Some Particular Cases

For an example of a pairwise correlation, consider the following two quantities that purport to measure the polarity of the solvent. Kosower (1958,1968) proposed a parameter, Z, based on the change in frequency of a charge-transfer transition of 1-ethyl-4-methoxycarbonyl-pyridinium iodide (1), which shows a pronounced increase of frequency when the polarity of the solvent is increased, because the ionic ground state is stabilized by a polar medium relative to the nonionic excited state. Dimroth and Reichardt (1969 Reichardt and Harbusch-Gomert, 1983) proposed another, E (30), or in normalized form, E, based on a visible transition of a pyridinium-V-phenoxide betaine dye (8, p, 40). (The label (30) is because the dye was the 30th of a number tried.) Eigure 4.5 shows the correlation of these two parameters for 40 solvents. 

The correlation does not appear to be very good, but in fact, by using a select set of solvents believed to be free from interferences such as specific interactions or purity problems, it has been used to obtain estimates of E.p(30) for solvents in which the dye is insoluble or reacts. With these additions . (30) became available for 271 solvents (Reichardt, 1994). 

Another example with a different result was the attempted correlation of the theoretically justified quantity p, defined by Dutkiewicz (1990) as (2N l3e (p lV) (proportional to the square of the dipole moment of the solvent molecule divided by its molar volume, with E . She found that, instead of lying close to a single line, the points fell into four fairly distinct classes (with a few outliers). See Eigure 4.6. 

FIGURE 4.5 Kosower s Z versus Reichardt s p(30). Data from the compilation of Abboud and Notario (1999). Points marked with crosses are omitted from the statistics. They are for compounds flagged by the last authors as uncertain and a group of esters and ethers that seem to have values of Z clustered about 64-65kcalmol , lying above the trend. No strong HBD solvents are included. 

Weakly dipolar and nonpolar, non-HBD (hydrogen-bond donor) molecules hydrocarbons and halogenated derivatives, ethers, esters, tertiary amines. 

---

In order to reply to some particular cases of NDT such that the detection two close defects and the fine thickness measure, we have developed techniques adapted to this type of problem. 

An additional regularity of the solution near the point xi takes place in some particular cases. For example, assume that the solution u satisfies the condition [u] =0 on B i n S , where B i is a ball centered at xi. Using the technique of Section 2.5, we can prove that the equation —Au = f holds in Bx, and consequently, u G Hf Bx,). In this case all the integrals in (4.98) make sense. Consequently, by (4.62), (4.93), the equality (4.97) follows. 

Electrodes. AH of the finished silver electrodes have certain common characteristics the grids or substrates used in the electrodes are exclusively made of silver, although in some particular cases silver-plated copper is used. Material can be in the form of expanded silver sheet, silver wire mesh, or perforated silver sheet. In any case, the intent is to provide electronic contact of the external circuit of the battery or cell and the active material of the positive plate. Silver is necessary to avoid any possible oxidation at this junction and the increased resistance that would result. 

It is easily seen that impact non-adiabatic relaxation of angular momentum proceeds exponentially in the Keilson-Storer approximation. Let us dwell upon some particular cases. 

Consider some particular cases, which are of interest to the understanding of the general flow properties in a heated capillary. First, we examine a simple case when the Peclet number is much less than unity. Since Zl < 1, the assumption Pcl 1 corresponds to the condition Pcl l C 1. In this case Eq. (9.62) reduces to... 

In contrast to (6)-(7), every equation with the number a is solved here on the whole interval tj < t < ij + i- It is interesting to note that in some particular cases solutions of problem (6)-(7) and problem (11)--(12) will coincide. This is certainly true in the situation wdren both / = 0 and La are independent of t. 

The kinetic problem for the intramolecular cross-linking reactions in general form was not yet solved. Only some particular cases, i.e. the cvclization of macromolecules, the intramolecular catalysis and diffusion-controlled collision of two reactive groups were studied theoretically bv Xorawetz, Sisido and Fixman... 

In some cases one can factor in two or more factors. This can be done, however, only when we use a molecular formulation of the binding constants, and only in some particular cases. We present two such cases, discussed also by Hill (1985). 

Equations (56) and (57) constitute the overall ML principle that we sought. This principle embodies all the physical and prior knowledge that we have about the object and noise. The specific In P( nm ) to be used will depend on the particular prior knowledge p( i,---, m) at hand. Specific In P( nm ) were derived in preceding sections. We shall carry through some particular cases next. 

