Very Similar Components

In this section, we consider a system of two components in the T, P, a b ensemble. Similar arguments and results apply to multicomponent systems. We have chosen the T, P, Na, ensemble because the isothermal-isobaric systems are the most common ones in actual experiments. 

Although the meaning of this statement is obvious, some care is necessary for a more precise definition. It is implicitly understood that the centers and the convention of the orientational angles have been chosen in the same way for the two molecules A and B. For instance, for H2O and D2O, we adopt the same convention for R and SI for the two molecules, in wich case the pair potential will be almost the same for H2O-H2O and D2O-D2O. It will differ markedly if we choose different conventions for the configurations of H2O and D2O. If the two components A and B are very different, say H2O and CH4, then the choice of a convention for X cannot be made in the same way for the two particles. In such cases, the concept of very similar is not expected to hold either, and the whole problem becomes irrelevant. 

The connection between the chemical potential and statistical mechanics is (see Section 3.5 and Appendix 9-F for more details) 

consider a system of N particles of type A only. The chemical potential for such a system (with the same P and T as before) is 

we have expressed the chemical potential in terms of only intensive parameters T, P, and 

Here we have expressed the chemical potential in terms of the intensive parameters T, P, and Xa. A system for which relation of the form (6.5.7) is obeyed by each component is called a symmetrical ideal solution. 

Relation (6.5.7) is important since it gives an explicit dependence of the chemical potential on the composition, the fruitfulness of which was recognized long ago. This relation has been obtained at the expense of the strong assumption that the two components are very similar. We know from experiment that a relation such as (6.5.7) holds also under much weaker conditions. 

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The nondispersive (filter-based) PAS detector consists of very similar components to the original setup used by Alexander Bell an IR light source, a chopper wheel and a measurement cell. In addition, optical filters have been added to improve selectivity, as has a pump to introduce the sample into the measurement cell. 

Consider two very similar components, propylene [1] and propane [3], with close boiling points. The ideal relative volatility is now defined as... 

Whereas the ideal solution model applies over the entire range of concentrations, but only for very similar components, the ideally dilute solution model applies to any solution, but only over a very limited range of concentrations. From a microscopic point of view, the ideally dilute solution holds as long as solute molecules are almost always completely surrounded by solvent molecules and rarely interact with other solute molecules. 

This method provides the exact solutions for ideal systems at constant temperature and pressure. It is successful in describing diffusion flow in (i) nearly ideal mixtures, (ii) equimolar counter diffusion where the total flux is zero (Nt = 0), (iii) diffusion of one component through a mixture of n — 1 inert components, and (iv) pseudo-binary case and the diffusion of two very similar components in a third. 

Very similar components A sufficient condition for SI solutions... 

For notational convenience, we shall discuss a two-component mixture of A and B. The generalization for multicomponent system is quite straightforward. We consider a system of two components in the T, P, NA, NB ensemble. We have chosen the T, P, NA, NB ensemble because the isothermal-isobaric systems are the most common ones in actual experiments. By very similar components, we mean, in the present context, that the potential energy of interaction among a group of n molecules in a configuration X is independent of the species we... 

Our simplified analysis indicates that a near-complete separation is possible, even for very similar components (Ka —> K ). In the latter case, the price to pay is that the flows of the auxiliary (y) and eluent (I) phases may become excessive. Before relating actual flows to specific separation problems, we can estimate the minimal flows qualitatively. For instance, to recover products at concentrations in the product streams in fom-section SMB-systems (Fig. 6),similar to their original (feed) concentrations, the eluent or desorbent flow should equal the feed stream Poorly soluble components, low capacities of the phases and near-identical partition coefficients, lead to large internal process streams and thereby to voluminous equipment and a substantial energy consiunption. 

These food components have the common property of being labile and so are degraded by light, heat, or oxygen, which limits the sample extraction and cleanup, and especially its great complexity, with dozens (or even hundreds) of very similar components or isomers, which sometimes makes it extremely difficult to separate the components of each family in the sample. An additional problem is the lack of standard compounds for most of them, which makes the identification very complicated. The difficulty is usually overcome by coupling LC with MS, or with MS/MS, to determine the structure of the separated compounds. 

Usually, peak capacity is defined as the number of peaks per unit time, see Eq. (3.5). The peak capacity as a separation criterion proves to be important when very similar components have to be separated, such as components of a homologous series, for example, ohgomers in this case, one expects equidistant spacing of the peaks. With similar components, we can hardly expect different interactions and thus a good selectivity. The situation is similar in the case of complex mixtures and/or a difficult matrix. Again, realistically here we do not come any further via selectivity. A separation in this case will be possible, when the many (similar ) components can be eluted distributed as narrow peaks throughout the chromatogram. 

A procedure called chromatography automatically and simply applies the principles of these fractional separation procedures. Chromatography can separate very complex mixtures composed of many very similar components. The various types of chromatographic instrumentation... 

In Section 4.4 we showed that in a mixture of very similar components, the chemical potential of each component has the form... 

Clearly, if a B were chosen to be equal to in (4.105), we would have two very similar components. We have made the two components different by choosing different diameters in (4.105). The value of A b computed for this case is about 0.76 hence, the two components are not similar. We can now change the value of EbbI T and follow the dependence of A b on this parameter. Figure 4.5 shows the dependence of A b on ebbI T, where all other parameters are as in (4.105). Note that at about bbI T == 0.70, we get A b = 0. Thus, the two components, which may seem to be quite different in the conventional sense, are considered to be similar according to our definition. This illustrates the idea that the energy and diameter parameters in the Lennard-Jones potential can be adjusted in such a way that the resulting value of Aj b will be zero. In other words, the dissimilarity in the can be compensated by a dissimi-... 

We shall see in section 6.8 that relations of the form (6.5.7) could be obtained under much weaker assumption on the similarity of the two components. In fact, relation (6.5.7) could not have been so useful had it been restricted to the extreme case of very similar components, such as two isotopes. 

In this section, I have demonstrated potential of FUV spectroscopy in the 120-300 nm region in the classification of commercial food wrap films (three types of polyethylene (PE), poly vinylidene chloride (PVDC), and polyvinyl chloride (PVC) see Table 5.4). Sato and I have measured FUV spectra in the 120-300 nm region of six types of commercial polymer wrap films [53]. FUV spectroscopy enables classification of polymer thin films in a straightforward manner by using raw spectral data. We also studied identification of three types of polyethylene PE films from different commercial companies. The FUV spectra of these PE films, which have very similar components and additives, are easily separated. The two types of PVDC films can also be identified. The present study has revealed that FUV spectroscopy is a very promising tool for the polymer film analysis. 

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