Elementary stability

Iooss, G. and Joseph, D. D. (1980). Elementary stability and bifurcation theory. Springer, New York. 

G. Iooss and D. D. Joseph, Elementary Stability and Bifurcation Theory (Springer-Verlag, New York, 1997). 

The forms of the equation and solution given by (6.335) and (6.337) also arise in the theory of non-chiral SmC liquid crystals when an applied electric field is tilted with respect to the planes of the smectic layers, as discussed by Schiller et al. [246]. They are also relevant when both electric and magnetic fields are applied parallel to the smectic layers across a sample of SmC, as envisaged by van Saarloos, van Hecke and Holyst [236]. In both these circumstances the coefficients a and b must obviously be different due to the tilt of the field or the inclusion of an additional magnetic field details may be found in References [236, 246]. Some further elementary stability analysis has been performed by van Saarloos et al [236] and Stewart [262]. 

An equation has been formulated to express the change in covalent radius (metallic radius) of an atom with change in bond number (or in coordination number, if the valence remains constant), the stabilizing (bond-shortening) effect of the resonance of shared-electron-pair bonds among alternative positions being also taken into consideration. This equation has been applied to the empirical interatomic-distance data for the elementary metals to obtain a nearly complete set of single-bond radii. These radii have been compared with the normal covalent... 

Pj release occurs at a relatively apparent slow rate (kobs = 0.005 s" ), so that the transient intermediate F-ADP-Pj in which P is non-covalently bound, has a life time of 2-3 minutes (Carlier and Pantaloni, 1986 Carlier, 1987). While the y-phosphate cleavage step is irreversible as assessed by 0 exchange studies (Carlier et al., 1987), the release of Pi is reversible. Binding of H2PO4 (Kp 10 M) causes the stabilization of actin filaments and the rate of filament growth varies linearly with the concentration of actin monomer in the presence of Pi (Carlier and Pantaloni, 1988). Therefore, Pi release appears as the elementary step responsible for the destabilization of actin-actin interactions in the filament. 

Ni catalysts u g ZrOj as a unique support that seems crucial to minimize coking under reaction conditions applied for CH4/CO2 reforming. For two successfiiUy developed catalysts, (Pt and Ni on Z1O2) the present contribution outlines the sequence of the elementary steps and the catalytic chemistry of the active metal and the support in order to explain catalysts activity and stability. 

The results here clearly demonstrate some of the important differences between reactions in the vapor phase and those in the aqueous phase. Water solvates the ions that form and thus enhances the heterolytic bond activation processes. This leads to more significant stabilization of the charged transition and product states over the neutral reactant state. The changes that result in the overall energies and the activation barriers of particular elementary steps can also act to alter the reaction selectivity and change the mechanism. 

The role of the Pt-Ba interaction in the mechanism of adsorption of NO species was also studied by a kinetic model reported in the literature [16]. The model, which consists of 10 elementary reversible steps, is based on the oxidation of NO to N02 over Pt and on the storage of N02 over Ba, and it was used to simulate the data collected over both the physical mixture and the ternary Pt-Ba/y-Al203 1/20/100 w/w sample. A spillover reaction between Pt and Ba oxide sites has also been included in the model to account for the observed lower thermal stability of Ba-nitrates in the presence of Pt [16]. Essentially, the model assumes that the adsorption of NO proceeds through the nitrate route and does not consider the nitrite route. 

The intermediate reaction complexes (after formation with rate constant, fc,), can undergo unimolecular dissociation ( , ) back to the original reactants, collisional stabilization (ks) via a third body, and intermolecular reaction (kT) to form stable products HC0j(H20)m with the concomitant displacement of water molecules. The experimentally measured rate constant, kexp, can be related to the rate constants of the elementary steps by the following equation, through the use of a steady-state approximation on 0H (H20)nC02 ... 

The important elementary reactions of coal liquefaction are the decomposition of coal structure with low bond dissociation energy, the stabilization of fragments by the solvent and the dissolution of coal units into the solution. 

Assumption 5. Transfer processes as within the cell have been regarded as quasistationary. The typical time of the processes in the electrode (time of a charging or discharging being Ur-I04 s) is longer than the time of the transitional diffusion process in the elementary cell tc Rc2/D 10"1 s (radius of the cell is Rc 10"5 m, diffusion coefficient of dissolved reagents is D 10"9 m2/s). Therefore, the quasistationary concentration distribution is quickly stabilized in the cell. It is possible to neglect the time derivatives in the transport equations. 

Elementary reaction mechanisms for nitrous oxide (N20) dissociation were studied on Fc"( i-0)( i-0H)Fc" + exchanged in ZSM-5, using density functional theory (DFT). The effect of the cluster size on the energetics and on the reaction routes of N20 dissociation were investigated over di-iron core inserted inside two different Z cluster (Z ) and (Z oh)- The results show that while the relative stability changes with the cluster termination, the height of the energetic barriers are similar. 

One of the most efficient approaches allowing us to investigate in a reasonable time a catalytic cycle on non-periodic materials in combination with reliable DFT functional is a cluster approach. The present study is devoted to the investigation of the effect of the cluster size on the energetic properties of the (p-oxo)(p-hydroxo)di-iron metal active site. As a first step, we have studied the stability of the [Fen(p-0)(p-0H)Fen]+ depending on the A1 position and cluster size. Then, we compared the energetics for the routes involving the first two elementary steps of the N20 decomposition catalytic process i.e. the adsorption and dissociation of one N20 molecule. 

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