Plate Theory of Chromatography

A large value of k for a given component usually means that a good separation will be achieved however, large k values imply long elution times. Values of k between 2 and 20 are generally considered useful. The capacity factor is related to the partition coefficient, K, of the solute (S) between the mobile and stationary phases, where K = [S]stationary/[S]mobiie and to the relative volumes of stationary and mobile phases at equilibrium, V% and Vjn, according to Eq. 14.4  

The relative retention of two components, a, may also be expressed in terms of K  

Theoretical plates are determined experimentally for a given solute, by measuring the retention time and the peak width at the base  

The plate theory assumes that complete equilibration occurs in each of the N segments of the column. This assumption is not applicable to all solutes, and because of this, the rate theory was developed. 

The rate theory takes into account the finite rate at which the solute can equilibrate between the mobile and stationary phases. The shapes of the bands that are predicted by the rate theory depend on the rate of elution, the diffusion of the solute along the length of the column, and the availability of different paths for the solute molecules to follow. The value of the HETP now depends on v, the volume flow rate of the mobile phase, according to the van Deemter equation, Eq. 14.7  

The expression for the transfer function is obtained directly from the Laplace transforms of these equations  

FIGURE S.12. Hypothetical uilibrium stage as represented in the plate theory of chromatography. 

The expression for the moments may be found from van der Laan s theo-[Eq. (8.35)]  

Comparison shows that Eqs. (8.40) and (8.48) are identical provided that N= L/l= vlflDi. N is then the number of theoretical plates (NTP) to which the column is equivalent. The definition of the height equivalent to a theoretical plate (HETP) follows naturally  

Combining this equation with Eq. (7.46) for yields an expression of the same general form as the van Deemter equation  

There are many levels at which chromatography theory can be discussed. A very rigorous and complete discussion is fieely available (Scott, www.chromatography-online.org). However, here a more intuitive and less rigorous treatment, which should be sufficient to permit informed design and use of chromatography in conjunction with mass spectrometry for trace analysis, is presented. 

Derivation of the Plate Theory requires a fair amount of algebra that is not intrinsically difficult, but may appear confusing as a result of the terminology and many symbols used. To alleviate this situation a list of symbols is provided at the end of this chapter. Moreover, the main mathematical development is presented in Appendix 3.1, so as to not interrupt the flow of the more intuitive description in the main text. 

The Plate Theory was first proposed by A.J.P. Martin and R.L.M. Synge in a paper (Martin 1941) that led to their sharing the Nobel Prize for Chemistry in 1952 for their invention of partition chromatography . Martin later went on to develop gas chromatography in collaboration with A.T. James (James 1952), although the idea had been suggested in the 1941 paper. 

As mentioned above, the Plate Theory is based upon a concept of sequential partitioning equilibria of anal54e between mobile and stationary phases, so it is convenient to define a partition coefficient for analyte A as  

Equation [3.1]. Of course, does vary with temperature and the chemical compositions of the two phases. Intuitively it is obvious that if increases, the analyte will take longer to elute from the column in view of the implied greater affinity for the stationary phase. This qualitative idea must now be made more rigorous. The present treatment is a simplification of a re-derivation (Said 1956) of the original theory (Martin 1941). It is emphasized that the approach assumes isocratic elution (fixed composition of mobile phase) and laminar flow (no turbulence). 

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For the solution of sophisticated mathematical models of adsorption cycles including complex multicomponent equilibrium and rate expressions, two numerical methods are popular. These are finite difference methods and orthogonal collocation. The former vary in the manner in which distance variables are discretized, ranging from simple backward difference stage models (akin to the plate theory of chromatography) to more involved schemes exhibiting little numerical dispersion. Collocation methods are often thought to be faster computationally, but oscillations in the polynomial trial function can be a problem. The choice of best method is often the preference of the user. 

GLUECKAUF, E. Trans. Faraday Soc. 51 (1955) 34. Theory of chromatography, Part 9 the theoretical plate concept in column separation. 

What is H anyway The original interpretation, taken from distillation theory, was height equivalent to a theoretical plate, or HETP. We have seen that this concept was inadequate, and the preceding discussion of the van Deemter equation has presented it as a measure of the extent of spreading of an analyte zone as it passes through a column. Thus, a more appropriate term might be column dispersivity. In fact, another, independent approach to the theory of chromatography defines H as... 

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. 

The plate theory of ion exchange chromatography was applied to the separation of some borate-glycol complexes. A theoretical elution equation was derived and tested. Equilibrium constants for the formation of the complexes were determined. The position of the peak of an eluted glycol could be predicted for any concentration of Na-borate 63). 

In this, we use the fundamental parameters of the kinetic and thermodynamic theory of chromatography plate number N, column hold-up volume Vj, and retention factor k. They all determine the peak volume, which is crucial for the concentration at the peak summit. Substituting Eq. 2.24 into 2.23, solving for and... 

Figure 3.2 The elution curve of a single component, plotted as the analyte concentration at the column exit (proportional to the detector response Rj,) as a function of V, the total volume flow of mobile phase that has passed through the column since injection of the analytical sample onto the column. (V is readily converted to time via the volume flow rate U of the mobile phase.) The objective of theories of chromatography is to predict some or all of the features of this elution curve in terms of fundamental physico-chemical properties of the analyte and of the stationary and mobile phases. Note that the Plate Theory addresses the position of the elution peak but does not attempt to account for the peak shape (width etc.). The inflection points occur at 0.6069 of the peak height, where the slope of the curve stops increasing and starts decreasing (to zero at the peak maximum) on the rising portion of the peak, and vice versa for the falling side the distance between these points is double the Gaussian parameter O. Modified from Scott, www.chromatography-online.org, with permission.

The theory of chromatography shows that there should be no noticeable difference between a recorded peak shape and a Gaussian profile so long as the number of theoretical plates of the column exceeds 100 [16]. The difference is small even for 25 plates, and careful experiments would be needed to demonstrate that difference. However, it is common to experience in practice peak profiles that are not truly Gaussian. 

There are two fundamental chromatography theories that deal with solute retention and solute dispersion and these are the Plate Theory and the Rate Theory, respectively. It is essential to be familiar with both these theories in order to understand the chromatographic process, the function of the column, and column design. The first effective theory to be developed was the plate theory, which revealed those factors that controlled chromatographic retention and allowed the... 

In a chromatographic separation, the individual components of a mixture are moved apart in the column due to their different affinities for the stationary phase and, as their dispersion is contained by appropriate system design, the individual solutes can be eluted discretely and resolution is achieved. Chromatography theory has been developed over the last half century, but the two critical theories, the Plate Theory and the Rate Theory, were both well established by 1960. There have been many contributors to chromatography theory over the intervening years but, with the... 

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