The Giddings Equation

The first alternative HETP equation to be developed was that of Giddings in 1961 [1] of which the Van Deemter equation appeared to be a special case. Giddings did not develop his equation because the Van Deemter equation did not fit experimental data. 

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It is seen that when u E, equation (1) reduces to the Van Deemter equation. 

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. 

When u E, this interstitial mixing effect was considered complete, and the resistance to mass transfer in the mobile phase between the particles becomes very small and the equation again reduces to the Van Deemter equation. However, under these circumstances, the C term in the Van Deemter equation now only describes the resistance to mass transfer in the mobile phase contained in the pores of the particles and, thus, would constitute an additional resistance to mass transfer in the stationary (static mobile) phase. It will be shown later that there is experimental evidence to support this. It is possible, and likely, that this was the rationale that explains why Van Deemter et al. did not include a resistance to mass transfer term for the mobile phase in their original form of the equation. 

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It is seen that the first term differs from the Giddings equation and now contains the mobile phase velocity to the power of one-half. However, when the first... 

A number of HETP equations were developed other than that of Van Deemter. Giddings developed an alternative form that eliminated the condition predicted by the Van Deemter equation that there was a finite dispersion at zero velocity. However, the Giddings equation reduced to the Van Deemter equation at velocities approaching the optimum velocity. Due to extra-column dispersion, the magnitude of which was originally unknown, experimental data were found not to fit the Van Deemter... 

In summary, it can be said that all the dispersion equations that have been developed will give a good fit to experimental data, but only the Van Deemter equation, the Giddings equation and the Knox equation give positive and real values for the constants in the respective equations. 

The Van Deem ter equation appears to be a special case of the Giddings equation. The form of the Van Deemter equation and, in particular, the individual functions contained in it are well substantiated by experiment. The Knox equation is obtained... 

It is seen that the first term differs from that in the Giddings equation, in that it now contains the mobile phase velocity to the power of one half. Nevertheless, again when ul/2 >> E, the first term reduces to a constant similar to the Van Deemter equation. The additional term for the resistance to mass transfer in the mobile phase is an attempt to take into account the... 

This can be examined further by considering the detailed expression for the first term of the Giddings equation. 

The expanded expression for the first term in the Giddings equation (7) is as follows ... 

Figure 6.14 Plot of the axial reduced plate height vs. the reduced mobile phase velocity, (a) Comparison of the results measured by three PFGNMR methods (PFGSE, PFGSTE and APGSTE, with S = 2 ms and A - 22 ms) and best fit of these experimental data (s3rmbols) to the Giddings equation (solid line). The individual contributions to this equation are represented by the dotted lines, (b) Best fit of a subset of these data limited to t/ < 15 to the van Deemter model and extrapolation (B = 1.35, A = 0.14, C = 0.11, n = 0). Reproduced with permission from U. Tallarek, E. Bayer, G. Guiochon, f. Am. Chem. Soc., 120 (1998) 1494 (Fig. 8). (c)1998 American Chemical Society.

Generally, if we plot the HETP against a linear velocity, we obtain a curved relationship with a minimum and a nearly linear increase of the HETP with linear velocity (Fig. 2.4) at high linear velocity. This relationship can be described by several equations the van Deemter equation, the Giddings equation, and the Knox equation. We will explore all of them in the following, starting with the van Deemter equation. 

Figure 2.8 shows an overlay of the van Deemter equation, Berdichevsky s version of the Giddings equation and the Knox equation using our coefficients. The graph shows the relationships up to a reduc velocity of 20, the typical... 

In reduced form, typical coefficients of Berdichevsky s version of the Giddings equation for a well-packed bed and small sample molecules are... 

An equation relating the height equivalent to a theoretical plate (HETP) to the linear velocity of the carrier gas for GAC was proposed by Giddings. The Giddings equation can be represented as... 

It is seen that the first term differs from the Giddings equation and now contains the mobile phase velocity to the power of one-half. However, when E, the first term reduces to a constant similar to the Van Deemter equation. The additional term for the resistance to mass transfer in the mobile phase is an attempt to take into account the turbulent mixing that takes place between the particles. Huber s equation, although not explicitly stated by the authors, implies that the mixing effect between the particles (that reduces the magnitude of the resistance to mass transfer in the mobile phase) only starts when the mobile phase velocity approaches the optimum velocity (as defined by the Van Deemter equation). In addition, the mixing effect is not complete until the mobile phase velocity is well above the optimum velocity. Thus, the shape of the HETP/u curve will be different from that predicted by the Van Deemter equation. The form of the HETP curve that is produced by the Huber equation is shown in Figure (1). 

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