Flow and Diffusion in the Mobile Phase

We should like to be able to obtain simple expressions like Eqs. 11.15 or 11.18 for the mobile phase. Unfortunately, mobile phase processes are complicated by flow. The consequences are discussed below. 

We have already dealt with stationary phase processes and have noted that they can be treated with some success by either macroscopic (bulk transport) or microscopic (molecular-statistical) models. For the mobile phase, the molecular-statistical model has little competition from bulk transport theory. This is because of the difficulty in formulating mass transport in complex pore space with erratic flow. (One treatment based on bulk transport has been developed but not yet worked out in detail for realistic models of packed beds [11,12].) Recent progress in this area has been summarized by Weber and Carr [13]. 

We can now express our scaling ideas in concrete form. First of all, most significant distances, such as those between adjacent channels or across channels, are scaled in proportion to mean particle diameter dn. For 

We now recall that the spreading of chromatographic peaks is due to different velocity states that molecules can occupy, allowing one molecule to get ahead of or behind another. Due to the complicated structure of a bed of packed particles, there are a number of ways in which these critical velocity increments or biases can arise [1]. These are  

Molecules travel faster at the center of narrow flow channels (e.g., those winding between adjacent particles) than at the outside edge. 

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Flow and diffusion in the mobile phase. As noted in Chapter 4, the microscopic flow process in a packed bed is very tortuous, each streampath suffering frequent changes in direction and velocity. A solute molecule carried in this flow will trace out a very uneven path. The randomness of the molecular path will be amplified by Brownian displacements of the molecule from one streampath to another (Figure 11.1). 

Calculate the plate height contributed by longitudinal diffusion in the mobile phase of a column for which y = 0.60 and in which the mean flow velocity is 2.0 cm/s. First assume that the column is a GC column with a typical solute diffusivity of Dm = 0.10 cm2/s second, assume a LC column with Dm = 1.0 x 10"5 cm2/s. 

Figure 4.1 Band-broadening processes in porous irregular microparticles, (a) eddy diffusion analyte molecules take different routes to circumnavigate the particles. They also move more quickly through wide channels than through narrow channels, (b) diffusion in the mobile phase. The short bracket indicates initial band width, the long bracket indicates final band width, (c) mass transfer. On the left is shown mass transfer in stagnant mobile phase in pores, and that due to the adsorption/desorption process. The narrow band represents initial band width, the broad band final band width. On the right is shown mobile phase mass transfer caused by laminar flow.

All molecules present in the mobile phase at time tm may diffuse in and against the flow direction. The contribution of the longitudinal diffusion in the mobile phase is described by Eq. (20) ... 

The terms 1/A,- and HC iV describe the effect of the eddy diffusion and nonequilibrium in the mobile phase, respectively. The contribution to zone broadening due to the eddy diffusion is dominating at high flow rates. The importance of both these terms is related with the particle size distribution (PSD) and with the homogeneity of the column packing or, in other words, with the relative differences in the flow... 

Glueckauf (24) studied the effect of four factors on the chromatographic process (1) diffusion in the mobile phase normal to the direction of flow, (2) longitudinal diffusion in the mobile phase, (3) diffusion into the particle, and (4) size of the particle. 

The solvent used was 5 %v/v ethyl acetate in n-hexane at a flow rate of 0.5 ml/min. Each solute was dissolved in the mobile phase at a concentration appropriate to its extinction coefficient. Each determination was carried out in triplicate and, if any individual measurement differed by more than 3% from either or both replicates, then further replicate samples were injected. All peaks were symmetrical (i.e., the asymmetry ratio was less than 1.1). The efficiency of each solute peak was taken as four times the square of the ratio of the retention time in seconds to the peak width in seconds measured at 0.6065 of the peak height. The diffusivities obtained for 69 different solutes are included with other physical and chromatographic properties in table 1. The diffusivity values are included here as they can be useful in many theoretical studies and there is a dearth of such data available in the literature (particularly for the type of solutes and solvents commonly used in LC separations). 

The form of the effective mobility tensor remains unchanged as in Eq. (125), which imphes that the fluid flow does not affect the mobility terms. This is reasonable for an uncharged medium, where there is no interaction between the electric field and the convective flow field. However, the hydrodynamic term, Eq. (128), is affected by the electric field, since electroconvective flux at the boundary between the two phases causes solute to transport from one phase to the other, which can change the mean effective velocity through the system. One can also note that even if no electric field is applied, the mean velocity is affected by the diffusive transport into the stationary phase. Paine et al. [285] developed expressions to show that reversible adsorption and heterogeneous reaction affected the effective dispersion terms for flow in a capillary tube the present problem shows how partitioning, driven both by electrophoresis and diffusion, into the second phase will affect the overall dispersion and mean velocity terms. 

B/Si is the molecular diffusion term and relates to diffusion of solute molecules within the mobile phase caused by local concentration gradients. Diffusion within the stationary phase also contributes to this term, which is significant only at low flow rates and increases with column length. As B is proportional to the diffusion coefficient in the mobile phase, the order of efficiency at low flow rates is liquids > heavy gases > light gases. 

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... 

One reason why small particles give better resolution is that they provide more uniform flow through the column, thereby reducing the multiple path term, A, in the van Deemter equation (23-33). A second reason is that the distance through which solute must diffuse in the mobile and stationary phases is on the order of the particle size. The smaller the particles, the less distance solute must diffuse. This effect decreases the C term in the van Deemter equation for finite equilibration time. The optimum flow rate for small particles is faster than for large particles because solutes diffuse through smaller distances. 

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