Columns longitudinal diffusion

FLOW. The rate at which zones migrate down the column is dependent upon equilibrium conditions and mobile phase velocity on the other hand, how the zone broadens depends upon flow conditions in the column, longitudinal diffusion, and the rate of mass transfer. Since there are various types of columns used in gas chromatography, namely, open tubular columns, support coated open tubular columns, packed capillary columns, and analytical packed columns, we should look at the conditions of flow in a gas chromatographic column. Our discussion of flow will be restricted to Newtonian fluids, that is, those in which the viscosity remains constant at a given temperature. 

The rate at which zones migrate down the column is dependent on equilibrium conditions and mobile-phase velocity on the other hand, how the zone broadens depends on flow conditions in the column, longitudinal diffusion, and the rate of mass transfer. Since various types of columns are nsed in GC—namely, open... 

To increase the number of theoretical plates without increasing the length of the column, it is necessary to decrease one or more of the terms in equation 12.27 or equation 12.28. The easiest way to accomplish this is by adjusting the velocity of the mobile phase. At a low mobile-phase velocity, column efficiency is limited by longitudinal diffusion, whereas at higher velocities efficiency is limited by the two mass transfer terms. As shown in Figure 12.15 (which is interpreted in terms of equation 12.28), the optimum mobile-phase velocity corresponds to a minimum in a plot of H as a function of u. 

Kovat s retention index (p. 575) liquid-solid adsorption chromatography (p. 590) longitudinal diffusion (p. 560) loop injector (p. 584) mass spectrum (p. 571) mass transfer (p. 561) micellar electrokinetic capillary chromatography (p. 606) micelle (p. 606) mobile phase (p. 546) normal-phase chromatography (p. 580) on-column injection (p. 568) open tubular column (p. 564) packed column (p. 564) peak capacity (p. 554)... 

Various mathematical concepts and techniques have been used to derive the functions that describe the different types of dispersion and to simplify further development of the rate theory two of these procedures will be discussed in some detail. The two processes are, firstly, the Random Walk Concept [1] which was introduced to the rate theory by Giddings [2] and, secondly, the mathematics of diffusion which is both critical in the study of dispersion due to longitudinal diffusion and that due to solute mass transfer between the two phases. The random walk model allows the relatively simple derivation of the variance contributions from two of the dispersion processes that occur in the column and, so, this model will be the first to be discussed. 

The dispersion of a solute band in a packed column was originally treated comprehensively by Van Deemter et al. [4] who postulated that there were four first-order effect, spreading processes that were responsible for peak dispersion. These the authors designated as multi-path dispersion, longitudinal diffusion, resistance to mass transfer in the mobile phase and resistance to mass transfer in the stationary phase. Van Deemter derived an expression for the variance contribution of each dispersion process to the overall variance per unit length of the column. Consequently, as the individual dispersion processes can be assumed to be random and non-interacting, the total variance per unit length of the column was obtained from a sum of the individual variance contributions. 

Thus, treating the diffusion process in a similar way to that shown in Figure 4 the total variance due to longitudinal diffusion in a column of length (1) is given by equation (7), viz.,... 

Equation (11) accurately describes longitudinal diffusion in a capillary column where there is no impediment to the flow from particles of packing. In a packed column, however, the mobile phase swirls around the particles. This tends to increase the effective diffusivity of the solute. Van Deemter introduced a constant (y) to account... 

Then, the contribution to the total variance per unit length for the column from longitudinal diffusion in the stationary phase will be... 

In summary, equation (13) accurately describes longitudinal dispersion in the stationary phase of capillary columns, but it will only be significant compared with other dispersion mechanisms in LC capillary columns, should they ever become generally practical and available. Dispersion due to longitudinal diffusion in the stationary phase in packed columns is not significant due to the discontinuous nature of the stationary phase and, compared to other dispersion processes, can be ignored in practice. 

