Plate, theoretical

In the case of a plate column the performance of a real plate is related to the performance of a theoretical one by the plate efficiency. In the case of a packed column the height equivalent to a theoretical plate HETP) gives a measure of the contacting efficiency of the packing. 

This is the ASTM D 2892 test method and corresponds to a laboratory technique defined for a distillation column having 15 to 18 theoretical plates and operating with a 5 1 reflux ratio. The test is commonly known as the TBP for True Boiling Point. 

The sample is distilled at predetermined and precisely controlled temperatures under conditions that give a fractionation equivalent to about one theoretical plate. 

Used in virtually all organic chemistry analytical laboratories, gas chromatography has a powerful separation capacity. Using distillation as an analogy, the number of theoretical plates would vary from 100 for packed columns to 10 for 100-meter capillary columns as shown in Figure 2.1. 

D 2892 Petroleum distillation method employing a 15 theoretical plate... 

Otherwise expressed, the number of theoretical plates required for a given separation increases when the reflux ratio is decreased, i.e., when the amount of condensed vapour returned to the colunm is decreased and the amount distilled off becomes greater. 

Three separate factors affect resolution (1) a column selectivity factor that varies with a, (2) a capacity factor that varies with k (taken usually as fej). and (3) an efficiency factor that depends on the theoretical plate number. 

In their original theoretical model of chromatography, Martin and Synge treated the chromatographic column as though it consists of discrete sections at which partitioning of the solute between the stationary and mobile phases occurs. They called each section a theoretical plate and defined column efficiency in terms of the number of theoretical plates, N, or the height of a theoretical plate, H where... 

A column s efficiency improves with an increase in the number of theoretical plates or a decrease in the height of a theoretical plate. 

Assuming a Gaussian profile, the extent of band broadening is measured by the variance or standard deviation of a chromatographic peak. The height of a theoretical plate is defined as the variance per unit length of the column... 

Gombining equations 12.13 through 12.15 gives the height of a theoretical plate in terms of the easily measured chromatographic parameters and w. 

The number of theoretical plates in a chromatographic column is obtained by combining equations 12.12 and 12.16. 

A chromatographic analysis for the chlorinated pesticide Dieldrin gives a peak with a retention time of 8.68 min and a baseline width of 0.29 min. How many theoretical plates are involved in this separation Given that the column used in this analysis is 2.0 meters long, what is the height of a theoretical plate ... 

Solving equation 12.12 for H gives the average height of a theoretical plate as... 

It is important to remember that a theoretical plate is an artificial construct and that no such plates exist in a chromatographic column. In fact, the number of theoretical plates depends on both the properties of the column and the solute. As a result, the number of theoretical plates for a column is not fixed and may vary from solute to solute. 

Equations 12.21 and 12.22 contain terms corresponding to column efficiency, column selectivity, and capacity factor. These terms can be varied, more or less independently, to obtain the desired resolution and analysis time for a pair of solutes. The first term, which is a function of the number of theoretical plates or the height of a theoretical plate, accounts for the effect of column efficiency. The second term is a function of a and accounts for the influence of column selectivity. Finally, the third term in both equations is a function of b, and accounts for the effect of solute B s capacity factor. Manipulating these parameters to improve resolution is the subject of the remainder of this section. 

If the capacity factor and a are known, then equation 12.21 can be used to calculate the number of theoretical plates needed to achieve a desired resolution (Table 12.1). For example, given a = 1.05 and kg = 2.0, a resolution of 1.25 requires approximately 24,800 theoretical plates. If the column only provides 12,400 plates, half of what is needed, then the separation is not possible. How can the number of theoretical plates be doubled The easiest way is to double the length of the column however, this also requires a doubling of the analysis time. A more desirable approach is to cut the height of a theoretical plate in half, providing the desired resolution without changing the analysis time. Even better, if H can be decreased by more than... 

Number of Theoretical Plates Needed to Achieve Desired Resolution for Selected Values of ks and a... 

To determine how the height of a theoretical plate can be decreased, it is necessary to understand the experimental factors contributing to the broadening of a solute s chromatographic band. Several theoretical treatments of band broadening have been proposed. We will consider one approach in which the height of a theoretical plate is determined by four contributions multiple paths, longitudinal diffusion, mass transfer in the stationary phase, and mass transfer in the mobile phase. 

Putting It All Together The net height of a theoretical plate is a summation of the contributions from each of the terms in equations 12.23-12.26 thus. 

Plot of the height of a theoretical plate as a function of mobile-phase velocity using the van Deemter equation. The contributions to the terms A B/u, and Cu also are shown. 

To increase the number of theoretical plates without increasing the length of the column, it is necessary to decrease one or more of the terms in equation 12.27 or equation 12.28. The easiest way to accomplish this is by adjusting the velocity of the mobile phase. At a low mobile-phase velocity, column efficiency is limited by longitudinal diffusion, whereas at higher velocities efficiency is limited by the two mass transfer terms. As shown in Figure 12.15 (which is interpreted in terms of equation 12.28), the optimum mobile-phase velocity corresponds to a minimum in a plot of H as a function of u. 

To minimize the multiple path and mass transfer contributions to plate height (equations 12.23 and 12.26), the packing material should be of as small a diameter as is practical and loaded with a thin film of stationary phase (equation 12.25). Compared with capillary columns, which are discussed in the next section, packed columns can handle larger amounts of sample. Samples of 0.1-10 )J,L are routinely analyzed with a packed column. Column efficiencies are typically several hundred to 2000 plates/m, providing columns with 3000-10,000 theoretical plates. Assuming Wiax/Wiin is approximately 50, a packed column with 10,000 theoretical plates has a peak capacity (equation 12.18) of... 

Microcolumns use less solvent and, because the sample is diluted to a lesser extent, produce larger signals at the detector. These columns are made from fused silica capillaries with internal diameters of 44—200 pm and lengths of up to several meters. Microcolumns packed with 3-5-pm particles have been prepared with column efficiencies of up to 250,000 theoretical plates. 

Open tubular microcolumns also have been developed, with internal diameters of 1-50 pm and lengths of approximately 1 m. These columns, which contain no packing material, may be capable of obtaining column efficiencies of up to 1 million theoretical plates.The development of open tubular columns, however, has been limited by the difficulty of preparing columns with internal diameters less than 10 pm. 

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