Differential equation for the elution

Recalling the basic differential equation for the elution curve given in chapter 2 is,... 

The integration of the differential equation that describes the rate of change of solute concentration within a plate to the volume flow of mobile phase through it. The integral of this equation will be the equation for the elution curve of a solute through a chromatographic column. 

Equation (9) is the basic differential equation that describes the rate of change of concentration of solute in the mobile phase in plate (p) with the volume flow of mobile phase through it. Thus, the Integration of equation (9) will provide the equation for the elution curve of a solute from any plate in the column. A simple algebraic solution to equation (9) is given below and the resulting equation for the elution curve from plate (p) is as follows -... 

The equation for the retention volume of a solute, that was derived by differentiating the elution curve equation, can be used to obtain an equation for the retention time of a solute (tr) by dividing by the flowrate (Q), thus,... 

The system of Eqs. 8.1a and 8.1b is the classical system of reducible, quasihnear, first-order partial differential equations of the ideal model of chromatography [1, 2,4r-6,9-17]. The properties of these equations have been studied in detail [4,9,10, 18-24], We discuss here those properties that are important for the xmderstanding of the solutions of the ideal model in the case of elution or displacement of a binary mixture. They are the existence of characteristic fines, called characteristics, the coherence condition, and the properties of the hodograph transform. 

A major criticism of the stochastic probability approach is that relatively slow secondary reactions, for which the near-equilibrium assumption does not apply, cannot be accommodated. In this situation, it is necessary to derive and solve simultaneous partial differential equations for mass conservation and obtain expressions for the first and second moments of the elution profile and the concomitant plate height arising from slow kinetics of secondary equilibrium. If, once again, the process can be represented as involving the reversible binding of two forms, the resolution of the interconverting species can be given by [59]... 

Liquid-solid adsorption based elution ifflro-matography First, we consider systems where the eluent is a liquid further, the mechanism of solute partitioning onto the solid particles is simply adsorption, e.g. adsorption on silica or alumina particles. We assume further that the eluent pressure drop along the bed does not influence the adsorption process. There are a number of ways by which elution chromatography in such a system may be described mathematically. The differential equation describing the solute adsorption-desorption process for a dilute liquid-phase system in a fixed bed of adsorbent particles continues to be equation (7.1.4) ... 

Once the elution-curve equation is derived, and the nature of f(v) identified, then by differentiating f(v) and equating to zero, the position of the peak maximum can be determined and an expression for the retention volume (Vr) obtained. The expression for (Vr) will disclose those factors that control solute retention. 

The opportunity to measure the dilute polymer solution viscosity in GPC came with the continuous capillary-type viscometers (single capillary or differential multicapillary detectors) coupled to the traditional chromatographic system before or after a concentration detector in series (see the entry Viscometric Detection in GPC-SEC). Because liquid continuously flows through the capillary tube, the detected pressure drop across the capillary provides the measure for the fluid viscosity according to the Poiseuille s equation for laminar flow of incompressible liquids [1], Most commercial on-line viscometers provide either relative or specific viscosities measured continuously across the entire polymer peak. These measurements produce a viscometry elution profile (chromatogram). Combined with a concentration-detector chromatogram (the concentration versus retention volume elution curve), this profile allows one to calculate the instantaneous intrinsic viscosity [17] of a polymer solution at each data point i (time slice) of a polymer distribution. Thus, if the differential refractometer is used as a concentration detector, then for each sample slice i. 

Now, by equating the second differential of the elution equation to zero and solving for v, an expression for the peak width at the points of inflexion can obtained ... 

The definition of the hold-up time is simple in gas chromatography because the interactiorrs between mobile and stationary phases are practically negligible. This is not so in liquid chromatography [163-165]. The density of the mobile phase is not the same in the bulk and in the monolayer in contact with the surface of the adsorbent. The situation is more complex in RPLC because the bonded layer swells when the proportion of the organic modifier in the mobile phase increases [32,166,167]. The organic modifier dissolves in the bonded layer and, when its concentration in this layer is sufficient, some molecules of water may also penetrate it. The hold-up time of a column is a fxmction of the nature and concentration of the organic modifier in an aqueous solution [168,169]. In order to predict accurately the elution band profiles, it might be necessary to account for the dependence of the hold-up volume on the mobile phase composition, which requires the use of a different mass balance equation in which the phase ratio has been left in the differential elements [170]. 

An SEC system with three different detectors has been applied for the characterization of copolymers EPM and EPDM an evaporation detector (ED) to measure the concentration AC, a differential refractive index detector (RI) to measure the refractive index difference An between the solution and the solvent (the mobile phase), and a LALLS detector to measure the corresponding molecular weight of the eluting solutes, in the effluent from a column [37]. All three detectors were interfaced with a microcomputer, and computations were based on the following equations ... 

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