Bimolecular reactions in liquids

Comparison of Bimolecular Reactions in Liquid and Solid Phases... 

In order to complete the set of equations describing polymerization reactions in dense and concentrated regimes, the rates kr and kd must be specified. In the stationary regime, in which the environmental responses to the microscopic motion of the polymer reactants can be assumed to follow the same regression throughout the reaction, these rates are time-independent. The theory of bimolecular reactions in liquids can then be applied at every reaction step. 

Although scan rates of few ten thousand volts per second (ie. 0/v 1 xs) seem to be a limit for quantitative analytical purposes (without using deconvolution procedures) for electrodes of few micrometers radii it is possible to use cyclic voltammetry in the low megavolt per second domain ( 0/v 25 ns) provided that the target is only identification of transient intermediates and determination of thermodynamic figures. Besides its intrinsic attractive value, such an order of magnitude of scan rates constitutes a milestone in electrochemical kinetics. Indeed bimolecular reactions in liquids cannot proceed faster than molecules encounter. Thus bimolecular rate constants are limited by diffusion limit rate... 

Diffusion of particles in the polymer matrix occurs much more slowly than in liquids. Since the rate constant of a diffusionally controlled bimolecular reaction depends on the viscosity, the rate constants of such reactions depend on the molecular mobility of a polymer matrix (see monographs [1-4]). These rapid reactions occur in the polymer matrix much more slowly than in the liquid. For example, recombination and disproportionation reactions of free radicals occur rapidly, and their rate is limited by the rate of the reactant encounter. The reaction with sufficient activation energy is not limited by diffusion. Hence, one can expect that the rate constant of such a reaction will be the same in the liquid and solid polymer matrix. Indeed, the process of a bimolecular reaction in the liquid or solid phase occurs in accordance with the following general scheme [4,5] ... 

Fourth, the model of a rigid cage for a bimolecular reaction in the polymer matrix helps to explain another specific feature. This model explains the simultaneous increase in activation energy and preexponential factor on transferring the reaction from the liquid (Eh At) to solid polymer matrix (Es, As). In the nonpolar liquid phase / obs = E = gas but in the polymer matrix [3,21] it is... 

A wide range of condensed matter properties including viscosity, ionic conductivity and mass transport belong to the class of thermally activated processes and are treated in terms of diffusion. Its theory seems to be quite well developed now [1-5] and was applied successfully to the study of radiation defects [6-8], dilute alloys and processes in highly defective solids [9-11]. Mobile particles or defects in solids inavoidably interact and thus participate in a series of diffusion-controlled reactions [12-18]. Three basic bimolecular reactions in solids and liquids are dissimilar particle (defect) recombination (annihilation), A + B —> 0 energy transfer from donors A to unsaturable sinks B, A + B —> B and exciton annihilation, A + A —> 0. 

The Collision Theory of Bimolecular Gaseous Reactions. This is the earliest theory of reaction rates. Since reaction between two species takes place only when they are in contact, it is reasonable to suppose that the reactant species must collide before they react. Since our knowledge of molecular collisions is more complete for the gaseous phase than for the liquid phase (in the latter case we speak of encounters rather than collisions), we will restrict our discussion to bimolecular reactions in the gaseous phase. 

If all the three particles (two reagents and an excitation quantum) meet simultaneously, the probability of the event Q and at the same time the reaction rate v will be proportional to [A] [B] q At where [A] and [B] are reagent concentrations, q is the probability density of the appearance of the suitable excitation quantum y in the place of a encounter between A and B at the corresponding time moment t and At is the time of the physical contact of reagents A and B in the bimolecular collision (in liquid At 10-13 s [21]). 

T. Arita, O. Kajimoto, M. Terazima, and Y. Kimura. Experimental verification of the smoluchowski theory for a bimolecular diffusion-controlled reaction in liquid phase. J. Chem. Phys., 120 7071-7074,2004... 

Because the forces giving rise to the formation of chemical bonds are very short-range forces, reactions in liquid solutions will require some sort of encounter or collision between reactant molecules. These encounters will differ appreciably from gas phase collisions in that they will occur in close proximity to solvent molecules. Indeed, in liquids any individual molecule will always be interacting with several surrounding molecules at the same time, and the notion of a bimolecular collision becomes rather arbitrary. Nonetheless, a number of approaches to formulating expressions for collision frequencies in the liquid phase have appeared in the chemical literature through the years. The simplest of these approaches presumes that the gas phase collision frequency expression is directly applicable to the calculation of liquid phase collision frequencies. The rationale for this approach is that for several second-order gas phase reactions that are also second-order in various solvents, the rate constants and preexponential factors are pretty much the same in the gas phase and in various solvents. For further discussion of the collision theory approach to reactions in liquids, consult the monograph by North (3). 

If the activation threshold Ajyw vanishes or is very low, just about every encotmter will lead to a reaction. Hence, it is not the height of this potential threshold but the frequency of collisions that then determines the conversion rate. In this case, the concentration of the transition complex can remain far below its equilibrium value because continued supply is stalled, while decomposition continues taking place. Reactions of this kind are said to be dijfusion-controlled (or diffusion-limited) because their collision frequency is dependent upon the diffusion rate (diffusion velocity) of the partners involved. Bimolecular reactions in water and similarly viscous liquids are of this type if the activatimi threshold sinks tmder the third or fourth rung of our potential ladder, meaning that Ajyw <20 kG (see Fig. 18.2). Because diffusion in solid substances proceeds incomparably slowly, almost all the bimolecular reactions in such an environment are diffusion-controlled. 

