Theoretical plate numbers, function

In CE, these two separation parameters, theoretical plate number and resolution, are functions of both the electrophoretic mobility of the analytes and EOF mobility. N is increased with increases in electrophoretic mobility and applied potential, but it decreases with an increase in the diffusion coefficient. R in turn increases with electrophoretic mobility and applied voltage but decreases with diffusion coefficient. In general, both efficiency and resolution are higher at higher voltages and in the presence of substances having small diffusion coefficients. 

Rel. (17a) indicates that n is proportional to the column length L and inversely proportional to H. The theoretical plate number n can be expressed as a function of length using a simple substitution of rel. (16) in rel. (17a) ... 

From the input values of F, the theoretical plate number, N is calculated by the relationship- approximately modified for thecolumn used (experimentally determined), though generally N is a function of linear velocity of the mobile phase. 

According to the plate theory, a concentration profile ( a band) of a solute at the outlet of the column is assumed to be the Gaussian distribution, if the column has the large number of plates. Consequently, the standard deviation, o, of the Gaussian distribution function can be expressed as 0 = V[j/ N. As the peak width at the base-line, obtained by drawing of tangents, is equal to 4o, the theoretical plate number is defined 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. 

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. 

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 hot feed enters the fractionator, which normally contains 30-50 fractionation trays. Steam is introduced at the bottom of the fractionator to strip off light components. The efficiency of separation is a function of the number of theoretical plates of the fractionating tower and the reflux ratio. Reflux is provided by condensing part of the tower overhead vapors. Reflux ratio is the ratio of vapors condensing back to the still to vapors condensing out of the still (distillate). The higher the reflux ratio, the better the separation of the mixture. 

Resolution The degree to which two peaks are separated. This is a function of the number of theoretical plates, N, in a column and the separation factor between the two components. 

Gilliland, E. R. (1940) Ind. Eng. Chem. 32, 1220. Multicomponent rectification, estimation of the number of theoretical plates as a function of the reflux ratio. 

Gilliland E.R (1940) Multicomponent Rectification - Estimation of the Number of Theoretical Plates as a Function of the Reflux Ratio, Ind Eng Chem, 32 1220. 

Correlate the degree of separation obtained as a function of column Peclet number and compare this to the number of theoretical plates required in simulation example CHROMPLATE for a similar degree of separation obtained. 

Solute equilibrium between the mobile and stationary phases is never achieved in the chromatographic column except possibly (as Giddings points out) at the maximum of a peak (1). As stated before, to circumvent this non equilibrium condition and allow a simple mathematical treatment of the chromatographic process, Martin and Synge (2) borrowed the plate concept from distillation theory and considered the column consisted of a series of theoretical plates in which equilibrium could be assumed to occur. In fact each plate represented a dwell time for the solute to achieve equilibrium at that point in the column and the process of distribution could be considered as incremental. It has been shown that employing this concept an equation for the elution curve can be easily obtained and, from that basic equation, others can be developed that describe the various properties of a chromatogram. Such equations will permit the calculation of efficiency, the calculation of the number of theoretical plates required to achieve a specific separation and among many applications, elucidate the function of the heat of absorption detector. 

Thus it can be seen that the observed HETP is not only a function of column packing but depends upon operating conditions and the properties of the solute. This is why different values of HETP (or different numbers of theoretical plates per unit column length) are obtained for various solutes. 

The efficiency of a column is a number that describes peak broadening as a function of retention, and it is described in terms of the number of theoretical plates, N. Two major theories have been developed to describe column efficiency, both of which are used in modern chromatography. The plate theory, proposed by Martin and Synge,31 provides a simple and convenient way to measure column performance and efficiency, whereas the rate theory developed by van Deemter et al.32 provides a means to measure the contributions to band broadening and thereby optimize the efficiency. 

Although the value of Rs for a given pair of peaks can be quickly transferred from one column to another by using the proportionality of Rx and Vn, this is not the case for the threshold criterion of eqn.(4.23). The problem is that if we know the boundaries of the area for which Rs min > 1 using a column of 10,000 plates, we only know the boundaries of the area for which Rs min > 0.5 for a column with 2,500 plates. We do not know what the boundaries for Rs min > 1 are in the latter case, because we have no information on how the value of Rs min changes with variations in the parameter settings. Only if the variation of the capacity factors as a function of the relevant parameters is known, can the boundaries of the area in which the resolution is adequate be calculated for different columns with different numbers of theoretical plates. Optimization methods in which this is the case (so-called interpretive methods ) will be discussed in section 5.5. 

Comtant reflux, varying overhead composition. The reflux is set at a predetermined value at which it is maintained for the entire run. Since the pot liquid composition is changing, the instantaneous composition of the distillate also changes. The progress of the distillate and pot compositions in a particular binary separation is illustrated in Fig. 13-120. The variation of the distillate composition for a multicomponent batch distillation is shown in Fig. 13-121 (these distillate product cuts have relatively low purity). The shapes of the curves are functions of volatility, reflux ratio, and number of theoretical plates. The distillation is continued until the average distillate... 

Band broadening within the chromatographic column is described as a function of mobile phase linear velocity ( ) by the well known van Deemter equation [5] that relates the height equivalent to the theoretical plate (HETP) to u the lower the HETP, the higher the plate number per unit length of a column. It follows that the highest efficiency is obtained for the shortest HETP. The van Deemter equation is ... 

Another convenient determination for N is by using the peak width at the half-height. From the same Gaussian function the peak width on the half-height is 2.355 times longer than the standard deviation of the same peak, and the resulting formula for the number of the theoretical plates will be... 

Prior to performing a formal validation, the analytical chemist should have performed some prevalidation during method development. The expectation is that a well-developed HPLC method should subsequently be validated with no major surprises or failures. Prior to validation, specificity and some degree of robustness should be demonstrated. In addition, some form of system suitability criteria will have been established. System suitability evaluates the capability of an HPLC system to perform a specific procedure on a given day. It is a quality check to ensure that the system functions as expected and that the generated data will be reliable. Only if the system passes this test should the analyst proceed to perform the specific analysis. System suitability can be based on resolution of two specified components, relative standard deviation, tailing factor, limit of quantitation or detection, expected retention times, number of theoretical plates, or a reference check. 

FIGURE 24-9 Number of theoretical plates required to attain a resolution of 1.0 as a function of partition ratio for values of relative retention ranging from 1.005 to 1.1 [Equation (24-35)]. (From Tang and Harris. )... 

Table 2.2 Number of theoretical plates required (Areq) to give baseline resolution (R = 1.5) as a function of the selectivity (a) and retention k )...

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