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X0(e-P - I)
Now, after each plate volume of charge has established, a new concentration in plate (I), its contents will be eluted through the column in the normal manner. If a total of (p) plate volumes of pure mobile phase are injected onto the column and the solute subsequently eluted by a further (v) plate volumes of equilibrlated mobile phase then, after (r) plate volumes of sample have been injected, the new concentration of solute in plate (1) will be eluted by a further (p-r) plate volumes of sample and (v) plate volumes of equilibriated mobile phase.
Thus the concentration of solute leaving the (n) th plate due to the (r) th volume of charge will be,
-(v+p-r-n)2 ( r e 2n
xn=x0 e-r-l -=—
v ' ^2rcn
It follows that the actual concentration of solute in the (n)th plate, (Xe) resulting from the injection of (p) plate volumes of pure mobile phase, followed by (v) plate volumes of equilibriated mobile phase carrying a concentration (X0) of solute, will be given by:-
Equation (1) can be used to calculate the elution curves that result from different volumes of pure mobile phase that have been injected onto a an equilibriated column. Values of (Xe) for such curves were calculated for a column having 500 theoretical plates and for sample volumes of 20, 50, 100, and 200 plate volumes respectively. The curves', relating solute concentration to plate volumes of mobile phase passed through the column, are shown in figure (1). It is seen that as the injection volume is increased, the retention volume of the peak also increases. The retention volume of the small negative peak produced by the smallest charge will be equal to that for the solute when Xj>Xo and also when chromatographed In the normal way with the column carrying pure mobile phase only This assumes that the presence of the low concentration of solute in the mobile phase does not
influence the retentive characteristics of the stationary phase it is seen that the concentration of solute in the eluted peak does not fail to zero until
Figure I Vacancy Elution Curves for Different Injection Volumes on a Column of 500 Theoretical Plates
Row ol mobile phase in plate volumes
the sample volume is in excess of 100 plate volumes, which is about five time the standard deviation of the normally loaded peak. Equation (1) can be developed further to provide a general equation for a column equilibriated
with (q) solutes at concentrations X1, X2, X3,.....Xq. For any particular solute
(S), if its normal retention volume is Rs on a column containing (n) plates, then the plate volume of the column (vs) is given by,
If the sample injected is contained in a volume (V) ml, then the charge measured in 'plate volumes' is V/vs and if the volume of mobile phase passed through the column is (y) ml then that will be equivalent to y/vs plate volumes If the sample injected onto the column contains solute (5) at a concentration Xs' then from equation (1) the concentration of this solute at the (n) th plate in the column (Xes) is given by:-
V - (— +— -r-n)2
r=— VS vs_
Thus for a chromatogram of (q) solutes
.(X +_L Vs vs
Xe = 2Xs * 2
S = 1 r = 1
If pure mobile phase is injected onto the column,then Xs'= 0 and equation (2) becomes,
V - (— + — -r-n)2 S-q ^ -JS-Jb-
Equation (3) is similar to that provided by Reilly et al (5) but the derivation is simpler, as those authors utilized the approximate binomial form of the elution curve in their derivation.
Vacancy chromatography has a number of applications areas in practice, none of which appear to have been extensively exploited. One particularly interesting application is that of quality control. If a particular product has a number of components present, and their relative composition must be kept constant as in, for example, a pharmaceutical product, Vacancy Chromatography can provide a particularly simple analytical procedure for quality control. The mobile phase is made up containing the components of the product in the specified proportions, but at a low concentration suitable for LC analysis. A sample of the product is dissolved In some pure mobile phase at the same total mass concentration as the standards In the mobile phase. A sample is then injected on the column. If the product contains the components in the specified proportion, no peaks will appear on the chromatogram as the sample and mobile phase will have the same composition. If any component is in excess, it will show a positive peak. If any component is present below specifications, it will show a negative peak The size of the peak will provide an accurate measure of the difference between the sample and that of the required standard.
The Resolving Power of an LC Column
It has already been stated that for two solutes to be resolved their peaks must be moved apart in the column and, at the same time, maintained sufficiently narrow to permit them to be eluted as discrete peaks. The criterion for two peaks to be resolved is arbitrary, but as stated in Chapter l, resolution is is usually defined as the ratio of the distance between the peaks, to the peak width at the points of inflection. The separation of a pair of solutes on columns of differing resolution is shown in figure (2).