Symmetrical decomposition

Fig. 2. a 60 MHz broad-line JH NMR spectrum of bulk polyethylene at 20 °C. b Two-component analyses, dotted line straight line decomposition [45] dashed line symmetrical decomposition [46,47]... 

The mechanism which was first written for the reaction involved symmetrical decomposition (10), conversion by a process (11) to hydroxyl radicals and decay of those radicals in an unspecified way (12). 

For the heterodinuclear [Li3 (l)( TiGa ] -, both decomposition pathways are observed. On the one hand, symmetric decomposition into the triscatechol gallium(III) and triscatechol titanium(IV) species occurs. Conversely, the asymmetric formation of a tetracatecholate titanium(IV) and a biscatecholate gallium(III) species takes place. Formation of a dicatecholate titanium derivative and a tetracatecholate gallium species was not observed. ... 

A more detailed analysis( ) shows that the special conditions which lead to this equation are (1) a branching coefficient independent of a (which is reasonable if this simply represents the number of individual nuclei in each microcluster, but surely less plausible in terms of a model in which branching is due to strain or cracking) (2) a termination coefficient which is equal to the branching coefficient times a/a (this should result in the same rate constant for acceleratory and decay periods, whereas they usually differ and sometimes even have different temperature coefficients) and (3) a symmetrical decomposition curve with a,- = 0.5 (potassium permanganate is the best example of this). Equation (34) presupposes the existence of some initial decomposition (nucleus formation) during the induction period integration between limits (ao, to) and (a, t) yields the Prout-Tompkins equation( )... 

Splitting of the operators can be quite tricky. In the case of Eq. (6.113), the splitting is motivated by the observation that it is easy to implement exp(—iv(r)) in real space (at all points on the mesh), but it is difficult to take double derivatives at all points on the mesh. This difficulty can be overcome by using the fact that the derivatives can be implemented trivially in Fourier space. So, if somehow the Laplacian part could be split from w(r) part, then the progress can be made. This is the motivation for splitting the operator in Eq. (6.113) into derivative and nonderivative parts by Baker-Campbell-Hausdorff formula. For most practical purposes, a symmetric decomposition is carried out so that within an error of the order dt ... 

So, in pseudospectral method, one has to switch between real space and Fourier space back and forth. Symmetric decomposition leads to the implementation of half time step in real space and the other half in Fourier space. That is the reason this method is also known as split-step method in the literature [84]. 

Wlien the atom-atom or atom-molecule interaction is spherically symmetric in the chaimel vector R, i.e. V(r, R) = V(/-,R), then the orbital / and rotational j angular momenta are each conserved tln-oughout the collision so that an i-partial wave decomposition of the translational wavefiinctions for each value of j is possible. The translational wave is decomposed according to... 

Chemical Properties. Diacyl peroxides (20) decompose when heated or photoly2ed (<300 mm). Although photolytic decompositions generally produce free radicals (198), thermal decompositions can produce nonradical and radical iatermediates, depending on diacyl peroxide stmcture. Symmetrical aUphatic diacyl peroxides of certain stmctures, ie, diacyl peroxides (20, = alkyl) without a-branches or with a mono-cx-methyl... 

Since the deformation tensor F is nonsingular, it may be decomposed uniquely into a proper orthogonal tensor R and a positive-definite symmetric tensor U by the polar decomposition theorem... 

Both symmetrical and unsymmetrical azo compounds can be made, so that a single radical or two different ones may be generated. The energy for the decomposition can be either thermal or photochemical. In the thermal decomposition, it has been established that the temperature at which decomposition occurs depends on the nature of the substituent groups. Azomethane does not decompose to methyl radicals and nitrogen until temperatures above 400°C are reached. Azo compounds that generate relatively stable radicals decompose at much lower temperatures. Azo compounds derived from allyl groups decompose somewhat above 100°C for example ... 

Although this material contains a small amount of the symmetrical dihydrazide, which is not easily eliminated on crystallization, it is entirely satisfactory for use as a reagent for the isolation of ketones. A purer product, m. p. 192°, with decomposition, can be obtained by adding the solution prepared from ethyl chloroacetate and trimethylamine to an alcoholic solution containing a considerable excess of the hydrazine hydrate. 

