Ion Chromatography - Dispersion by Resistance to Mass Transfer in the Stationary Phase 3
The simplified HETP equation is a hyperbolic function of the mobile phase linear velocity (u) and the various components of which are shown in figure 7. The multi-path term is, for all practical operating conditions, independent of the linear velocity, and is represented by a straight line parallel to the velocity axis and is shown green in figure 7. The second term is a reciprocal function of the mobile phase linear velocity and is shown as the orange curve in figure 7. Theory suggests that the resistance to mass transfer has two parts which, in gas chromatography with packed columns it certainly does. However, when the mobile phase is a liquid, due to the tortuous path followed by the mobile phase between the particles of the packing, the streams of mobile phase are continually changing direction with the consequent formation of many eddies. These eddies greatly increase the effective diffusivity of the solute in the mobile phase. This large increase in (Dm) reduces the mobile phase resistance to transfer term to virtually zero (equation (9)) leaving the predominant contribution to resistance mass transfer to be the mass transfer in the stationary phase in which the solute diffusivity is very low. (c.a.10-5cm2s-1). An interesting situation occurs in capillary column gas chromatography where the converse applies. There is no packing and the flow of gas is Newtonian with the expected parabolic velocity profile, thus, the resistance to mass transfer in the mobile phase will be very large. However, the stationary phase is spread as a thin film on the walls of the tube and, thus, (df) is normally very small so in the case of capillary column gas chromatography the dominant contribution to resistance to mass transfer is from the mobile phase whereas the resistance to mass transfer in the stationary phase is very small. The combined resistance to mass transfer terms are shown as the purple linear curve in figure 7.
The combination of the three terms provide the hyperbolic curve shown as red in figure 7. It is seen that the curve exhibits a minimum called the minimum HETP (Hmin) and this occurs at the optimum mobile phase velocity (uopt) At the optimum velocity the column will have the maximum number of theoretical plates and, thus, the highest possible efficiency under the given operating conditions (mobile phase composition, temperature etc.). It must be emphasized that operating the the column at the optimum mobile phase velocity may or may not produce the separation in the minimum time as a different phase system may provide a better selectivity and, thus, can achieve the separation with a shorter column with consequent reduced elution time.
Differentiating equation (13) and equating to zero, expressions for (Hmin). and (uopt) can be obtained.
Equating to zero.
Substituting for (uopt) in equation (19) and expression for (Hmin). Can be obtained.
The dispersion theory can be extended considerably further and for those interested they are recommended to read Chromatographic Theory in this series. The theory given so far should be adequate for the analyst employing ion chromatography as a separation technique.