Relation to peak width

If you look at the nmr spectra of many different kinds of organic compounds, you will notice that some resonances are sharp and others are broad. In a few spectra, all of the peaks may be broad as the result of poor spectrometer performance, but this is not true for the spectra of Figures 9-29 (p. 312) and 24-2 (p. 1173) where, within a given spectrum, some resonances will be seen to be sharp and others broad. We can understand these differences by consideration of the lifetimes of the magnetic states between which the nmr transitions occur.1 The lifetimes of the states can be related to the width of the lines by the Heisenberg uncertainty principle. 

It is important to note that regardless of the mechanism of double-peak formation, the horizontal and vertical readouts in the superstoichiometric regions yield useful information, related either to peak width (W) or to peak height (//) (Fig. 2.24). Additionally, the distance Ar between equivalent points is the basis of FIA titrations, since the end points (Eq) are located at the apexes of the respective peaks, where the equivalent conditions are met. 

The rate of the initial decay of P (t) is related to the width of the overall absorption envelope for the 8727 to 8927 cm > frequency range of the experimental spectrum. If a Fourier transform were taken of a Lorentzian /(w) for only the principal peak in the spectrum, the decay rate would be significantly smaller. 

Recently, we have obtained better NMR data for the allomorphs with a Bruker instrument at 200 MHz for protons. The Cl signal of IIIj seemed to be a singlet, but its profile was broadened unsymme-trically to the lower ppm side and suggested additional weak peaks. The half-width was almost the same as that of cellulose I. The common characteristics related to half-width of the Cl signal within each families were clearly evident. 

Because the first deviation from linearity of retention diagrams is hard to detect, T, may be determined by using any parameter related to the width or asymmetry of the peak, penetration in polymer bulk causing tailing before any modification of retention volume is noticeable [218]. For example the ratio of peak half width and retention time exhibits a pronounced increase at T, [220]. 

The breadth of the envelope of the absorption spectrum in Figure 6(d) is, as in the case of direct dissociation, related to the width of the first peak of the autocorrelation function. It basically reflects the number of vibrational states in the upper manifold that have appreciable FC overlap with the initial-state wave function I gr., and are therefore excited,... 

Crystallite size represents the spatial extent of coherent domains in a solid. We see that the size of the coherent domain is related to the width of the diffraction peaks in reciprocal space. The Scherrer equation, which relates this size to peak width, assumes an extremely simple form in reciprocal space ... 

The constant phase-angle model does not give the best fit to experimental results in all cases. Based on the Cole-Cole model, Cosenza et al. (2009) found that the position of the maximum phase peak in the phase versus frequency plot shows a proportionality to the grain size and that the exponent is related to the width of the grain-size distribution. 

In experimental measurements, sueh sharp 5-funetion peaks are, of eourse, not observed. Even when very narrow band width laser light sourees are used (i.e., for whieh g(co) is an extremely narrowly peaked funetion), speetral lines are found to possess finite widths. Let us now diseuss several sourees of line broadening, some of whieh will relate to deviations from the "unhindered" rotational motion model introdueed above. 

Recalling that a separation is achieved by moving the solute bands apart in the column and, at the same time, constraining their dispersion so that they are eluted discretely, it follows that the resolution of a pair of solutes is not successfully accomplished by merely selective retention. In addition, the column must be carefully designed to minimize solute band dispersion. Selective retention will be determined by the interactive nature of the two phases, but band dispersion is determined by the physical properties of the column and the manner in which it is constructed. It is, therefore, necessary to identify those properties that influence peak width and how they are related to other properties of the chromatographic system. This aspect of chromatography theory will be discussed in detail in Part 2 of this book. At this time, the theoretical development will be limited to obtaining a measure of the peak width, so that eventually the width can then be related both theoretically and experimentally to the pertinent column parameters. 

To day peak widths are rarely used in chromatographic analysis except for the purpose of calculating peak areas. Peak widths, however, can provide a means of measuring the diffusivity of a solute which is a function of the molecular weight. Consequently, if a reliable relationship between diffusivity and molecular weight can be identified, then the molecular weight of the solute can be assessed. Peak widths of solutes eluted from an open tube can give very precise values of diffusivity. There are a number of equations that purport to relate diffusivity to... 

The quantitative determination of a component in gas chromatography using differential-type detectors of the type previously described is based upon meas urement of the recorded peak area or peak height the latter is more suitable in the case of small peaks, or peaks with narrow band width. In order that these quantities may be related to the amount of solute in the sample two conditions must prevail ... 

The peak-shaped response of differential-pulse measurements results in unproved resolution between two species with similar redox potentials, hi various situations, peaks separated by 50 mV may be measured. Such quantitation depends not only upon the corresponding peak potentials but also on the widths of the peak. The width of the peak (at half-height) is related to the electron stoichiometry ... 

Most detectors are concentration sensitive devices and thus the peak height will be proportional to the maximum concentration in the peak, which, in turn, will be proportional to the total area of the peak. The total area of the peak is proportional to the total mass of solute contained in the peak providing it is not excessively tailing. As the peak height is inversely related to the peak width, then, if peak heights are to be used for analytical purposes, all parameters that can affect the peak width must be held constant. This means that the capacity factor of the solute (k ) must remain constant and, consequently, the solvent... 

X-Ray diffraction has an important limitation Clear diffraction peaks are only observed when the sample possesses sufficient long-range order. The advantage of this limitation is that the width (or rather the shape) of diffraction peaks carries information on the dimensions of the reflecting planes. Diffraction lines from perfect crystals are very narrow, see for example the (111) and (200) reflections of large palladium particles in Fig. 4.5. For crystallite sizes below 100 nm, however, line broadening occurs due to incomplete destructive interference in scattering directions where the X-rays are out of phase. The two XRD patterns of supported Pd catalysts in Fig. 4.5 show that the reflections of palladium are much broader than those of the reference. The Scherrer formula relates crystal size to line width ... 

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