Similarity methods

S is the path length between the points a and b. The Euler equation to this variation problem yields the condition for the reaction path, equation (B3.5.14). A similar method has been proposed by Stacho and Ban [92]. 

A term that is nearly synonymous with complex numbers or functions is their phase. The rising preoccupation with the wave function phase in the last few decades is beyond doubt, to the extent that the importance of phases has of late become comparable to that of the moduli. (We use Dirac s terminology [7], which writes a wave function by a set of coefficients, the amplitudes, each expressible in terms of its absolute value, its modulus, and its phase. ) There is a related growth of literatm e on interference effects, associated with Aharonov-Bohm and Berry phases [8-14], In parallel, one has witnessed in recent years a trend to construct selectively and to manipulate wave functions. The necessary techifiques to achieve these are also anchored in the phases of the wave function components. This bend is manifest in such diverse areas as coherent or squeezed states [15,16], elecbon bansport in mesoscopic systems [17], sculpting of Rydberg-atom wavepackets [18,19], repeated and nondemolition quantum measurements [20], wavepacket collapse [21], and quantum computations [22,23], Experimentally, the determination of phases frequently utilizes measurement of Ramsey fringes [24] or similar" methods [25]. 

Laurino examined a similar method in which methanesulfonanilides were alkylated with bromoacetaldehyde diethyl acetal and then cyclized with TiCU[4J. 1 hese methods presumably involve generation of an electrophilic intermediate from the acetal functionality, followed by an intramolecular Friedel-Crafts reaction. As a consequence, the cyclization is favoured by ER substituents and retarded by EW groups on the benzene ring. 

A vast amount of research has been undertaken on adsorption phenomena and the nature of solid surfaces over the fifteen years since the first edition was published, but for the most part this work has resulted in the refinement of existing theoretical principles and experimental procedures rather than in the formulation of entirely new concepts. In spite of the acknowledged weakness of its theoretical foundations, the Brunauer-Emmett-Teller (BET) method still remains the most widely used procedure for the determination of surface area similarly, methods based on the Kelvin equation are still generally applied for the computation of mesopore size distribution from gas adsorption data. However, the more recent studies, especially those carried out on well defined surfaces, have led to a clearer understanding of the scope and limitations of these methods furthermore, the growing awareness of the importance of molecular sieve carbons and zeolites has generated considerable interest in the properties of microporous solids and the mechanism of micropore filling. 

Before we look more closely at specific gravimetric methods and their applications, let s take a moment to develop a broad survey of gravimetry. Later, as you read through the sections of this chapter discussing different gravimetric methods, this survey will help you focus on their similarities. It is usually easier to understand a new method of analysis when you can see its relationship to other similar methods. 

What we seek next is a quantitative relationship between the extent of the polymerization reaction, the composition of the monomer mixture, and the point of gelation. We shall base our discussion on the system described by reaction (5.U) other cases are derived by similar methods. To further specify the system we assume that A groups limit the reaction and that B groups are present in excess. Two parameters are necessary to characterize the reaction mixture ... 

A similar method, ASTM D3107, has been developed for measuring stretch and stretch recovery of woven fabrics made in whole or in part from stretch yams. The term stretch yams refers to thermoplastic filament or spun yams having a high degree of potential elastic stretch and rapid recovery. These yams are characterized by a high degree of yam cud. 

Fastness to Crocking. Crocking is defined as the transfer of color from the surface of a dyed fabric to another surface by mbbing. AATCC test method no. 8 is a method by which a colored test fabric swatch is fastened to the base of a Crockmeter and mbbed against a white crock test cloth under controlled conditions. Color transfer to the white cloth is evaluated by comparison with the AATCC Chromatic Transference Scale. A similar method, AATCC 116, uses a Rotary Vertical Crockmeter, which requires a smaller area of test fabric than the Crockmeter. 

All known commercial syntheses are based on this approach with improvements in preparations of the three components (23). Shordy after the first synthesis, similar methods were pubUshed employing other C -halo compounds, such as l,l,3-tribromo-2-propanone, 2,2,3-tribromopropanal (24), 2,2,3-trichloropropanal, and l,l,3-trichloro-2-propanone (23). 

