Reversible bimolecular reaction

Reversible bimolecular reactions such asA + B C + D can be solved exactly by the method of separation of variables and the ordinary differential equations in the variable s are Lame equations. This makes the evaluation of the Fourier-type coefficients very difficult since derivative formulas and orthogonality conditions do not seem to exist or at least are not easily used. In addition to this, even if such formulas did exist, it seems unlikely that numerical results could be easily obtained. It does turn out, however, that these reversible bimolecular processes can be solved exactly and conveniently in the equilibrium limit, and this was done by Darvey, Ninham, and Staff.14... 

Approximate treatment of the many-particle effects in reversible bimolecular reactions has been undertaken in several papers (see for a review [78]) we would like also to note here pioneering studies of Ovchinnikov s group [79-82] and Kang and Redner s paper [83]. The former approach was discussed above in Section 2.1.2.3 where the kinetics of the approach to equilibrium for the simple reaction A B + B (dissociation and association of molecules A) was shown to approach the equilibrium as t 3/2. Note also that in the paper [84] a new elegant quantum-field formalism has been developed for the first time and applied to the diffusion-controlled reactions in the fluctuation regime its results agree completely with the phenomenological estimate (2.1.61). 

The mathematical theory of a reversible bimolecular reaction in an explosion. 

This expression coincides quite well with the equation for a reversible bimolecular reaction, which we used earlier. In particular, the rate of decomposition remains proportional to the square of the nitric oxide concentration, and the reaction rate is zero at the equilibrium concentration. The latter statement follows direction from the theory of non-branching chains, since for such chains the chemical energy of the centers cannot be utilized... 

As long as the oxygen concentration remains constant throughout the cooling of the explosion products, the previously developed similarity theory and the mathematical theory of a reversible bimolecular reaction in an explosion remain valid in their entirety. 

There are many kinetic evidences for the fact that the nitronium cation and the aromatic substrate are involved in a reversible bimolecular reaction to form a o-complex, which, being a strong acid, undergoes fast deprotonation (Scheme 2). 

In the simplest case, the protein-ligand interaction can be represented as, or modeled as, a reversible bimolecular reaction such as depicted by P-i-l.ol L. The change in Gibbs free energy (AG) for the interaction in the direction indicated is related to the standard free energy change (AG") by the following equation ... 

When a reaction is reversible, the overall rate expression must be thermodynamically consistent. For a reversible bimolecular reaction where all species are competitively adsorbed, the net rate of reactant consumption, r, is... 

They state, however, that the obeerved reaeti m veloeitiee do not fit in with the curves cf either a simple or a reversible bimolecular reaction, and criticize Cohen s statement that transamination velocities can be described by bimoleeular velocity constants. 

When the perturbation is small, the reaction system is always close to equilibrium. Therefore, the relaxation follows generalized first-order kinetics, even if bi- or trimolecular steps are involved (see chapter A3.41. Take, for example, the reversible bimolecular step... 

Based on this work, it has been proposed that a specifically solvated carbene (Scheme 4.6, Reaction 2) nndergoes bimolecular reactions at slower rates than a free carbene (Scheme 4.6, Reaction 1). Other alternatives that mnst be considered are participation of rapid and reversible ylide formation with the ylide acting as a... 

In initial rate studies no products need be present in the feed, and the terms in the rate expression involving the partial pressures of these species may be omitted under appropriate experimental conditions. The use of stoichiometric ratios of reactants may also cause a simplification of the rate expression. If one considers a reversible bimolecular surface reaction between species A and ,... 

Reactions and reactivity of nucleophiles with thiolsulfonates 137 Nucleophilic substitutions of sulfenyl derivatives general considerations 139 Bimolecular substitution at sulfenyl sulfur stepwise or concerted 140 Reversibility in reactions of nucleophiles with cyclic thiolsulfonates 145 Other reactions of thiolsulfonates 147... 

The absence of dimer radical cation formation by diphenyl selenide under the pulse radiolysis conditions is in contrast to bimolecular reactions believed to occur under electrochemical conditions/ In these experiments, a rotating disk electrode was used in combination with commutative voltammetry under anhydrous conditions. The results led to the conclusion that reversible one-electron oxidation is followed by disproportionation, then reaction of the resulting dication with diphenyl selenide or an external nucleophile, with the likely intermediacy of the dimer dication (Fig. 33). As expected, the dihydroxy selenane is formed when water is present. Based on the kinetics of the electrochemical reaction, the authors believe the diselenide dication, not the radical cation, to be the intermediate that reacts with the nucleophile. 

