Some illustrative examples

In this section, we present some examples of singlet generalized MDFs. We confine ourselves to spherical particles in two dimensions. All illustrations given in this section were obtained by a Monte Carlo computation on a two-dimensional system consisting of 36 Lennard-Jones particles for which the pair potential is presumed to have the form 

We use (7 as our length unit, and hence the densities refer to particles per unit area a . 

FIGURE 5.24. The singlet functions Xc K) for spherical particles interacting via a Lennard-Jones potential (a = 1.0 and s/kT - 0.5 . The number density is indicated next to each curve. 

Suppose that the intermolecular potential for simple particles can be represented by 

FIGURE 5.25. The singlet functions Xb(v) as a function of density (indicated next to each curve) for spherical particles with parameters j = 1.0, s/kT = 0.5. 

In this section, we will attempt to illustrate the various principles and techniques introduced in the previous section with several simple examples. Special emphasis will be on the stretching modes of a molecule. 

The molecule N2F2 can exist in two geometrical forms, namely, cis and trans. Here we are only concerned with the trans isomer with C2h symmetry. The methodical derivation of the symmetry species of the vibrational modes is best conducted in tabular form, as shown below. 

The vibrational modes and the corresponding frequencies of lrans- 2 2- Note that only iq, iq, and U4 are stretching modes. 

The vibrational modes and their frequencies of CF4. Note that only one component is shown for the degenerate modes. 

This is a tetrahedral molecule with 7d symmetry. The derivation of the vibrational modes is summarized below. 

Traditionally, mixtures of two (or more) components were characterized and studied by examining the excess thermodynamic functions. These offer a kind of global view of the system. In this book, we have developed the local point of view of mixtures. This view consists of the KB integrals, the local composition, the PS, and the solvation thermodynamic quantities. All of these convey local information of the system. Therefore, the study of these quantities provides new and complementary information on the local behavior around each molecule in the mixture. We present, in this last section, a few illustrative examples of systems for which such a complete set of local quantities is available. 

1 Lennard-Jones particles with the same e but different diameter a 

The first example is a mixture of two kinds of Lennard-Jones (LJ) particles. These particles are defined by the pair potentials 

The pair correlation functions for this system were calculated by solving the Percus-Yevick equations as described in section 2.9, and Appendices D and E. 

The calculations were done for mixtures at eleven mole fractions xA = 0.01, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.99, and a fixed volume density tj = 0.4. The functions gij(R) were calculated at 100 points in the range of (0, Rmax) where Rmax = 12ctaa- The KB integrals were calculated as 

This observation is a striking example of the effect of MW on selectivity, because the less polar mechanism involving radical reactions is now disappearing in comparison with the more polar S Ar mechanism. 

A second example is the competition between Diels-Alder cycloaddition and Michael addition during the reaction of 2-methoxythiophene III with DMAD under solvent-free conditions [73] (Fig. 4.12). 

Microwaves were shown to affect both reactivity and selectivity. The effect on yield is rather limited in the Diels-Alder reaction and found to be higher in the Michael addition. This process was even more favored by use of acetic acid as the solvent. These results can be explained by the natural assumption that the transition state leading to Michael addition (M) much more polar than that leading to Diels-Alder cycloaddition (DA). 

These assumptions were confirmed by calculations performed on the common GS and each TS, and consideration of the dipole moments. When one considers the relative dipole moments of the TS, the polarity is higher for that leading to Michael addition. Consequently, under MW activation, one can expect enhancement in M compared with DA. The most important factor affecting the selectivity is the solvent a protic solvent stabilizes the more polar TS, thus strengthening the tendency revealed in its absence. 

To illustrate these trends, we now present some typical illustrative examples. These were selected because strict comparisons of microwave and classical heating activation were made under similar conditions (time, temperature, pressure, etc. ..) for the same reaction medium and using, preferably, a single mode system equipped with stirring. They mostly involve reactions performed under solvent-free conditions or, occasionally, in a nonpolar solvent, because these conditions also favor observation of microwave effects. 

