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a = |^ (2.43)
which demonstrates that separation in liquid chromatography arises from differences in partition coefficient (KD).
2.6.2 Resolution fRJ
Defining separation in terms of the selectivity factor (equations (2.41)-(2.43)) ignores the effects of peak width and band broadening. The resolution factor (Rs) is defined by equation (2.44), which takes into account both the difference in retention as well as the average peak widths of the two peaks (0.5(vvbj + wa>i)) (Knox, 1977).
r -_____tr’2 ~ tr-x_ (2 44)
Ks 0.5(wbjl+wb,2) ' ' ^
Figure 2.5 shows that relative concentration (or for analytical separations chromatographic response) must also be taken into account when considering resolution. In particular it should be noted that the position of the valley between the peaks is shifted to longer times as the relative concentration of the second peak is decreased. When the peak separation is 3.2a, then Rs = 0.8; and no valley is detected if the ratio of the concentrations of the two peaks is 10:1. When the separation of the peak maxima is equal to 6a, then Rs = 1.5, and the degree of peak overlap is less than 1%.
220.127.116.11 Factors influencing resolution. If the difference in retention times of the two peaks is small (^6a), then the resolution factor may be approximated by
Rs * tri~kl (2.45)
By substituting equations (2.26), (2.41), and (2.42) into equation (2.45), equation (2.46) may be obtained
Figure 2.6 Effect of selectivity (a), column efficiency (N) and retention (k) on resolution (adapted from Snyder and Kirkland (1986)). The curves were simulated using equations (2.4), (2.24) and (2.22) and the following parameters: (a) k\ = 3.0; to = 100s; a = 1.13; N = 5000; Rs = 1.53; (b) êi = 3.0; /0 = 100 s; a = 1.13; N= 1000; Rs = 0.68; (a) k, = 3.0; t0 = 100 s; a = 1.05; N= 5000; Rs = 0.63; (b) k\ = 0.5; t0 = 100 s; a = 1.13; TV = 1000; Rs = 0.67.
which allows resolution to be described into terms of a retention factor (k'/(k' + 1), a selectivity factor (a-1) and a column efficiency factor (N°'5). If the assumption that the two peak widths are unequal is not made, then a very similar expression (equation (2.47)) is obtained
r =IVN____________—____(a~D- (2.47)
s 4 (/c'i +1) a y ’
Of the three factors governing resolution (equations (2.46) and (2.47)), selectivity is the most important and is the one that is most easily manipulated to optimize separations. However, Figure 2.6 shows that poor resolution can arise as a result of poor column efficiency (Figure 2.6b), low selectivity (Figure 2.6c) or inadequate retention (Figure 2.6d). Thus an important part of any optimization strategy is to recognize which parameter is the main contributor to poor resolution.
Figure 2.7 shows that the optimum range for k' is between 2 and 10 and ideally the solvent strength should be adjusted so that the retention of the peak of interest lies within this range. Clearly, decreasing the solvent strength to increase k! to value of greater than 10 has an insignificant effect on resolution. The curves in Figure 2.7 have been drawn using a relatively conservative value of 5000 for N. It is clear that with this modest level of column efficiency baseline resolution (Rs = 1.5) of two peaks can be achieved with a selectivity of 1.10 and k' = 10. However, if à = 1.05 then baseline resolution is not achievable.
Increasing column efficiency (N) is the least attractive method of increasing resolution because it can only be achieved with packed columns at the expense of increased pressure and longer analysis times (see section 2.5.4). Furthermore, equations (2.46) and (2.47) show that resolution is proportional to the square root of the column efficiency. Therefore, doubling the column efficiency by doubling the length or halving the particle diameter only increases resolution by a factor of 1.414. Table 2.2 further illustrates the role of column efficiency in determining resolution showing the number of theoretical plates required for baseline resolution as a function of k' and a. Conventional packed liquid chromatography columns will given column efficiencies in the range 5000 to 15 000 and open-tubular systems are generally needed for higher plate counts.
2.6.3 Effect of peak asymmetry on column efficiency and separation