Chromatography equations describing separations

The theory of chromatography has been reasonably well established with a kinetic or rate theory that describes the broadening of the bands or zones of separated components on a stationary-phase bed, describes their time of appearance at any particular point, and provides details of the separation power or resolution of the particular system employed. Numerous equations describing zone broadening have been proposed, the simplest being the general form of the van Deemter equation as derived for gas-liquid chromatography,... 

In chromatography, the separation efficiency of any single separation method is limited by the efficiency and selectivity of the separation mode, that is, the plate count of the column and the phase of the selected system. Adding more columns will not overcome the need to identify more components in a complex sample, due to the limitation of peak capacities. The peak capacity in an isocratic separation can be described, following Grushka (1970), as given in Equation (17.1) ... 

Describe the principle of separation involved in elution chromatography and derive the retention equation ... 

Subsequently four different CE modes are described in the sections Capillary Zone Electrophoresis, Capillary Gel Electrophoresis, Capillary Isoelectric Focussing, and Micellar Electrokinetic Chromatography (MEKC), respectively. The fundamental principles of the specific separation modes are briefly explained, using appropriate equations where required. In Table 3 all equations are listed. In addition, the influence of both instrumental parameters and electrolytic solution parameters on the optimization of separations is described. 

Before paired-ion chromatography is discussed, it is illustrative to consider another solubility-based separation which relies on a similar approach. This separation is ion-pair extraction. Ion-pair extraction uses two immiscible liquids (aqueous and organic) often in a separatory funnel. An ionized compound (Aij) that is water soluble can be made to favor solution into the organic phase during an extraction by using a suitable counter ion (Baq) to form a neutral ion pair. Since the ion pair behaves as though it is a nonionic, neutral species, it will prefer to reside in the organic liquid layer, and the entire process can be described by the equation ... 

The separation efficiency of a column for liquid chromatography and the relation with the mobile-phase velocity (u) can be described by the Van Deemter equation, which in lumped terms reads [2]... 

The hydrodynamic aspects of chromatography play a major role in selecting the appropriate particle size. In contemporary HPLC, flow rates higher than the optimum on the H-u plots predicted by Eq. (1.10) or Eq. (1.11) are used to allow shorter separation times without significant loss of resolution. The minimum velocity, Mmm, for the H u plots described by the van Deemter equation (1.10) can be calculated from Eq. (1.12) [12] ... 

RP chromatography is, by far, the most widely used LC mode for the separation of complex mixtures based on different lipophilicities of sample compounds.The effect of the volume fraction q> of the organic solvent in a binary aqueous-organic mobile phase on the retention factors k in RP chromatography can be very often described by simple equation (Eq. 3) ... 

The mobile phase in RP chromatography contains water and one or more organic solvents, most frequently acetonitrile, methanol, tetrahyrofuran, or propanol. By the choice of the organic solvent, selective polar interactions (dipole-dipole, proton-donor, or proton-acceptor) with analytes can be either enhanced or suppressed, and the selectivity of separation can be adjusted. Binary mobile phases are usually well suited for the separation of a variety of samples, but ternary or, less often, quaternary mobile phases may offer improved selectivity for some difficult separations. The retention times are controlled by the concentration of the organic solvent in the aqueous-organic mobile phase. Equation 1 is widely used to describe the effect of the volume fraction of methanol or acetonitrile

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This chapter introduces fundamental aspects and basic equations for the characterization of chromatographic separations. Starting from the simple description of an analytical separation of different compounds the influences of fluid dynamics, mass transfer and thermodynamics are explained in detail. The important separation characteristics for preparative and process chromatography, e.g. the optimization of resolution and productivity as well as the differences compared with chromatography for analytical purposes, are described. Especially, the importance of understanding the behavior of substances in the nonlinear range of the adsorption isotherm is highlighted. 

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... 

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