Peak widths

For a le couple, the Nemstian behavior predicts a peak full width at half-height (FWHH) of 90.6 mV. Real peak FWHH usually differs from that value. This 

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From equation 12.1 it is clear that resolution may be improved either by increasing Afr or by decreasing wa or w-q (Figure 12.9). We can increase Afr by enhancing the interaction of the solutes with the column or by increasing the column s selectivity for one of the solutes. Peak width is a kinetic effect associated with the solute s movement within and between the mobile phase and stationary phase. The effect is governed by several factors that are collectively called column efficiency. Each of these factors is considered in more detail in the following sections. 

Now that we have defined capacity factor, selectivity, and column efficiency we consider their relationship to chromatographic resolution. Since we are only interested in the resolution between solutes eluting with similar retention times, it is safe to assume that the peak widths for the two solutes are approximately the same. Equation 12.1, therefore, is written as... 

Explain the difference in the retention times, the peak areas, and the peak widths when switching from a split injection to a splitless injection. 

Resolving power (mass). The ability to distinguish between ions differing slightly in mass-to-charge ratio. It can be characterized by giving the peak width, measured in mass units, expressed as a function of mass, for at least two points on the peak, specifically for 50% and for 5% of the maximum peak height. 

Measurement of Residual Stress and Strain. The displacement of the 2 -value of a particular line in a diffraction pattern from its nominal, nonstressed position gives a measure of the amount of stress retained in the crystaUites during the crystallization process. Thus metals prepared in certain ways (eg, cold rolling) have stress in their polycrystalline form. Strain is a function of peak width, but the peak shape is different than that due to crystaUite size. Usually the two properties, crystaUite size and strain, are deterrnined together by a computer program. 

The preceding eqiiations are accurate to within about 10 percent for feed injections that do not exceed 40 percent of the final peak width. For large, rec tangiilar feed injections, the baseline width of the response peak is approximated by ... 

Maximum continuous speed, 105 percent = Initial (lesser) speed at. 707 x peak amplitude (critical) Wg = Final (greater) speed at.707 x peak amplitude (critical) W2-/v,= Peak width at the half-power" point... 

Spectral resolution. Consider a Mn Ka photon, which has a namral peak width (the foil width at half maximum, or FWHM) of approximately 1... 

In the concepts developed above, we have used the kinematic approximation, which is valid for weak diffraction intensities arising from imperfect crystals. For perfect crystals (available thanks to the semiconductor industry), the diffraction intensities are large, and this approximation becomes inadequate. Thus, the dynamical theory must be used. In perfect crystals the incident X rays undergo multiple reflections from atomic planes and the dynamical theory accounts for the interference between these reflections. The attenuation in the crystal is no longer given by absorption (e.g., p) but is determined by the way in which the multiple reflections interfere. When the diffraction conditions are satisfied, the diffracted intensity ft-om perfect crystals is essentially the same as the incident intensity. The diffraction peak widths depend on 26 m and Fjjj and are extremely small (less than... 

The CRO pitch sample (Fig. 9) and the PVC samples (Fig. 10) show well formed (002) peaks which first broaden, and then sharpen, as the heating temperature is increased. The KS pitch sample shows a very similar result. Diamond [35] noticed this effect in his work on carbonization of coals. Figures 9 and 10 show that the widtli and position of the (002) peaks do not change dramatically upon heating in this temperature range for the pitch and PVC samples. These peak widths are consistent with stacks of order 5 to 7 layers accordmg to the Scherrer equation assuming d,oo2) is about 3.5A. 

The peak width (w) is the distance between each side of a peak measured at 0.6065 of the peak height. The peak width measured at this height is equivalent to two standard deviations (2o) of the Gaussian curve and, thus, has significance when... 

The peak width at the base (wb) is the distance between the intersections of the tangents drawn to the sides of the peak and the peak base geometrically produced. The peak width at the base is equivalent to four standard deviations (4a) of the... 

Recalling that a separation is achieved by moving the solute bands apart in the column and, at the same time, constraining their dispersion so that they are eluted discretely, it follows that the resolution of a pair of solutes is not successfully accomplished by merely selective retention. In addition, the column must be carefully designed to minimize solute band dispersion. Selective retention will be determined by the interactive nature of the two phases, but band dispersion is determined by the physical properties of the column and the manner in which it is constructed. It is, therefore, necessary to identify those properties that influence peak width and how they are related to other properties of the chromatographic system. This aspect of chromatography theory will be discussed in detail in Part 2 of this book. At this time, the theoretical development will be limited to obtaining a measure of the peak width, so that eventually the width can then be related both theoretically and experimentally to the pertinent column parameters. 

Starting with the Poisson form of the elution equation, the peak width at the points of inflexion of the curve (which corresponds to twice the standard deviation of the normal elution curve) can be found by equating the second differential to zero and solving in the usual manner. Thus, at the points of inflexion, ... 

The peak width at the points of inflexion of the elution curve is twice the standard deviation of the Poisson or Gaussian curve and thus, from equation (8), the variance (the square of the standard deviation) will be equal to (n), the total number of plates in the column. 

Let the distance between the injection point and the peak maximum (the retention distance on the chromatogram) be (y) cm and the peak width at the points of inflexion be (x) cm. If a computer data acquisition and processing system is employed, then the equivalent retention times can be used. 

Using equation (10), the efficiency of any solute peak can be calculated for any column from measurements taken directly from the chromatogram (or, if a computer system is used, from the respective retention times stored on disk). The computer will need to have special software available to identify the peak width and calculate the column efficiency and this software will be in addition to that used for quantitative measurements. Most contemporary computer data acquisition and processing systems contain such software in addition to other chromatography programs. The measurement of column efficiency is a common method for monitoring the quality of the column during use. 

Equation (10) also allows the peak width (2o) and the variance (o ) to be measured as a simple function of the retention volume of the solute but, unfortunately, does not help to identify those factors that cause the solute band to spread, nor how to control it. This problem has already been discussed and is the basic limitation of the plate theory. In fact, it was this limitation that originally invoked the development of the... 

The measurement of efficiency is important, as it is used to monitor the quality of the column during use and to detect any deterioration that might take place. However, to measure the column efficiency, it is necessary to identify the position of the points of inflection which will be where the width is to be measured. The inflection points are not easily located on a peak, so it is necessary to know at what fraction of the peak height they occur, and the peak width can then be measured at that height. 

Reiterating the conditions for a chromatographic separation once again, for two solutes to be resolved their peaks must be moved apart in the column and maintained sufficiently narrow for them to be eluted as discrete peaks. However, the criterion for two peaks to be resolved (usually defined as the resolution) is somewhat arbitrary and is usually defined as the ratio of the distance between the peak maxima to half the peak width (a) at the points of inflection. To illustrate the various degrees of resolution that can be obtained, the separation of a pair of solutes 2o, 3o, 4o, 5o and 6o apart are shown in Figure 12. Although, for baseline resolution, it is clear that the peak maxima should be separated by at least 6o for most quantitative analyses. 

Although not apparent at this time, it will become clear that two adjacent peaks from solutes of different chemical type, or significantly different molecular weight, are not likely to have precisely the same peak widths (i.e., exhibit the same efficiency). Nevertheless, in most cases, the difference will be relatively small and, in fact, likely to be negligible. As a consequence, the widths of closely adjacent peaks will, at this time, be assumed to be the same. 

If the widths of the two peaks are the same, then the peak width in volume flow of mobile phase will be... 

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