On the statistical independence of various column contributions to band broadening. Part 3: Independent treatment of longitudinal diffusion in the plate model

It is shown theorectically that the classical formula for calculating the theoretical plate number from squared first central moment, t , and second central moment, σ², according to n _theor = t /σ² is valid only when the capacity ratio, k approaches infinity. The general relation between the classical experimental HETP value, H = L/nm _theor, and the underlying true theoretical plate height, ΔL, is found to be when (σ′)² is the total column contribution to band broadening, L is the column length, D _m is the average diffusion coefficient of the sample component in the mobile phase, D _s is its value in the stationary phase, and u is the average linear velocity of the mobile phase. The mobile phase displacement, as well as the mass exchange process, is assumed to be continuous, but the application of the plate concept conditions leads to a mass balance equation that can be interpreted as belonging to a modified discontinuous plate model. The contributions 2D _m/u and k 2 D _s/u from longitudinal sample diffusion in the mobile and stationary phases, respectively, are consistent with the assumption that the processes are statistically independent, although the common solution technique of the differential equations does not take full account of this independence.     

On the statistical independence of various column contributions to band broadening. Part 2: The non-equilibrium contribution predicted by a slow,statistically independent relaxation of concentrations

The mass balance changes of Said's so-called “stage” model, based on the movement of the mobile phase with mean velocity ū (=L/t _m), are synchronized by introduction of the relaxation time of Giddings, t_r=1/(k_m+k_s) where k_m and k_s are the general overall mass rate constants for sample transfer to and from the stationary phase, respectively. This makes the “stage” length equal to the true theoretical plate height, ΔL, related to the classical HETP contribution due to non-equilibrium, H_(α), according to the “discontinuous-ΔL” relation Here k = (t _ms ? t _m)/t _m is the central moment-based capacity ratio, L the column length, and σ²_(α) the second moment contribution from the non-equilibrium only. Correct application of the relaxation-time model to chromatography requires that the real sample concentration in the stationary phase at a given position and time, C_s,l,t, is in a continuous equilibrium with the real sample concentration in the mobile phase, C_m,l+ΔL/2,t at that time displaced down the column by a distance This leads to the classical HETP contribution obtained from various other continuous models, which implies that ΔL is a good estimation of the true theoretical plate height.