Theoretical plate equation

When N is calculated from a chromatogram, L and W must be converted to time-based units, terms cancel, and the theoretical plate equation is simply... 

Because the squares of the variances are additive, the contributions to band broadening from injection and diffusion can be inserted into the theoretical plate equation ... 

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. 

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

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... 

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. 

An equation showing the effect of the mobile phase s flow rate on the height of a theoretical plate. 

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... 

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 ... 

Solution of the model equations shows that, for a linear isothermal system and a pulse injection, the height equivalent to a theoretical plate (HETP) is given by... 

Equivalent Theoretical Plate. The separative capacity of a theoretical plate in a continuous process can be obtained in the same manner. By equating the separative capacity of the unit to the net increase in value of the four streams handled (eq. 8) ... 

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. 

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 ... 

Equation (5) was examined by Scott and Reese [1] employing mixtures of nitrobenzene and fully deuterated nitrobenzene as the test sample. Their retention times were 8.927 min. and 9.061 min., respectively, giving a difference of 8.04 seconds. The separation ratio of the two solutes was 1.023 and the efficiencies of the front and rear portions of the peaks were 5908 and 3670 theoretical plates, respectively. The detector was, not surprisingly, found to have the same response to both solutes, i.e., a = (3. Thus, inserting these values in equation (5),... 

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 efficiency of a column (n), in number of theoretical plates, has been shown to be given by the following equation,... 

Equation (18) displays the relationship between the column efficiency defined in theoretical plates and the column efficiency given in effective plates. It is clear that the number of effective plates in a column is not aii arbitrary measure of the column performance, but is directly related to the column efficiency as derived from the plate theory. Equation (18) clearly demonstrates that, as the capacity ratio (k ) becomes large, (n) and (Ne) will converge to the same value. 

Equation (24) shows that when the charge is placed on the first plate, (Xn) can never equal zero and pure mobile phase free of solute will never elute from the column. However, in practice, it is almost impossible to place the sample exclusively on the first plate, and there will be a finite volume of mobile phase that will occupy a finite number of theoretical plates when it is injected onto the column. 

The sum expressed by equation (25) also lends itself to a digital solution and can be employed in an appropriate computer program to calculate actual peak profiles for different volumes of pure mobile phase that have been injected onto an equilibrated column. The values of (Xg) were calculated for a column having 500 theoretical plates and for sample volumes of 20, 50, 100 and 200 plate volumes, respectively. The curves relating solute concentration (Xe) to plate volumes of mobile phase passed through the column are shown in Figure 17. 

Equation (33) shows that the maximum capacity ratio of the last eluted solute is inversely proportional to the detector sensitivity or minimum detectable concentration. Consequently, it is the detector sensitivity that determines the maximum peak capacity attainable from the column. Using equation (33), the peak capacity was calculated for three different detector sensitivities for a column having an efficiency of 10,000 theoretical plates, a dead volume of 6.7 ml and a sample concentration of l%v/v. The results are shown in Table 1, and it is seen that the limiting peak capacity is fairly large. 

In order to examine the thermal changes that take place in a column, it is necessary to derive an equation that describes the temperature change in a theoretical plate, in terms of its physical properties of the plate and the volume flow of mobile phase that passes through it. 

Equation (54) is an explicit expression that defines the temperature change of the detector in terms of the initial concentration of the solute placed on the column and the volume of mobile phase that passes through it. It can be used, with the aid of a computer, to synthesize the different shaped curves that the detector can produce. Employing a computer in the manner of Smuts et al. [23], Scott [24] calculated the relative values of (0) for (v= 74 to 160) with a column of 100 theoretical plates, and for (Ca) ranging from 0.25 to 4 and (4>) ranging from 0.01 to 1.25. The curves are shown in Figure 24. 

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. 

The next equation of importance is the relationship between the column length (L) and the height of the theoretical plate (H),... 

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