Shapes of Chromatographic Peaks

In the experiments on accelerators the required short-lived products of nuclear reactions are converted into volatile compounds and separated by gas-solid chromatography techniques in a continuous regime. Open columns of a meter in length and a few millimeters in diameter are used. The linear velocity of the carrier gas has varied from centimeters to meters per second. To optimize the separation process, it is important to understand how the experimental conditions, the properties of the separated species and other factors affect the shape and position of the resulting adsorption zone. 

In general the term zone profile - plc(z) or pTC(z) - used below means the (normalized) distribution of the adsorbate as a function of the column longitudinal coordinate after EOB — either found experimentally or expected from theory. With small number of atoms and short lifetimes, the meaning of the term might need to be refined. The experimental chromatogram is now usually a histogram of the number of decay events observed within short sections of the column. The events may be identified in the course of the run or after EOB. 

For long-lived y-active nuclides obtained with sufficient yield, which is the case for lighter homologs of TAEs, the profiles can be revealed by spectrometric measurements from outside the column. The profiles of short-lived a-activities can be measured only when the column is a channel formed by surfaces of electronic particle detectors. Such measurements are done in real time of course, they also work well for s.f. nuclides. The latter can also be registered in open columns made of the materials which can serve as solid-state track detectors of fission fragments (fused silica, mica) see Sect. 1.2.2. 

The shape of the zone, at least a characteristic of its width, is of major concern in all chromatographic techniques. Jonsson [4] and Giddings [5] seem to discuss the case and peculiarities of open columns in more details than do other general monographs on chromatography. 

Necessarily, the zone profiles within the column and the more common elution curves do not have the same statistical parameters (see below about the statistical moments). Indeed, even if the internal zone profile, which broadens with time, is completely symmetrical, the elution curve must be skewed because the later fractions of the elution peak spend a longer time in the column [4] than the earlier ones. It is preferable to discuss the internal zone profiles because they provide a common basis for considering the regularities of IC and TC. 

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The shapes of chromatographic peaks often resemble the Gaussian distribution, though distorted to a variable degree. Theory seldom finds shape functions for chromatographic peaks in explicit form. More frequently, only the Laplace transform of the appropriate differential equation can be solved. Then, even if the solution cannot be inversed, it allows evaluation of the statistical moments of the distribution [4],... 

Rate law The empirical relationship describing the rate of a reaction in terms of the concentrations of participating species. Rate theory A theory that accounts for the shapes of chromatographic peaks. 

The plate theory successfully accounis for the Gaussian shape of chromatographic peaks and their rate of movemeni down a column. The theory was ul-limaicly abandoned in favor of the rate theory however, because it fails to account for peak broadening in a mechanistic way. Neverihekss, the original terms for efficiency have been carried over lo ihe rate iheory. This nomenclature is perhaps unfortunate because it tends to perpetuate Ihe myth ihat a column contains plates where equilibrium conditions exist. In fact, the equilibrium state can never be reah/cd with the mobile phase in constant motion. 

For given values of the retention volumes above, the shape of chromatographic peaks depends on the value of the kinetic parameter k jF. If the only slow process is equilibration of injected substance vapour in the polymer bulk, the simplest theories of chromatographic rate processes [113] give for 7c, ... 

Since the shape of chromatographic peaks is essentially the same as a standard error plot in statistics, the peak width at the base may be written in terms of standard deviation. 

In a typical pulse experiment, a pulse of known size, shape and composition is introduced to a reactor, preferably one with a simple flow pattern, either plug flow or well mixed. The response to the perturbation is then measured behind the reactor. A thermal conductivity detector can be used to compare the shape of the peaks before and after the reactor. This is usually done in the case of non-reacting systems, and moment analysis of the response curve can give information on diffusivities, mass transfer coefficients and adsorption constants. The typical pulse experiment in a reacting system traditionally uses GC analysis by leading the effluent from the reactor directly into a gas chromatographic column. This method yields conversions and selectivities for the total pulse, the time coordinate is lost. 

Results with high bias—Coelution with other target or non-target analytes affects the quantitation of the target analytes due to the degradation of chromatographic peak shape. 

Another factor that contributes to Rs is the plate count N. However, we have seen in section 1.5 that optimization through an increase in N is expensive, not only in terms of equipment and columns, but also in terms of analysis time. Therefore, as long as the shape of the peaks and the plate height (length of the column divided by N) are satisfactory, we should not rely on the number of plates for optimization, unless as a last resort. Methods which may be used to optimize the chromatographic system with respect to the required number of plates will be described in chapter 7. 

Chapter 4 starts with some basic equations, which relate the molecular-kinetic picture of gas-solid chromatography and the experimental data. Next come some common mathematical properties of the chromatographic peak profiles. The existing attempts to find analytical formulae for the shapes of TC peaks are subject to analysis. A mathematical model of migration of molecules down the column and its Monte Carlo realization are discussed. The zone position and profile in vacuum thermochromatography are treated as chromatographic, diffusional and simulation problems. 

We now return to the simulated curves, in order to show how to extract the area, position, and width from a chromatographic peak. Several simple methods are available for symmetrical peaks (and even more for a special subset of these, Gaussian peaks), but since chromatographic peaks are often visibly asymmetric (and in that case obviously non-Gaussian), we will here use a method that is independent of the particular shape of the peak. It is a standard method that, in a chromatographic context, is described, e.g., by Kevra etal. in J. Chem. Educ. 71 (1994) 1023. 

In practice, it is seldom necessary to calculate the value of resolution. More frequently, the chromatographer simply looks at the shape of the peaks and estimates the Rs value from the peak shape. Of course, the concentrations and/or responses of the components in a mixture are rarely equal. Figure 7 illustrates the significance of the resolution value for two components with different relative concentrations. 

The shape of the partition isotherms in the chromatographic process has a decisive influence on the shape of the peak. Let us now differentiate between two possible cases ... 

The information given by the positional-shape criteria is interesting, not only when the chromatographic peaks are asymmetric or have low efficiencies, but also with symmetric peaks which are very close to each other. Poorly defined optima obtained with a positional criterion often become clearer when the shape of the peaks is considered. 

Figure 6.7 Sketch illustrating the importance of the number of mass spectrometric data points (either full spectra or SIM or MRM) acquired across a chromatographic peak, for adequate representation of the shape of the peak in order to calculate accurate peak areas and/or heights. Usually 10 scans (or SIM or MRM acquisition windows) are considered necessary to provide an adequate representation of the peak centroid (and thus retention time) and peak area.

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