Elution curve equation

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. 

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

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... 

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. 

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. 

Equation (10) is the basic elution curve equation it is a Poisson function, but when n is large, the function approximates to a normal error function or Gaussian function. In practical chromatography systems, n is always greater than 100 and, thus, all chromatographic peaks will be Gaussian or nearly Gaussian in shape. 

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