Plate height overall

Although the above listing of contributions to the colunn plate height is not coq>rehensive, it encompasses the major bandbroadening factors and the overall plate height can be expressed as their sun, equation (1.30). 

To better understand the heightened resolving power of 2D systems, we need some measure by which 2D and ID separations can be compared. Not all criteria of separation power lend themselves to ready comparison. The resolution of a specific pair is not a suitable criterion because this resolution varies widely for different separation mechanisms irrespective of ID or 2D configurations. Plate height and plate number are not directly comparable because these are defined only for a single dimension. While H or N values can be found for each axis, it is not immediately obvious how to combine them for both axes in order to compare the overall separation effectiveness with that of a ID system. 

We see, then, two distinct kinds of plate height terms for the mobile phase a term HD proportional to flow velocity v, valid when diffusion terminates a molecule s velocity bias, and a velocity-independent term Hfy valid when flow terminates the bias. The question yet unanswered is how HD and Hf combine in contributing to the overall experimental plate height. Plate heights are usually additive because variances (for independent processes) are additive it is tempting to apply the additive rule here. However, additivity does not apply to HD and Hf. We see this in simple physical terms by looking at two extremes of flowrate. 

The coupling equations shown above are applicable to one kind of velocity bias, such as that between adjacent channels. However, the theory is more or less the same for the other kinds of velocity biases listed previously, but the constants (a>fi9 a>A, (oa) are different, yielding different a> and A in the foregoing coupling expressions. Each kind of velocity bias generates its own independent random walk, leading to an additive plate height term H,. The overall mobile phase term is therefore the sum... 

The last two equations provide a good summary of how the various plate height terms are assembled into an overall plate height equation for chromatography. Whether or not all the constants (y, qy A,) of the equations are known, these expressions show the nature of the dependence of H on the controllable system parameters vy dp, d, Ry Dmy and Ds. The expressions consequently have profound consequences for guiding practical chromatography. These consequences will be discussed in the next chapter. 

When the terms contributing to band broadening are collected and equilibrium spreading is assumed to be negligible, the following general expression for overall plate height is obtained ... 

In this equation the first factor is called the eddy diffusion, the second and third are molecular diffusion, and the last two are called resistance-to-mass-transfer terms. All the terms include the mobile-fluid velocity as a variable that is proportional to the flow rate in some, and inversely proportional in others. The overall relation between plate height and flow velocity of the mobile phase is the statistical resultant of the five terms and is usually depicted in the form of a Van Deemter plot. ° Such a diagram shows that an optimum flow velocity for minimum band spreading exists for a given chromatographic column. 

FIGURE 24-2 Comparison of curves for plate height against velocity for the individual terms (upper) and for the overall value Oower) of Equation (24-14). Left, the values for a liquid mobile phase right, values for a gaseous mobile phase (liquid stationary phases in both cases). The numbered curves represent eddy diffusion (1), molecular diffusion in the mobile phase (2), and resistance to mass transfer in the stationary (4) and mobile (5) phases. The contribution of the term for molecular diffusion in the stationary phase is negligible at velocities near the optimum for both liquid and gas systems. 

EXAMPLE 24-3 Clalculate and plot the curves for plate height against velocity for each of the terms and the overall h in (24-14), using the following values for the lO- cm ... 

The influence of the different mass transfer parameters on the overall efficiency of a column is shown in Fig. 2.11, where the efficiency represented by the plate height is plotted versus the mobile phase velocity. 

More precise relationships for /Jdisp were discussed earlier, in the previous section (see Eqs. 6.91a, 6.91b, and 6.91f). A and B in the equation above are characteristic of the packing material and Pe — udp/Dm is the particle Peclet niunber, with u the interstitial velocity, hi the chromatographic literature, the particle Peclet number is frequently named the reduced velocity. Typical values of A and B in a well-packed column are 1.5 and 1.6, respectively [56]. For such a column, used at a moderate to high Peclet munber, hdisp is a small contribution to the overall reduced plate height of the column. 

In order to optimize for a minimum plate height it is convenient to write the individual contributing components of the overall plate height Equation 7.6 in terms of those physical quantities which are most easily controlled e g. dp, dp, DM and Ds, thus ... 

Contributions from longitudinal diffusion are greatest as the flow rate approaches zero at which point, of course, any separation would cease and the band would merely spread out gradually. Eddy diffusion (A) has no flow-related component but does contribute an absolute limit on the how small the overall plate height can be. Finally, the mass transfer term (C w) contributes increasing amounts of bandbroadening as the mobile phase flowrate increases. 

This effect of Eddy diffusion on the overall plate height can be expressed as the A term... 

The overall effect of these factors can be combined in the van Deemter Equation, which relates plate height (column efficiency) to flow rate (p). 

There is also an analogous contribution to the overall plate height due to diffusive spreading in the stationary phase and using similar reasoning as above this may be expressed as... 

The factors affecting the contribution of this effect to the overall plate height are as those experienced for the mobile phase mass transfer term. Thus, the mobile phase mass transfer term may be expressed as... 

Giddings [3,4] has shown that the overall plate height arising from these kinetic factors can be expressed mathematically as... 

It can be shown that the contribution to the overall plate height from these extra column effects is a function of a number of instrumental parameters, i.e. 

Figure 21.6. Plot illustrating the flow-velocity dependence of overall plate height H and various contributions to H (Eqn. 21.8).

The overall plate height arising from the above factors is described by the Van Deemter equation as a function of carrier-gas velocity v. 

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