Peakshapes

Chromatograms and spectra are normally considered to consist of a series of peaks, or lines, superimposed upon noise. Each peak arises from either a characteristic absorption or a characteristic compound. In most cases the underlying peaks are distorted for a variety of reasons such as noise, blurring, or overlap with neighbouring peaks. A major aim of chemometric metiiods is to obtain the underlying, undistorted, information. 

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

Sometimes die width at a different percentage of the peak height is cited rather than the half-width. A further common measure is when the peak has decayed to a small percentage of die overall height (for example 1 %), which is often taken as the total width of the peak, or alternatively has decayed to a size that relates to die noise. 

In many cases of spectroscopy, peakshapes can be very precisely predicted, for example from quantum mechanics, such as in NMR or visible spectroscopy. In other situations, the peakshape is dependent on complex physical processes, for example in chromatography, and can only be modelled empirically. In the latter situation it is not always practicable to obtain an exact model, and a number of closely similar empirical estimates will give equally useful information. 

Three common peakshapes cover most situations. If these general peakshapes are not suitable for a particular purpose, it is probably best to consult specialised literature on the particular measurement technique. 

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The mathematical formulations of the diffusion problems for a micropippette and metal microdisk electrodes are quite similar when the CT process is governed by essentially spherical diffusion in the outer solution. The voltammograms in this case follow the well-known equation of the reversible steady-state wave [Eq. (2)]. However, the peakshaped, non-steady-state voltammograms are obtained when the overall CT rate is controlled by linear diffusion inside the pipette (Fig. 4) [3]. 

Figure 4.14b and c illustrate the possibility of using convolution (Section 1.3.2) to transform all the voltammograms, whether they are plateau- or peakshaped, into a plateau-shaped wave. Measuring the height of this plateau allows determination of the kinetic constant, showing that this does not necessarily require that the raw current-potential curve be plateau-shaped. The standard potential, FpQ, may also be determined this way. 

The voltammetric behavior of the first-order catalytic process in DDPV for different values of the kinetic parameter Zi(= ( 1 + V) Ti) at spherical and disc electrodes with radius ranging from 1 to 100 pm can be seen in Fig. 4.25. For this mechanism, the criterion for the attainment of a kinetic steady state is %2 > 1-5 (Eq. 4.232) [73-75]. In both transient and stationary cases, the response is peakshaped and increases with j2. h is important to highlight that the DDPV response loses its sensitivity toward the kinetics of the chemical step as the electrode size decreases (compare the curves in Fig. 4.25a, c). For the smallest electrode (rd rs 1 pm, Fig. 4.25c), only small differences in the peak current can be observed in all the range of constants considered. Thus, the rate constants that can... 

In both cases, it is clear that the response can be expressed as the sum of the solution for planar electrodes given by Eq. (6.33) and a contribution related to the electrode size (the second addend in the right-hand side of Eqs. (6.40) and (6.41)). When the electrode radius decreases, the current evolves from the transient peakshaped response to a sigmoidal stationary one in the same way as observed for a simple charge transfer process (see Sects. 5.2.3.2 and 5.2.3.3). For small values of the electrode radius, the planar term in (6.40) and (6.41) becomes negligible and the current simplifies to... 

Nevertheless, to a first approximation the Gaussian peakshape can be assumed for a chromatographic peak. If/(r) detector response) as a function of time and tR is the retention time of the peak, then a Gaussian peak can be described by... 

These peakshapes are common in most types of chromatography and spectroscopy. A simplified equation for a Gaussian is... 

Gaussian and Lorentzian peakshapes of equal half-heights... 

The main difference between Gaussian and Lorentzian peakshapes is that the latter has a bigger tail, as illustrated in Figure 3.2 for two peaks with identical half-widths and heights. 

Asymmetric peakshapes often described by a Gaussian/Lorentzian model, (a) Tailing left is Gaussian and right Lorentzian. (b) Fronting left is Lorentzian and right Gaussian... 

The peakshape in the frequency domain relates to the decay curve (or mechanism) in the time domain. The time domain equivalent of a Lorentzian peak is... 

In die Fourier transform of a real time series, die peakshapes in the real and imaginary halves of die spectrum differ. Ideally, the real spectrum corresponds to an absorption lineshape, and die imaginary spectrum to a dispersion lineshape, as illustrated in Figure 3.20. The absorption lineshape is equivalent to a pure peakshape such as a Lorentzian or Gaussian, whereas die dispersion lineshape is a little like a derivative. 

There are a variety of solutions to this problem, a common one being to correct this by adding together proportions of the real and imaginary data until an absorption peakshape is achieved using an angle 0 so diat... 

The time series decays more slowly than the original, but there is not much increase in noise. The peakshape in the transform is almost as narrow as that obtained using a single exponential, but noise is dramatically reduced. 

Ratios of peak intensities for the case studies (a)-(d) assuming ideal peakshapes and peaks detectable over an indefinite region... 

The normal distribution curve is not only a probability distribution but is also used to describe peakshapes in spectroscopy and chromatography. 

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