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Two Solutes Separated on Columns of Different Resolving
It is seen that for base-line resolution the peak maxima must be six standard deviations (6o) apart. But for accurate quantitative analysis, employing peak heights measurements, a separation of (4o) is usually quite adequate. Even when peak area measurements are employed, a separation of (4o) will usually provide adequate accuracy, particularly if computer data acquisition and processing is employed with modern software. Therefore, throughout this book, whenever dealing with resolution, or column design, a resolution of (4o) will be assumed.
It should be pointed out that two adjacent peaks from solutes of different chemical type or significantly different molecular weight will not necessarily have precisely the same peak widths. However, the difference will be relatively small and, In the vast majority of cases, will be
negligible. Consequently, the peak widths of closely adjacent peaks will be considered the same.
Consider the two peaks depicted in the now well recognized figure (3). The difference between the two peaks, for solutes (A) and (B), measured in volume flow of mobile phase will be,
n(vm +Kbvs) - n(vm +Kavs) = ï(Êâ + Ka)vs--------------(4)
Now, because It can be assumed that the widths of the two peaks are the same, then the peak width in volume flow of mobile phase will be,
2o = 2Vn(vm +KAvs)------------------- (5)
A Chromatogram Showing the Separation of Two Solutes
Where «a is the distribution coefficient of the first of the eluted pair of solutes between the two phases. Taking the already discussed criterion that resolution is achieved when the peak maxima of the pair of soiutes are (4o) apart then
4Vn(vm +Kavs) = ï(Êâ + Ka)vs
Dividing through by (vm),
ã 4 (1+k.)
Now as (a) is defined as, a = -~
4(l+k ) A
. 2 9 k, (a-1)
Equation (6) is extremely Important and was first developed by Purnell (6) in 1959. It allows the necessary efficiency to achieve a given separation to be calculated from a knowledge of the capacity factor of the first eluted peak of the pair and their separation ratio. It is used extensively throughout this book and is particularly important in the theory of column design.
It is of interest to determine from equation (6) how the required efficiency to achieve a separation varies with the separation ratio (a) and the capacity factor of the first eluted peak of the pair (ê'ä). In figure (4), curves relating (n) and (ê'ä) are shown for a column separating solute pairs having separation ratios of 1.02, 1.05 and 1.10. It is seen that the necessary efficiency (n) increases as the separation becomes more difficult. That is, when the peaks are closer together and (a) Is small. This to be expected, but what is not so obvious is the dramatic increase in (n) as the value of the capacity factor becomes small. It follows, that to reduce the number of
theoretical plates needed, and thus reduce the necessary column length and analysis time, the phase system should be chosen such that the pair of solutes that are closest together in the chromatogram, are not eluted at very low (Þ values.
Graph of Log. Efficiency against Capacity factor for Solute Pairs having Different Separation Ratios
The Effective Plate Number
The concept of the effective plate number was introduced and employed in the late nineteen fifties by Purnell (7), Desty (8) and others. Its introduction arose directly as a result of the development of the capillary column, which, even in I960, could be made to produce efficiencies of up to a million theoretical plates (9). It was noted, however, that these high efficiencies were were only realized for solutes eluted close to the column dead volume, that is, at very low k' values. Furthermore, they in no way reflected the increase in resolving power that would be expected from such high efficiencies on the basis of the performance of packed columns. This poor performance, relative to the high efficiencies produced, can be shown theoretically ( and indeed will be, later in this book) to result from the high phase ratio of capillary columns made at that time. That is the ratio of the mobile phase to the stationary phase in the column. The high phase ratio was
due to the fact that there was very little stationary phase in the capillary column (the film was very thin). It has already been shown that the corrected retention volume of a solute is directly proportional to the amount of stationary phase there is in the column and, consequently, solutes were eluted from a capillary column at relatively low k' values. The thin films gave rise to very high efficiencies but, as was shown in the previous section, at low k’ values very high efficiencies are needed to achieve relatively simple separations.
To compensate for, what appeared to be very misleading efficiencies values, the 'effective plate number' was introduced The 'effective plate number' uses the corrected retention distance, as opposed to the total retention distance to calculate the efficiency. Otherwise the calculation is the same as that used in the normal calculation of theoretical plates. In this way the 'effective plate number' becomes significantly smaller than the true number of theoretical plates for solutes eluted at low k' values At high k' values, the the two measures of efficiency tends to converge In this way the 'effective plate number' appears to more nearly correspond to the column resolving power. In fact, it is an indirect way of trying to define resolution in terms of the number of 'effective plates' in the column