Chromatography Gaussian curves

Gaussian profiles are also utilized to approximate peak shapes observed in different types of spectroscopy. Again, we need to stress that the actual molecular processes behind a spectroscopically observed transition are very complex and do not strictly follow Gaussian curves. However, here too, Gaussian curves can serve as useful approximations. 

It is a guiding principle of this book to generate all the data that subsequently are analysed by the methods developed for the task. Thus, we need functions for the generation of spectra and chromatographic concentration profiles. Generally, we use Gaussians or linear combinations of Gaussians for the purpose. This is a matter of convenience rather than a necessity. 

A Gaussian curve is characterised by peak position, peak height and peak width. Commonly, the half width is used, i.e. the width at half peak height. We accommodate this convention. In statistics, the Gaussian distribution is usually normalised to unit integral, however, this is not useful in the present context. 

The following function gauss. m creates a Gaussian curve with a given width and centre and a peak maximum of one, which is more convenient for our purposes. 

Multiplication with a scalar height then defines the amplitude of the peak. This is shown in the example Gauss Curve. m below  

---

The absorption spectra are modelled by Gaussians (see 3.2 Chromatography / Gaussian Curves). 

Gaussian curve and, thus, also has significance when dealing with chromatography theory. 

In this system the bands (zones) broaden because of diffusion effects and nonequilibrium. This broadening mechanism is fairly symmetrical and the resulting elution bands approach the shape of a Gaussian curve. This system best explains liquid or gas partition chromatography. The system may be viewed in two ways ... 

One may study zone broadening in gas chromatography by observing the shape of the elution peak which is Gaussian in ideal systems. The base width of the Gaussian curve is measured in standard deviation units, therefore... 

In chromatography, represents the width of the peak at half-height — 2.35a) and a the variance of the peak. The width of the peak at the base is labelled w and is measured at 13.5 per cent of the height. At this position, for the Gaussian curve, u) = 4ahj definition. 

Numerical calculations involving the Error curve are certainly made very easy by means of the spreadsheet. This applies not only to statistical considerations, but to simulation of chromatograms as well. Explanation of band shapes is elucidated by attributing their characteristics to those of the Gaussian curve. Thus, bandwidth is related to a, the standard deviation, V., the retention volume to p, the mean, as is well known in contemporary discussion of chromatography. Similarly, one can... 

Figure 21-3 Schematic gas chromatogram showing measurement of retention time (0 and width at half-height (wi/2). The width at the base (w) is found by drawing tangents to the steepest parts of the Gaussian curve and extrapolating down to the baseline. The standard deviation of the Gaussian curve is a. In gas chromatography, a small volume of CH injected with the 0.1- to 2-fxL sample is usually the first component to be eluted.

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 (4o) of the Gaussian curve and, thus, also has significance when dealing with chromatography theory. 

The shape ofthe elution curve for a pulse injection can be approximated by the Gaussian error curve for AT > 100, which is almost the case for column chromatography [2]. The value of N can be calculated from the elution volume Vg (m ) and the peak width W (m ), which is obtained by extending tangents from the sides of the elution curve to the baseline and is equal to four times the standard deviation (Ty (m ) = as shown in Figure 11.7. 

A chromatography column of 10 mm i.d. and 100 mm height was packed with particles for gel chromatography. The interparticle void fraction e was 0.20. A small amount of a protein solution was applied to the column and elution performed in an isocratic manner with a mobile phase at a flow rate of 0.5 cm min. The distribution coefficient A of a protein was 0.7. An elution curve of the Gaussian type was obtained, and the peak width W was 1.30 cm . Calculate the Hs value of this column for this protein sample. 

This is an even function that has a maximum of 0.399 for x = 0. The function also has two inflection points x — 1 with a corresponding ordinate value of 0.242, which represents 60.6% of the maximum value of the function. The width of the function at the inflection points is approximately 2er (er = 1). In modern chromatography, Wj/2 represents the width of the peak at the half-height (wi/2 = 2.35er) and a2 the variance of the peak. The width of the peak at the base, w, is measured at 13.5% of the height. At this position, if the curve is Gaussian, w = 4a by definition. 

SEC by itself is not an absolute MW determination method but the analysis of the elution peak has been used extensively for estimating the molecular purity of dendrimers. If the shape of the elution peak of a size exclusion chromatography experiment is Gaussian, the total dispersion, oT of the curve is given by the sum of squares [47] ... 

Equation (10) is the basic elution curve equation it is a Poisson function, but when n is large, the function approximates to a normal error function or Gaussian function. In practical chromatography systems, n is always greater than 100 and, thus, all chromatographic peaks will be Gaussian or nearly Gaussian in shape. 

Driven by the concentration gradient, solutes naturally diffuse when contained in a fluid. Thus, a discrete solute band will diffuse in a gas or liquid, and because the diffusion process is random, it will produce a concentration curve that is Gaussian in form. This diffusion effect occurs in the mobile phase of both packed GC and liquid chromatography (LC) columns. The longer the solute band remains in the column, the greater will... 

Figure l. A computer simulated example of the need for super-resolution in chromatography. The coa site curve, shown as a solid line, is the sum of three gaussian components occurring at time channels 45, 50 and 58,... 

---