete- and continuous-flow models. They are derived by calculating the mean and variance of a chromatographic peak as a function of the capacity factor k. 

The debate as to the exact model to describe the ion-pair phenomena will no doubt continue. Difficulties in devising a model arise from conflicting conclusions based on a large amount of experimental data. However, it is important to emphasise that theory guides experimentation. Therefore the importance of having a model is to understand the factors that control chromatographic retention, and thus, to aid in the prediction of the separating ability of a mobile phase. 

The ideal model should be applied to get information about the thermodynamic behavior of a chromatographic column. Through work by Lapidus and Amundson (1952) and van Deemter et al. (1956) in the case of linear isotherms and by Glueckauf (1947, 1949) for nonlinear isotherms, considerable progress was made in understanding the influences of the isotherm shape on the elution profile. This work was later expanded to a comprehensive theory due to improved mathematics. Major contributions come from the application of nonlinear wave theory and the method of characteristics by Helfferich et al. (1970, 1996) and Rhee et al. (1970, 1986, 1989), who made analytical solutions available for Eqs. 6.41 and 6.42 for multi-component Langmuir isotherms. 

In addition to preparing the model catecholate complexes of rhodium and chromium, the analogous enterobactin complexes were also prepared and their CD spectra recorded (62). From examination of molecular models it is apparent that either the A-cis or A-cis diastereomers of a metal enterobactin complex are structurally possible. In theory, these diastereomers should be separable by chromatographic techniques analogous to those used for the hydroxamates (vide supra) however, under a variety of conditions only one chromatographic fraction is obtained. We conclude that one isomer predominates to the exclusion of the other. 

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