The curves represent a plot of log (h ) (reduced plate height) against log (v) (reduced velocity) for two very different columns. The lower the curve, the better the column is packed (the lower the minimum reduced plate height). At low velocities, the (B) term (longitudinal diffusion) dominates, and at high velocities the (C) term (resistance to mass transfer in the stationary phase) dominates, as in the Van Deemter equation. The best column efficiency is achieved when the minimum is about 2 particle diameters and thus, log (h ) is about 0.35. The optimum reduced velocity is in the range of 3 to 5 cm/sec., that is log (v) takes values between 0.3 and 0.5. The Knox... 

If it is assumed that the column is operated at relatively high velocities, such that the contribution from longitudinal diffusion is no longer significant, then to a first approximation. 

Van Deemter considered peak dispersion results from four spreading processes that take place in a column, namely, the Multi-Path Effect, Longitudinal Diffusion, Resistance to Mass Transfer in the Mobile Phase and Resistance to Mass Transfer in the Stationary Phase. Each one of these dispersion processes will now be considered separately... 

The dispersion described in figure 2 shows that the longer the solute band remains in the column, the greater will be the extent of longitudinal diffusion. Since the length of time the solute remains in the column is inversely proportional to the mobile phase velocity, so will the dispersion be inversely proportional to the mobile phase velocity. Van Deemter et al derived the following expression for the... 

The A term represents the contribution from eddy diffusion, the B term the contribution from longitudinal diffusion, and the C terms the contributions from mass transfer in the mobile and stationary phases to the total column plate height. By differentiating equation (1.31) with respect to the mobile phase velocity and setting the result equal to zero, the optimum values of mobile phase velocity (u ) and plate height (HETP ) can be obtained. 

Longitudinal diffusion will become more serious the longer the solute species spend in the column, so this effect, unlike flow dispersion is reduced by using a rapid flow rate of mobile phase. 

You can see that these dispersion mechanisms are affected in different ways by the flow rate of mobile phase. To reduce dispersion due to longitudinal diffusion we need a high flow rate, whereas a low flow rate is needed to reduce dispersion due to the other two. This suggests that there will be an optimum flow rate where the combination of the three effects produces minimum dispersion, and this can be observed in practice if N or H (which measure dispersion) are plotted against the velocity or flow rate of the mobile phase in the column. The shape of the graph is shown in Fig. 2.3f. 

The total column dispersion is due to the combined effects of flow dispersion, longitudinal diffusion and mass transfer. 

An additional contribution to molecular spreading is provided by longitudinal diffusion. Whether the mobile phase within the column is moving or at rest, sample molecules tend to diffuse randomly in all... 

Lapidus, L. and Amundson, N. R. J. Phys. Chem. 56 (1952) 373. Mathematics of adsorption in fixed beds — The rate determining steps in radial adsorption analysis ibid 56 (1952) 984. The effect of longitudinal diffusion in ion exchange and chromatographic columns. 

First, we look at isocratic separations. Let us assume that the analysis can be accomplished within a retention factor of 10. We also suppose that the analysis is carried out with a typical reversed-phase solvent and a sample with a typical molecular weight of a pharmaceutical entity. In order to manipulate the analysis time, we will keep the mobile phase composition the same and vary the flow rate. The maximum backpressure that we will be able to apply is 25MPa (250 bar, 4000psi). In Figure 1, we have plotted the plate count as a function of the analysis time for a 5 J,m 15-cm column. We see that the column plate count is low at short analysis times and reaches a maximum at an analysis time of about 1 h. A further increase in analysis time is not useful, since the column plate count declines again. This is the point where longitudinal diffusion limits the column performance. The graph also stops at an analysis time of just under 5 min. This is the point when the maximum allowable pressure drop has been reached. 

The B-term in the equation is the contribution to the plate height resulting from longitudinal diffusion (molecular diffusion in the axial direction) and arises from the tendency of the solute band to diffuse away from the band center as it moves down a column. It is proportional to the time that the sample spends in the column and also to its diffusion... 

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