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02 = (jj2 + oc2
where, o2 is the variance of the eluted peak, oj2 is the variance of the eluted sample, and oc2 is the variance due to column dispersion.
Now the maximum increase in band width that can be tolerated due to any extraneous dispersion process is obviously a matter of choice but Kllnkenberg (5) suggested a 5% increase in standard deviation (ora 10%
Increase in peak variance) was the maximum extra-column dispersion that could be tolerated without serious loss in resolution. This criteria is now generally accepted.
Consider a volume (Vi), injected onto a column and forming a rectangular distribution at the front of the column. The variance of the final peak will be the sum of the variances of the sample volume plus the normal variance of a peak for a small sample. Now the variance of the rectangular
distribution of sample volume ate the beginning of the column is
assuming the peak width is increased by 5% due to the sample volume,
Thus, =n(vm + Kvs)2(l.052-l)
= n(vm + Kvs)2 0.102
*2 2 Consequently, \J~ = n(vm + Kvs) 1 23
Vt =^(vm + Kvs)l.l Bearing in mind that, Vr = n(vm + Kvs)
It is seen that the that the maximum sample volume that can be tolerated can be calculated from the retention volume of the solute concerned and the the efficiency of the column. A knowledge of the maximum sample volume that can be place on a column is important where the column efficiency available is only just adequate and the compounds of interest are minor components of the mixture to be analyzed and are only partly resolved.
Consider the situation where an LC column is fed with a mobile phase containing a solute at a given concentration until equilibrium is achieved, and an injection of pure mobile phase, devoid of the solute, is placed on the first plate of the column. This will result in a fall in concentration of the solute in the first plate which, mathematically, will represent the injection of a charge of negative concentration. This negative concentration profile of sample will pass through the column in exactly the same way as a positive concentration profile and will be eluted at the same retention time or volume but will be recorded as a negative peak by the detector
The generation of a chromatogram of negative peaks by the injection of a sample of pure mobile phase Into a mobile phase containing solutes at constant and known concentrations has been given the term Vacancy Chromatography . Vacancy chromatography has interesting properties and has a number of useful applications, few of which have been exploited to date. The processes involved in Vacancy Chromatography will be now considered theoretically employing the equations derived from the plate theory.
If a mobile phase, carrying a constant concentration of solute (X0), is fed continuously onto a chromatographic column and equilibrium is allowed to be established, the eluent from the column will also contain the solute at a concentration (X0) If a sample of the same solute, dissolved in the mobile phase at a concentration of (Xj), is now injected onto the column where either (X0<Xj) ,or (X0>Xj), then this will result in a perturbation on the concentration (X0) and, from the plate theory, this perturbation will pass through the column and be sensed by the detector at the end of the column
The equation describing this perturbation of solute concentration at the end of the column will, from the plate theory, be given by:-
X(a)=(Xj-X0)^ where X(n), is the concentration of the solute in the mobile phase
the frx'S+K rsl'ratck l^CJV ItiU Cl l\, Ml/Ill
v, is the volume passed through the column in plate volumes, and n, is the number of theoretical plates in the column.
!f the sample injected is solely mobile phase with no solute present, then, Xj=0, and,
Thus, the actual concentration leaving the (n)th plate and entering the detector, (Xe), will be,
XE — X(rD + X0 — X0
Employing the Gaussian form of the elution equation, this can be put in the form,
XE — X(n) + X0 — X0
Furthermore, at the peak maximum when v=n , and w=0,
It is seen from equation (I) that under the conditions considered, where the charge is place on the first plate, (Xn) can never equal zero and pure mobile phase will never elute from the column. However, the sample is rarely injected solely on the first plate. As it occupies a finite volume of mobile phase when it is injected onto the column, it will also occupy a finite number of theoretical plates. If (p) plate volumes of pure mobile phase are injected onto a column that has been equilibriated with mobile phase containing a concentration (X0) of a solute, then, on the application of a charge of pure mobile phase,
For p=l The change in concentration of solute on the first plate will be,
X0(e ' - i)
for p=2 , The change in concentration of charge on the first plate will be,
and for p=p The change in concentration of charge on the first plate will be,