Francis Rouessac - Chemical Analysis

Like any equilibrium constant, it is a function of temperature. It is related to variations in free enthalpy, standard enthalpy, and entropy of the exchange reaction by means of the following relationships:

(1.8)

The experimental determination of K Tat 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 Tat 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.

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 σ 2(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 Rand, 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 Rand 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 Rand 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 Rthe retention time of the compound. t Rand 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 4 σ 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 = 4 σ ), 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 0.1designates the width of the peak measured at 10% of its height ( Figure 1.5).

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 does not take into account the hold‐up time t Mspent by the compound in the mobile phase. The three equations then become:

(1.17)

(1.18)

Figure 1.7 Retention factors and separation factor (or selectivity factor) between two adjacent compounds. Each compound has its own retention factor. On this figure, the separation factor is around 1.3. α alone is not enough to determine whether separation is really possible.

(1.19)

Currently, it is considered that these last three equations are not very useful.

The height equivalent to a theoretical plate H , as already defined ( Eq. (1.4)), is calculated for reference compounds to permit a comparison of columns of different lengths. However, H does not behave as a constant; its value depends upon the compound chosen and upon the experimental conditions.