The theoretical plate number

In a chromatographic process, a solute is equilibrated many times between the mobile and stationary phases during passage through the column. Each equilibration is equivalent to one equilibrium stage or one theoretical plate. Though in SEC, the concept of the so-called stationary phase, as in other chromatographic separation modes, is not definitive, the theoretical plate number N derived from the plate theory is still used as a measure of column efficiency. 

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 

The peak width at half height is exclusively used instead of the peak 

The larger the plate number is, the higher the column efficiency is. The theoretical plate number gives the relative ability of a given column to provide narrow bands (small values of W) and improved separations. 

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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. 

FIGURE 14.1 Determination of the theoretical plate number N by the half-height method. 

A particular column can be used for different types of LC by changing the eluent components. For example, a column packed with RP-18 bonded silica gel can be used for SEC with THF, NPLC with n-hexane, and RPLC with aqueous acetonitrile. When separation cannot be achieved by improving the theoretical plate number of a column, it may be achieved by selection of an appropriate stationary phase material and/or eluent. 

The height equivalent to a theoretical plate (HETP) is the length of the column divided by the theoretical plate number. 

The theoretical plate number (AO is inversely related to the amount of zone broadening occurring in a column. The greater the value for N, the more efficient is the column but differences of less than 25% are not very significant. 

This equation indicates that the particle size, dp, is the main contributor to the H value. The smaller the particles, the higher the theoretical plate number. The optimum condition is obtained by the relationship between the theoretical plate height and the flow velocity. 

The theoretical plate number N of peak B can be calculated from the chromatogram given in Figure 1.3 by the following equation ... 

Commercial instruments have a reasonable balance between the recommended column size and the volume of the column and connecting tubing (XY). However, the theoretical plate number of a single column may give different values on different instruments, and even on replacement of the components and parts of a single instrument. Such discrepancies can be understood in terms of differences in the mechanics of the instruments and the design of their parts. 

The connecting tubing should be as short and narrow as possible. The volume of a 20 cm x 0.5 mm i.d. tube is 39.3 p. That of a 20 cm x 0.25 mm i.d. tube is 9.8 /A, and that of a 20 cm x 0.125 mm i.d. tube is 2.5 [A. Some detectors are equipped with a heat exchanger that consists of a metal block containing a capillary tube. The volume of this tube also affects the theoretical plate number. If highly sensitive operation is not required, the heat exchanger can be removed or bypassed. 

Apart from the above-discussed parameters for HPLC optimization of chiral resolution on antibiotic CSPs, some other HPLC conditions may be controlled to improve chiral resolution on these CSPs. The effect of the concentrations of antibiotics (on stationary phase) on enantioresolution varied depending on the type of racemates. The effect of the concentrations of teicoplanin has been studied on the retention (k), enantioselectivity (a), resolution (Rs), and theoretical plate number (N) for five racemates [21]. An increase in the concentration of teicoplanin resulted in an increase of a and Rs values. The most surprising fact is that the theoretical plate number (N) increases with the increase in the concentration of teicoplanin. It may be the result of the resistance of mass transfer resulting from analyte interaction with free silanol and/or the linkage chains (antibiotics linked with silica gel). This would tend to trap an analyte between the silica surface and the bulky chiral selector adhered to it. This is somewhat... 

Screen-printed carbon electrodes were also evaluated in the Topas microchip for separation and detection of pAP and AsA. The theoretical plate number (AO, half-peak width (wy2), peak current and resolution are also shown in Table 34.2. Peak current at SPEs was higher than those obtained with gold and platinum wires. 

A gold film was also evaluated using pAP as analyte model. The theoretical plate number (AO was 5990 m-1 with a half-peak width (wy2) of 4.1s for pAP. Peak current at gold-film electrode was lower than at SPEs but higher than at gold and platinum wire. 

