The selectivity in chromatography is the ability of the chromatographic system to chemically distinguish between two different analytes, based on their interactions with the stationary and mobile phases.
1. Mathematical definition #
Quantitatively, selectivity (\alpha) is defined as the ratio of the retention factors (k) of two analytes:
Equation 1.19: \alpha_{j,i}=\frac{k_{j}} {k_{i}}
By convention, \alpha_{j,i} is always equal or larger than 1. In other words, the largest retention factor is always divided by the smaller one. A value of \alpha_{j,i} = 1, implies that the analytes co-elute fully. A high value of \alpha_{j,i} indicates that the analytes can be separated easily.
2. Thermodynamics #
It is also possible to rewrite Equation 1.19 in terms of distribution coefficients (K_{\text{d}}; see Module 1.1) using Equation 1.2, to obtain
Equation 1.20: \alpha_{j,i}=\frac{K_{\text{d},j}}{K_{\text{d},i}}=\frac{k_{j}} {k_{i}}
Equation 1.20 is not really used in practice, but does serve to illustrate that the selectivity is purely affected by thermodynamic factors. The selectivity is exclusively influenced by the analyte and the phase system (i.e. mobile and stationary phase chemistry), temperature, and, to a lesser extent, by the pressure.
3. Minimum required selectivity #
Equation 1.39 in the book is one of the most useful ones in chromatography. It described the relationship between the resolution (R_{\text{S},j,i}) of two successive peaks (i, eluting first, and j, eluting last), the number of theoretical plates (N), the selectivity (\alpha_{j,i}) and the average retention factor (\bar{k}=(k_i+k_j)/2). The equation reads
Equation 1.39: R_{\text{S},j,i}=\frac{\sqrt{N}}{2} \cdot \frac{\alpha_{j,i}-1}{\alpha{j,i}+1} \cdot \frac{\bar{k}}{1+\bar{k}}
Reshuffling yields an equation that describes the minimum required selectivity ([\alpha_{j,i}]_{\text{min}}) as a function of the required resolution (R_{\text{S,req}}), the number of plates available and the average retention factor.
Figure 1 illustrates the minimum required selectivity for a required resolution of 1.5 as a function of the available number of plates, for various values of \bar{k} (indicated in the legend).
Figure 1. Required selectivity as a function of plate number for different retention factors.
It is seen larger \bar{k} values (pink, solid lines) lead to a drastic reduction in the minimum required selectivity (or a drastic reduction in the number of plates required at a given value for \alpha.
At larger \bar{k} values the effect of the retention factor diminishes, as is clear from the marginal differences between the two lowest lines (\bar{k} = 5 and \bar{k} = 10). With 10 000 plates available, a typical range for LC, [\alpha_{j,i}]_{\text{min}} values between 1.13 and 1.07 are found to be required for the range 1<\bar{k}<10. For 100 000 plates, as can be achieved in GC, values of 1.04<[\alpha_{j,i}]_{\text{min}}<1.02 suffice for the range 1<\bar{k}<10.
