System peaks theory

Figure 5. Mass chromatogram of photoinitiators in a polymer system. Peak 1 benzoin, molecular weight 212.24 (theory), 212.6 (MC-2). Peak 2 hydrobenzoin, molecular weight 214.26 (theory),...

We have discussed the theory of system peaks in linear chromatography [20]. The discussion is based on the use of the equilibrium-dispersive model. The mass balance equations are written for the n components of the sample and for the p additives ... 

Equation 13.22ab permits the calculation of the relative response factor of each component in indirect detection (see next section). The mass balance of the additive permits the prediction of the sign and size of the area of its system peak. Thus, theory permits a quantitative explanation of the system peak phenomenon and an accurate prediction of the chromatograms obtained, provided that an acceptable... 

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. 

At this point, it is important to stress the difference between separation and resolution. Although a pair of solutes may be separated they will only be resolved if the peaks are kept sufficiently narrow so that, having been moved apart (that is, separated), they are eluted discretely. Practically, this means that firstly there must be sufficient stationary phase in the column to move the peaks apart, and secondly, the column must be constructed so that the individual bands do not spread (disperse) to a greater extent than the phase system has separated them. It follows that the factors that determine peak dispersion must be identified and this requires an introduction to the Rate Theory. The Rate Theory will not be considered in detail as this subject has been treated extensively elsewhere (1), but the basic processes of band dispersion will be examined in order to understand... 

The conspicuous separation between the cathodic and anodic peak potentials was initially interpreted in terms of the simple theory for redox polymers as a kinetic effect of slow heterogeneous charge transfer the thermodynamic redox potential of the whole systems was calculated from the mean value between Ep and Ep ... 

According to the quantum transition state theory [108], and ignoring damping, at a temperature T h(S) /Inks — a/ i )To/2n, the wall motion will typically be classically activated. This temperature lies within the plateau in thermal conductivity [19]. This estimate will be lowered if damping, which becomes considerable also at these temperatures, is included in the treatment. Indeed, as shown later in this section, interaction with phonons results in the usual phenomena of frequency shift and level broadening in an internal resonance. Also, activated motion necessarily implies that the system is multilevel. While a complete characterization of all the states does not seem realistic at present, we can extract at least the spectrum of their important subset, namely, those that correspond to the vibrational excitations of the mosaic, whose spectraFspatial density will turn out to be sufficiently high to account for the existence of the boson peak. 

Another study (Bedani et al., 2006) starts from the multidimensional sampling theory (Murphy et al., 1998a), which is discussed in Chapter 2. This sampling theory states that one needs to sample the first dimension separation system at least three to four times per peak width for maximum resolution. Bedani et al. then equate the second-dimension total analysis time to the first-dimension narrowest peak standard deviation. This defines the second-dimension operational parameters. All other parameters can be derived from this balance and Bedani s study goes through this and discusses how the rest of these variables are obtained. 

Scattering and Disorder. For structure close to random disorder the SAXS frequently exhibits a broad shoulder that is alternatively called liquid scattering ([206] [86], p. 50) or long-period peak . Let us consider disordered, concentrated systems. A poor theory like the one of Porod [18] is not consistent with respect to disorder, as it divides the volume into equal lots before starting to model the process. He concludes that statistical population (of the lots) does not lead to correlation. Better is the theory of Hosemann [158,211], His distorted structure does not pre-define any lots, and consequently it is able to describe (discrete) liquid scattering. The problems of liquid scattering have been studied since the early days of statistical physics. To-date several approximations and some analytical solutions are known. Most frequently applied [201,212-216] is the Percus-Yevick [217] approximation of the Ornstein-Zernike integral equation. The approximation offers a simple descrip-... 

As yet, the number of applications is limited but is likely to grow as instrumentation, mostly based on existing CE systems, and columns are improved and the theory of CEC develops. Current examples include mixtures of polyaromatic hydrocarbons, peptides, proteins, DNA fragments, pharmaceuticals and dyes. Chiral separations are possible using chiral stationary phases or by the addition of cyclodextrins to the buffer (p. 179). In theory, the very high efficiencies attainable in CEC mean high peak capacities and therefore the possibility of separating complex mixtures of hundreds of... 

Yet at 0.1 V/s the anodic peak due to the oxidation of the radical cation does not exhibit the shape characteristic of stripping of a solid phase. At faster scan rates the anodic peak broadens considerably and splits into two peaks the same behavior is noticeable in Figure 1. We do not have an explanation for this phenomenon. A recent theoretical treatment of redox molecules attached to electrode surfaces predicts that under certain conditions an anodic surface wave can broaden and split with increasing scan rate in a manner shown in Figure 3 (16). However the same theory predicts that the corresponding cathodic peak normalized to constant scan rate will increase with increasing scan rate. The latter prediction is not observed in our system. 

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