Normalized resolution product

On the other hand, if it is planned that the optimization column can be a different length (typically shorter) than the final separation column, another CRF can be used the time-corrected normalized resolution-product, r tc, given by... 

Shown in Figure 5 are the calculated response surfaces for both the threshold separation (CRF-4) and time-corrected normalized resolution product (CRF-5). As expected, the threshold response surface is discontinuous, with 3 local optima in contrast, the response surface for CRF-5 is smooth, with no apparent local optima. 

Figure 5. Comparison of the threshold separation and time-corrected, normalized, resolution-product response surfaces for the eight component sample (Table ID). Response surfaces calculated via equations 9 and 10 using the isothermal retention surfaces of Figure 4.

Criteria for entire chromatograms sum criteria (section 4.3.1), product criteria (section 4.3.2), normalized resolution product (r, eqn.4.19), calibrated normalized resolution product (r, eqn.4.21) and minimum resolution (section 4.3.3). 

Because both Rs and Rs are proportional to VN, the normalized product of Rs values is equal to that of the S values, and both are independent of the number of plates. The normalized resolution product (r) will vary from zero, in the case where one or more pairs of peaks show no resolution, to one, if the resolution is equal for all the pairs of peaks in the chromatogram. Therefore, in choosing r as the optimization criterion the aim is to achieve an equal distribution of the peaks over the chromatogram. 

In section 4.3.2 we have seen that the normalized resolution product criterion r aims at achieving a chromatogram in which all peaks appear at constant resolution intervals from the first one. If r is used instead of r, then the regular intervals start at an imaginary peak at t=t0. A chromatogram for which r = 1 is one of a series for which the constant intervals can be found. Once the absolute value of S, the number of peaks and the plate number are known, the chromatogram is defined unambiguously. 

When a hypothetical peak is assumed at t = t0 the calibrated normalized resolution product becomes ... 

In these equations, p is again the number of peaks of interest. Hence, the product includes only the relevant S values, while the sum is taken over all pairs of peaks. If the sum is divided by the number of relevant peaks (p -1 or p), then a value of r = 1 or r = 1 can only be reached if all irrelevant peaks appear nowhere in the chromatogram, i.e. coincide with the imaginary peak at t= t0. If we divided the sum by the total number of peaks (n), then the resulting values for r would not be restricted to the range 0< r< 1, and we would no longer be able to refer to r as a normalized resolution product. 

The normalized resolution product (r or r ) can be obtained in an analogous way. In terms of S the product reads ... 

Finally, the time-corrected normalized resolution product can be found from... 

Figure 5.2 (a) Pseudo-isometric three-dimensional response and (b) iso-response contour plot for a two-parameter optimization problem. Parameters (in triangular representation) quaternary mobile phase composition. Criterion normalized resolution product (see section 4.3.2). O, is the location of the optimum. For further details see section 5.5.2. Figure taken from ref. [502]. Reprinted with permission. 

For the important case of the optimization of the mobile phase composition in reversed phase LC (RPLC), a typical two-dimensional response surface tends to be much less rugged, especially if the number of sample components is relatively small (n<10). A typical example is shown in figure 5.5. The selection of the normalized resolution product (r, eqn.4.19) as the criterion has also contributed to the smoother appearance of figure 5.5 relative to figure 5.1. Note that the criterion r has been recommended in chapter 4 for optimization processes in which the dimensions of the column are to be optimized after completion of the procedure (table 4.11). Therefore, the grid search approach is more appropriate for this kind of optimization than for optimization processes on the final analytical column. 

Such an iterative procedure has been worked out in detail by Drouen et al. [576]. Refinements of the method using the phase selection diagram discussed above include the use of normalized resolution products (see section 4.3.2), shifted compositions and confidence ranges. 

Figure 5.32 Initial phase selection diagrams for three possible ternary mobile phase systems applied to the separation of five diphenyl amines. Top (Initial) retention lines. Bottom (initial) response line. Criterion normalized resolution product (r eqn.4.19 drawn line) Also shown is the response surface using the product resolution criterion (IIeqn.4.18 dashed line). The required chromatograms are shown in figure 5.33 (a, b and c). Figure taken from ref. [576]. Reprinted with permission.

The non-linearity of the retention lines is apparent from this figure. The response lines have been drawn for two different criteria the normalized resolution product r (drawn line eqn.4.19) and the product resolution function FIRS (dashed line eqn.4.18). It is seen that the product resolution criterion would in fact have guided us to a completely different optimum at a composition of 24.1% methanol and 25.2% THF. The chromatogram that we would have obtained at this composition is shown in figure 5.33h. Clearly, this chromatogram is less attractive than the one of figure 5.33g. Obviously, the normalized resolution product is to be preferred to the resolution product itself (see the discussion in section 4.3.2). 

There are also other, more complex criteria, including the normalized resolution product, r [11],... 

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