Reaction invariant

Chapter 5 describes a study of the effect of micelles on the Diels-Alder reaction of 1 with 2. Literature studies on micellar catalysis of Diels-Alder reactions invariably failed to reveal significant accelerations. These results are unexpected, since most Diels-Alder reactants have a high affinity for... 

The columns of the orthonormal matrix Vp are linear combinations of reaction invariants . In fact, the only invariants for the batch reaction being analyzed can be stoichiometric coefficients. Hence the matrix Vp may be interpreted as containing the stoichiometric information (Waller and Makila (1981)) and its rank Nr can be considered to be equal to the number of independent... 

Photoassisted catalytic hydrogenation reactions invariably involve promoted loss of a ligand and generation of coordinatively unsaturated species. Examples in the literature are becoming increasingly common. 

Liquid water arrives in the CCL via transport through the PEM or it is generated in the electrochemical reaction. Invariably, PEECs require a medium that is highly effective in transforming liquid water into water vapor otherwise, liquid water will clog pores and channels in gas diffusion layers and flow fields that are needed for the gaseous supply of reactants. 

An activated complex containing three species (other than solvent or electrolyte), which attends a third-order reaction, is not likely to arise from a single termolecular reaction involving the three species. Third- (and higher-) order reactions invariably result from the combination of a rapid preequilibrium or preequilibria with a rds, often unidirectional. Such reactions are... 

These quantities are preserved like atoms in the given reactions and hence are called reaction invariants (ref. 16). In this example we found 4 linearly independent reaction invariants. It does not mean, however, that the species Ml, 2,. .. M6 are built necessarily from 4 atoms. In fact, introducing the species M = CH4, 2 — O2, 3 = M4 = 5 = 2 9 and M — 2 2... 

K.V. Waller and P.M. Makila, Chemical reaction invariants and variants and their use in in reactor modeling, simulation and control,... 

Notice that the number NA of atoms is usually small compared to the number NS of species, and hence the RAND algorithm is very effective in terms of computational effort. The rank of the atom matrix, however, must be equal to the number NA of atoms. At this point it is interesting to remark that instead of the atom matrix we can use a virtual atom matrix, i.e., the matrix of reaction invariant coefficients if the atom matrix is not available or we are interested in a restricted equilibrium. For details see Section 1.8.1. 

The law of mass action (Equation 15-2) is always stated as applying to a given temperature, and it appears not to have temperature involved in its statement. Yet the rates of chemical reaction invariably increase markedly with increase in temperature. Because concentrations will be negligibly affected by temperature, the temperature-sensitive factor in the law of mass action must be the rate constant, 1. As a good approximation, we say that k is proportional to the fraction of molecules (or collisions) that have the required enthalpy of activation ... 

In most reactions that proceed spontaneously, the enthalpy of the system decreases. If no work is done, the system gives off heat to the surroundings. But in some spontaneous reactions, heat is absorbed in the absence of work and the enthalpy of the system increases. Such reactions invariably show an increase in the entropy of the system. 

With substituted 1,4-dienes, such as (15), (18) and (22), the isomerization/complexation reaction invariably produces mixtures of complexes. Although these can often be separated chromatographically, it... 

Complex reactions invariably involve intermediates which are formed in some steps, removed in others, and have a wide range of lifetimes. Longer lifetimes can result in build-up to significant intermediate concentrations during reaction, but these intermediates must also be sufficiently reactive to allow the subsequent reactions to occur. Intermediates can also be so short lived that they are removed almost as soon as they are formed, resulting in very, very low steady state concentrations. The lifetimes of intermediates and their concentrations have profound effects on the analysis of the kinetics of the reactions in which they occur. 

It has been demonstrated that hydrated-electron reactions invariably proceed by the transfer of an electron from its site in the solvent into a vacancy in the acceptor molecule, irrespective of the charge of the latter... 

Equation (9) describes a linear stoichiometric variety of dimension R. A reactive projection is defined as a multiple orthogonal projection from a C-dimensional space to a (C - R - 1)-dimensional subspace, where the directions of the R projection rays follow the directions of (qi, q>,. .., q ) defined in Eq. (10). Such projection causes the stoichiometric variety to disappear, leaving a reaction-invariant projection. The set of canonical coordinates defining the projective subspace can be found by substituting Eq. (10) into Eqs. (2)—(4) [7]. 

Figure 11.6 shows an example of the phase diagram for a reactive system, in which a compound C is formed from components A and B. An isothermal cut and the polythermal projection are also shown. Such a phase diagram can be obtained via a reaction invariant projection of a higher-dimensional simple eutectic phase diagram. AS and BS are binary nonreactive eutectics, since their presence is not affected by the reaction, while ACSb and BCSa are ternary reactive eutectics. Similar... 

These reactions invariably afford -substituted vinyliodonium salts in the Z-configura-tion (i.e. anti-introduction of the nucleophile and proton) and are generally regarded as nucleophilic additions. However, it seems plausible in the case of HX in methanol that addition is initiated by protonation of the triple bond. 

Several approaches to what are formally Si-0 and O-Si-O heterocycles have been reported, although these reactions invariably involve the use of the Si-O bond only as a temporary tether. The coupling and RCM of dissymmetric alcohols was reported by Eustache and co-workers <01TL239> and this method was applied in the synthesis of attenol A <020L4105>. Denmark and Yang performed the RCM of Si-O tethered dienes in tandem with silicon-assisted cross-... 

The previous discussion might lead to the conclusion that solvolytic reactions invariably obey the reactivity- selectivity principle. In fact, a considerable amount of data has recently been presented in which the expected inverse relationship was not observed. 

Fjeld, M., O. A. Asbjornsen and K. J. Astrom, Reaction invariants and their importance in the analysis of eigenvectors, state observability and controllability of the continuous stirred tank reactor, Chem. Eng. Sci., 29, 1917-1926 (1974). 

Glasser et al. (1987) and Hildebrandt et al. (1990) demonstrated this two-dimensional approach on a number of small reactor network problems, with better results than previously reported. Moreover, Omtveit and Lien (1993) were able to consider higher-dimensional problems as well through projections in concentration space that allow a complete two-dimensional represention. These projections were accomplished through the principle of reaction invariants (Fjeld et al., 1974) and the imposition of system specific constraints. 

Ihe interaction of polynuclear metal complexes with pyrrole-type molecules has also attracted some attention, and it is interesting in connection with possible interactions with surfaces, where vicinal metal atoms may simultaneously interact with the pyrrole molecule. Cluster compounds containing intact coordinated pyrrole ligands are not known, since this reaction invariably involve N-H and/or C-H bond activation to yield derivatives most frequently containing bridging pyrrolyl ligands, as exemplified in Eig. 6.2. 

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