Chemical Analysis. Francis Rouessac

      The experimental determination of K_T at two different temperatures enables us to calculate these variations, if we assume that the standard enthalpy and entropy variations remain virtually the same between these two temperatures. We can write:

(1.9)

If we know K_T at two temperatures, Eq. (1.9) helps us calculate the terms a and b.

      (1.10)

      and

      (1.11)

      In general, the standard enthalpy variation is negative. The same goes for the standard entropy variation, which corresponds to an increase in order when the solute is fixed on the stationary phase.

1.6 COLUMN EFFICIENCY

      1.6.1 Theoretical Efficiency (Number of Theoretical Plates)

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).

      1.6.2 Number of Effective Plates (Real Efficiency)

To compare the performance of columns of different design for a given compound – or to compare, in gas chromatography, the performances between a capillary column and a packed column – more realistic values are obtained by replacing the total retention time t_R, which appears in Eqs. (1.13)–(1.15), with the adjusted retention time (Figure 1.7), which