Bimolecular reactions may be classified into two major groups direct metathesis and association reactions. The latter also are related to the reverse of the unimolecular reactions discussed above. However, as we shall see, significant differences would exist when there are energy transfer limitations. It is also convenient further to classify bimolecular reactions as radical-molecule, radical-radical, and molecule-molecule reactions. The application of TST to bimol ular reactions, described symbolically by... 

The QRRK approach illustrated above also constitutes the basis to analyze the behavior of the reverse, i.e., association, reactions that proceed through chemically activated transition states. Recently Dean (1985) reformulated the unimolecular quantum-RRK method of Kassel and devised a practical method for the proper description of the fall-off behavior of bimolecular reactions, including reactions when multiple product channels are present. The method developed was shown to describe the behavior of a large variety of bimolecular reactions with considerable success (Dean and Westmoreland, 1987 Westmoreland et ai, 1986). 

Intramolecular general base catalysis of hycholysis (21a) was unexpected since the ester has a phenolic leaving group. Felton and Bruice (1968, 1969) reasoned that, if nucleophilic attack occurred, the leaving phenolate ion group would be properly positioned to attack the intermediate acylimidazole and thereby reverse the reaction. The normally less efficient general base reaction then becomes the favoured pathway, as in hydrolysis of acetyl salicylate (see Section 4). Likewise, Fife and McMahon (1970) explained bimolecular general base catalysis by imidazole (21b) in hydrolysis of o-(4-nitrophenylene) carbonate 3 49) by reversibility... 

The above discussion shows that several possible pathways for the interconversion of sulfur rinp exist. However, none of these alone can explain all the experimental observations. It therefore seems likely that several of them are effective simultaneously. Unimolecular dissociation reactions as discussed under (a) and (d) will dominate at high temperatures due to the increase in entropy. At lower temperatures, however, bimolecular reactions like the dimerization (c) may be most important, at least in case of the small rings (Sg, S, Sg) whose unimolecular dissociation is strongly endothermic. Larger rings will probably decompose according to mechanism (d), which in a way is the reversal of the dimerization (c). 

Bimolecular reactions of two molecules, A and B, to give two products, P and Q, are catalyzed by many enzymes. For some enzymes the substrates A and B bind into the active site in an ordered sequence while for others, bindingmay be iii a random order. The scheme shown here is described as random Bi Bi in a classification introduced by Cleland. Eighteen rate constants, some second order and some first order, describe the reversible system. Determination of these kinetic parameters is often accomplished using a series of double reciprocal plots (Lineweaver-Burk plots), such as those at the right. 

Reversible chemical reactions. In any reversible process, we must consider rate constants for both the forward and the reverse reactions. At equilibrium a reaction proceeds in the forward direction at exactly the same velocity as in the reverse reaction so that no change occurs. For this reason there is always a relationship between the equilibrium constant and the rate constants. For Eq. 9-9, /c is the bimolecular rate constant... 

In particular, it is useful to define the critical point through F(nc) = 0 (the stationary state). Since multicomponent chemical systems often reveal quite complicated types of motion, we restrict ourselves in this preliminary treatment to the stable stationary states, which are approached by the system without oscillations in time. To illustrate this point, we mention the simplest reversible and irreversible bimolecular reactions like A+A —> B, A+B -y B, A + B —> C. The difference of densities rj t) = n(t) — nc can be used as the redefined order parameter 77 (Fig. 1.6). For the bimolecular processes the... 

A new principal element of the Waite-Leibfried theory compared to the Smoluchowski approach is the relation between the effective reaction rate K(t) and the intermediate order parameter x = Xab (r,i). In its turn, the Smoluchowski approach is just an heuristic attempt to describe the simplest irreversible bimolecular reactions A + B- B,A + B- B and A + B -> 0 and cannot be extended for more complicated reactions. The Waite-Leibfried approach is not limited by these simple reactions only it could be applied to the reversible reactions and reaction chains. However, in the latter case the particular linearity in the joint correlation function x = Xab (r, ) does not always mean linearity of equations since additional non-linearity caused by particle densities can arise. 

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