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Cyclization reactions effected by intramolecular attack of the heteroatom on a nitrile group provide a useful source of 2-amino heterocycles. Some illustrative examples are depicted in Scheme 16. 

Ring closures which depend on the conversion of the heteroatom into an electrophile are mostly associated with the formation of thiophene, selenophene and tellurophene rings and some illustrative examples are shown in Scheme 17. The last example which concerns the conversion of reaction with isocyanides is of particular interest since it appears to entail the attack of an electrophilic nitrogen species on the aryl ring. 

The synthetic application of reactions based upon the intramolecular addition of a carbanion or its enamine equivalent to a carbonyl or nitrile group has been explored extensively. One class of such reactions, namely the Dieckman, has already been discussed in Section 3.03.2.2, since ring closure can often occur so as to form either the C(2)—C(3) or C(3)—C(4) bond of the heterocyclic ring. Some illustrative examples of the application of this type of ring closure are presented in Scheme 46. 

In contrast to the previous section, there is little information available on C NMR spectra of pyrazoles and indazoles in books or reviews. Consequently, this section will treat the literature results in some detail, giving the principal references and some illustrative examples. 

Halogen-substituted 2-pyridones are key intermediates for further metal-catalyzed coupling reactions and the halogenation of these scaffolds has already been described in previous sections. In the following section, a variety of C - C and C - N cross-coupling reactions under microwave-assisted conditions are described with some illustrative examples. 

The products of incomplete combustion may be associated with particulate matter before their discharge into the atmosphere, and these may ultimately enter the aquatic and terrestrial environments in the form of precipitation and dry deposition. It is therefore essential to ensure total destruction of the contaminants, generally by raising the temperature. The spectrum of compounds that have been examined is quite extensive, and several of them are produced by reactions between hydrocarbons and inorganic sulfur or nitrogen constituents of air. Some illustrative examples involving other types of reaction include the following ... 

The energy for the fission of the covalent bond in organic contaminants is normally supplied thermally using thermodynamically accessible chemical or biochemical reactions, or by the introduction of catalysts to lower the activation energy of the reactions. There has been interest, however, in using electrical energy in a number of forms to carry out these reactions. A selection of processes for the destruction of contaminant is noted with some illustrative examples. 

The role of temperature may be of particular significance for mixed cultures of organisms in the natural environment, or when they are used in laboratory studies. Temperature may effect important changes both in the composition of the microbial flora and on the rates of reaction. Greatest attention has hitherto been directed to hydrocarbons and PCBs. Some illustrative examples of important issues that are especially relevant to bioremediation include the following ... 

In addition to their direct role in the bioremediation of contaminants, plants can play an important role in their dissemination through consumption by ruminants. The detoxification and metabolism of higher plants has been reviewed by Sandermann (1994), and some illustrative examples include the following ... 

As for waste from the production of chemicals, the array of structures represented by agrochemicals is truly enormous. Only some illustrative examples are provided, and it is important to emphasize that not only the original compound, but also potential metabolites should be considered. The pathways for biodegradation of many of the structures have been presented in Chapter 9 and reference should be made to these for details. There is increased interest in the degradation of agrochemicals after application, and abiotic reactions including photochemical degradation that are important on the soil surface are discussed in Chapter E... 

Process economics for biological products was discussed by Harrison et al. (op. cit., pp. 334—369) and Datar and Rosen [in Asenjo (ed.). Separation Processes in Biotechnology, DeUcer, New York, 1990, pp. 741-793] at length, and also by Ladisch (op. cit., pp. 401—430). They provided some illustrative examples with cost an yses. Bioprocess design software can also prove helpful in the overall design process (Harrison et al., op. cit., pp. 343-369). 

This section on flowsheets basically aims to provide some illustrative examples of the use of the various mineral processing unit operations that have been described. A general flowsheet involving almost all the unit operations pertinent to mineral processing is shown in Figure 2.32. The others refer specifically to beach sands, lead-zinc concentration, molybdenum, and the rare earths. 

