Schlogl reaction

Now consider the evolution in a situation as in fig. 34. There are three stationary macrostates 0fl, 0j>, c, of which 0a and 0C are locally stable and (j)b is unstable. Of course, even the pure macroscopist would not regard (j)b as a realizable state, on the ground that a system in (j>b would be caused to move into either 0a or 0C by the smallest perturbation. Systems having a macroscopic characteristic as in fig. 34 are called bistable . There are numerous examples the ones that occur most often in the literature are the laser (section 9 below), the tunnel diode 0, and the Schlogl reaction (X.3.6). The macroscopic rate equation for this reaction is... 

Exercise. One-step processes have been approximated by diffusion equations in VIII.5, but the danger has been pointed out in XI.5. Show for the Schlogl reaction (X.3.6) that it is incorrect to compute the relative stability (1.11) by using the potential function V in (XI.5.6) rather than U. 

Exercise. Find the equilibrium fluctuations at the critical point of the Schlogl reaction (X.3.6). 

A further discussion of the Schlogl reaction provides an illustration of an important difference in the behavior of steady states in deterministic systems and in systems subject to stochastic fluctuations. In contrast to the deterministic Schlogl system analyzed earlier in terms of conventional chemical concentrations, the analysis of the stochastic system is carried out in terms of the probability that there are n x t) molecules in the system at time t in addition to nA = a and n-Q = b molecules of A and B that are held constant. Figure 6.2 illustrates the individual pathways and rates by which a system with n molecules could undergo a transition to a system containing either n -f 1 or n — 1 molecules. This leads to the master equation for each of the... 

Fig. 6.2. Kinetic pathways for the Schlogl reaction. There are four pathways by which the system can either increase or decrease the number of x molecules. The rates depend on the numbers of the interacting molecules and the rate constants as shown...

The two-cell stochastic model of the Schlogl reaction describing first-order phase transition has been carefully studied (Borgis Moreau, 1984b Moreau Borgis, 1984). 

Borgis, D. Moreau, M. (1984b). Two cell stochastic model of Schlogl reaction with small diflfusional coupling, J. Stat. Phys., 34, 631-51. 

As a simple illustration of how reactions may be treated stochastically on mesoscopic scales, we show how one may construct a mesoscopic Markov process model for the Schlogl reaction (Eq. [3]). As before, we assume that species A and B in the Schlogl mechanism are pool species and their concentrations do not change in time Their effect is incorporated into the rate constants of the reaction. A Markov model of the reactions is a birth-death process, where chemical species are born in some reactions (X is born in the reactions (1) A -> X and (3) 2X - - B 3X) and dies in others (X dies in the reactions (2) X A and (4) 3X 2X - - B). Suppose there are ttx molecules of species X in a cell. Then the probability that a reaction X — A occurs and changes the number of molecules in the cell from x to — 1 is just... 

AN ELEMENTARY BISTABLE REACTION THE SCHLOGL MODEL 2.1 Schlogl reaction... 

The nucleation rate of the Schlogl reaction has been explicitly... 

Blume R, Havecker M, Zafeiratos S, Teschner D, Kleimenov E, Knop-Gericke A, Schlogl R, Barinov A, Dudin P, Kiskinova M. 2006. Catalytically active states of Ru(OOOl) catalyst in CO oxidation reaction. J Catal 239 354. 

Stohr, B., Boehm, H.P., and Schlogl, R. Enhancement of the catalytic activity of activated carbons in oxidation reactions by thermal treatment with ammonia or hydrogen cyanide and observation of a superoxide species as a possible intermediate. Carbon 29, 1991 707-720. 

Schlogl R, Hess C. Characteristics of selective oxidation reactions. In Hess C, Schlogl R, editors. Nanostructured catalysts Selective oxidation reactions. Cambridge RSC Nanoscience Nanotechnology 2011. p. 355-397. 

Celaya Sanfiz A, Hansen TW, Girgsdies F, Timpe O, Rodel E, Ressler T, Trunschke A, Schlogl R. Preparation of phase-pure Ml MoVTeNb oxide catalysts by hydrothermal synthesis-influence of reaction parameters on structure and morphology. Topics in Catalysis. 2008 50(l t) 19-32. 

Bron M, Teschner D, Knop-Gericke A, Steinhauer B, Scheybal A, Havecker M, Wang D, Fodisch R, Honicke D, Wootsch A, Schlogl R, Claus P. Bridging the pressure and materials gap in-depth characterisation and reaction studies of silver-catalysed acrolein hydrogenation. Journal of Catalysis. 2005 234(l) 37-47. 