The efficiency remains high even with high concentrations of methanol. With acetonitrile as the modifier, the efficiency was significantly reduced as the concentration of the modifier was increased. For this study the theoretical plate number (N) was estimated using the equation of Foley and Dorsey (5M. 

In capillary gel electrophoresis, one of the major contributors to band broadening, besides the injection and detection extra-column effects, is the longitudinal diffusion of the solute molecules in the capillary tube [14], The theoretical plate number (N) is characteristic of column efficiency ... 

Ruhr s group studied the separation of double- and single-stranded DNA restriction fragments in capillary electrophoresis with polymer solutions under alkaline conditions in epoxy-coated capillaries and found that at pH 11 the theoretical plate numbers exceeded several millions [96], At pH 12, single-stranded DNA molecules were still well separated in entangled hydroxyethylcellulose (HEC) solutions, but the resolution decreased significantly in dilute polymer solutions. 

ID X 150 mm L by means of a high-pressure pump under a pressure of 300kg/cm. The theoretical plate number of the packed column was 4470. 

When packed into chromatography columns, TRIM beads imprinted with Boc-L-Phe were shown to have column efficiencies and separation abilities superior to ground and sieved bulk material [5]. The theoretical plate number was approximately double that obtained with conventional crushed polymer under the same conditions and the resolution of a racemate was also slightly enhanced. The difference, however, was not that great considering the additional preparation time and effort involved. 

This parameter is very useful in chromatography for the characterization of peak broadening per unit length of the column. In addition to H, the peak broadening characterization in a column can be done using the theoretical plate number n. For a column of length L, n is defined as... 

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) ... 

In addition to the theoretical plate number, an effective plate number N is defined by substituting tp in rel. (17c) with tp. The formula for N will be... 

Rel. (18) shows how N depends on chromatographic retention time tp, and since tp is compound related (index i" omitted), it also shows that N (as well as n) are compound dependent. Both rel. (17c) and (18) can be used to measure the theoretical plate number or effective plate number based on experimental data obtained with a given column. This measurement is useful in practice to select columns (higher n gives lower peak broadening) and also to assess the loss in performance of a column after a certain period of usage. 

The ionization and net charge of the sample components are affected greatly by changes in the pH. As a result, the migration rate, the solubility, the theoretical plate number, and peak height could be all affected. 

The theoretical plate number for the separation of these model proteins has a maximal value at pH 7. However, the peak capacity increases with pH because... 

Eor the projection of an extraction unit, the practical theoretical plate number is determined by dividing the theoretical plate number by the plate efficiency value ... 

A column in which the ascending vapour is in contact with the refluxing liquid is used for this purpose. The reflux is generated by an overhead condenser. Due to the phase transfer in the column, a height equivalent of theoretical plate HETP can be defined where the two phases, liquid and vapour, are in thermodynamic equilibrium. The theoretical plate number is defined as... 

From the experimental data [44], the selectivity of the adsorbents was calculated as the relative retention volume of the separated species. It can be seen in Table 9 that Cs-CPM adsorbent possesses better selectivity than the Ca-CPM one with respect to the separation of the isomers of xylene and ethylbenzene with like properties. The results enabled us to conclude that the separation of the isomers of m- and p-cresol up to the purity degree of 99% can be achieved using the columns with the theoretical plates number (TPN) not exceeding 200, and that for the o- and m-isomers - employing the packed intermediate efficiency columns with TPN of 2000-2500. To separate the same pairs of the species using the Ca-CPM, the capillary columns with the TPN of 20000 were reqnired. 

The above-mentioned equation for n in fact is only valid for isocratic separations and if the peaks are symmetric the peak capacity is larger with gradient separations. Tailing decreases the peak capacity of a column. In real separations the theoretical plate number is not constant over the full k range. However, it is even more important to realize that a hypothetical parameter is discussed here. It is necessary to deal with peaks that are statistically distributed over the accessible time range. The theory of probabilities allows us to proceed from ideal to near-real separations. Unfortunately, the results are discouraging. 

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