The number of theoretical plates

A linear value (e.g. in mm) for plate height Hs may be calculated from Equation 2.18. It should be understood, however, that it is in reality a small volume dV in which a distribution equilibrium of the substance molecules is established between the stationary and the mobile phases. 

If the entire column length L is considered as being divided into sections of the length H5 (in the following referred to simply as H), we obtain the theoretical plate number N  

The plate number N may be derived from the chromatogram if the total (uncorrected) retention time t s is related to the base width W5. 

In order to determine N it is merely necessary to measure the total uncorrected retention time tf s of Iho peak and its base width, both, of course, in the same unit of measurement. Since it is frequently not possible to determine the precise position of the inflectional tangents which delimit Wg on the base line, the relationship between Wq.5 and W5 is often used and N is calculated thus  

The higher the plate number N in a separation system, the greater is the separation efficiency. In Equations 2.20 and 2.21, the total retention time t s still includes the dead time, which makes no contribution towards separation. 

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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. 

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. 

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. 

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

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... 

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. 

Efficiency The efficiency of capillary electrophoresis is characterized by the number of theoretical plates, N, just as it is in GC or ITPLC. In capillary electrophoresis, the number of theoretic plates is determined by... 

Sometimes the height equivalent to a theoretical plate (HETP) is employed rather than and to characterize the performance of packed towers. The number of heights equivalent to one theoretical plate required for a specified absorption job is equal to the number of theoretical plates,... 

The required number of actual plates, A/p, is larger than the number of theoretical plates, because it would take an infinite contacting time at each stage to estabhsh equihbrium. The ratio is called the overall column efficiency. This parameter is difficult to predict from theoretical... 

For linear equiHbrium and operating lines, an expHcit expression for the number of theoretical plates required for reducing the solute mole fraction... 

This is the one case where the overall column efficiency can be related analytically to the Murphree plate efficiency, so that the actual number of plates is calculable by dividing the number of theoretical plates through equation 86 ... 

The simplest efficiency is the overaH column efficiency which is the number of theoretical plates in a column divided by the number of actual plates ... 

Example This equation is obtained in distillation problems, among others, in which the number of theoretical plates is required. If the relative volatility is assumed to be constant, the plates are theoretically perfect, and the molal liquid and vapor rates are constant, then a material balance around the nth plate of the enriching section yields a Riccati difference equation. 

The design of a plate tower for gas-absorption or gas-stripping operations involves many of the same principles employed in distillation calculations, such as the determination of the number of theoretical plates needed to achieve a specified composition change (see Sec. 13). Distillation differs from gas absorption in that it involves the separation of components based on the distribution of the various substances between a gas phase and a hquid phase when all the components are present in Doth phases. In distillation, the new phase is generated From the original feed mixture by vaporization or condensation of the volatile components, and the separation is achieved by introducing reflux to the top of the tower. 

Although Eq. (14-31) is convenient for computing the composition of the exit gas as a function of the number of theoretical stages, an alternative equation derived by Colburn [Tran.s. Am. Jn.st. Chem. Eng., 35, 211 (1939)] is more useful when the number of theoretical plates is the unknown ... 

The left-hand side of Eq. (14-55) represents the efficiency of absorption of arw one component of the feed-gas mixture. If the solvent oil is denuded of solute so that Xo = 0, the left-hand side is equal to the fractional absorption of the component from the rich feed gas. When the number of theoretical plates N and the hquid and gas rates L i and G, f have been fixed, the uractional absorption of each component may be computed directly and the operating lines need not be placed by trial and error as in the graphic approach described earlier. 

Thus, the variance of the peak is inversely proportional to the number of theoretical plates in the column. Consequently, the greater the value of (n), the more narrow the peak, and the more efficiently has the column constrained peak dispersion. As a result, the number of theoretical plates in a column has been given the term Column Efficiency. From the above equations, a fairly simple procedure for measuring the efficiency of any column can be derived. 

The separation of the critical pair would require a minimum column efficiency and, so, the number of theoretical plated produced by the column must be reported. The... 

Knowing the number of theoretical plates that are required, the length of the column (L) is defined as the product of the number of plates and the variance per unit length of the column (H), i.e.. 

Starting with the same basic equation of Purnell (chapter 6) which is applicable to all forms of chromatography, and allows the number of theoretical plates required to separate the critical pair of solutes to be calculated. 

Now, the column length (L) can be defined as the product of the minimum plate height and the number of theoretical plates required to complete the separation as specified by the Purnell equation. 

The alternative expression for resolution given in equation (7) demonstrates that the plate resolution, as in other forms of chromatography, depends on the number of theoretical plates, the selectivity and the capacity ratio of the solute for the particular plate concerned. In practice, however, the expression given in equation (7) appears to be the more practically useful for TLC. separations. 

The plate to plate type calculation is a fundamental procedure wherein the tower is assumed to be composed of theoretical equilibrium plates. The actual plates required are determined from the number of theoretical plates using a predicted overall tower efficiency. The starting point for a tower calculation is usually a specified feed composition, feed temperature, and tower operating pressure. The procedure involves defining the compositions and temperamres on each plate in the tower and subsequently the resultant compositions and temperatures of the product streams. The actual computations, which involve trial... 

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