Resistance to Mass Transfer in the Stationary Phase

At the start, molecules 1 and 2, the two closest to the surface, will enter the mobile phase and begin moving along the column. This will continue while molecules 3 and 4 diffuse to the interface at which time they will enter the mobile phase and start following molecules 1 and 2. All four molecules will continue their journey while molecules 5 and 6 diffuse to the mobile phase/stationary phase interface. By the time molecules 5 and 6 enter the mobile phase, the other four molecules will have been smeared along the column and the original 6 molecules will have suffered dispersion. 

Marcel Dekker, Inc. 270 Madison Avenue, New York, New York 10016 

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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 caused by the resistance to mass transfer in the stationary phase is exactly analogous to that in the mobile phase. Solute molecules close to the surface will leave the stationary phase and enter the mobile phase before those that have diffused further into the stationary phase and have a longer distance to diffuse back to the surface. Thus, as those molecules that were close to the surface will be swept along in the moving phase, they will be dispersed from those molecules still diffusing to the surface. The dispersion resulting from the resistance to mass transfer in the stationary phase is depicted in Figure 8. 

Figure 6. Resistance to Mass Transfer in the Stationary Phase...

The composite curve from the Huber equation is similar to that obtained from that of Van Deemter but the individual contributions to the overall variance are different. The contributions from the resistance to mass transfer in the stationary phase and... 

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... 

Substituting for (h ) the expression for the resistance to mass transfer in the stationary phase,... 

It is seen that equations (13) and (15) are very similar to equation (10) except that the velocity used is the outlet velocity and not the average velocity and that the diffusivity of the solute in the gas phase is taken as that measured at the outlet pressure of the column (atmospheric). It is also seen from equation (14) that the resistance to mass transfer in the stationary phase is now a function of the inlet-outlet pressure ratio (y). 

It follows that, for all solutes eluted at a (k") value of unity or more, the resistance to mass transfer in the stationary phase can be ignored and equations (28) and (29) reduce to... 

It should be pointed out, however, that the diffusivity of the solute in the mobile phase can be changed in two ways. The solute that is chromatographed can be changed, in which case the above assumptions are clearly valid, as (Ds) is likely to change linearly with (Dm)- However, the solute diffusivity can also be changed by the employing a different mobile phase. In this case, (Dm) will be changed but (Ds) will remain the same. In the second case, the above assumptions are not likely to be precisely correct. Nevertheless, if the resistance to mass transfer in the stationary phase makes only a small contribution to the overall value of (H) (i.e., because df dp (see equation (l)),then the assumption Dm = eDg will still be approximately... 

In Figure 7, the resistance to mass transfer term (the (C) term from the Van Deemter curve fit) is plotted against the reciprocal of the diffusivity for both solutes. It is seen that the expected linear curves are realized and there is a small, but significant, intercept for both solutes. This shows that there is a small but, nevertheless, significant contribution from the resistance to mass transfer in the stationary phase for these two particular solvent/stationary phase/solute systems. Overall, however, all the results in Figures 5, 6 and 7 support the Van Deemter equation extremely well. 

Thus as (y) will always be greater than unity, the resistance to mass transfer term in the mobile phase will be, at a minimum, about forty times greater than that in the stationary phase. Consequently, the contribution from the resistance to mass transfer in the stationary phase to the overall variance per unit length of the column, relative to that in the mobile phase, can be ignored. It is now possible to obtain a new expression for the optimum particle diameter (dp(opt)) by eliminating the resistance to mass transfer function for the liquid phase from equation (14). 

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... 

On page 6, it was shown that in the front half of the peak, there will be a net transfer of solute from the mobile phase to the stationary phase and thus the resistance to mass transfer in the mobile phase will dominate. At the rear half of the peak there is a net transfer of solute from the stationary phase to the mobile phase and in this case the resistance to mass transfer in the stationary phase will dominate. Then if the resistance to mass transfer in the stationary phase is greater than that for the mobile phase, the rear part of the peak will be broader than the front half. In which case,... 

The possibility of obtaining significant improvements in performance by using semi-packed and open tubular columns is clearly illustrated by the values for the separation impedance in Table 1.17. Variation of the reduced plate height with the reduced velocity for an open tubular column is given by equation (1.82), assuming that the resistance to mass transfer in the stationary phase can be neglected... 

In figure (5) the values of (Hmin.) are plotted against solute diffusivity and it is seen that the independence of (Hmin.) on diffusivity is largely confirmed. Close examination, however,shows that neither of the lines for the two solutes are completely horizontal with the baseline, but the dependance of (Hmin) on diffusivity Is extremely small for the solute benzyl acetate. The slight slope of the line for the solute hexamethylbenzene might well result from the fact that either the (A) term is not completely independent of the diffusivity (Dm) as shown by the results in figure 3, or the resistance to mass transfer in the stationary phase does make a small but significant contribution to the the value of (H). 

Returning to equation (1), it is seen that the Van Deemter equation predicts that the total resistance to mass transfer term must also be linearly related to the reciprocal of the solute diffusivity. Furthermore, it is seen from equation(l), that, if there is a significant contribution from the resistance to mass transfer in the stationary phase, the curves will show a positive intercept. 

Figure 8 shows the predicted linear relationship between the resistance to mass transfer term and the square of the particle diameter. The linear correlation is extremely good and it is seen that there is, indeed an intercept on the (C) term axis, at zero particle diameter, which confirms the existence of a small, but significant, contribution from the resistance to mass transfer in the stationary phase. 

The plate theory assumes that an instantaneous equilibrium is set up for the solute between the stationary and mobile phases, and it does not consider the effects of diffusional effects on column performance. The rate theory avoids the assumption of an instantaneous equilibrium and addresses the diffusional factors that contribute to band broadening in the column, namely, eddy diffusion, longitudinal diffusion, and resistance to mass transfer in the stationary phase and the mobile phase. The experimental conditions required to obtain the most efficient system can be determined by constructing a van Deemter plot. 

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