Bimolecular Model

A hypothetical bimolecular model was proposed by Lefever (1968). For certain values of parameters this model exhibits an oscillatory solution. This oscillatory solution is represented by a limit cycle. 

This reaction scheme consists of a set of reactions between the two intermediate substances X and Y as follows  

Rate equations lead to two nonlinear differential equations for X and Y concentrations. The nonlinearity of the system is due to simple power terms such as x y. 

The determinant of the coefficient matrix vanishes, A2 = 0, thus this condition is satisfied for A = 0. 

There are in fact two solutions after the singular solution, X = A, Y — BjA bifurcates, the new solution is a stable limit cycle. Lefever and Nicolis (1971) give these solutions for the point with parameter values A = 1,5 = 3, lying in the parameter plane region corresponding to a limit cycle and an unstable singular point, simultaneously. 

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As discussed in section 3.3, there is no reason to expect that measures of "overall" redox conditions (such as Pt electrode potentials or concentrations of dissolved H2) will ever provide an improved basis for quantitatively predicting rates of environmental redox reactions. However, extensions and refinements to the simplified bimolecular model can be made when sufficient data are available. 

TABLE 19.2. Values for Fractional Occupancy (FO) Versus Concentration or log Concentration for the Bimolecular Model... 

Theoretical analysis of models of transition states in bimolecular and termolecular reactions of NO with F2 indicates that the termolecular rate would be eight orders of magnitude lower than the observed rate even if the activation energy of the termolecular reaction were zero. The bimolecular model yields a pre-exponential factor which is in agreement with the experimental results. Therefore, the reaction between NO and F2 is bimolecular even at dry-ice temperatures. 

Gillespie and Mangel (1981) proposed a stochastic formulation of chemical kinetics and presented an explanation for the limit cycle of the bimolecular model. 

Fig. 11.5 (pp. 479-480). Reaction order for reactivation of several enzymes, determined from initial slopes (A) GDPH (Rudolph et aL, 1977b) (B) LDH-H4 (Rudolph et a/., 1977a) for both enzymes plots were calculated for irreversible unimolecular bimolecular model [C) LDH-M4 reaction order n = 2.0 0.1 (Rudolph et a/., 1976) (D) TIM reaction order n = 2.0 + 0.2, at high protein concentration the first-order reaction becomes significant (Jaenicke, 1979) (E) Zn liver alcohol dehydrogenase, n = 1.9 0.2 (Jaenicke 1979). 

The results from studies of the kinetics of refolding of various oligomeric proteins are summarized in Table 11.1. Also listed are the kinetic constants for renaturation of the different enzymes studied by Jaenicke and co-workers as well as the activity of the monomeric unit (if any) fitting with the irreversible unimolecular-bimolecular model (Jaenicke, 1979 Jaenicke and Rudolph, 1980). 

Flere, we shall concentrate on basic approaches which lie at the foundations of the most widely used models. Simplified collision theories for bimolecular reactions are frequently used for the interpretation of experimental gas-phase kinetic data. The general transition state theory of elementary reactions fomis the starting point of many more elaborate versions of quasi-equilibrium theories of chemical reaction kinetics [27, M, 37 and 38]. 

Smoluchowski theory [29, 30] and its modifications fonu the basis of most approaches used to interpret bimolecular rate constants obtained from chemical kinetics experiments in tenus of difhision effects [31]. The Smoluchowski model is based on Brownian motion theory underlying the phenomenological difhision equation in the absence of external forces. In the standard picture, one considers a dilute fluid solution of reactants A and B with [A] [B] and asks for the time evolution of [B] in the vicinity of A, i.e. of the density distribution p(r,t) = [B](rl)/[B] 2i ] r(t))l ] Q ([B] is assumed not to change appreciably during the reaction). The initial distribution and the outer and inner boundary conditions are chosen, respectively, as... 

Many additional refinements have been made, primarily to take into account more aspects of the microscopic solvent structure, within the framework of diffiision models of bimolecular chemical reactions that encompass also many-body and dynamic effects, such as, for example, treatments based on kinetic theory [35]. One should keep in mind, however, that in many cases die practical value of these advanced theoretical models for a quantitative analysis or prediction of reaction rate data in solution may be limited. 

The simple difhision model of the cage effect again can be improved by taking effects of the local solvent structure, i.e. hydrodynamic repulsion, into account in the same way as discussed above for bimolecular reactions. The consequence is that the potential of mean force tends to favour escape at larger distances > 1,5R) more than it enliances caging at small distances, leading to larger overall photodissociation quantum yields [H6, 117]. 

