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As the analyte migrates through the column, it occupies a continually expanding zone (Figure 1.6). This linear dispersion σ_l, measured by the variance σ_l ², increases with the distance of migration. When this distance reaches L, the total column length, the variance will be:

(1.12)

In line with the plate theory model of distillation, this approach also leads to the value of the height equivalent to one theoretical plate H and to the number N of theoretical plates (N = L/H).

Therefore, for any chromatogram that shows an elution peak of a compound with the temporal variance σ² (σ = σ_L / υ where υ is the mean elution speed of a solute), we can determine the theoretical efficiency N for this compound (Eq. (1.13)) as a function of retention time t_R and, by deduction, we can also get the value of H, knowing that H = L/N.

(1.13)

These two parameters are indirectly accessible from the elution peak of the compound. We measure t_R and σ, whose ratio is identical to that of L over σ_L (Eq. (1.13)).

Figure 1.6 Dispersion of a solute in a column. Left, a graph corresponding to the isochronic image of the concentration of an eluted compound at a particular instant. Right, a chromatogram revealing the variation of the concentration at the column outlet, as a function of time. t_R and σ have the same ratio as L and σ_L. The efficiency N can therefore be calculated from the chromatogram by measuring σ directly. On the graph, we find about 100 theoretical plates.

On the chromatogram, σ represents the half‐width of the peak at 60.6% of its height and t_R the retention time of the compound. t_R and σ should be measured in the same units (time, distance or eluted volume if the flow rate is constant). If σ is expressed in units of volume (using flow rate), then 4σ corresponds to the volume of the peak, which corresponds to the eluent volume containing 95% of the initially injected compound. By consequence of the properties of the Gaussian curve (ω = 4σ), Eq. (1.14) results. However, because of the distortion of most peaks at their base, that equation is rarely used and Eq. (1.15) is preferred.

N is a relative parameter, since it depends upon both the solute chosen and the operating conditions adopted. Generally, a compound that appears towards the end of the chromatogram is selected in order to get a reference value, when it is unknown whether the column will successfully achieve a given separation.

(1.14)

(1.15)

For asymmetric peaks (which is often the case), we sometimes encounter the empirical Eq. (1.16) for calculating the column’s efficiency:

(1.16)

where ω_0.1 designates the width of the peak measured at 10% of its height (Figure 1.5).