Elemental criterion

In section 4.2. we will define other criteria that may be used to characterize the separation between a pair of adjacent peaks in a chromatogram (so-called elemental criteria). 

The resolution between two peaks has been defined in chapter 1 and this definition has been reviewed in section 4.1.1. In this section we will define and investigate various other criteria that may be used to quantify the extent of separation between a pair of adjacent peaks in a chromatogram. We will refer to these criteria as elemental criteria . Later in this chapter the elemental criteria will serve as the basis of criteria for judging the extent of separation in entire chromatograms. 

Figure 4.2 Three definitions for peak-valley ratios as elemental criteria to quantify the extent of separation between a pair of adjacent peaks in a chromatogram, (a) Peak-valley ratio (P eqn.4.3) according to Kaiser, (b) median peak-valley ratio (Pm eqn.4.4) according to Schupp and (c) (opposite page) the valley-to-top ratio (P eqn.4.5) according to Christophe. 

A comparison of various elemental criteria has been reported by Knoll and Midgett [412] and by Debets et al. [413]. Figure 4.4 shows the variation of some of the criteria for the separation of pairs of chromatographic peaks as a function of the time difference between the peak tops (At = t2 — t,). By definition, Rs (and hence S) varies linearly with At. The peak-valley ratios (P) and the fractional overlap both increase rapidly with increasing At at first, but level off towards At 4 ct to reach the limiting value of 1. At high values of At, Rs and S will keep increasing, while the other criteria will not. 

Figure 4.4 Variation of some elemental criteria as a function of the difference in retention times between the two solutes. Data calculated for Gaussian peaks of equal height. Courtesy of Anton Drouen [405].

Characteristics of different elemental criteria for measuring the extent of separation of a pair of chromatographic peaks. 

Recommendations for the use of different elemental criteria for measuring the extent of separation of a pair of chromatographic peaks. The preferred criteria are given, while possible alternatives appear in brackets. 

We will base our discussion about criteria by which to judge the quality of an entire chromatogram on the elemental criteria for pairs of chromatographic peaks, which have been defined in chapter 1 (Rs and a) and in the previous section. We will look at several ways of combining the numbers for all individual pairs of peaks into a single number. We will then discuss the influence of other parameters, such as the analysis time and the number of peaks on the proposed criteria. 

A second major category of proposed criteria to express the extent of separation in an entire chromatogram is that in which the product is takerfof the values for all pairs of peaks of one of the elemental criteria defined before. Taking the products of these criteria is equivalent to taking the sum of the logarithms, for instance... 

Data for capacity factors, elemental criteria and for criteria judging the extent of separation in the entire chromatograms. Chromatograms are shown in figure 4.8. Criteria for pairs of peaks separation factor (S, eqn.4.15), resolution (Rs, eqn.4.14) and peak-valley ratio (P, eqn.4.10). 

The use of Pv in sum or product criteria creates a special problem, because two values for Pv can be calculated for each peak in the chromatogram. This applies to isolated as well as to ill-resolved peaks and it also applies to the first and the last peaks observed in the chromatogram. The number of P, P Rs, or S values in an entire chromatogram usually equals n-1 (where n is the number of peaks). If an imaginary peak is assumed to be present at t = t0, the number of values for the elemental criteria becomes n. We can deal with this problem in three ways. 

In this section we will generally use C for some function of the elemental criteria (Rp S or P), for instance one of the optimization criteria in table 4.8. C refers to a criterion which has been corrected for the number of peaks in the chromatogram, while C, refers to a time-corrected criterion. 

If peak-valley ratios are used as elemental criteria, then the separation between the first peak and the (imaginary) preceding one, as well as the separation between the last peak and the (imaginary) following one, may readily be characterized by a P value of one. The retention time of the last peak, which may be used in combination with a product of P values (see table 4.13) refers to the last appearing peak in the chromatogram, no matter whether or not this is a relevant peak. 

The quality of the eventually selected separation depends on the appropriateness of the criterion and it is therefore important to select the criterion well. This selection consists of several steps. The first is to select what is called an elemental criterion. which describes the separation quality of two substances, the second is to derive from the elemental criterion value a global criterion. There are two types of elemental criteria. They have been called p- and. v-criteria. The p criteria compare in some way the height of the peaks with the valley in between them. Several variants have been proposed. Two of them are shown in Fig. 6.3 (the valley-to-peak ratio F, . = 1 — v/hi and the... 

The optimization results for different criteria may be conflicting in the sense that they show optima at different values of the factors. One does not need to find the optimum of the two (or more) responses separately, but rather an adequate compromise. There are several ways of doing this. The most usual, but not necessarily the best is to combine (elemental) criteria in some way to obtain what have been called global criteria. Again several such criteria have been propo.sed. for instance the COF (chromatographic optimization function) 5] given by ... 

The concentrations of elements listed by the EPA for toxic elements each average less than the allowable toxic elements criteria for toxic hazardous waste. 

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