Using a guided ion beam instrument the translational energy dependent reaction cross sections of endothemiic fragmentation processes can be detemiined [32]. Modelling these cross sections ultimately yields their energy tln-esholds and a great deal of valuable themiochemical infomiation has been derived with this teclmique. Precision of 0.2 eV can be obtained for reaction tln-esholds. Bimolecular reactions can also be studied and reaction enthalpies derived from the analysis of the cross section data. 

The catalytic effect on unimolecular reactions can be attributed exclusively to the local medium effect. For more complicated bimolecular or higher-order reactions, the rate of the reaction is affected by an additional parameter the local concentration of the reacting species in or at the micelle. Also for higher-order reactions the pseudophase model is usually adopted (Figure 5.2). However, in these systems the dependence of the rate on the concentration of surfactant does not allow direct estimation of all of the rate constants and partition coefficients involved. Generally independent assessment of at least one of the partition coefficients is required before the other relevant parameters can be accessed. 

Figure 5.2. Kinetic analysis of a bimolecular reaction A + B 7 C according to the pseudophase model.

In retrospect, this study has demonstrated the limitations of two commonly accepted methods of analysing solubilisation and micellar catalysis, respectively. It has become clear that solubilisate ririg-current induced shifts need to be interpreted with due caution. These data indicate a proximity of solubilisate and parts of the surfactant and, strictly, do not specify the location within the micelle where the encounter takes place. Also the use of the pseudophase model for bimolecular reactions requires precaution. When distribution of the reactants over the micelle is not comparable, erroneous results are likely to be obtained... 

Given the molecular formula CgHnBr construct a molecular model of the isomer that is a pnmary alkyl bromide yet relatively unreactive toward bimolecular nucleophilic substitution... 

Phospholipids e.g. form spontaneously multilamellar concentric bilayer vesicles73 > if they are suspended e.g. by a mixer in an excess of aqueous solution. In the multilamellar vesicles lipid bilayers are separated by layers of the aqueous medium 74-78) which are involved in stabilizing the liposomes. By sonification they are dispersed to unilamellar liposomes with an outer diameter of 250-300 A and an internal one of 150-200 A. Therefore the aqueous phase within the liposome is separated by a bimolecular lipid layer with a thickness of 50 A. Liposomes are used as models for biological membranes and as drug carriers. 

It would be reasonable to expect that the decomposition of the N,N-dimethylimino ester chlorides proceeds via a bimolecular mechanism already demonstrated for the thermal decomposition of simple imino ester salts (79). In the carbohydrate series, where an isolated secondary hydroxyl group is involved, such a process would result in chlorodeoxy sugar derivatives with overall inversion of configuration, provided that the approach of the chloride ion is not sterically hindered. Further experiments are in progress in this laboratory utilizing additional model substance to establish the scope and stereochemical course of the chlorination reaction. 

Michael reactions and, 895 Beta-keto ester, 851 alkylation of, 859-860 cyclic, 892-893 decarboxylation of, 857, 860 Michael reactions and. 895 pKd of, 852 synthesis of, 892-893 Beta-lactam antibiotics, 824-825 Beta oxidation pathway, 1133-1137 mechanism of, 1133-1136 Beta-pleated sheet (protein), 1038 molecular model of, 1039 secondary protein structure and, 1038-1039 Betaine, 720 Bextra. structure of, 544 BHA, synthesis of, 629 BHT, synthesis of. 629 Bicycloalkane. 129 Bijvoet. J. M., 299 Bimolecular, 363... 

A device model to describe two-carrier structures is basically similar to that used for one carrier structures except that continuity equations for both earner types are solved. The additional process that must be considered is charge carrier recombination. The recombination is bimolecular, R=y(np), where the recombination coefficient is given by 43)... 

To explain the observed magnitude of E and other kinetic features of reaction, a homogeneous bimolecular interaction between neighbouring CIO4 ions in the crystal structure was postulated and application of the activated complex theory to this model gave good agreement with the experimental observations. 

On the other hand, Doblhofer218 has pointed out that since conducting polymer films are solvated and contain mobile ions, the potential drop occurs primarily at the metal/polymer interface. As with a redox polymer, electrons move across the film because of concentration gradients of oxidized and reduced sites, and redox processes involving solution species occur as bimolecular reactions with polymer redox sites at the polymer/solution interface. This model was found to be consistent with data for the reduction and oxidation of a variety of species at poly(7V-methylpyrrole). This polymer has a relatively low maximum conductivity (10-6 - 10 5 S cm"1) and was only partially oxidized in the mediation experiments, which may explain why it behaved more like a redox polymer than a typical conducting polymer. 

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