Multipath

It is also seen that, at very low velocities, where u E, the first term tends to zero, thus meeting the logical requirement that there is no multipath dispersion at zero mobile phase velocity. Giddings also introduced a coupling term that accounted for an increase in the effective diffusion of the solute between the particles. The increased diffusion has already been discussed and it was suggested that a form of microscopic turbulence induced rapid solute transfer in the interparticulate spaces. 

In 1967, Huber and Hulsman [2] introduced yet another HETP equation having a very similar form to that of Giddings. Their equation included a modified multipath term somewhat similar in form to that of Giddings and a separate term describing the resistance to mass transfer in the mobile phase contained between the particles. The form of their equation was as follows ... 

In order to relate the value of (H) to the solute diffusivity and, consequently, to the molecular weight according to equation (11), certain preliminary calculations are necessary. It has already been demonstrated in the previous chapter (page 303) that the dynamic dead volume and capacity ratio must be used in dispersion studies but, for equation (11) to be utilized, the value of the multipath term (2Xdp) must also be... 

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. 

The multipath dispersion on a thin layer plate is the process most likely to be described by a function similar to that in the van Deemter equation. However, the actual mobile phase velocity is likely to enter that range where the Giddings function (3) applies. In addition, as the solvent composition is continually changing (at least in the vast majority of practical applications) the solute diffusivity is also altered and thus, the mobile phase velocity at which the Giddings function applies will vary. 

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. 

There are four basic dispersion processes that can occur in a packed column that will account for the final band variance. They are namely, The Multipath Effect, Longitudinal Diffusion, the Resistance to Mass Transfer in the Mobile Phase and the Resistance to Mass Transfer in the Stationary Phase. All these processes are random and essentially noninteracting and, therefore, provide individual contributions of variance that can be summed to produce the final band variance. Each process will now be discussed individually. 

In a packed column the solute molecules will describe a tortuous path through the interstices between the particles and obviously some will travel shorter paths than the average and some longer paths. Consequently, some molecules will move ahead of the average and some will lag behind thus causing band dispersion. The multipath effect is illustrated in figure (4)... 

The Multipath effect can also be used to demonstrate the use of the Random Walk Model. 

Dividing by the column length, (l), the variance per unit length or, the multipath contribution (Hm) to the overall height of the theoretical plate (H), is obtained,... 

The upper curve, which is the result of a curve fitting procedure to the points shown, is the HETP curve. The column was 25 cm long, 9 mm in diameter and packed with 8.5 micron (nominal 10 micron) Partisil silica gel. The mobile phase was a solution of 4.8 Sw/v ethyl acetate in n-decane. The minimum of the curve is clearly indicated and it is seen that the fit of the points to the curve is fairly good. As a result of the curve fitting procedure the values of the Van Deemter constants could be determined and the separate contributions to the curve from the multipath dispersion, longitudinal dispersion and the resistance to mass transfer calculated. 

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. 

It is seen that the composite curve obtained from the Huber equation is indeed similar to that obtained from that of Van Deemter but the individual contributions to the overall variance are different. Although the contributions from the resistance to mass transfer in the mobile phase and longitudinal diffusion are common to both equations, the (A) term from the Huber equation increases with mobile phase flow-rate and only becomes a constant value, similar to the multipath term in the Van Deemter equation, when the mobile velocity is sufficiently large. In practice, however, it... 

It is seen that the Golay equation produces a curve identical to the Van Deemter equation but with no contribution from a multipath term. It is also seen that, the value of (H) is solely dependent on the diffusivity of the solute in the mobile phase and the linear mobile phase velocity, It is clear that the capillary column can, therefore, provide a simple means of determining the diffusivity of a solute in any given liquid. 

In addition it is seen, from equation (12), that the expression for uopt is very similar to that for a packed column but the expression for Hmin. is much simpler as it is devoid of the (A) term from the multipath effect. 

An example of the results they obtained for the solute benzyl acetate chromatographed with the solvent mixture 5.4%(w/w) ethyl acetate in n-hexane is shown in figure (I). The curve through the points is the fitted curve to the Van Deemter equation and the contributions from the multipath... 

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

In a similar manner to the design process for packed columns, the physical characteristics and the performance specifications can pe calculated theoretically for the open tubular columns Again, the procedure involves the use of a number of equations that have been previously derived and/or discussed (1). However, it will be seen that as a result of the geometric simplicity of the open tubular column, there are no packing factors and no multipath term and so the equations that result are far less complex and easier to manipulate and to understand. 

A multipath term in van Deemter equation As surface area... 

Hm contribution to H from multipath dispersion Hl contribution to H from longitudinal diffusion ion mobile phase Hl(S) contribution to H from longitudinal diffusion ion stationary phase K distribution coefficient K(a) distribution coefficient of solute (A)... 

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