Analogy procedure

An analogous procedure can be applied to a plane surface. The surface can be roughened by the successive application of one or another recipe, just as was done for the line in Fig. VII-6. One now has a fractal or self-similar surface, and in the limit Eq. VII-20 again applies, or... 

A third approach is suggested by Hugo s formulation of material balances at the limit of bulk diffusion control, described in Section 11.3. Hugo found expressions for the fluxes by combining the stoichiometric conditions and the Stefan-Maxvell relations, and this led to no inconsistencies since there are only n - 1 independent Stefan-Maxwell relations for the n fluxes. An analogous procedure can be followed when the diffusion is of intermediate type, using the dusty gas model equations in the form (5.10) and (5.11). Equations (5.11), which have the following scalar form ... 

The analogous procedure for a multivariate problem is to obtain many experimental equations like Eqs. (3-55) and to extract the best slopes from them by regression. Optimal solution for n unknowns requires that the slope vector be obtained from p equations, where p is larger than n, preferably much larger. When there are more than the minimum number of equations from which the slope vector is to be extracted, we say that the equation set is an overdetermined set. Clearly, n equations can be selected from among the p available equations, but this is precisely what we do not wish to do because we must subjectively discard some of the experimental data that may have been gained at considerable expense in time and money. 

The computation of mesopore size distribution is valid only if the isotherm is of Type IV. In view of the uncertainties inherent in the application of the Kelvin equation and the complexity of most pore systems, little is to be gained by recourse to an elaborate method of computation, and for most practical purposes the Roberts method (or an analogous procedure) is adequate—particularly in comparative studies. The decision as to which branch of the hysteresis loop to use in the calculation remains largely arbitrary. If the desorption branch is adopted (as appears to be favoured by most workers), it needs to be recognized that neither a Type B nor a Type E hysteresis loop is likely to yield a reliable estimate of pore size distribution, even for comparative purposes. 

Thin-Film Oven Test (ASTMD1754). This test has the purpose of determining the hardening effect of heat and air on a static film of asphalt when exposed in a thin film. An analogous procedure is the Rolling Tbin-Film Test (ASTM D2872) which has the same purpose but utilizes a moving film exposed for 75 min at 163 °C. 

Jui y trials represent a form of decision making. In statistics, an analogous procedure for making decisions falls into an area of statistical inference called hypothesis testing. 

A combination of the preceding type of synthesis and of cyclization of 4-amino-5-arylazopyrimidine can be seen in the novel procedure of Richter and Taylor. Proceeding from phenylazomalonamide-amidine hydrochloride (180), they actually close both rings in this synthesis. The pyrimidine ring (183) is closed by formamide, the triazole (181) one by oxidative cyclization in the presence of cupric sulfate. Both possible sequences of cyclization were used. The synthetic possibilities of this procedure follow from the combination of the two parts. The synthesis was used for 7-substituted 2-phenyl-l,2,3-triazolo[4,5-d]-pyrimidines (184, 185). An analogous procedure was employed to prepare the 7-amino derivatives (188) from phenylazomalondiamidine (186). 

Bruce and Sutcliffe obtained l-acetyl-2-methyl-3-phenylindole (123) by the action of acetyl chloride on 2-methyl-3-pheiiylindole magnesium iodide in ether.These authors were able to obtain l-benzoyl-2-benzyl-3-phenylindole (124) but not l-acetyl-2-benzyl-3-phenylindole (125) from 2-benzyl-3-phenylindole magnesium iodide by analogous procedures.3-Acetyl-2-phenylindole (126) and 3-propionyl-2-pheny]indole (127) have recently been prepared in fair yield by the acylation of 2-phenylindole magnesium iodide with acetyl and propionyl chloride, respectively. Le ete obtained a mixture of l-acetyl-3-ethylindole (128) and 2-acetyl-3-ethylindole (129) by the interaction of acetyl chloride with 3-ethylindole magnesium iodide in ether. 

The following trialkylcarbinols (Table 12.1) may be prepared by an analogous procedure with the time required for the absorption of carbon monoxide as shown. For liquid products, the dilution with water is followed by extraction with pentane, the pentane solution is dried, and the solvent is removed (rotary evaporator), affording the pure product. 

The following dialkyl ketones may be prepared by an analogous procedure with the time required for the absorption of carbon monoxide as shown (Table 12.2). 

Note that the analogous procedure carried out for NDSTGs does not, in general, yield the mimmal graph. 

By an entirely analogous procedure, the thermochemical equation for the formation of HC1 from the elements (Example 8.3) is found to be... 

An analogous procedure can equate the empirical pA2 to pKB for noncompetitive antagonists. Utilizing the equation for agonist response in the presence of a noncompetitive antagonist (Equation 6.10), equiactive concentrations with a dose ratio of 2 in the presence and absence of antagonist is given by ... 

An analogous procedure was realized in the selenium series 4H-selenopyran 136 (R = H) was converted with LDA at -77°C to a lithium intermediate (R = Li), which gave ethyl ester 183 by successive treatments with C02 and MeCHN2 (90AG450). 

Allyllithium reagents have also been used in the synthesis of (Z)-y-alkoxyallylboronates 23 2 5. Stereoselectivity is excellent in these reactions since the (Z)-y-alkoxyallyl carbanions prepared by metalation of allyl ethers are stabilized by chelation. The (Z)-y-alkoxyallyl(diisopinocam-pheyl)boranes are prepared at low temperature by an analogous procedure and must be used at — 78 "C otherwise reaction diaslereoselectivity suffers owing to the facile isomerization to the -isomer26. 

The (5)-enantiomer of this amino acid is available by an analogous procedure starting from (5)-2-hydroxy-l,2,2-triphenylethyl acetate81. 

It is thought that the chlorination proceeds through a ir-com-plex between cupric chloride and anthracene, and that this complex then undergoes homolytic dissociation. Hence aromatic rings subject to attack by chlorine atoms can be chlorinated in this way. Thus one can convert pyrene to 1-chloropyrene (90% yield), but phenanthrene is not chlorinated. Analogous procedures using cupric bromide lead to 9-bromoanthracene (99% yield) and 1-bromopyrene (94% yield).7... 

It is possible to treat ketones with allyl alcohol and an acid catalyst to give y,5-unsaturated ketones directly, presumably by initial formation of the vinylic ethers, and then Claisen rearrangement.In an analogous procedure, the enolates (126) of allylic esters [formed by treatment of the esters with lithium isopropylcyclohex-... 

Following an analogous procedure, we now operate on both sides of equation (6.30) with the operator Bx to obtain... 

A perfectly analogous procedure can be performed for the RIS model on the 2nnd lattice, using an expression for Z developed with the statistical weight matrix in Eq. (4.4) or (4.5) [153]. The replacement for Eq. (4.6) becomes ... 

The subscript M refers to metal ions. An analogous procedure to that... 

The following compounds have also been prepared by analogous procedures ... 

The chromium, tungsten, and vanadium complexes can be prepared by an entirely analogous procedure. 

The formation of pyrazoles from reactions of suitably substituted 2-arylazirines and molybdenum hexacarbonyl has been discussed earlier in this section (see Schemes 89, 90)47 an analogous procedure depicting the transformation of 2-formyl-3-phenyl-2H-azirine into 3-phenylisoxazole is illustrated in Scheme 109.47... 

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