General Applicability

The principal advantage of the DDA is that it is completely flexible regarding the geometry of the scatterer, being limited only by the need to use the dipole size d small compared to both any structural 

It implies that the number of dipoles N increase with both size and h. The expected accuracy of cross sections is then several percents, if h belongs to the well-tested range, approximately described as 

For larger n the accuracy of the simulation with default dipole size deteriorates, and smaller, hence more dipoles must be used to improve it. However, there is no strict limitation on n -accurate results can be obtained for a wide range of n, given enough computational resources. In this respect state-of-the-art DDA formulations (Sec. 2.4.3) can be of great help. 

For particles smaller than A, e.g. nanoparticles, Eq. (2.79) is definitely not relevant. In this case the main requirement for d is that it should be small enough to resolve fine details of particle shape. For instance, for a compact shape (like sphere or cube) it is recommended to use at least 10 dipoles along the smallest dimension, no matter how small the particle is. Therefore, for nanoparticles N is almost independent of size. However, N does depend on h, which is critical due to the wide range of n used in plasmonic applications. In more details these issues are discussed in Sec. 2.6. 

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The method used here is based on a general application of the maximum-likelihood principle. A rigorous discussion is given by Bard (1974) on nonlinear-parameter estimation based on the maximum-likelihood principle. The most important feature of this method is that it attempts properly to account for all measurement errors. A discussion of the background of this method and details of its implementation are given by Anderson et al. (1978). 

Bucherer reaction Bucherer discovered that the interconversion of 2-naphthol and 2-naphthylamine through the action of alkali and ammonia could be facilitated if the reaction was carried out in the presence of (HSO3]" at about 150 C. This reaction is exceptional for the ease with which an aromatic C —OH bond is broken. It is not of general application, it is probable that the reaction depends upon the addition of [HSO3]" to the normally unstable keto-form of 2-naphthol, and subsequent displacement of —OH by —NH2. 

The reaction is of general application and of great importance, and a large number of syntheses have been effected by its use. 

This method has a very general application range analysis for metals in crude oils, in their various distillation cuts, and in their residues as well as for metals contained in spent lubricating oils, water, lubricants, etc. 

The importance of the solid-liquid interface in a host of applications has led to extensive study over the past 50 years. Certainly, the study of the solid-liquid interface is no easier than that of the solid-gas interface, and all the complexities noted in Section VIM are present. The surface structural and spectroscopic techniques presented in Chapter VIII are not generally applicable to liquids (note, however. Ref. 1). There is, perforce, some retreat to phenomenology, empirical rules, and semiempirical models. The central importance of the Young equation is evident even in its modification to treat surface heterogeneity or roughness. ... 

Students are sometimes puzzled at the extraordinarily general application of the process of recrystallisation, since it may appear to them to depend on the assumption that the impurities are more soluble than the... 

Distillation. This process is carried out in the apparatus shown in Figs. 36, 37 and 38. The distillation apparatus (Fig. 36) is of very general application and is made in one piece. 

Another valuable drying agent of general application is anhydrous calcium sulphate, marketed under the name of Drierite. It is rapid in its action, but has only a limited drying capacity because it forms the hydrate 2CaS04,H20 and thus combines with only 6-6 per cent, of its weight of water. The vapour pressure of the system ... 

P -f lOROH -f 5Br, — 2H3PO, -f lORBr -f 2H,0 The reaction is of general application with primary alcohols (n propyl to n hexadecyl) the yields are over 90 per cent, of the theoretical, but with secondary alcohols the yields are 50-80 per cent. in the latter case a small quantity of high boiling point by-product is also formed which can, however, be readily removed by fractional distillation. The reaction is conveniently carried out in a special all glass apparatus. 

The reaction is of general application the yields of primary alcohols approach the theoretical values, and for secondary alcohols are 85-95 per cent. The process is best carried out with the aid of a special apparatus. 

The method is generally applicable when other modes of esterification are either slow, inefficient, or likely to cause isomerisation it is, however, time-consuming and expensive. Small quantities of acid impurities are sometimes produced, hence it is advisable to wash the ester with saturated sodium bicarbonate solution. The silver salt can usually be prepared by dissolving the acid in the calculated quantity of standard ammonium hydroxide solution and... 

Molecular mechanics methods are not generally applicable to structures very far from equilibrium, such as transition structures. Calculations that use algebraic expressions to describe the reaction path and transition structure are usually semiclassical algorithms. These calculations use an energy expression fitted to an ah initio potential energy surface for that exact reaction, rather than using the same parameters for every molecule. Semiclassical calculations are discussed further in Chapter 19. 

