Number of theoretical plates A

Figure 1.5 Influence of (a) the relative retention (o), (b) the (average) capacity factor (k) and (c) the number of theoretical plates (A/) on the resolution (Rv) according to eqn.(1.22). In each case the two other parameters are kept constant, k and N are assumed not to equal zero, and a not to equal one.

One might expect that three covalent interactions should be especially favourable for imprinting and for subsequent molecular recognition. However, it has been shown that three boronic acid interactions [2,3,58], three different covalent interactions [36], or three or more non-covalent interactions [75,76] all lead to poorer resolving power (that is, a smaller number of theoretical plates A, h) for the polymers. Here the main factor must be the low rate of formation of polymer-substrate complexes. 

In the elucidation of retention mechanisms, an advantage of using enantiomers as model templates is that non-specific binding, which affects both enantiomers equally, will cancel out. Therefore the separation factor (a) uniquely reflects the contribution to binding from the enantioselectively imprinted sites. As an additional comparison, the retention on the imprinted phase is compared with the retention on a non-imprinted reference phase. The efficiency of the separations is routinely characterised by estimating a number of theoretical plates (A), a resolution factor (Rs) and a peak asymmetry factor (As) [10]. These quantities are affected by the quality of the packing and mass transfer limitations, as well as the amount and distribution of the binding sites. 

In addition to the EMG fimction, many other mathematical models have been suggested to accoimt for the profiles of experimental peaks and to determine characteristic shape-parameters, such as a number of theoretical plates, a skew and an excess. These parameters are related to the second, third and fourth moments (Eq. 6.77) of the peak, respectively. For example, the Gram-Charlier series (GC) [96,114,115] and the Edgeworth-Cramer series (EC) [115,116] have been... 

N = number of theoretical plates a = separation factor k = peak capacity... 

Sometimes the height equivalent to a theoretical plate (HETP) is employed rather than HG and HL 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, A p. 

The efficiency of a chromatographic column is measured by the number of theoretical plates (A ) to which the column is equivalent. The term was originally used to describe the process of distillation and can be visualised as a series of hypothetical layers in which the solute concentrations in the relevant phases are assumed to be in... 

The number of theoretical plates (A). This is a measure of the efficiency of the column in carrying out a particular separation. It is important to realise that this factor is dependent upon the test system... 

The reflux ratio is defined as the ratio of the number of drops of distillate that return to the distillation flask compared to the number of drops of distillate collected. In an efficient column, the reflux ratio should equal or exceed the number of theoretical plates. A high reflux ratio ensures that the column will achieve temperature equilibrium and achieve its maximum efficiency. This ratio is not easy to determine in fact, it is impossible to determine when using a Hickman head, and it should not concern a beginning student. In some cases, the throughput, or rate of takeoff, of a column may be specified. This is expressed as the number of milliliters of distillate that can be collected per unit of time, usually as mL/min. 

The preceding equations identify three independent factors that contribute to SEC resolution (1) the number of theoretical plates (A/) (2) the difference in the distribution coefficients of adjacent peaks (AA p), which is related to calibration curve slope (m) and (3) the phase ratio or porosity or ). 

The number of theoretical plates (A ) characterizes the quality of a column the larger the value of N, the more complicated is the sample mixture that can be separated using the column. The value of N can be calculated from the following equations ... 

This equation shows that resolution is a function of three factors, namely, selectivity (a), number of theoretical plates (A), and capacity factor k ). As discussed above, selectively relates to the ability to separate zone centers (difference in Rf values), whereas the number of theoretical plates measures zone spreading throughout the chromatographic system. The capacity factor describes retention of a component by the stationary phase. In HPLC, k is stated in terms of column volumes, and values of 1 to 10 are normal. 

Resolution (RJ in a chromatographic system is a function of three essentially independent variables the capacity factor (fc ) (defined generally by the solvent strength), the number of theoretical plates (A/), and the selectivity (a) according to the well-known equation given below ... 

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... 

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

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. 

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... 

Fig. 25. McCabe-Thiele diagram, (a) Number of theoretical plates, 5 (b) number of actual plates, 8.

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 ... 

Favorable Vapoi Liquid Equilibria. The suitabiHty of distiUation as a separation method is strongly dependent on favorable vapor—Hquid equiHbria. The absolute value of the key relative volatiHties direcdy determines the ease and economics of a distillation. The energy requirements and the number of plates required for any given separation increase rapidly as the relative volatiHty becomes lower and approaches unity. For example given an ideal binary mixture having a 50 mol % feed and a distillate and bottoms requirement of 99.8% purity each, the minimum reflux and minimum number of theoretical plates for assumed relative volatiHties of 1.1,1.5, and 4, are... 

In the example, the minimum reflux ratio and minimum number of theoretical plates decreased 14- to 33-fold, respectively, when the relative volatiHty increased from 1.1 to 4. Other distillation systems would have different specific reflux ratios and numbers of theoretical plates, but the trend would be the same. As the relative volatiHty approaches unity, distillation separations rapidly become more cosdy in terms of both capital and operating costs. The relative volatiHty can sometimes be improved through the use of an extraneous solvent that modifies the VLE. Binary azeotropic systems are impossible to separate into pure components in a single column, but the azeotrope can often be broken by an extraneous entrainer (see Distillation, A7EOTROPTC AND EXTRACTIVE). 

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. 

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