Thus, in hydrogen-transfer reactions, most of the catalysts do prefer the outer-sphere mechanism instead of the MPV or the insertion mechanisms. For instance, the high stability of the intermediate formed, alkoxide in the case of carbonyl hydrogenation, is a major drawback for the inner-sphere mechanism. Nevertheless, in some particular cases, the inner-sphere mechanism may be competitive with the outer-sphere one. In these cases, some requirements must be accomplished, such as the high lability of one of the metal ligands in order to allow easily the substrate coordination or the formation of not very stable intermediates. 

Other classical ligands such as triphenylphosphine or 1,2-diphenylphosphinoethane have been considered. However, no good results were obtained under these conditions since a mixture of ArH and ArAr was produced. Moreover, the use of acetonitrile as solvent does not lead to positive results. In addition to the above-described general case, some particular cases have been examined, such as in the presence of bromothiophene. 

Attempts have been made to find a general composition rule for this equation, and some interesting results have even be derived by Rica da Silva. It is relatively easy to show that in some particular cases, for some solutions, this... 

The hydroxy groups of pyrrolizidine amino-alcohols are readily replaced by chlorine atoms upon treatment with thionyl chloride (see, e.g., refs. 101 and 103). In this reaction, the allylic hydroxyl group is more reactive and can be selectively replaced by chlorine.79 In some particular cases, methoxyl groups can also be substituted for halogen e.g., l/3-methoxymethyl-8a-pyrrolizidine, when treated with hydro-bromic acid, gives rise to the corresponding bromo derivative.104... 

On the whole, it must be admitted that etching by fluoric acid, though practised to some extent, and in some particular cases with marked success, has not been found effective for general purposes. It is usefully applied, however, to form the scales on thermometer tubes, and written or printed inscriptions on glass vessels. The attempt to make it subservient to typographical purposes has been a failure. 

Electrodes. All of the finished silver electrodes have certain common characteristics the grids or substrates used in the electrodes are made exclusively of silver, although in some particular cases silver-plated copper is used,... 

Generally speaking, the concept of forward and reverse reaction rates is inapplicable to many-route reactions, but it can be applied in some particular cases. Such is the case of a double-route reaction with one of basic routes being at equilibrium (35). 

In some particular cases, however, the analytical solution to this problem can be attained by the methods of the generalized circuit theory [49, 20]. Within this approach, the CGF for a mesoscopic connector between two reservoirs is expressed in terms of the distribution p(T) of the transparencies of the conduction channels,... 

From Eq. (4.120), some particular cases of interest can be deduced ... 

Exact solutions of equation (F.18) can be found in some particular cases. For example, if the tensor of velocity gradients is symmetrical, equation (F.18) has an exact simple solution... 

The effect of the pressure on a and on k is usually negligible. Only in some particular cases (e.g. in supercritical fluid chromatography, SFC see section 3.4) will it be a relevant parameter. 

While numerical methods come into question for solutions involving variable D, D can be assumed to be constant or practically constant for most cases of practical interest. In addition, simplified solutions for diffusion along the x-axis can be used instead of the general solution, except for some particular cases which will be pointed out later. This greatly simplifies presentation of the problem and the resulting equation for diffusion is ... 

We shall return to the derivation of the individual terms of the effective Hamiltonian listed in (7.43) for some particular cases later in this chapter. To conclude this section, we now consider some of the general properties of the operator Xeff (0) and its eigenfunctions. Each of the terms listed in (7.43) is composed of products of the three operators Pq, Qo and X7. Although each of these operators is individually Hermitian (i.e. the operator is self-adjoint, pj, = P0), a product of any of them is not necessarily Hermitian. In fact, a term is only Hermitian when it is palindromic, that is, when it reads the same forwards as backwards. Inspection of equation (7.43) reveals that 3Ceff(0) is Hermitian up to and including terms in X2 but that there are non-Hermitian terms in the /.3 and higher-order contributions. The nature ofthese non-Hermitian properties can... 

In summary, we must say that unfortunately there is as yet no generally acknowledged opinion about the origin of the non-Debye dielectric response. However, there exist a significant number of different models which have been elaborated to describe non-Debye relaxation in some particular cases. In general these models can be separated into three main classes ... 

Classically, protein identification requires total or partial determination of their sequences by the method of Edman degradation. In some particular cases, protein identification can... 

In addition, Koga and coworkers claimed a spectacular rate enhancement adding either 1,1,4,7,10,10-hexamethyltriethylenetetramine or ureas (such as Af,Af -dimethyl-N,N -propylene urea or iV -dimethyl-A iV -ethylene urea) as a substitute for HMPA during the alkylation of tetralone lithium salt in DME399. Interestingly, in the case of secondary alkyl halides, a kinetic resolution occurred in some particular cases with fair400 to excellent enantioselectivities401. 

---