Elementary elimination

TliCf riiiite difl erence fomiulatioii of steady heal conduction problems usu ally results in a system of iV algebraic equations in /V unknown nodal temperatures that need to be solved siiiiullaneously. When Af is small (such as 2 or 3), we can use the elementary elimination method to eliminate ail unknowns except one and then solve for that unknown (sec Example 5-1). The other unknowns are then determined by back substitution. When W is large, which is usually Uie case, the elimination luelliod is not practical and we need to use a more systematic approach that can be adapted to computers. 

Gaussian Elimination, hi the most elementary use of Gaussian elimination, the first of a pair of simultaneous equations is multiplied by a constant so as to make one of its coefficients equal to the corresponding coefficient in the second equation. Subtraction eliminates one term in the second equation, permitting solution of the equation pair. 

Reaction mechanism A sequence of steps by which a reaction occurs at the molecular level, 307,318-319q elementary steps, 307 intermediates, elimination of, 309-311 rate expression for, deducing, 308-309 slow steps, 307... 

Elementary considerations indicate that with appropriate substitutions some of the reactions mentioned above can be eliminated. Indeed, when 5-methyl-2-vinyl-furan was used, no alkylation was observed, the positions C-3 and C-4 being rather unreactive16, and the polymer was a mixture of linear chains with polyunsaturations and linear saturated chains, i.e. only structures like 21, 23 and 26 were present, with a 5-methyl ring instead of the 5-unsubstituted one. When 2-isopropenylfuran was used, no hydride transfer took place since this requires a hydrogen atom in the a-position to the ring, which this monomer does not have the polymers were white and gave electronic spectra transparent down to 280 nm. Alkylation at C-5, how-... 

The reliability of the results depends in large measure on how well deviations from the (ideal) linear relationship between log / and dry weight per unit area can be eliminated or allowed for. As is well known, this can be accomplished by the comparative method (3.10), provided that standard (reference system) and unknown, identical in mass, shape, and elementary composition, are exposed to the same x-ray beam. In the cytological investigations, these conditions are difficult to meet, not only because the samples are complex in composition, but also because they are very small, as is clear from the units employed (micromicrograms per square micron or 10 12 gram per 10 8 sq cm). 

Solution Under the assumption of intrinsic kinetics, all mass transfer steps are eliminated, and the reaction rate is determined by Steps 4-6. The simplest possible version of Steps 4-6 treats them all as elementary, irreversible reactions ... 

Although the foregoing reactions involve dehalogenation by reduction or elimination, nucleophilic displacement of chloride may also be important. This has been examined with dihalomethanes using HS at concentrations that might be encountered in environments where active anaerobic sulfate reduction is taking place. The rates of reaction with HS exceeded those for hydrolysis and at pH values above 7 in systems that are in equilibrium with elementary sulfur, the rates with polysulfide exceeded those with HS. The principal product from dihalomethanes was the polythio-methylene HS (CH2-S) H (Roberts et al. 1992). 

From these data, some key information can be drawn in both cases, the couple methane/pentane as well as the couple ethane/butane have similar selectivities. This implies that each couple of products (ethane/butane and methane/pentane) is probably formed via a common intermediate, which is probably related to the hexyl surface intermediate D, which is formed as follows cyclohexane reacts first with the surface via C - H activation to produce a cyclohexyl intermediate A, which then undergoes a second C - H bond activation at the /-position to give the key 1,3-dimetallacyclopentane intermediate B. Concerted electron transfer (a 2+2 retrocychzation) leads to a non-cychc -alkenylidene metal surface complex, C, which under H2 can evolve towards a surface hexyl intermediate D. Then, the surface hexyl species D can lead to all the observed products via the following elementary steps (1) hydrogenolysis into hexane (2) /1-hydride elimination to form 1-hexene, followed by re-insertion to form various hexyl complexes (E and F) or (3) a second carbon-carbon bond cleavage, through a y-C - H bond activation to the metallacyclic intermediate G or H (Scheme 40). Under H2, intermediate G can lead either to pentane/methane or ethane/butane mixtures, while intermediate H would form ethane/butane or propane. 

