Elution curve

FIG. 16-33 Elution curves under trace linear equilibrium conditions for different feed loading periods audN = 80. Solid lines, Eq. (16-172) dashed line, Eq. (16-174) for Tp= 0.05. 

FIG. 16-34 Elution curves under traee linear equilibrium eouditious with a pulse feed from Eq. (16-174). 

In the present work it has been shown that on-line coupling of flowthrough fractionation in RCC with ICP-EAS detection enables not only the fast and efficient fractionation of trace elements (TE) in environmental solids to be achieved but allows real-time studies on the leaching process be made. A novel five-step sequential extraction scheme was tested in on-line mode. The optimal conditions for the fractionation were chosen. Investigating elution curves provides important information on the efficiency of the reagents used, the leaching time needed for the separation of each fraction, and the potential mobility of HM forms. 

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... 

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. 

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

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. 

As (n) is large and virtually the whole elution curve for any solute is contained between w = -2 and w = 2 (Le, contained within four standard deviations... 

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 ... 

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,... 

Starting with the Poisson form of the elution equation, the peak width at the points of inflexion of the curve (which corresponds to twice the standard deviation of the normal elution curve) can be found by equating the second differential to zero and solving in the usual manner. Thus, at the points of inflexion, ... 

The peak width at the points of inflexion of the elution curve is twice the standard deviation of the Poisson or Gaussian curve and thus, from equation (8), the variance (the square of the standard deviation) will be equal to (n), the total number of plates in the column. 

Consequently, the composite elution curve for the total sample originally dispersed over the first three plates will be given by,... 

It follows that the elution curve for a sample, initially dispersed over (r) plates, will be... 

Figure 16. Elution Curves Resulting from the Injection of a Charge that Occupied the First Twenty-One Plates of a Column...

Figure 17. Vacancy Elution Curves from Different Injection Volumes on a Column of 500 Theoretical Plates...

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

The form of equation (27) is very similar to that obtained by Reilly et ai. [11] but the derivation is simpler, as those authors utilized the approximate binomial form of the elution curve in their procedure. 

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

Now, all the curves are describing the same chromatogram thus, by simple proportion, the ratios of the variance of each elution curve to the square of the retention (in the respective units in which the variables are defined) will all be equal. 

Figure 11. Elution Curves Presented as a Function of the Normalized Tube Length...

In addition, as mentioned before, the visual identification of the spot width is also open to considerable error but, if the plate is scanned, and the elution curve obtained, then both the peak width (w) or the peak width at the base (wb) can be obtained fairly accurately (assuming appropriate software is available). 

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