Peakshapes In Chromatography And Spectroscopy

A typical chromatogram or spectrum consists of several peaks, at different positions, of different intensities, and sometimes of different shape. Each peak either corresponds to a characteristic absorption in spectroscopy or a characteristic compound in chromatography. In most cases the underlying peaks are distorted for a variety of reasons such as noise, poor digital resolution, blurring, or overlap with neighbouring peaks. A major aim of chemometric methods is to obtain the underlying, undistorted information, which is of direct interest to the chemist.

In some areas of chemistry such as NMR or vibrational spectroscopy, peak shapes can be predicted from very precise physical models and calculations. Indeed, detailed information from spectra often provides experimental verification of quantum mechanical ideas. In other areas, such as chromatography, the underlying mechanisms that result in experimental peak shapes are less well understood, being the result of complex mechanisms for partitioning between mobile and stationary phases, so the peaks are often modelled very empirically.

4.7.1 Principal Features

Peaks can be characterized in a number of ways, but a common approach, for symmetrical peakshapes, as illustrated in Figure 4.20, is to characterize each peak by:

1. a position at the centre (e.g. the elution time or spectral frequency);

2. a width, normally at half height;

The relationship between area and height is dependent on the peak shape, although heights are often easiest to measure experimentally. If all peaks have the same shape, the relative heights of different peaks are directly related to their relative areas. However, area is usually a better measure of chemical properties such as concentration so it is important to have some idea about peak shapes before using heights as indicators of concentration. In chromatog-raphy, widths of peaks may differ from column to column or even from day to day using a similar column according to instrumental performance or even across a chromatogram, but the relationship between overall area and concentration should remain stable, providing the detector sensitivity remains constant. In spectroscopy peak shapes are normally fairly similar on different instruments, although it is important to realize that filtering can distort the shape and in some cases filtering is performed automatically by the instrumental hardware, occasionally without even telling the user. Only if peak shapes are known to be

Width

Figure 4.20 Characteristics of a symmetrical peak

Position of centre

Figure 4.20 Characteristics of a symmetrical peak fairly constant in a series of measurements can heights be safely substituted for areas as measurements of chemical properties.

Sometimes the width at a different proportion of the peak height is reported rather than the half width. Another common measure is when the peak has decayed to a small percentage of the overall height (for example 1 %), which is often taken as the total width of the peak, or using triangulation, which fits the peak (roughly) to the shape of a triangle and looks at the width of the base of the triangle.

Three common peak shapes are important, although in some specialized cases it is probably best to consult the literature on the particular measurement technique, and for certain types of instrumentation, more elaborate models are known and can safely be used.

4.7.2 Gaussians

These peak shapes are common in most types of chromatography and spectroscopy. A simplified formula for a Gaussian is given by:

where A is the height at the centre, x0 is the position of the centre and s relates to the peak width.

Gaussians crop up in many different situations and have also been discussed in the context of the normal distribution in statistical analysis of data (Section 3.4). However, in this chapter we introduce these in a different context: note that the pre-exponential factor has been simplified in this equation to represent shape rather than an area equal to 1. It can be shown that:

• the width at half height of a Gaussian peak is given by A^2 = 2s(ln2)1/2;

• and the area by (n)l/2A s (note this depends on both the height and the width).

4.7.3 Lorentzians

The Lorentzian peak shape corresponds to a statistical function called the Cauchy distribution. It is less common but often arises in certain types of spectroscopy such as NMR. A simplified formula for a Lorentzian is given by:

where A is the height at the centre, x0 is the position of the centre and s relates to the peak width.

It can be shown that:

• the width at half height of a Lorentzian peak is given by Ai/2 = 2s;

• and the area by nAs (note this depends on both the height and the width).

The main difference between Gaussian and Lorentzian peak shapes is that the latter has a bigger tail, as illustrated in Figure 4.21 for two peaks with identical heights and half widths.

Figure 4.21 Gaussian and Lorentzian peak shapes

4.7.4 Asymmetric Peak Shapes

In many forms of chemical analysis, especially chromatography, it is hard to obtain symmetrical peak shapes. Although there are a number of quite sophisticated models that can be proposed, a very simple first approximation is that of a Lorentzian/Gaussian peak shape. Figure 4.22(a) represents a tailing peak shape. A fronting peak is illustrated in Figure 4.22(b): such peaks are much rarer. The way to handle such peaks is to model the sharper half by a Gaussian (less of a tail) and the broader half by a Lorentzian, so, for example, the right-hand half of the peak in Figure 4.22(a) is fitted as a 'half' Lorentzian peak shape. From each half a different width is obtained, and these are used to describe the data.

Figure 4.22 (a) Tailing and (b) fronting peaks

Figure 4.22 (a) Tailing and (b) fronting peaks

Although several very sophisticated approaches are available for the analysis of asymmetric peak shapes, in practice, there is normally insufficient experimental evidence for more detailed modelling, and, as always, it is important to ask what further insights this extra information will give, and what cost in obtaining better data or performing more sophisticated calculations. The book by Felinger [6] provides a more extensive insight into a large number of peak shape functions especially in chromatography.

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