Elution equation

Using this equilibrium relation and the previous set of equations, elution curves by means of methanol eluant from the aaivated carbon column which has adsorbed o-chlorobenzoic acid can be calculated The rate of desorption is also dependent on alcohol content and the amount adsorbed, but as a rough estimation simple treatment of constant... 

Now that we have defined capacity factor, selectivity, and column efficiency we consider their relationship to chromatographic resolution. Since we are only interested in the resolution between solutes eluting with similar retention times, it is safe to assume that the peak widths for the two solutes are approximately the same. Equation 12.1, therefore, is written as... 

The behavior predicted by this equation is illustrated in Fig. 16-33 with N = 80. Xp = (Evtp/L)/[il — )(p K -i- )] is the dimensionless duration of the feed step and is equal to the amount of solute fed to the column divided by tne sorption capacity. Thus, at Xp = 1, the column has been supplied with an amount of solute equal to the station-aiy phase capacity. The graph shows the transition from a case where complete saturation of the bed occurs before elution Xp= 1) to incomplete saturation as Xp is progressively reduced. The lower cui ves with Xp < 0.4 are seen to be neany Gaussian and centered at a dimensionless time - (1 — Xp/2). Thus, as Xp 0, the response cui ve approaches a Gaussian centered at Ti = 1. 

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... 

If the corrected retention volume in the pure strongly eluting solute is very small compared with the retention volume of the solute in the other pure solvent, i.e., V"a V"b, which is very often the case in practical LC, then equation (12)... 

It should be recalled that the distribution coefficients are referenced to the solvent mixture and not the stationary phase and are thus the inverse of the distribution coefficient employed in the chromatography elution equation. 

It is seen from equation (22) that there will, indeed, be a temperature at which the separation ratio of the two solutes will be independent of the solvent composition. The temperature is determined by the relative values of the standard free enthalpies of the two solutes between each solvent and the stationary phase, together with their standard free entropies. If the separation ratio is very large, there will be a considerable difference between the respective standard enthalpies and entropies of the two solutes. As a consequence, the temperature at which the separation ratio becomes independent of solvent composition may well be outside the practical chromatography range. However, if the solutes are similar in nature and are eluted with relatively small separation ratios (for example in the separation of enantiomers) then the standard enthalpies and entropies will be comparable, and the temperature/solvent-composition independence is likely be in a range that can be experimentally observed. 

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 (3) merely sums the two peaks to produce a single envelope. Providing retention times can be measured precisely, the data can be used to determine the composition of a mixture of two substances that, although having finite retention differences, are eluted as a single peak. This can be achieved, providing the standard deviation of the measured retention time is small compared with the difference in retention times of the two solutes. Now, there is a direct relationship between retention volume measured in plate volumes and the equivalent times, which is depicted in Figure 6. 

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. 

Equation (16) was first developed by Purnell [3] in 1959 and is extremely important. It can be used to calculate the efficiency required to separate a given pair of solutes from the capacity factor of the first eluted peak and their separation ratio. It is particularly important in the theory and practice of column design. In the particular derivation given here, the resolution is referenced to (Ra) the capacity ratio of the first... 

Equation (22) allows the maximum sample volume that can be used without seriously denigrating the performance of the column to be calculated from the retention volume of the solute and the column efficiency. In any separation, there will be one pair of solutes that are eluted closest together (which, as will be seen in Part 3 of this book, is defined as the critical pair) and it is the retention volume of the first of these that is usually employed in equation (22) to calculate the maximum acceptable sample volume. 

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... 

Employing the alternative Gaussian form of the elution equation. 

Equation (24) shows that when the charge is placed on the first plate, (Xn) can never equal zero and pure mobile phase free of solute will never elute from the column. However, in practice, it is almost impossible to place the sample exclusively on the first plate, and there will be a finite volume of mobile phase that will occupy a finite number of theoretical plates when it is injected onto the column. 

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. 

Peak capacity can be very effectively improved by using temperature programming in GC or gradient elution in LC. However, if the mixture is very complex with a large number of individual solutes, then the same problem will often arise even under programming conditions. These difficulties arise as a direct result of the limited peak capacity of the column. It follows that it would be useful to derive an equation that... 

Equation (33) shows that the maximum capacity ratio of the last eluted solute is inversely proportional to the detector sensitivity or minimum detectable concentration. Consequently, it is the detector sensitivity that determines the maximum peak capacity attainable from the column. Using equation (33), the peak capacity was calculated for three different detector sensitivities for a column having an efficiency of 10,000 theoretical plates, a dead volume of 6.7 ml and a sample concentration of l%v/v. The results are shown in Table 1, and it is seen that the limiting peak capacity is fairly large. 

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