Substitution of equivalents

More complete discussions of Kraitchman s equations can be found in standard treatises [16]. In addition, analogous equations are available for multiple isotopic substitution of equivalent atoms, e.g., H2O to D2O [17]. 

If the generation of a connectivity stack and the permutation or substitution of equivalent nodes are carried out without effective algorithms, the number of generated stacks might be astronomical and the calculation time is very time-consuming. It is clear that the connectivity stack method also includes methods for avoiding these problems. 

Note that we have taken a cosine rather than a sine function for our solution. Substitution of either Eq. (4-2) or the equivalent sine function into Eq. (4-1) gives a true statement (with certain restrictions on co) therefore, both are solutions. Moreover, the sum or difference... 

Methane ethane and cyclobutane share the common feature that each one can give only a single monochloro derivative All the hydrogens of cyclobutane for example are equivalent and substitution of any one gives the same product as substitution of any other Chlorination of alkanes m which the hydrogens are not all equivalent is more com plicated m that a mixture of every possible monochloro derivative is formed as the chlo rmation of butane illustrates... 

Substitutes and Equivalents. In the past 25 years, many fats have been developed to replace part or all of the added cocoa butter ia chocolate-flavored products. These fats fall into two basic categories commonly known as cocoa butter substitutes and cocoa butter equivalents. Neither can be used ia the United States ia standardized chocolate products, but they are used ia small amounts, usually up to 5% of the total weight of the product, ia some European countries. 

Cocoa butter substitutes of all types enjoy widespread use ia the United States chiefly as ingredients ia chocolate-flavored products. Cocoa butter equivalents are not widely used because of their higher price and limited supply. 

Cocoa butter substitutes and equivalents differ greatly with respect to their method of manufacture, source of fats, and functionaHty they are produced by several physical and chemical processes (17,18). Cocoa butter substitutes are produced from lauric acid fats such as coconut, palm, and palm kernel oils by fractionation and hydrogenation from domestic fats such as soy, com, and cotton seed oils by selective hydrogenation or from palm kernel stearines by fractionation. Cocoa butter equivalents can be produced from palm kernel oil and other specialty fats such as shea and ilHpe by fractional crystallization from glycerol and selected fatty acids by direct chemical synthesis or from edible beef tallow by acetone crystallization. 

The hydrauhc diameter method does not work well for laminar flow because the shape affects the flow resistance in a way that cannot be expressed as a function only of the ratio of cross-sectional area to wetted perimeter. For some shapes, the Navier-Stokes equations have been integrated to yield relations between flow rate and pressure drop. These relations may be expressed in terms of equivalent diameters Dg defined to make the relations reduce to the second form of the Hagen-Poiseulle equation, Eq. (6-36) that is, Dg (l2SQ[LL/ KAPy. Equivalent diameters are not the same as hydraulie diameters. Equivalent diameters yield the correct relation between flow rate and pressure drop when substituted into Eq. (6-36), but not Eq. (6-35) because V Q/(tiDe/4). Equivalent diameter Dg is not to be used in the friction factor and Reynolds number ... 

The values in Fig. 23-5 have to be corrected in the case of differing resistivity values because of the dependence of mutual inductance on the soil resistivity. This is achieved by substituting an equivalent distance, a instead of the real distance, a, giving ... 

Furthennore, only one monobromination product of benzene was ever obtained, which suggests that all the hydrogen atoms of benzene are equivalent. Substitution of one hydrogen by bromine gives the sane product as substitution of any of the other hydrogens. 

Lithium 1,2,4-triazolate with [Rh2( j,-Ph2PCH2PPh2)(CO)2( j.-Cl)]PFj. gives the A-framed complex 177 (L=L = CO) (86IC4597). With one equivalent of terf-butyl isocyanide, substitution of one carbon monoxide ligand takes place to yield 177 (L = CO, L = r-BuNC), whereas two equivalents of rerr-butyl isocyanide lead to the product of complete substitution, 177 (L = L = r-BuNC). The starting complex (L = L = CO) oxidatively adds molecular iodine to give the rhodium(II)-rhodium(II) cationic species 178. 

By using basicity data, Ridd and Smith- showed that 5-nitro- and 5-chloro-benzimidazole and their 2-methyl analogs exist essentially as mixtures of equivalent amounts of 29 and 30, and, in agreement with this ratio, 5-substituted benzimidazoles form comparable amounts of 1- and 3-derivatives on alkylation,- showing earlier alkylation ratios- to be erroneous. There are, however, other factors which can lead to the predominance of one tautomeric form. Basicity measurements indicate that 31 is preferred to the alternative non-hydro-... 

The substitution of Ip and L resulted in a spring constant of 0.71 X 10 chain elasticity, which was almost equivalent to the experimentally obtained slope value. 

The introduction of vectors of constant displacement length to represent the individual elements, which actually vary in length, is rendered more plausible by inquiry into the effect of incorporating this artifice in the treatment of the freely jointed chain. In this case V = m H. Upon substitution of this expression together with n nlm in Eq. (17), the previous expression for / , Eq. (6), is recovered. Hence the calculated distribution is unaff ected by an arbitrary subdivision of the chain in this manner. We conclude that the value chosen for m in the reduction of the real chain to an equivalent freely jointed chain likewise is inconsequential (within the limits on m stated above). 

In the case of a heterogeneous polymer comprising molecular species differing in size but otherwise equivalent chemically, substitution of Eq. (12) for ASm gives... 

Allylic substitutions catalysed by palladium NHC complexes have been studied and the activity and selectivity of the catalysts compared to analogous Pd phosphine complexes. A simple catalytic system involves the generation of a Pd(NHC) catalyst in situ in THF, from Pdj(dba)j, imidazolium salt and Cs COj. This system showed very good activities for the substitution of the allylic acetates by the soft nucleophilic sodium dimethyl malonate (2.5 mol% Pdj(dba)3, 5 mol% IPr HCl, 0.1 equiv. C (CO ), THF, 50°C) (Scheme 2.22). Generation of the malonate nncleophile can also be carried out in situ from the dimethyhnalonate pro-nucleo-phile, in which case excess (2.1 equivalents) of Cs COj was used. The nature of the catalytic species, especially the number of IPr ligands on the metal is not clear. 

---