Number of transitions

Direct splitting requires temperatures above 977°C. Yields of around 30% at 1127°C are possible by equiUbrium. The use of catalysts to promote the reaction can lower the temperature to around the 327—727°C range. A number of transition metal sulfides and disulfides are being studied as potential catalysts (185). Thermal decomposition of H2S at 1130°C over a Pt—Co catalyst with about 25% H2 recovery has been studied. 

Propane, 1-propanol, and heavy ends (the last are made by aldol condensation) are minor by-products of the hydroformylation step. A number of transition-metal carbonyls (qv), eg, Co, Fe, Ni, Rh, and Ir, have been used to cataly2e the oxo reaction, but cobalt and rhodium are the only economically practical choices. In the United States, Texas Eastman, Union Carbide, and Hoechst Celanese make 1-propanol by oxo technology (11). Texas Eastman, which had used conventional cobalt oxo technology with an HCo(CO)4 catalyst, switched to a phosphine-modified Rh catalyst ia 1989 (11) (see Oxo process). In Europe, 1-propanol is made by Hoechst AG and BASE AG (12). 

Wilkinson Hyd.rogena.tion, One of the best understood catalytic cycles is that for olefin hydrogenation in the presence of phosphine complexes of rhodium, the Wilkinson hydrogenation (14,15). The reactions of a number of olefins, eg, cyclohexene and styrene, are rapid, taking place even at room temperature and atmospheric pressure but the reaction of ethylene is extremely slow. Complexes of a number of transition metals in addition to rhodium are active for the reaction. 

Fixed-Bed Behavior The number of transitions occurring in a fixed bed of initially uniform composition before it becomes saturated by a constant composition feed stream is generally equal to the variance of the system. This introductory discussion will be limited to single transition systems. 

Results of quantum-mechanical calculations (Dunitz and Orgel, 1957) have given values for the OSPEs of a number of transitional metal ions and the degree of inversion of mixed spinels... 

A number of transition-metal complexes of RNSO ligands have been structurally characterized. Three bonding modes, r(A,5), o-(5)-trigonal and o (5 )-pyramidal, have been observed (Scheme 9.1). Side-on (N,S) coordination is favoured by electron-rich (et or j °) metal centers, while the ff(S)-trigonal mode is preferred for less electron-rich metal centres (or those with competitive strong r-acid co-ligands). As expected ti (N,S)... 

First, consider the case in which the spin system and the lattice are in equilibrium at temperature T in field Hq. This means that the number of transitions upward must equal the number downward. If Wu is the probability of an upward transition per unit time and is the probability of a downward transition, then =... 

Let P a a ) be the probability of transition from state a to state a. In general, the set of transition probabilities will define a system that is not describ-able by an equilibrium statistical mechanics. Instead, it might give rise to limit cycles or even chaotic behavior. Fortunately, there exists a simple condition called detailed balance such that, if satisfied, guarantees that the evolution will lead to the desired thermal equilibrium. Detailed balance requires that the average number of transitions from a to a equal the number of transitions from a to a ... 

HarM63a Harrison, M A. The number of transitivity sets of Boolean functions. SIAM J. 11 (1963) 886-878. 

For most real systems, particularly those in solution, we must settle for less. The kinetic analysis will reveal the number of transition states. That is, from the rate equation one can count the number of elementary reactions participating in the reaction, discounting any very fast ones that may be needed for mass balance but not for the kinetic data. Each step in the reaction has its own transition state. The kinetic scheme will show whether these transition states occur in succession or in parallel and whether kinetically significant reaction intermediates arise at any stage. For a multistep process one sometimes refers to the transition state. Here the allusion is to the transition state for the rate-controlling step. 

Polysulfides have been prepared with many different types of cations, both monoatomic Hke alkah metal ions and polyatomic Hke ammonium or substituted ammonium or phosphonium ions. In this chapter only those salts will be discussed in detail which contain univalent main-group cations although a large number of transition metal polysulfido complexes have been prepared [7-9]. 

The latter reactions are catalyzed by a number of transition metal ions which can exist in several oxidation states in aqueous solution, e.g.,... 

This result can be simplified considerably by observing that, except for end effects, the number of transitions from X to Y along the polymer chain (i.e., structures like XY ) must equal the number of transitions from Y to X (i.e., structures like YX ). This requires that... 

In chemical kinetic studies the most relevant attributes are the counts of the various species present and the numbers of transitions of various types that occur during each iteration. For example, in a study of three types of reacting ingredients, A, B, and C, the numbers of each species will change with time, and this variation can reveal important information about the kinetics of the reactions involved. Also informative will be the numbers of transitions, say, from A B and A C, that take place in each iteration. 

The number of transitions between upper and lower states depends not only on the populations and of the two states, but also on the efficiency of relaxation, more often called the transition probability W, as well as on the temperature of the surrounding lattice. Each NMR transition will be accompanied by a corresponding change in the lattice energy. 

Although the path (a) has been verified by a stoichiometric reaction [23], the details of exact reaction mechanism remain unsettled. Triggered by this publication [and the Pd-catalyzed doublethiolation of alkynes described in Eq. (7.7) in Section 7-3], a number of transition metal-catalyzed additions of S-X or Se-X bonds to C-C unsaturated organic compounds started to be published. In 1994, BackvaU et al. applied the Pd(OAc)2-catalyzed hydrothiolation to conjugated enynes and obtained 17,... 

On the other hand, the enantioselective 1,4-addition of carbanions such as enolates to linear enones is an interesting challenge, since relatively few efficient methods exist for these transformations. The Michael reaction of p-dicarbonyl compounds with a,p-unsaturated ketones can be catalysed by a number of transition-metal compounds. The asymmetric version of this reaction has been performed using chiral diol, diamine, and diphosphine ligands. In the past few years, bidentate and polydentate thioethers have begun to be considered as chiral ligands for this reaction. As an example, Christoffers et al. have developed the synthesis of several S/O-bidentate and S/O/S-tridentate thioether... 

The quantity bm 2 represents the probability of the transition m - n. Clearly, the number of transitions per unit time depends on the intensity of the incident radiation, which is proportional to < J 2, and the square of the matrix element (m px n). The latter determines the selection rules for spectroscopic transitions (see the following section). 

---