Elution curve,equation for

Equation (9) 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. The integration of equation (9) will provide the elution curve equation for any solute eluted from any plate in the column. A simple method for the integration of equation (9) is given in Appendix 1, where the solution, the elution curve equation for plate (p), is shown to be... 

Thus, the elution curve equation for the last plate in the column, the (n) th plate (that is, the equation relating the concentration of solute in the mobile phase entering the detector to volume of mobile phase passed through the column) is given by... 

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. 

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. 

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,... 

Thus, for a chromatogram of (q) solutes, the elution curve equation will be given by. 

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

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 elution curves, calculated from equation (10), for a solute eluted from three columns having 4, 9, and 15 plates respectively are shown in figure 3. With the exception of having a different number of theoretical plates all three columns have identical physical properties. 

So far the Plate Theory has been used to determine the equation for the retention volume of a solute, calculate the capacity factor of a solute and identify the dead volume of the column and how it should be calculated. However, the equation for the elution curve of a solute that arises directly from the Plate Theory can do far more than that to explain the characteristics of a chromatogram. The equation will now be used in a variety of ways to expand our knowledge of the chromatographic process. 

In the development of the plate theory and the derivation of the equation for the elution curve of a solute, it was assumed that the initial charge was located In the first plate of the column. In practice, this is difficult to achieve, and any charge will, in fact, occupy a finite column volume and consequently a specific number of the first theoretical plates of the column. Consider the situation depicted in figure 1 where the initial charge is distributed over (r) theoretical plates. 

Since PqQ is a linear isotherm, (Ki) is constant for all values of (Xs)then from equation (2) all concentrations in the band will travel at the same velocity and a symmetrical elution curve will be produced. This symmetrical curve is depicted as the normal peak (a) in figure (4). The symmetrical nature of the elution curve is to be expected from the plate theory. 

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