Gaussian distributions chromatograms

Column Efficiency. Under ideal conditions the profile of a solute band resembles that given by a Gaussian distribution curve (Fig. 11.1). The efficiency of a chromatographic system is expressed by the effective plate number defined from the chromatogram of a single band. 

Figure 1.3—Chromatographic peaks, a) Retention time b) Distribution of the peak c) Significance of the three basic parameters and features of a Gaussian distribution d) Example of a real chromatogram that shows the elution of components leading to peaks that resemble Gaussian distributions.

The features of an ideal chromatogram are the same as those obtained from a normal distribution of random errors (Gaussian distribution equation (1.2), cf. 21.3). In keeping with the classical notations, fi would correspond to the retention time of the eluting peak, a to the standard deviation of the peak (a2 represents the variance) and y represents the signal, as a function of time, from the detector located at the end of the column (see Fig. 1.3). 

On the chromatogram, a represents the width of the peak at 60.6% of its height and /R the retention time. tR and a have to be measured using the same units (time, distance, or eluted volume if the flow is constant). If a is expressed in units of volume (using the flow), then 4Gaussian distribution, equation (1.8) is preferred. Equation (1.9) is less used because of the distortion of most peaks at the base. 

Real chromatograms (Fig. 2.6-3) take into account the thermodynamic influences as well as the kinetics of mass transfer and fluid distribution. A rectangular concentration profile of the solute at the entrance of the column soon changes into a bell-shape Gaussian distribution, if the isotherm is linear. Figure 2.7a shows this distribution and some characteristic values, which will be referred to in subsequent chapters. With mass transfer resistance or nonlinear isotherms the peaks become asym-... 

J. Normal distribution Generally, the peak profiles in the chromatograms are assumed to be symmetrical, and are treated by the well-known normal or Gaussian distribution equation ... 

Figure 1.3 Chromatographic peaks, (a) The concept of retention time. The hold-up time is the retention time of an unretained compound in the column (the time it took to make the trip through the column) (b) Anatomy of an ideal peak (c) Significance of the three basic parameters and a summary of the features of a Gaussian curve (d) An example of a real chromatogram showing that while travelling along the column, each analyte is assumed to present a Gaussian distribution of concentration.

In the discussion of the mechanism of chromatographic separation, it was tacitly assumed that the distribution coefficient is a constant, which, in other words, means occurence of a linear relationship between the concentration of the sample molecules in the stationary and mobile phases irrespective of the quantity of the sample. An elution peak with a Gaussian distribution in the chromatogram is taken to represent such a situation, namely, that the isotherm of the relationship ... 

The chromatogram contains a wealth of information. It will contain one or more peaks, usually of a slightly asymmetrical, normal Gaussian) distribution (more on this later), indicating when each sample component eluted from the column and was detected. Each component has a retention time tj that is specific for that component, and is reasonably consistent from one day to the next, run under a set of specific chromatographic conditions. The is time of the maximum of the peak and is slighdy longer than the true elution time because of the time delay between when a component actually elutes from the column and... 

The models used to predict peak shape, based on Gaussian distributions, have the advantage of using very intuitive parameters, related to properties which can directly be measured on the chromatograms (position and height of the maxima, and width of the peaks). The equation describing a pure Gaussian peak is ... 

Figure 1 shows the chromatogram of a medium product range petroleum distillate recovered from fire debris. This is characteristic of a petroleum distillate in that the predominant peaks are associated with a homologous series of n-alkanes in a Gaussian distribution with abundant but less significant isoparaffinic, cycloparaffinic, and aromatic compounds also present. These less significant... 

It follows from the general theory of chromatography that the elution curve F(V) (chromatogram) of a uniform solute can be approximated by a Gaussian distribution function ... 

For a successful inversion of Eq. 5, it is important to adequately define matrix G. First, it is recommendable to adjust g(V, E) with a continuous analytical expression, and then to calculate the heights of the individual g(E) functions from that expression. Many analytical functions (e.g., a Gaussian distribution) never strictly drop to zero, and this would produce full G matrixes with positive and nonzero elements. Instead, it is preferable to set to 0 all of the almost-null elements of G (e.g., those smaller than 1% of the maximum). Also, choose G of minimal dimensions, in the sense that 1) its p columns must strictly cover the range of the corrected chromatogram [Ei - Vp and 2) its m rows must strictly cover the range of the measured chromatogram [Ei-E ]. Since, in general, the BB functions are skewed and non-uniform, it is convenient to specify each individual g(E) to contain (c + 1 + if) non-zero points, where c and d are the number of points before and after E, respectively. Thus, the number of columns of G results p = m — c — d, and the matrix is defined as follows ... 

As can be seen by observation of actual thin-layer chromatograms, many experimental concentration profiles can in fact be described by the Gaussian distribution curves. 

In a real chromatogram the peaks often have profiles that are non-Gaussian. There are several reasons for this. Besides the accepted approximations, such as invariance of the distribution coefficient K with concentration, there are irregularities of concentration in the injection zone at the head of the column. Furthermore, the speed of the mobile phase is zero at the walls and maximum at the centre of the column. The asymmetry observed in the peak shape is measured by a parameter called the skewing factor, which is calculated at 10% of the peak height (Fig. 1.4) ... 

Modern HPLC is a routine tool in any analytical laboratory. Standard HPLC system represents a separation output in the form of chromatogram (typical modern chromatogram is shown in Figure 1-6). Each specific analyte in the chromatogram is represented by a peak. In the absence of the strong specific analyte interactions with the stationary phase and at relatively low analyte concentration, peaks are symmetrical and resemble a typical Gaussian (normal distribution) curve. 

As previously indicated, this discussion is organized for chromatograms from very narrow polymer standards for which we can consider that the effect of molecular weight distribution is negligible and for which the unique separation process is size exclusion. With these limitations, the contribution to band broadening is conveniently separated into extra column effects, eddy dispersion, static dispersion, and mass transfer. In the most classical chromatographic interpretation, extra-column effects are not discussed and the three other contributions are considered as Gaussian, so there is simply the addition of their variances. The number of theoretical plates is defined as N = VJaY and the influence of v, the linear velocity of the eluent, is summarized by the so-called Van Deemter equation ... 

Two computer-generated chromatograms are shown In Figure 1. Both are simulations of the random distribution of 160 Gaussian components, each of which has a standard deviation a of eight seconds, over a total separation space of 175 minutes. The amplitude range Is 100 1800 and Is scaled In terms of ADC units as described above. The total baseline peak capacity In both simulations Is 219 (a = 0.731). The noise In the second simulation Is Gaussian with a standard deviation of thirty ADC units. 

Peaks representation of the detector signal resulting from elution of component molecules obtained by continuously plotting the signal as the analysis proceeds. The distribution of molecules within an eluted band results in a Gaussian peak see chromatogram. 

Apart from the use of uniform (or almost uniform) standards, other methods for determining the BB function have been developed. For example, by assuming a uniform and Gaussian BB function with a linear molar mass calibration, it is possible to use the mass and molar mass chromatograms for simultaneously estimating the standard deviation of the BB function and the calibration coefficients.Alternatively, if the shape of the MMD is known (e.g., it is a Poisson distribution on a linear molar mass axis), then the BB function can be estimated from the difference between the (mass or molar mass) chromatogram and its theoretical prediction in the absence of BB. Finally, the BB function can be theoretically predicted from a representative fractionation model. " Unfortunately, however, this approach is so far unfeasible due to the difficulty in determining the associated physicochemical parameters. 

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