Nature gives us some illustrative examples of iterative methodologies in its biochemical mechanisms. The fatty acid-polyketide biosynthesis is one of them. The assembly of acyl units by sequential Claisen-type condensations to form a polyketide or fatty acid takes place at a multi-enzyme complex, at which the initial molecule is lengthened by one C2-unit per pass of a reaction cycle (Fig. 2). 

Some illustrative examples of the application of 14N NMR spectroscopy in sulfur-nitrogen chemistry include (a) studies of the (NSC1)3<->3NSC1 equilibrium in solution28 and (b) identification of the S-N species present in solutions of sulfur in liquid ammonia.29... 

TMC ATRA reactions can also be conducted intramolecularly when alkyl halide and alkene functionalities are part of the same molecule. Intramolecular TMC ATRA or atom transfer radical cyclization (ATRC) is a very attractive synthetic tool because it enables the synthesis of functionalized ring systems that can be used as starting materials for the preparation of complex organic molecules [10,11], Furthermore, halide functionality in the resulting product can be very beneficial because it can be easily reduced, eliminated, displaced, converted to a Grignard reagent, or if desired serve as a further radical precursor. The use of copper-mediated ATRC in organic synthesis has been reviewed recently and some illustrative examples are shown in Scheme 3 [10,11,31,32,33],... 

A graphical representation of the multilevel approach is shown in Fig. 4. All three models are now commonly accepted and are widely used by a number of research groups (both academic and industrial) around the world. In a recent paper, we have given an overview of the three models as they are employed at the University of Twente, together with some illustrative examples (Van der Hoef et al., 2004). In this chapter, we will focus on the technical details of each of the models, much of which has not been published elsewhere. The development of detailed closure relations from the simulations, as indicated in Fig. 3, is still ongoing. Some preliminary results for both the drag-force closures and solid pressure will be presented in the Sections II and III. In this chapter, we will... 

Rather than to draw up an exhaustive list of pH sensing fluorescent probes which can be found in several reviews 112,119) we will try to define different classes of compounds whose fluorescence is sensitive to protonation and give some illustrative examples. 

From all these projects, there are several worth mentioning as a result of their achievements, however our present overview only addresses the salient points of some illustrative examples. Emphasis, as stated earlier, is placed on those projects where materials issues took centre stage in the research activities or material innovations were considered instrumental for further progress. A more... 

Some illustrative examples of all these considerations are given below ... 

Some illustrative examples from the field of polyquinanes are the synthesis of some derivatives of bicyclo[3.3.0]octane 6 (Scheme 6.7) [12] [15] -which have been used in the total syntheses of coriolin, hirsutic acid and quadrone- and the synthesis of triquinacene 7 and some of its derivatives. The retrosynthetic analysis of perhydrotriquinacene-l,4,7-trione (7a) is shown in Scheme 6.7bis. In the actual synthesis the hydroxy groups must be protected either as trialkylsilyl ethers or more conveniently as benzyl ethers [16] [17]. 

Numerous examples have been reported of transition metal alkyl complexes which can be converted into carbene complexes by a-hydride abstraction [429-431], This process can also proceed intramolecularly by oxidative insertion of the metal into the a-C-H bond. Figure 3.8. shows some illustrative examples of iron, rhenium, and... 

Inter- and intramolecular (cyclometallation) reactions of this type have been ob-.served, for instance, with titanium [408,505,683-685], hafnium [411], tantalum [426,686,687], tungsten [418,542], and ruthenium complexes [688], Not only carbene complexes but also imido complexes L M=NR of, e.g., zirconium [689,690], vanadium [691], tantalum [692], or tungsten [693] undergo C-H insertion with unactivated alkanes and arenes. Some illustrative examples are sketched in Figure 3.37. No applications in organic synthesis have yet been found for these mechanistically interesting processes. 

On the one hand, this particular feature makes it more difficult to distinguish between reactions involving radical cations, free radicals or carbenium ions, but on the other hand the chemist acquires an additional tool to control the course of the intended reaction. Some illustrative examples of cyclization reactions that utilize cleavage of the radical cations, primarily generated by single-electron oxidation, will be given in the following sections. 

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