The Plate Theory

Said [1] developed the Martin concept [2] to derive the elution curve equation in the following way. 

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Consider the equilibrium conditions that are assumed to exist in each plate, then 

Note that (K) is defined with reference to the stationary phase, Le., 

the larger the value of (K), the more the solute will be distributed in the stationary phase. (K) is a dimensionless constant and, in gas/liquid and liquidAiquid systems, (Xs) and (Xm) can be measured as mass of solute per unit volume of phase. In gas/solid and liquid/solid systems, (Xs) and (Xm) can be measured as mass of solute per unit mass of phase. 

Primarily the Plate Theory provides the equation for the elution curve of a solute. Such an equation describes the concentration of a solute leaving a column, in terms of the volume of mobile phase that has passed through it. It is from this equation, that the various characteristics of a chromatographic system can be determined using the data that is provided by the chromatogram. The Plate Theory, for example, will provide an equation for the retention volume of a solute, show how the column efficiency can be calculated, determine the maximum volume of charge that can be placed on the column and permit the calculation of the number of theoretical plates required to effect a given separation. 

The Plate Theory, in whatever form, assumes that the solute is, at all times, in equilibrium with both the mobile and stationary phase. Due to the continuous exchange of solute between the mobile and stationary phases as it progresses down the column, equilibrium between the phases is. In fact, never actually achieved. As a consequence, to develop the Plate Theory, the column is considered to be divided Into a number of cells or plates. Each cell is allotted a finite length, and thus, the solute spends a finite time in each cell. The size of the cell is such that the solute is considered to have sufficient time to achieve equilibrium with the two phases. Thus, the smaller the plate, the more efficient the solute exchange between the two phases in the column and consequently the more plates there are in a given columa This is why the number of Theoretical Plates in a column is termed 

Said (2) developed the equation of the elution curve In the following way. Consider the equilibrium existing in each plate, then 

(Xm) and (Xs) are the concentrations of the solute in the mobile and stationary phases respectively and (K) Is the distribution coefficient of the solute between the two phases. 

Equation (1) merely states that the general distribution law applies to the system and that the adsorption isotherm is linear. At the concentrations normally employed in liquid chromatographic separations this will be true. It will be shown later that the adsorption isotherms must be very close to linear if the system is to have practical use, since nonlinear isotherms produce asymmetrical peaks. 

For half a century different theories have been and continue to be proposed to model chromatography and to explain the migration and separation of analytes in the column. The best known are those employing a statistical approach (stochastic theory), the theoretical plate model or a molecular dynamics approach. 

To explain the mechanism of migration and separation of compounds on the column, the oldest model, known as Craig s theoretical plate model is a static approach now judged to be obsolete, but which once offered a simple description of the separation of constituents. 

These successive equilibria provide the basis of plate theory according to which a column of length L is sliced horizontally into N fictitious, small plate-like discs of same height H and numbered from 1 to n. For each of them, the concentration of the solute in the mobile phase is in equilibrium with the concentration of this solute in the stationary phase. At each new equilibrium, the solute has progressed through the column by a distance of one disc (or plate), hence the name theoretical plate theory. 

The height equivalent to a theoretical plate (HETP or H) will be given by equation (1.5)  

This employs the polynomial approach to calculate, for a given plate, the mass distributed between the two phases present. At instant I, plate J contains a total mass of analyte which is composed of the quantity of the analyte that has 

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Molecular Interactions, the Thermodynamics of Distribution, the Plate Theory and Extensions of the Plate... 

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... 

The concentration profiles of the solute in both the mobile and stationary phases are depicted as Gaussian in form. In due course, this assumption will be shown to be the ideal elution curve as predicted by the Plate Theory. Equilibrium occurs between the mobile phase and the stationary phase, when the probability of a solute molecule striking the boundary and entering the stationary phase is the same as the probability of a solute molecule randomly acquiring sufficient kinetic energy to leave the stationary phase and enter the mobile phase. The distribution system is continuously thermodynamically driven toward equilibrium. However, the moving phase will continuously displace the concentration profile of the solute in the mobile phase forward, relative to that in the stationary phase. This displacement, in a grossly... 

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... 

Figure 1. The Elution Curve of a Single Peak The Plate Theory...

Recalling the plate theory, it must be emphasized that (Vm) is not the same as (Vm)-(Vm) is the moving phase and a significant amount of (Vm) will be static (e.g., that contained in the pores). It should also be pointed out that the same applies to the volume of stationary phase, (Vs), which is not the same as (Vs), which may include material that is unavailable to the solute due to exclusion. 

In this chapter, the elution curve equation and the plate theory will be used to explain some specific features of a chromatogram, certain chromatographic operating procedures, and some specific column properties. Some of the subjects treated will be second-order effects and, therefore, the mathematics will be more complex and some of the physical systems more involved. Firstly, it will be necessary to express certain mathematical concepts, such as the elution curve equation, in an alternative form. For example, the Poisson equation for the elution curve will be put into the simpler Gaussian or Error function form. 

Equation (1) is the well-known Gaussian form of the elution curve equation and can be used as an alternative to the Poisson form in all applications of the Plate Theory. 

There is an interesting consequence to the above discussion on composite peak envelopes. If the actual retention times of a pair of solutes are accurately known, then the measured retention time of the composite peak will be related to the relative quantities of each solute present. Consequently, an assay of the two components could be obtained from accurate retention measurements only. This method of analysis was shown to be feasible and practical by Scott and Reese [1]. Consider two solutes that are eluted so close together that a single composite peak is produced. From the Plate Theory, using the Gaussian form of the elution curve, the concentration profile of such a peak can be described by the following equation ... 

Equation (10) also allows the peak width (2o) and the variance (o ) to be measured as a simple function of the retention volume of the solute but, unfortunately, does not help to identify those factors that cause the solute band to spread, nor how to control it. This problem has already been discussed and is the basic limitation of the plate theory. In fact, it was this limitation that originally invoked the development of the... 

Equation (18) displays the relationship between the column efficiency defined in theoretical plates and the column efficiency given in effective plates. It is clear that the number of effective plates in a column is not aii arbitrary measure of the column performance, but is directly related to the column efficiency as derived from the plate theory. Equation (18) clearly demonstrates that, as the capacity ratio (k ) becomes large, (n) and (Ne) will converge to the same value. 

Thus, from the plate theory, (Rr) the concept of resolution as introduced by Giddings, will be given by... 

In most mathematical processes, including the derivation of the plate theory, the assumption is made that the initial charge is placed on the first plate of the column. This is difficult to achieve in practice, as the charge must occupy a finite portion of... 

Vacancy chromatography has some quite unique properties and a number of potentially useful applications. Vacancy chromatography can be theoretically investigated using the equations derived from the plate theory for the elution of... 

Now, from the plate theory, this transient concentration change will be eluted through the column as a concentration difference and will be sensed as a negative or positive peak by the detector. The equation describing the resulting concentration profile of the eluted peak, from the plate theory, will be given by... 

Equation (25) can be extended to provide a general equation for a column equilibrated with (q) solutes at concentrations Xi, X2, X3,...Xq. For any particular solute (S), if its normal retention volume is Vr(S) on a column containing (n) plates, then from the plate theory, the plate volume of the column for the solute (S), i.e., (vs) is given by... 

From the Plate Theory, the peak width at the base is given by... 

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