The plate number

The efficiency of a chromatographic system (i.e. the column plus the instrument) is usually expressed in terms of the number of theoretical plates (N), which may be defined as follows  

Here the peak shape is assumed to be Gaussian, with a being the standard deviation. It follows from figure 1.4 that eqn.(1.16) can also be written as 

Another convenient equation can be derived for instruments that provide information on both the peak height (h) and the peak area (A)  

In applying eqn.(1.17) one should be aware of the units involved. N is dimensionless, so that if tR is expressed in seconds and h in mV, A should be expressed in mV.s. 

From the number of plates in the column, the height equivalent of a theoretical plate (HETP), usually abbreviated to plate height (H), can easily be calculated  

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When only a few solutes are separated, they may occupy only a small portion of the total column volume at any given instant. In such cases, the productivity is improved by cyclic feed injections, timed so that the most strongly retained component from an injection elutes just before the least strongly retained component from the following injection (see Fig. 16-57). For a mixture of two components with k > 1, when the same resolution is maintained between bands of the same injections and bands of successive injections, the cycle time tc and the plate number requirement are ... 

The plate number for the same column also depends on the eluent, e.g., a permitted operation for some styrene-divinylbenzene columns is to change the eluent from tetrahydrofuran (THF) to dimethylacetamide (DMAC) and then return to THF. The plate number in DMAC is considerably lower than in THF. After the replacement of DMAC by THF the old N value is obtained again. 

The result of this equation describes the quality of the separation on the basis of an ideal size exclusion mechanism with a given pore volume distribution. The quality of the packing is deliberately excluded from this consideration. This parameter should be measured separately and judged by the plate number. The ASTM standard method for HPSEC of polystyrene (4) contains the following equation for resolution (R,) ... 

Giddings pointed out (32) that separated compounds must remain resolved throughout the whole process. This situation is illustrated in Figure 1.5, where two secondary columns are coupled to a primary column, and each secondary column is fed a fraction of duration Ar from the eluent from the first column. The peak capacity of the coupled system then depends on the plate number of each individual separation and on At. The primary column eliminates sample components that would otherwise interfere with the resolution of the components of interest in the secondary columns. An efficient primary separation may be wasted, however, if At is greater than the average peak width produced by the primary column, because of the recombination of resolved peaks after transfer into a secondary column. As At increases, the system approaches that of a tandem arrangement, and the resolution gained in one column may be nullified by the elution order in a subsequent column. 

The column performance (efficiency) is measured either in terms of the plate height (H), the efficiency of the column per unit length, or the plate number (N), i.e. the nnmber of plates for the column. This number depends upon the column length (L), whereas the plate height does not. The mathematical relationships between the nnmber of plates, the retention time of the analyte and the width of the response is shown in the following equations ... 

The plate number in equation (4.56) corresponds to the value when the effective value of the capacity factor (equal to k when the band is at the column midpoint) is equal to the capacity factor in isocratic elution for the same column. The effective value of the capacity factor, k, is simply 1/1.15b. In most cases k, will be large and equation (4.57) is simplified by equating l/k, to zero. The resolution between two adjacent bands in a gradient program, again analogous to isocratic elution, is e q>ressed by equation (4.58)... 

The more efficient the column, the smaller will be at a given value of Vr. To measure efficiency, we use quantities called the plate number (N) or the plate height (H) of the column, which are defined as follows ... 

As the column becomes more efficient, N gets larger and H gets smaller. The plate number is dimensionless, but H has units of length, and is usually measured in mm or fim. There are other ways in which N can be measured, depending on whereabouts on the peak we choose to measure its width. Another way is ... 

In Section 2.3.3 you worked out the effect pf extra column dispersion on the peak of an unretained solute, using a column with a plate number of 10 000. The extra-column dispersion will decrease the plate number that wer actually observe for this column. The table below contains the retention volume, peak widths and plate number for an unretained solute on this column. 

Neither the capacity factor nor the separation factor take into account the effect of dispersion, which is measured by the plate number or the plate height of the column. These were defined in Section 2.3.2. 

The tailing is probably caused by a mixed mechanism, for instance adsorption on active silica sites that are not end-capped. To reduce this, we can try adding a salt to the water. To get better resolution we need to change the selectivity, a, which means changing the chemistry of the mobile phase, or increasing the plate number of the column, or both. 

The resolution of the three oestrogens still has to be improved, so to proceed further we can either work on the selectivity, a, by using, instead of methanol, a different water-soluble solvent such as acetonitrile, tetrahydrofuran or dioxane, or we can try to improve the separation by increasing the plate number of the column. If we change the solvent, we cannot be sure about what will happen to the selectivity, and we may have to do a lot more experimental work to get any improvement. Increasing the plate number, if it can be done, is the easier of the two options. Fig. 4.2f shows the improvement that results when two 30 cm columns are used in series, with a flow rate of 1 cm3 min-5. The oestrogens are separated from the excipients and are also separated reasonably well from one another. The separation is complete in about 20 minutes. 

When the column is ready to be used, the chromatogram of a suitable test mixture should be obtained. The plate number and retention times of the test solutes should be noted, and the peaks should have a satisfactory shape (minimal tailing). For measurement of the plate number, the recorder should be used at a high chart speed. Fig. 5.1b(i) and (ii) show test chromatograms for a C-18 column prepared by the above method, and Fig. 5.1c and 5.Id show the data that you should report with the chromatogram. The retention for an unretained peak is taken as the small baseline disturbance just before the first peak. 

In each case the plate numbers are calculated using Eq. 2.3a and the actual peak width wj. Thus, for the first column ... 

The parameter N is universally referred to as the plate number, but an alternative means of quoting efficiency is in terms of opiate height, H or HETP. Plate number and plate height are inversely related by the equation... 

Two compounds were separated by HPLC with an Rs value of 0.75, the plate number for the second compound being 4500. Calculate the number of plates required to obtain resolutions of (a) 1.0 and (b) 1.5. 

G m is the molar rate of flow per unit area of inert gas, it refers to the plate numbered from the bottom upwards (and suffix n refers to material leaving plate n),... 

The physical and chemical aspects of liquid chromatography, in addition to mechanical aspects, are briefly described in this chapter. Theoretical approaches are explained in detail in later chapters. The effect of stationary phase materials on the chemical selectivity is described in Chapter 3, and the influence of the eluent components is covered in Chapter 4. The plate number theory is discussed in Chapter 5. Quantitative optimization is explained in Chapter 6. 

FIGURE 4.1 Effect of the plate number (N), the separation factor (a ), and the retention factor (k) on resolution (Rs). (Adapted from Sandra, P.J. 1989. High Resolut. Chromatogr. 12 82-86. With permission.)... 

Assuming an exclusion volume of 5 ml per column allows construction of Table I from Equation 1. Table I lists the bandwidth in microliters as a function of column plate number and the number of columns in series. The data assume that the plate number may be generated at total exclusion, as well as at total permeation actual measurements made using the smaller pore size column si ibstantiate this. 

Equation (2) shows that the resolution is a function of three different factors (1) the resolving power of the column as measured by the plate number that expresses the relative width of bands (2) the relative retention of the two compounds that measures how far apart the bands are from each other and (3) the magnitude of retention, as separation is a result of retention. The relative influence of these factors has been discussed by Snyder (72,13) in a form very easy to use in practice. 

If the bandwidth at the column outlet is expressed in length unit, we have an expression for the plate number, which is similar to Eq. (3), and given... 

Both Eqs. (43) and (44) show that the minimum pressure is proportional to the plate number and inversely proportional to the square of the particle diameter. 

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