Longitudinal diffusion dispersion

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. 

Dispersion equations, typically the van Deemter equation (2), have been often applied to the TLC plate. Qualitatively, this use of dispersion equations derived for GC and LC can be useful, but any quantitative relationship between such equations and the actual thin layer plate are likely to be fraught with en or. In general, there will be the three similar dispersion terms representing the main sources of spot dispersion, namely, multipath dispersion, longitudinal diffusion and dispersion due to resistance to mass transfer between the two phases. 

HETP of a TLC plate is taken as the ratio of the distance traveled by the spot to the plate efficiency. The same three processes cause spot dispersion in TLC as do cause band dispersion in GC and LC. Namely, they are multipath dispersion, longitudinal diffusion and resistance to mass transfer between the two phases. Due to the aforementioned solvent frontal analysis, however, neither the capacity ratio, the solute diffusivity or the solvent velocity are constant throughout the elution of the solute along the plate and thus the conventional dispersion equations used in GC and LC have no pertinence to the thin layer plate. 

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

I where the subscripts extr, dif. Joule, cone, and ads indi- cate the contributions from extracolumn dispersion, longitudinal diffusion. Joule self-heating, concentration... 

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. 

Diffusion plays an important part in peak dispersion. It not only contributes to dispersion directly (i.e., longitudinal diffusion), but also plays a part in the dispersion that results from solute transfer between the two phases. Consider the situation depicted in Figure 4, where a sample of solute is introduced in plane (A), plane (A) having unit cros-sectional area. Solute will diffuse according to Fick s law in both directions ( x) and, at a point (x) from the sample point, according to Ficks law, the mass of solute transported across unit area in unit time (mx) will be given by... 

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. 

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... 

Equations in Table IX are written per unit of bed volume (A )g is a time averaged, mean axial bed conductivity. A is a longitudinal diffusivity and Ai allows for particle to particle conductivity. Not all the terms in the model as given in the table are important. For example, Wu et al. (1995, 19%) and Xiao and Yuan (1996) neglect the accumulation and dispersion terms in Eq. (30) and the accumulation and conduction terms in Eq. (28). 

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. 

In practice, because of the slow diffusion rates in liquids, dispersion due to longitudinal diffusion becomes important only at very low velocities. Because the dispersion increases only slowly with increasing mobile phase velocity, flow rates used in hplc are considerably higher than the value corresponding to minimum dispersion. This gives us fast separations without too much loss of efficiency. 

The three contributions to dispersion are also shown as separate curves in figure 1. It is seen that the major contribution to dispersion at the optimum velocity, where the value of (H) is a minimum, is the multipath effect. Only at much lower velocities does the longitudinal diffusion effect become significant. Conversely, the mobile phase velocity must be increased to about 0.2 cm/sec before the resistance to mass transfer begins to become relatively significant compared to that of the multipath effect. 

Maximum Column Inlet Pressure Extra Column Dispersion Multipath Packing Factor Longitudinal Diffusion Packing Factor Column Mobile Phase Fraction... 

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