An example is the transformation of the technological iron catalyst (Greeley and Mavrikakis, 2005 Herzog et al., 1996 Schlogl, 1991) used for ammonia synthesis (Figure 5). This catalyst has been shown recently to be a sub-nitride under catalytic reaction conditions. At ambient pressure the chemical potential of ammonia is only sufficient to form a phase Fe N being a sub-nitride of Fe2N. This structure, which is distinctly different... 

Because the physics of XRD requires strong compromises in the choice of the sample environment, it is not realistic to apply actual catalytic reaction conditions in these measurements. Unavoidably, some extrapolations have to be made to link experimental observations to the state of the functioning catalyst in a reactor optimized for performance in the sense of production of desired products. The gas flow and energy transport conditions, for example, must often be chosen to be different in the catalyst testing and in XRD experiments. The catalytic performance is often analyzed in terms of a model of the kinetics, and the results may be used to make a connection with the conditions of the XRD experiment. A discussion of the issues has been published (Schlogl and Baerns, 2004). 

For industrial catalysts made by careful reduction of magnetite fused with nonreducible oxide promoters the important role of the (111) face seems to be confirmed [154]. However, the question whether the active industrial catalyst exposes mostly (111) faces remains unresolved. If not, further improvements of the catalyst are at least theoretically possible [155]. A critical evaluation of our present knowledge of the reaction mechanism was recently made by Schlogl [156]. 

For each set of constant input and output concentration constraints a system of linear chemical reactions has a unique steady state. For a network of nonlinear biochemical reactions, however, there could be several steady states compatible with a given set of constraints. The number and character of these steady states are determined by the structure of the network including the extent of nonlinearity, the number and connectivity of the individual chemical reactions and the values of the reaction rate constants and the concentrations of the reactants. The higher the order of a chemical reaction, the more steady states may be compatible with a given set of chemical constraints. The simple trimolecular reaction system of Schlogl [13] illustrates how a third-order chemical reaction can have two stable steady states compatible with a single set of chemical constraints ... 

F. Schlogl, Chemical reaction models for non-equilibrium phase transition. Z. Physik. 253, 147-161 (1972)... 

Schlogl and coworkers were also among the first to investigate Cgg interaction with oxygen [60], although in connection with degradative chemical reactions... 

Ferrocene behaves like an aromatic compound activated for electrophilic substitution reactions. Thus, only minor modifications of experimental procedures developed for aromatics are necessary to obtain ferrocene derivatives (a useful review on general methods is given by Schlogl and Falk [42]). For central chiral ferrocenes, resolution of the racemate is a frequently applied technique. Traditionally, resolutions are best achieved by salt formation between a chiral acid or base and the... 

X. Bao, B. Pettinger, G. Ertl, and R. Schlogl, In-situ Raman Studies of Ethylene Oxidation at Ag(lll) and Ag(llO) Under Catalytic Reaction Conditions, Ber. Bun-sen-Ges. Phys. Chetn. Chem. Phys. 97, 322 (1993)... 

In 1957, Schlogl reported the synthesis and characterization of several ferrocene amino acids, including ferrocenylala-nine (Fer, 14 Scheme 18). Other organometallic amino acids include alanine, phenylalanine (Phe), glycine derivatives, as well as further ferrocene-based amino acids, which are discussed below. The Schlogl paper also describes the reaction of ferrocene carboxylic acid and ferrocene carbaldehyde with amino acids. The chemistry was later picked up in numerous publications, and ferrocene carboxylic acid amides or ferrocene imines with basically all amino acids and many dipeptides were reported. Such conju tes may serve as ligands for other transition metal ions. " Kraatz has prepared ferrocene carboxylic acid conju tes with short peptides that serve as enzyme inhibitors. ... 

The original version of the first Schlogl model contains a fourth step, the back reaction C —> B -I- U. We assume that the product C is immediately removed from the system. We consider two different ways of nondimensionalizing this rate equation, (i) Set t = k2 PnP ) t and p = 2/°/( iPa k- P, ). This is acceptable for all situations where k p > k p. Then we obtain the following nondimensionalized version of (1.66) ... 

As the previous models, the second Schlogl model relies on the pool chemical assumption to account for the inflow of reactants. If we assume instead that this reaction scheme occurs in a CSTR, we obtain the Gray-Scott model [170, 171] ... 

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