In principle, the task of solving a linear algebraic systems seems trivial, as with Gauss elimination a solution method exists which allows one to solve a problem of dimension N (i.e. N equations with N unknowns) at a cost of O(N ) elementary operations [85]. Such solution methods which, apart from roundoff errors and machine accuracy, produce an exact solution of an equation system after a predetermined number of operations, are called direct solvers. However, for problems related to the solution of partial differential equations, direct solvers are usually very inefficient Methods such as Gauss elimination do not exploit a special feature of the coefficient matrices of the corresponding linear systems, namely that most of the entries are zero. Such sparse matrices are characteristic of problems originating from the discretization of partial or ordinary differential equations. As an example, consider the discretization of the one-dimensional Poisson equation... 

The surface concentration Cq Ajc in general depends on the electrode potential, and this can affect significantly the form of the i E) curves. In some situations this dependence can be eliminated and the potential dependence of the probability of the elementary reaction act can be studied (called corrected Tafel plots). This is, for example, in the presence of excess concentration of supporting electrolyte when the /i potential is very small and the surface concentration is practically independent of E. However, the current is then rather high and the measurements in a broad potential range are impossible due to diffusion limitations. One of the possibilities to overcome this difficulty consists of the attachment of the reactants to a spacer film adsorbed at the electrode surface. The measurements in a broad potential range give dependences of the type shown in Fig. 34.4. 

It was assumed that C—C bond cleavage passes through an elementary step of p-alkyl transfer. The mechanism of hydroisomerization passes also by a p-alkyl transfer step, but in this case the P-H elimination-olefin reinsertion occurs rapidly and a skeletal isomerization also occurs. 

Activation Energy Considerations. Activation energy considerations can provide a basis for eliminating certain elementary reactions from a sequence of reactions. Unfortunately, the necessary activation energy data is seldom available, and one must estimate these parameters by empirical rules and generalizations that are of doubtful reliability. 

Since the activation energy of an endothermic elementary reaction cannot be less than AH, the reaction with the significantly lower activation energy will occur much more frequently, other factors being equal. Similar arguments permit one to eliminate the HBr dissociation reaction from consideration. 

Oxidative coupling via la —> 2a and the reductive elimination routes, that commence from 4a as the precursor, involve different stereoisomers along the most feasible pathway. Accordingly, the conversions of the terminal allylic groups of the [Nin(octadienediyl)L] complex represent indispensable elementary processes. 

Each step includes elementary acts that require different properties of the metal, for example, sufficiently low ionization potential to favor oxidative addition, sufficiently weak metal-carbon bonds, tendency to form square-planar complexes and to reach pentacoordination to allow insertion, a sufficiently high electron affinity to allow reductive elimination, and so on. Some properties are conflicting and a compromise has to be reached. 

The derivation of a rate law from a postulated mechanism is a useful application of reaction mechanisms. It shows how the kinetics of the elementary reaction steps are reflected in the kinetics of the overall reaction. The following example illustrates this for a simple, gas-phase reaction involving an open sequence. The derivations typically employ the stationary-state hypothesis (SSH) to eliminate unknown concentrations of reactive intermediates. 

The catalytic asymmetric hydrogenation with cationic Rh(I)-complexes is one of the best-understood selection processes, the reaction sequence having been elucidated by Halpern, Landis and colleagues [21a, b], as well as by Brown et al. [55]. Diastereomeric substrate complexes are formed in pre-equilibria from the solvent complex, as the active species, and the prochiral olefin. They react in a series of elementary steps - oxidative addition of hydrogen, insertion, and reductive elimination - to yield the enantiomeric products (cf. Scheme 10.2) [56]. 

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