Taste and Odor. The measurement of taste and odor is somewhat subjective and depends on the personal judgements of individuals. Panels of not less than five observers, and preferably more than ten, are used. The sample is diluted with odor-free water until a ratio at which the odor is just perceptible is determined this ratio is called the threshold odor number (TON). A similar method is used to detect a distinct taste in water (see Flavor characterization). ... 

M). A similar method, using smaller samples and horizontal grids has also been developed and referred to as the Tumer-NewaH classification. 

Mixed esters containing the dicarboxylate moiety, eg, cellulose acetate phthalate, are usually prepared from the partially hydroly2ed lower aUphatic acid ester of cellulose in acetic acid solvent by using the corresponding dicarboxyhc acid anhydride and a basic catalyst such as sodium acetate (41,42). Cellulose acetate succinate and cellulose acetate butyrate succinate are manufactured by similar methods as described in reference 43. 

Recent Developments. A considerable amount of cellulose acetate is manufactured by the batch process, as described previously. In order to reduce production costs, efforts have been made to develop a continuous process that includes continuous activation, acetylation, hydrolysis, and precipitation. In this process, the reaction mixture, ie, cellulose, anhydride, catalyst, and solvent, pass continuously through a number of successive reaction zones, each of which is agitated (92,93). In a similar process, the reaction mass is passed through tubular zones in which the mixture is forced through screens of successively small openings to homogenize the mixture effectively (94). Other similar methods for continuous acetylation of cellulose have been described (95,96). 

Polyheterocycles. Heterocychc monomers such as pyrrole and thiophene form hiUy conjugated polymers (4) with the potential for doped conductivity when polymerization occurs in the 2, 5 positions as shown in equation 6. The heterocycle monomers can be polymerized by an oxidative coupling mechanism, which can be initiated by either chemical or electrochemical means. Similar methods have been used to synthesize poly(p-phenylenes). 

QSAJi Methods for Fluid Solubility Prediction. Several group contribution methods for predicting Hquid solubiHties have been developed. These methods as weU as other similar methods are often called quantitative stmcture-activity relationships (QSARs). This field is experiencing rapid development. 

By similar methods, ethyleae dibromide [106-93-4] can also be obtained. This compound is used ia gasoliae as an antiknocking additive. 

The best laboratory synthesis of thymine (947) is probably from 3-methylmalic acid (945) which gives 2-formylpropionic acid (946 R = H) in situ by decarboxylation and oxidation in fuming sulfuric acid prior to condensation with urea (46JA912) a similar method from ethyl 2-formylpropionate (946 R = Et) is also described (68IZV918). 

An alternative, similar method of naming fused systems due to Stelzner B-31MI10200) was used quite extensively in the early-to-mid twentieth century literature however, its disadvantages in comparison with the present system have rendered it virtually obsolete. 

Phenylthiomethyl dithioacctals (RSCH2SC6H5) were prepared and cleaved by similar methods. ... 

How can we categorise in a more precise way the oxidation-resistance of materials If we can do so for oxidation, we can obviously follow a similar method for sulphidation or nitrogenation. 

Recently, Gray, Reinhoudt and coworkers have prepared a number of furan-con-taining crowns by similar methods ". It is worthwhile noting that these workers prepared considerably larger macrocycles than those prepared by Timko and Cram and the complexation profile for these was obviously different. 

Tomoi and coworkers adopted a somewhat more direct approach in their synthesis of 16-crown-5 derivatives bearing a single alkenyl residue. They hoped to obtain precursors to polymers which could be used as phase transfer catalysts. In this approach I,I-bis-chloromethylethylene (a-chloromethallyl chloride) was allowed to react with the dianion of tetraethylene glycol (NaH/THF). By this method, methylene-16-crown-5 could be isolated in 66% yield after vacuum distillation. Ozonolysis led, in almost quantitative yield, to the formation of oxo-16-crown-5 as shown in Eq. (3.38). These authors prepared a number of other, closely related species by similar methods. 

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