Syntheses of alkenes with three or four bulky substituents cannot be achieved with an ylide or by a direct coupling reaction. Sterical hindrance of substituents presumably does not allow the direct contact of polar or radical carbon synthons in the transition state. A generally applicable principle formulated by A. Eschenmoser indicates a possible solution to this problem //an intermolecular reaction is complex or slow, it is advisable to change the educt in such a way. that the critical bond formation can occur intramolecularly (A. Eschenmoser, 1970). 

In synthetic target molecules esters, lactones, amides, and lactams are the most common carboxylic acid derivatives. In order to synthesize them from carboxylic acids one has generally to produce an activated acid derivative, and an enormous variety of activating reagents is known, mostly developed for peptide syntheses (M. Bodanszky, 1976). In actual syntheses of complex esters and amides, however, only a small selection of these remedies is used, and we shall mention only generally applicable methods. The classic means of activating carboxyl groups arc the acyl azide method of Curtius and the acyl chloride method of Emil Fischer. 

Unique chemistry is associated with the cyclopentenone all five carbon atoms can be functionalized, and the endo-methyl groups of the acetonide assure clean stereoselective addition of the alkenylcopper reagent from the convex side. The use of the acetonide group to control enolate regioselectivity and to mask alcohols should be generally applicable. 

A variation of Hantzsch s synthesis, using thioureas in conjunction with a-diazoketones in place of a-halogenoketones, has proved to be generally applicable. In this manner, King and MUler (310) obtained 2-amino-4-phenylthiazole in 67% yield. A wide range of 4-substituted 2-aIkyl (or aryl) aminothiazoles and 2-arylimino-3,4-diarylthiazolines have been prepared by Hampel and Muller (627, 665, 666). 

The reactivity of the halogens decreases m the order F2 > CI2 > Br2 > I2 Fluo rme is an extremely aggressive oxidizing agent and its reaction with alkanes is strongly exothermic and difficult to control Direct fluonnation of alkanes requires special equip ment and techniques is not a reaction of general applicability and will not be discussed further... 

The comparison plot offers a particularly simple and direct means of comparing the shapes of a pair of isotherms but for more general applications which involve a numt>er of samples of a solid covering a wide range of specific surface, the a,-method is preferable. The j-curve represents a convenient way of recording and using the reference isotherm. 

Although many quantitative applications of acid-base titrimetry have been replaced by other analytical methods, there are several important applications that continue to be listed as standard methods. In this section we review the general application of acid-base titrimetry to the analysis of inorganic and organic compounds, with an emphasis on selected applications in environmental and clinical analysis. First, however, we discuss the selection and standardization of acidic and basic titrants. 

As with acid-base and complexation titrations, redox titrations are not frequently used in modern analytical laboratories. Nevertheless, several important applications continue to find favor in environmental, pharmaceutical, and industrial laboratories. In this section we review the general application of redox titrimetry. We begin, however, with a brief discussion of selecting and characterizing redox titrants, and methods for controlling the analyte s oxidation state. 

If may be apparenf fo fhe reader af fhis sfage fhaf, when lasers are used as specfroscopic sources, we can no longer fhink in terms of generally applicable experimenfal mefhods. A wide variefy of ingenious techniques have been devised using laser sources and if will be possible fo describe only a few of fhem here. 

A number of reductive procedures have found general applicability. a-Azidoketones may be reduced catalytically to the dihydropyrazines (80OPP265) and a direct conversion of a-azidoketones to pyrazines by treatment with triphenylphosphine in benzene (Scheme 55) has been reported to proceed in moderate to good yields (69LA(727)23l). Similarly, a-nitroketones may be reduced to the a-aminoketones which dimerize spontaneously (69USP3453279). The products from this reaction are pyrazines and piperazines and an intermolecular redox reaction between the initially formed dihydropyrazines may explain their formation. Normally, if the reaction is carried out in aqueous acetic acid the pyrazine predominates, but in less polar solvents over-reduction results in extensive piperazine formation. 

From the foregoing discussion it is evident that the most general methods for the synthesis of pyrazines, quinoxalines and phenazines fall into type A and type B categories, but other methods do exist. Although most of these are not of such general applicability, they are worthy of comment. 

Synthesis of heterocycles by forming C—X bonds by radical reactions is not a generally applicable method, and seems not to be useful for making small rings. However, the attack of thiol radicals on double bonds can be a practical synthetic route, such as in the conversion of 1-hexene-7-thiol to thiepane (Section 5.17.3.3.1). 

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