Plate height relationships

There is some disagreement on the correct equation for describing the relationship between plate height and mobile-phase velocity. In addition to the van Deemter equation (equation 12.28), another equation is that proposed by Hawkes... 

The column performance (efficiency) is measured either in terms of the plate height (H), the efficiency of the column per unit length, or the plate number (N), i.e. the nnmber of plates for the column. This number depends upon the column length (L), whereas the plate height does not. The mathematical relationships between the nnmber of plates, the retention time of the analyte and the width of the response is shown in the following equations ... 

Figure 7.5 Left, variation of the average plate height of fine-and coeurse-particle layers as a function of the solvent aigration distance and method of developaent. Right, relationship between the optiauB plate height and solvent migration distance for forced-flow development.

The plate height, and thus the total number of theoretical or effective plates, depends on the average linear carrier gas velocity (van Deemter relationship) and, for a particular carrier gas, the efficiency will maximize at a particular flow rate. Only at the optimum carrier gas flow rate are n, N, and HETP Independent of the column length. The efficiency will also depend on the column diameter (see section 1.7.1) where typical values for n, N, and HETP for different column types can also be found. Values for n, N, and HETP are reasonably independent of temperature but may vary with the substance used for their determination, particularly if the test substance and statioKary phase are not compatible. 

The diffusivities thus obtained are necessarily effective diffusivities since (1) they reflect a migration contribution that is not always negligible and (2) they contain the effect of variable properties in the diffusion layer that are neglected in the well-known solutions to constant-property equations. It has been shown, however, that the limiting current at a rotating disk in the laminar range is still proportional to the square root of the rotation rate if the variation of physical properties in the diffusion layer is accounted for (D3e, H8). Similar invariant relationships hold for the laminar diffusion layer at a flat plate in forced convection (D4), in which case the mass-transfer rate is proportional to the square root of velocity, and in free convection at a vertical plate (Dl), where it is proportional to the three-fourths power of plate height. 

FIGURE 12.2 Van Deemter plot illustrating evolution of particle sizes and resulting changes of relationship of plate height and linear velocity. Source From Swartz, M., J. Liq. Chromatogr. Rel. Technol., 2005, 28, 1253. With permission from Taylor Francis Group.)... 

This equation indicates that the particle size, dp, is the main contributor to the H value. The smaller the particles, the higher the theoretical plate number. The optimum condition is obtained by the relationship between the theoretical plate height and the flow velocity. 

H is the plate height (cm) u is linear velocity (cm/s) dp is particle diameter, and >ni is the diffusion coefficient of analyte (cm /s). By combining the relationships between retention time, U, and retention factor, k tt = to(l + k), the definition of dead time, to, to = L u where L is the length of the column, and H = LIN where N is chromatographic efficiency with Equations 9.2 and 9.3, a relationship (Equation 9.4) for retention time, tt, in terms of diffusion coefficient, efficiency, particle size, and reduced variables (h and v) and retention factor results. Equation 9.4 illustrates that mobile phases with large diffusion coefficients are preferred if short retention times are desired. 

As in gas chromatography, there is a relationship between the reduced velocity of the mobile phase and the reduced plate height. It is the Knox equation (4) ... 

The van Deemter equation is an empirical formula that describes the relationship between linear velocity (flow rate) and plate height (or column efficiency) (van Deemter et al., 1956). The particle size is one of the variables used in the van Deemter equation. As illustrated in Fig. 4.1, as the particle size decreases to less than 2.5 p,m, the efficiency is increased. Furthermore, when using smaller size particles, the efficiency is not affected with increasing flow rates or linear velocities. 

As stated previously, additional information on the sample is obtainable from band broadening (plate height) measurements. If plate height is measured as a function of flow velocity at a fixed retention level, a linear relationship is obtained, as predicted by Equation 7. The slope of the line yields the diffusivity D of the sample, and the intercept provides the polydispersity a. . The D value translates into a value for the average particle diameter d via the Stokes-Einstein relationship... 

