Gaussian peak model

In the deconvolution using Gaussian peak model, all peaks have a deviation of less than 1%. These are even close to 0%, regardless of whether the peaks overlap. 

For example, a Gaussian peak model is used to derive many fluently used fundamental equations in chromatography (17), but the Gaussian function rarely provides an accurate model for real chromatographic peaks. Convectional effects introduced by the flow cell can cause asymmetry in chromatographic peaks even if a Gaussian profile has been established prior to detection (18). One model diat is well accepted to represent a real chromato phic peak is an exponentially modified Gaussian (EMG) function. The FIA profile is characterized primarily by a dispersion coefficient (19) which offers information about die manifold, but direct information about the peak parameters such as second moment or variance can not be readily obtained. Recently, the EMG was used to describe FIA profiles to obtain the second moment and characterize the peak (20-22). 

Most GPC columns are provided with vendor estimates of the plate count of the column and a chromatogram of a series of test peaks. These plate count estimates are usually obtained using small molecule analytes that elute at the total permeation volume (Vp) of the column. The Gaussian peak shape model... 

Figure 5-21. N(ls) core level spectra of the iiniim model compound PC20X adsorbed on ITO. The upper curve corresponds to a thick film, the central curve to an intermediate thick film, and the lower curve to an ultra thin Him, essentially a mono-layer in thickness. The bold solid lines are the filled curves and the thin solid and dolled lines are the Gaussian peak components lor physisorbed and chemisorbed PC20X, respectively.

A multicomponent 2D chromatogram is considered as a series of 2D peaks with random position and height. For the sake of simplicity, here we assume that the peaks are modeled with bidimensional Gaussian peaks, thus the signal is expressed as... 

Model Gaussian Peaks. If all the terms on the right-hand side of Eq. (8.13) can be modeled by Gaussians, the square of the integral breadth of the observed peak... 

The success of this modeling can be ascertained by the ability to replicate the observed peak shapes using Gaussian peaks centered at peak positions suggested by the modeling (Fig. 3). For the case where n = 3/2, five peaks were needed 1 for the monomeric A1 and 2 peaks each for each of the two dinuclear A1 species. These peaks were combined to successfully replicate the observed NMR peaks recorded for this sample. When n = 2, the data were reproduced using only 4 peaks, two each for each of the two dinuclear A1 species. Our earlier predictions (11) showed that HCl could combine with either of the dinuclear Al-species in three different positions, which showed different acid strengths. 

A chromatogram without noise and drift is composed of a number of approximately bell-shaped peaks, resolved and unresolved. It is obvious that the most realistic model of a single peak shape or even the complete chromatogram could be obtained by the solution of mass transport models, based on conservation laws. However, the often used plug flow with constant flow velocity and axial diffusion, resulting in real Gaussian peak shape, is hardly realistic. Even a slightly more complicated transport equation... 

The model is directly related to the widely used theoretical plate concept, which, in principle, is only valid for Gaussian peaks. 

Whether a staggered model will reproduce the torsion sensitive distance distribution with sufficient accuracy for intermediate barriers, or a weighted sum over Gaussian peaks has to be applied, will also depend on the total change in the torsion dependent distance compared to the u-values of the said distance. 

The majority of chromatographic separations as well as the theory assume that each component elutes out of the column as a narrow band or a Gaussian peak. Using the position of the maximum of the peak as a measure of retention time, the peak shape conforms closely to the equation C = Cjjjg, exp[-(t -1] ) The modelling of this process, by traditional descriptive models, has been extensively reported in the literature. 

Since the retention of the template on the imprinted polymers strongly depends on the sample load (see later Fig. 5.25), the theoretical models describing the various dispersion processes are not applicable. Nevertheless, on an imprinted CSP for L-PA, the least retained enantiomer, D-PA, elutes as a fairly Gaussian peak (Fig. 5.3), which should be governed by the same band broadening mechanisms that are considered in the models. A measure of the dispersion of a Gaussian peak is the deviation from its mean value, which is reflected in the reduced plate height as ... 

O Spreadsheet Summary Chapter 15 of Applications of Microsoft si Excel in Analytical Chemistry begins with an exercise treating the resolution of overlapped Gaussian peaks. The overlapped chromatogram, the response, is modeled as the sum of Gaussian curves. Initial estimates are made for the model parameters. Excel calculates the residuals, the difference between the response and the model, and the sum of the squares of the residuals. Excel s Solver is then used to minimize the sum of the squares of the residuals, while displaying the results of each iteration. 

The retention time tRjin and the second moment for the Gaussian profile (Eq. 6.61) have been replaced by variables indexed with g . These parameters tg and og must be optimized by curve fitting. Equation 6.143 is only suitable for symmetric peaks. Analytical solutions of, for example, the transport dispersive model (which describes asymmetric band broadening only for a very low number of stages) are not suited to describing the asymmetry often encountered in practical chromatograms. Thus, many different, mostly empirical functions have been developed for peak modeling. A recent extensive review by Marco and Bombi (2001) lists over 90 of them. 

Jeansonne, M. S. Foley, J. P. Review of the exponentially modified Gaussian chromatographic peak model since 1983,/. Chromatogr. Sci, 1991, 29, 258-266. 

The effect of a volumetric or electronic time constant can be modeled as a convolution of a Gaussian peak g(t) with a filter function (or transfer function) h(T). This convolution generates the output signal f(t) defined by ... 

Many spectrometric peaks can be described reasonably well in terms of one of two model peak shapes, those of a Gaussian and of a Lorentzian (or Cauchy) peak. Leaving out the normalizing factors as immaterial in the present context, the Gaussian peak can be described as... 

Clarke has examined the thermodynamic equation of state and the specific heat for a Lennard-Jones liquid cooled through 7 at zero pressure. He found that drops with decreasing temperature near where the selfdiffusion becomes very small. Wendt and Abraham have found that the ratio of the values of the radial distribution function at the first peak and first valley shows behavior on cooling much like that observed for the volume of real glasses (Fig. 6), with a clearly defined 7. Stillinger and Weber have studied a Gaussian core model and find a self-diffusion constant that drops essentially to zero at a finite temperature. They also find that the ratio of the first peak to the first valley in the radial distribution function showed behavior similar to that found by Wendt and Abraham" for Lennard-Jones liquids. However, the first such evidence for a nonequilibrium (i.e. kinetic) nature of the transition in a numerical simulation was obtained by Gordon et al., who observed breakaways in the equation of state and the entropy of a hard-sphere fluid similar to those in real materials. 

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