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The Plate Theory, however, does little to explain how the efficiency of a column may be changed or, what causes peak dispersion in a column in the first place. It does not tell us how dispersion is related to column geometry, properties of the packing, mobile phase flow-rate, or the physical properties of the distribution system. Nevertheless, it was not so much the limitations of the Plate Theory that provoked Van Deemter et al (3) (who were chemical engineers and mathematicians) to develop, what is now termed the Rate Theory for chromatographic dispersion, but more to explore an alternative mathematical approach to explain the chromatographic process. Virtually all basic chromatography theory evolved over the twenty five years between 1940 and 1965 and it was in the middle of this period that Van Deemter and his colleagues presented their Rate Theory concept in (1956). Since that time, other Rate Theories have been presented, together with accompanying dispersion equations and in due course each will be discussed, but most were very similar in form to that of Van Deemter et at. It is interesting to note, however, that, even after thirty five years of chromatography development, the equation that best describes band
dispersion in practice is still the Van Deemter equation. This is particularly true for columns operated around the mobile phase optimum velocity where the maximum column efficiency is obtained.
The purpose of the Rate Theory is to aid in the understanding of the processes that cause dispersion in a chromatographic column and to identify those factors that control it. Such an understanding will allow the best column to be designed to effect a given separation in the most efficient way. However, before discussing the Rate Theory some basic concepts must be introduced and illustrated.
The Summation of Variances
The width of the band of an eluted solute relative to its proximity to its nearest neighbor determines whether two solutes are resolved or not. The ultimate band width as sensed by the detector is the results of a number of individual dispersion processes taking place in the chromatographic system, some of which take place in the column itself and some in the sample valve, connecting tubes and detector. In order to determine the ultimate dispersion of the solute band it is necessary to be able to calculate the final peak variance. This is achieved by taking into account all the individual dispersion processes that take place in a chromatographic system. It is not possible to sum the band widths resulting from each individual dispersion process to obtain the final band width, but it is possible to sum all the respective variances. However, the summation of all the variances resulting from each process is only possible if each process is non-interacting and random in nature. That is to say, the extent to which one dispersion process progresses is independent of the development and progress of any other dispersion process.
Thus, assuming there are (n) non-interacting, random dispersive processes occurring in the chromatographic system, then any process (p) acting alone
will produce a Gaussian curve having a variance î2,
9 9 9 9 9
Hence, î, +o ^................î ^= a
where, î2 is the variance of the solute band as sensed by the detector.
The above equation is the algebraic enunciation of the principle of the summation of variances and is fundamentally important. If the individual dispersion processes that are taking place in a column can be identified, and the variance that results from each dispersion determined, then the variance of the final band can be calculated from the sum of all the
individual variances. This is how the Rate Theory provides an equation for the final variance of the peak leaving the column. As an example the principle of the summation of variances will be applied to extra column dispersion
Extra Column Dispersion
Extra column dispersion is that contribution to the total band dispersion that arises from spreading processes taking place outside the column. There are five major sources of extra column dispersion and these are as follows:-
1/ Dispersion resulting from the sample having a finite volume which can be considered to contribute a variance,(0^)
2/ Dispersion occurring in the channels of the sample valve itself which can be considered to contribute a variance,(o^)
3/ Dispersion occurring in the sample valve/column and column/detector connections.These can be considered to contribute a variance, (ot2)
4/ Dispersion occurring in the detector cell, or in the sensing volume of the detector, which can be considered to contribute a variance, (o^)
5/ Finally, there will be dispersion resulting from the response time of the electronic system and sensor of the detector contributing a variance, (oR)
Thus, summing all the variances,
2 2 2 2 2 2 oE=os+ov+ot+oD+oR
Where (o^) is the the variance of the eluted peak.
It follows that the allocation of all the permitted extra column dispersion to sample volume dispersion, as defined by Klinkenberg (4) and suggested on page (54), is not permissible . Other sources of dispersion must be taken into account and take a share of the permitted 10% increase in column variance.