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and the retention volume (Vr), given by:-
Vp = n(vm + Kvs)
_ n (Vm + Kvs) _ 4yn(vm + Kvs) 4
(S), is the number of peaks, having the same width as the last peak, that can be fitted Into the chromatogram up to and Including the last peak. This function does not, of course, give a true vaiue for the peak capacity, as ail the peaks that are eluted before the last, will have significantly smaller widths. Consequently, a considerably greater number of peaks can be fitted into the chromatogram than the value of (S), calculated in this way, suggests. In order to evaluate a true number for the peak capacity of a column a different approach must be used.
Consider the chromatogram In figure (5), which diagrammatical ly represents (ë) resolved peaks, each peak being separated from its neighbor by (4o).
The base width of the (r)th peak will be,
Diagram of a Chromatogram having (r) Resolved Peaks
Consider a point where the last two peaks merge. At this point the retention volume of the last peak, minus half the peak width at the base, will equal the retention volume of the last but one peak, plus half its peak width at the base.
n(vm+l<'(r-l)vs)+2Vr'(vm+K(r-1)vs) = n(vm+Krvs)+44/ri(vrn +K/-Vs)
and, (n+2^n )(vm +K(/-_1)vs) = (n-2^n)(vm +Krvs)
(vm +K(/-_j)Vs) = 7—^~r\ (Vm +K/-VS)
Therefore, for peak width (w( r-1)) of peak (r-1),
w (ë-l) = 4/n(vm +Ê(Ë_1 )V s) = 4V n (Vm+K/-Vs)
In a similar way It can be shown that the width (w(/--2)) of peak (r-2) will be:-
w(r~l) = (vm +K(/-_t)Vs) = 4,m ^ ^Vm +K/"Vs^
Thus, if the number of peaks that can be fitted into the chromatogram between the ‘dead time’ and the time for the complete elution of the last peak is (r), then,
(n+2^nj(vm +K/-vs) -
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, Ìóé| + ^n+2^n J )
(n-2^n)2 + n-2^n) t |n+2^nj n+2^
4^n(vm +K/-VS )+nvm
Noting that —-=k‘, and rearranging vm
JL-A 1+k' Vn
Replacing the geometric series by the expression for its sum,
Rearranging to provide an expression for (r)
ã \l r\ yn k‘ n-2vn — +0.5 I--V
Equation (12) is very similar to that produced by Giddings (12). However, Giddings makes certain assumptions in his derivation, not made in the above argument, that render the peak capacities quoted by him for liquid chromatographic systems somewhat less than those given by equation (13). However, the difference will not be practically significant. Equation (13) shows that the peak capacity depends on the column efficiency and the capacity ratio of the last eluted peak. Employing equation (13), the peak capacity of a series of columns having different efficiencies were calculated for a range of peak capacity factors and the results are shown as curves relating peak capacity to capacity ratio in figure (6)
Graph îÃ Peak Capacity against Capacity Factor
Capacity Factor (k‘)
It is seen that, as one would expect, the peak capacity increases with the column efficiency, but the overriding factor is the capacity ratio of the last eluted peak. It follows that any limitation to the value of (Þ for the last peak will also limit the peak capacity. Davis and Giddings (13) have shown that the theoretical peak capacity will be an exaggerated value of the true peak capacity due to the statistically irregular distribution of the individual (k ) values of each solute in a realistic multi-component mixture. In fact, they pointed out that solutes do not array themselves conveniently along the chromatogram four standard deviations apart to provide the maximum peak capacity. Nevertheless, the actual nature of the distribution of (k‘) values for any given solute mixture is unpredictable and will vary from mixture to mixture depending on the source of the sample. Consequently, the values for the theoretical peak capacity of a column given by equation (13) will be a good basis for the comparison of the peak capacities obtainable from different columns, albeit the theoretical values obtained will be In excess of the peak capacities realized in practice.
Now any characteristic of the column system that places a limit on the maximum value of (k), will also place a limit on the maximum peak capacity. One factor that controls the maximum (k') value of the last eluted
peak is the detector sensitivity. As the (k') of a solute increases, the peak becomes more dispersed, and consequently, the peak height is reduced, and eventually at a given (k‘) the peak will disappear into the detector 'noise'. If it is merely necessary to unambiguously identify the presence of a peak, then the peak maximum would need to be about five time the noise level. The detector sensitivity or the minimum detectable concentration (Xp) is usually defined as equal to twice the noise level and consequently the peak height at the maximum (k') value must be 2.5.Xp. Now,the concentration of solute at the peak maximum can be taken as twice the average concentration. Thus if a a mass (m) is placed on the column and the peak width is 4i/n(vm+Kvs),