Here we review the basic rules governing random walks and their relationship to plate height. First, the concentration (probability) profile generated by a random walk is a Gaussian with a variance [Pg.254]

The optimization procedure outlined above is independent of the specific plate height equations chosen to represent chromatography. It depends only on the validity of the scaling relationships used to define reduced plate height and reduced velocity, and the emergence of universal curves from the scaling relationships. 

Liquid Distribution on Gas Chromatographic Support Relationship to Plate Height, J. C. Giddings, Anal. Chem., 34, 458 (1962). 

These results demonstrated that cartridge performance in trace enrichment is flow-rate dependent and, therefore, is a variable that should be controlled (remember the relationship of plate height versus linear velocity). 

Although the B and C terms exhibit opposite relationships with analyte diffusion, the C-term relationship is mainly of interest because resistance to mass transfer is the dominant form of band-spreading at the faster velocities that are desired. Equations (17-9) and (17-10) imply that speeding up diffusion will increase mass transfer and help decrease plate height. The Wilke-Chang equation [9] shows that diffusivity is directly proportional to temperature and inversely proportional to viscosity ... 

The plate height depends on various experimental conditions. The most simple expression describing the relationship between H and the velocity of the mobile phase, u, is the well-known van Deemter equation (21 ... 

Packed columns. Much has been written on the relationships between linear velocity (or flow rate) and column efficiency. The relationship of most practical utility is the Knox equation (1977) (equation (2.32)), which describes the relationship between the reduced plate height (A) and the reduced velocity (v) ... 

Figure 2.4 Relationship between reduced plate height (ft) and reduced velocity (v) according to the Knox equation (ft = + Bjv + Cv) (equation 2.32)). (a) Optimum values for the...

The kinetic contributions to zone broadening are evaluated by fitting data for the column plate height, as a function of the mobile-phase velocity, to a mathematical model describing the relationship between the two parameters. Several models have been used in the above experiment, but those by de Ligny and Remijnsee and Knox and Pryde, and developed by Guiochon and Siouffi are most widely used and, at least for a first approximation, allow for comparison and determination of the differences between TLC and column chromatography... 

L/N) of this relationship, which is called the plate height (H) or the height equivalent of a theoretical plate (HETP). Thus ... 

For GC the relationship between the plate height and flow velocity is given by the van Deemter equation ... 

Experimental Determination of the Number of Plates in a Column The number of theoretical plates, N, and the plate height, H, are widely used in the literature and by instrument manufacturers as measures of column performance. Figure 30-12 shows how N can be determined from a chromatogram. Here, the retention time of a peak t and the width of the peak at its base W (in units of time) are measured. It can be shown (see Feature 30-4) that the number of plates can then be computed by the simple relationship... 

Over the past 40 years, an enormous amount of theoretical and experimental effort has been devoted to developing quantitative relationships describing the effects of the experimental variables listed in Table 30-5 on plate heights for vai ious types of... 

As shown by Equation 30-27, the contribution of longitudinal diffusion to plate height is inversely proportional to the linear velocity of the eluent. Such a relationship is not surprising, inasmuch as the analyte is in the column for a briefer period when the flow rate is high. Thus, diffusion from the center of the band to the two edges has less time to occur. 

Giddings [29], Huber [49], and Horvath and Lin [50] have used alternate models to accoimt for the relationships between the rate of variation of the solute concentration in the stationary phase, its mobile phase concentration, and the various parameters characterizing the chromatographic system used. This explains the differences in the plate height equations they derived, as we see in the next section. 

Frey et al. [61] compared the plate height equations under linear conditions for chromatography with traditional and with perfusion columns. In the latter case, convection takes place in large macropores inside the packing particles. For a column packed with a material having a uniform porosity, in the pores of which there is no convection, the reduced plate height is expressed by the following relationship [29-31]... 

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