Theory of Linear Chromatography

One can distinguish between two basic types of assumption common to all theoretical efforts to understand chromatography (16). First, it may be assumed that the equilibrium isotherm is linear alternatively, one might acknowledge that, under the experimental conditions selected, this isotherm 

Similarly, a decision must be made whether or not to take into account the influence on band profiles of such phenomena as axial dispersion (dispersion in the direction of the concentration gradient in the column) and resistance to mass transfer (i.e.. the fact that equilibration between mobile and stationary phases is not instantaneous). These phenomena are responsible for the finite efficiency of actual columns. Neglecting them and assuming the column to have infinite efficiency leads to a model of ideal chromatography. Taking them into account results in one of the models of nonideal chromatography. 

These models assume that the chromatographic column can be divided into a series of a finite number of identical plates. Each plate contains volumes and of the mobile and stationary phases, respectively. The sample is introduced as a solution of known concentration in the mobile phase used to fill the required number of plates. Plate models are essentially empirical, and cannot be related to first principles. Depending upon whether one assumes continuous or batch operation. two plate models can be considered The Craig model and the Martin and Synge model. 

In this model [17], mobile phase is transported through the column by withdrawing it from the last plate, moving the remaining mobile phase from each plate to the next one in succession, and refilling the first plate. The process is repeated as often as necessary to elute the material introduced. It is easily shown that the fraction R of the sample in the mobile phase is expressed by 

When N, exceeds about 50 plates, this profile cannot be distinguished from a Gaussian profile with a retention volume 

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Figure 2.3a illustrates the typical condition used most often in the theory of linear chromatography (Chapter 6), for the sake of convenience — the Dirac pulse injection or 6 fimction. Although it may correspond to a finite amotmt injected, its width is 0, so the concentration of even a small size injection is imrealistically... 

Use the theory of linear chromatography with very small pulses to analyze chromatography systems... 

Villermaux, J. Theory of Linear Chromatography, Percolation Processes Theory and Applications (Rodriguez, A. E. and Toudeur, D., eds., Alphen eau den (Rijn, the Netherlands, Sijthoff Noordhoff, 1981, p. 83). 

It is these two points of view which are used in the Chapter 2 to discuss the theory of gas chromatography. Linear, nonideal chromatography may be visualized by the relationships shown in Figures 1.12 and 1.13. 

Rhee et al. developed a theory of displacement chromatography based on the mathematical theory of systems of quasi-linear partial differential equations and on the use of the characteristic method to solve these equations [10]. The h- transform is basically an eqmvalent theory, developed from a different point of view and more by definitions [9]. It is derived for the stoichiometric exchemge of ad-sorbable species e.g., ion exchange), but as we have discussed, it can be applied as well to multicomponent systems with competitive Langmuir isotherms by introducing a fictitious species. Since the theory of Rhee et al. [10] is based on the use of the characteristics and the shock theories, its results are comprehensive e.g., the characteristics of the components that are missing locally are supplied directly by this theory, while in the /i-transform they are obtained as trivial roots, given by rules and definitions. 

Glueckauf E. 1949. Theory of the chromatography VII The general theory of two solutes following non-linear isotherms. Discuss Faraday Soc. 7,12. 

The results presented in the papers [19,20] indicate a correlation of Rp(eiythro) > Rv threo) for 50 of 52 diastereomeric pairs studied. The qualitative analysis of the experimental data by means of Snyder theory of linear adsorption chromatography [31] suggests that the aforementioned correlation between Rp values of erythro and threo diastereoisomers on silica gel can be used for the assignment of the relative configurations of other nonionic diastereomeric compounds of type I, which do not possess intramolecular hydrogen bonds or which have bonds of the types OH... OH orOH...N [20]. 

Casassa, E. R and Tagami, Y. An equilibrium theory for exclusion chromatography of branched and linear polymer chains, Macromolecules, 2, 14, 1969. 

The theory of chromatography of linear and cyclic polymers with functional groups has also been developed. A linear in-pore partition function Q 1) was created for polymeras with one specific functional group a. Q 1) can be expressed by... 

The theory of chromatography depends on the assumption of a linear system with diffusivity independent of concentration over the relevant range. The validity of this assumption may be conveniently tested by varying the pulse (or step) size. 

The minor disturbance or perturbation method relies on equilibrium theory too and was first suggested by Reilley et al. (1962). As known from linear chromatography, the retention time of a small pulse injected into a column filled with pure eluent can be used to obtain the initial slope of the isotherm. This approach is expanded to cover the whole isotherm range. For the example of a single-component system (Fig. 6.24) the procedure is as follows The column is equilibrated with a concentration ca and, once the plateau is established, a small pulse is injected at a time fstart a and a pulse of a different concentration is detected at the corresponding retention time tR a. 

Kucera, E. Contribution to the theory of chromatography/linear non-equilibrium elution,... 

As discussed already in Chapter 2 (Section 2.2.6), Giddings [10] has developed a nonequilibrium theory of chromatography and showed that the influence of the kinetics of mass transfers can be treated as a contribution to axial dispersion. As illustrated in Chapter 6, this approximation is excellent in linear chromatography, as long as the column efficiency exceeds 20 to 30 theoretical plates. 

We have discussed the theory of system peaks in linear chromatography [20]. The discussion is based on the use of the equilibrium-dispersive model. The mass balance equations are written for the n components of the sample and for the p additives ... 

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