Reference to the decomposition of KMn04 has already been made in the discussion of chain branching reactions (Chap. 3, Sect. 3.2) in which the participation of a highly reactive intermediate was postulated. This work provided a theoretical explanation of the Prout—Tompkins rate equation [eqn. (9)]. Isothermal decomposition in vacuum of freshly prepared crystals at 473—498 K gives symmetrical sigmoid a time curves which are described by the expression... 

Asymmetric fission is observed in the spontaneous decomposition of sCf1Ji(15M and other very heavy nuclei. We may ask when the transition to symmetric fission would begin. The next elongated core, in the series represented in Figs. 11 and 12, would contain 31 spherons, and the transition to it should occur for 28 spherons in the core of the undistorted nucleus, that is, at N = 163 (calculated with use of Eq. 1). We conclude that lftf,Lw,(i,20 and adjacent nuclei should show both asymmetric and symmetric fission. 

The effect has been most commonly encountered in the decomposition of symmetrical diacyl peroxides where it is easily recognized since the symmetrical radical dimer, for which Ag must be zero, is formed and shows net polarization. Clearly, studies of such systems are capable of providing valuable information on the dynamics of radicals and radical pairs in solution, the polarization process providing a time base for events (see Section V,B). 

The rates of radical-forming thermal decomposition of four families of free radical initiators can be predicted from a sum of transition state and reactant state effects. The four families of initiators are trarw-symmetric bisalkyl diazenes,trans-phenyl, alkyl diazenes, peresters and hydrocarbons (carbon-carbon bond homolysis). Transition state effects are calculated by the HMD pi- delocalization energies of the alkyl radicals formed in the reactions. Reactant state effects are estimated from standard steric parameters. For each family of initiators, linear energy relationships have been created for calculating the rates at which members of the family decompose at given temperatures. These numerical relationships should be useful for predicting rates of decomposition for potential new initiators for the free radical polymerization of vinyl monomers under extraordinary conditions. 

Predictive equations for the rates of decomposition of four families of free radical initiators are established in this research. The four initiator families, each treated separately, are irons-symmetric bisalkyl diazenes (reaction 1), trans-phenyl, alkyl diazenes (reaction 2), tert-butyl peresters (reaction 3) and hydrocarbons (reaction 4). The probable rate determining steps of these reactions are given below. For the decomposition of peresters, R is chosen so that the concerted mechanism of decomposition operates for all the members of the family (see below)... 

Equation 6 would hold for a family of free radical initiators of similiar structure (for example, the frarw-symmetric bisalkyl diazenes) reacting at the same rate (at a half-life of one hour, for example) at different temperatures T. Slope M would measure the sensitivity for that particular family of reactants to changes in the pi-delocalization energies of the radicals being formed (transition state effect) at the particular constant rate of decomposition. Slope N would measure the sensitivity of that family to changes in the steric environment around the central carbon atom (reactant state effect) at the same constant rate of decomposition. 

In Table I are listed the radical products (R )(column 2), AE(x) values (column 3), EA values (column 4) and the experimental temperatures for the one- and ten hour half life rates for the decomposition of trona-symmetric bisalkyl diazenes (columns 5 and 6), (rona-phenyl,alkyl diazenes (columns 7 and 8), peresters (columns 9 and 10) and hydrocarbons (columns 11 and 12). 

The quality of fit to the linear equation 7 is excellent for the radical forming decompositions of Irons-symmetric bisalkyl diazenes (reaction 1 - Table II) and Irons-phenyl, alkyl diazenes (reaction 2 - Table II). The quality of fit to equation 7 is not as high for the radical forming decompositions of lerl-butyl peresters (reaction 3 - Table II) and hydrocarbons (reaction 4 - Table II). This suggests that transition state arguments may be used to rationalize the rates of reactivity very well for reactions 1 and 2, and fairly well for reactions 3 and 4. 

Since the quantum chemical calculations used to parameterize equations 6 and 7 are relatively crude semiempirical methods, these equations should not be used to prove or disprove differences in mechanisms of decomposition within a family of initiators. The assumption made in the present study has been that the mechanism of decomposition of initiators does not change within a particular family of initiators (reactions 1-4). It is generally accepted that trow5-symmetric bisalkyl diazenes (1) decompose entirely by a concerted, synchronous mechanism and that trans-phenyl, alkyl diazenes (2) decompose by a stepwise mechanism, with an intermediate phenyldiazenyl radical (37). For R groups with equal or larger pi-... 

The linear free energy equations generated in this study are useful for reasonably accurate predictions of rates of radical forming decompositions of Irons-symmetric bisalkyl diazenes (1),... 

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