Chromatographic theory isotherms

There is a fundamental relationship described in chromatographic theory between the retention volume of a elution peak and the mid-point of a breakthrough curve achieved by operating the column under frontal analysis conditions (41 ). In the Henry s Law region of the adsorption isotherm, the net retention volume and its measurement can be used to describe the variation of sorbate breakthrough volume as illustrated in Figure 8. Utilizing the experimental apparatus described in the last section, retention volumes were measured as a function of pressure at 40°C (T =... 

The equilibrium theory of binary and multicomponent isothermal adsorption systems appears to have been first developed by Glueckauf. More general and comprehensive treatments have been developed by Rhee, Aris, and Amundson and by Helfferich and Klein. The former treatment exploits the analogy between chromatographic theory and the theory of kinematic waves. The detailed quantitive theory has been developed only for ideal Langmuir systeiris without axial dispersion or mass transfer resistance and in which the initial and boundary conditions represent constant steady states. Subject to these constraints the treatment is sufficiently general to allow the dynamic behavior to be predicted for systems with any number of components, provided only that the separation factors are known. In the... 

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. 

The behavior of chromatographic columns operated in gradient elution, under linear conditions i.e., assuming linear isotherms for all the solutes) has been studied theoretically by numerous authors [2,4-10]. The most comprehensive treatment is that based on the linear solvent strength (LSS) theory of Snyder et al. [5,6]. This theory has formd widespread acceptance [7,8] and has been extended to include the contributions of the various mass transfer resistances to band broadening [9-11]. It assumes the injection of infinitesimal pulses of a feed and a linear gradient of the volume fraction of a mobile phase modifier, cf). 

The effect of excessive sample mass on the chromatographic process is extremely complex. The theory of mass overload is also intricate [5-7] and requires a considerable amount of basic physical chemical data, such as the adsorption isotherms of each solute measured over a wide range of concentration, before it can be applied to a practical problem. Unless the production size, and the production economy, will support the necessary basic data gathering, the problem of mass overload is more conveniently and thriftily approached from a simple experimental stance. 

Even when the isotherm relation is nonlinear, chromatographic technique involves very small concentration change and the isotherm relation governing a small pulse may be considered to be locally linear (Fig. 6.7). Then, by utilizing the local slope of the isotherm as an apparent equilibrium constant, the theory developed for linear isotherm systems can be applied to the analysis of the behavior of the small concentration pulse introduced to the column kept at this point of the equilibrium relation. With this in mind, chromatographic measurement was made by introducing a small adsorbable tracer into the carrier streams which contain the same adsorbable components of different concentration levels. The detection of concentration perturbation at the outlet of a column is sometimes accompanied by fluctuation of a base... 

The Equilibrium Theory of chromatography is a very powerful tool to study and understand the dynamics of chromatographic columns for single component, binary and multi-component systems, whose retention behavior is described by any type of isotherm. The mathematical model equations are solved using the method of characteristics, and in the case of the Langmuir isotherm one finds out... 

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