Formation of pyrene excimer (a complex between a photoexcited and a ground-state pyrene molecule Scheme 4) is an extensively characterized and well-understood bimolecular process (35). Because the process is known to be diffusion controlled in normal liquid solutions, it serves as a relatively simple model system for studying solvent effects on bimolecular reactions. In fact, it has been widely employed in the probing of the solute-solute clustering in supercritical fluid solutions (40-42,46,47,160,166-168). (See Scheme 4.)... 

Transition state theory applies to all mono-, bi-, trimole-cular processes, in the gas phase as well as in liquid phase (and equally to heterogeneous processes), in contrast to elementary collision theory, which is limited to bimolecular reactions in the gas phase. 

In gases an elementary act of the bimolecular reaction occurs by the collision of two particle-reactants if an excessive energy of colliding particles exceeds the activation barrier and the configuration of the formed pair is convenient for the reaction. In liquid (solution) the bimolecular act occurs in somewhat different way. At first particle-reactants diffuse in solution and get into the same cage, neighboring for some time, they collide in it and undergo transformation in one of the collisions if the same conditions are fulfilled as those for the bimolecular reaction in the gas phase. The situation is somewhat more difficult when molecular complexes or electrostatic interactions appear between molecules in solution, which will be considered in the next... 

Due to the described above process of encounter and collision of two particles in liquid (and, in general, in the condensed phase), the following general scheme is valid for the bimolecular reaction in solution ... 

These energies can be estimated, first, by the partition coefficient of the substance between two liquid phases and, second, by the critical temperature of mutual dissolution of two phases. The ratio of the number of neighbors in the cage n to the number of moles of the solvent S (in mol/1) is usually close to unity. According to the encounter theory, the rate constant of bimolecular reaction k z, where z is the frequency factor of bimolecular collisions. In liquid a molecule is surrounded by n molecule-neighbors, vibrates in this cage with the frequency n, and collisions with each neighbor 6v times and with one molecule 6v/n times. The vibration of a particle sur-... 

Table 6.1. Rate constants of bimolecular radical reactions in liquid (H2O) and gas...

It was pointed out that a bimolecular reaction can be accelerated by a catalyst just from a concentration effect. As an illustrative calculation, assume that A and B react in the gas phase with 1 1 stoichiometry and according to a bimolecular rate law, with the second-order rate constant k equal to 10 1 mol" see" at 0°C. Now, assuming that an equimolar mixture of the gases is condensed to a liquid film on a catalyst surface and the rate constant in the condensed liquid solution is taken to be the same as for the gas phase reaction, calculate the ratio of half times for reaction in the gas phase and on the catalyst surface at 0°C. Assume further that the density of the liquid phase is 1000 times that of the gas phase. 

Instead of concentrating on the diffiisioii limit of reaction rates in liquid solution, it can be histnictive to consider die dependence of bimolecular rate coefficients of elementary chemical reactions on pressure over a wide solvent density range covering gas and liquid phase alike. Particularly amenable to such studies are atom recombination reactions whose rate coefficients can be easily hivestigated over a wide range of physical conditions from the dilute-gas phase to compressed liquid solution [3, 4]. 

The chlorination of hydrocarbons proceeds via the chain mechanism [195]. Chlorine atoms are generated photochemically or by the introduction of the initiator. However, liquid-phase chlorination occurs slowly in the dark in the absence of an initiator. The most probable reaction of thermal initiation in RH chlorination is the bimolecular reaction... 

In addition to the bimolecular reactions of organic compounds with dioxygen, free radicals are generated in an oxidized substrate in the liquid phase by the trimolecular reaction [3,8,9]... 

The rate constant of this reaction is lower when rotational diffusion is slower. The experimental data given above prove that the medium of the polymer matrix influences on the bimolecular reaction quite differently than in liquid. 

Rapid bimolecular reactions are limited by diffusion of reactants in the liquid and solid phases. Diffusion occurs in polymers much more slowly than in liquids. Hence, such rapid reactions as recombination of free radicals occurs in polymers with rate constants of a few order of magnitude more slowly than in solution. For example, the reaction of sterically hindered phenoxyl with the peroxyl radical... 

Reactions described earlier were not limited by rotational diffusion of reactants. It is evident that such bimolecular reactions can occur that are limited not by translational diffusion but by the rate of reactant orientation before forming the TS. We discussed the reactions of sterically hindered phenoxyl recombination in viscous liquids (see Chapter 15). We studied the reaction of the type radical + molecule, which are not limited by translational diffusion in a solution but are limited by the rate of reactant orientation in the polymer matrix [28]. This is the reaction of stable nitroxyl radical addition to the double bond of methylenequinone. 

The mechanism of antioxidant action on the oxidation of carbon-chain polymers is practically the same as that of hydrocarbon oxidation (see Chapters 14 and 15 and monographs [29 10]). The peculiarities lie in the specificity of diffusion and the cage effect in polymers. As described earlier, the reaction of peroxyl radicals with phenol occurs more slowly in the polymer matrix than in the liquid phase. This is due to the influence of the polymeric rigid cage on a bimolecular reaction (see earlier). The values of rate constants of macromolecular peroxyl radicals with phenols are collected in Table 19.7. 

The rates of many chemical reactions does not appear to depend on the solvent. This is because the activation energy for the process of diffusion in a liquid is nearly 20 kJ mol1 whereas for chemical reactions it is quite large. Thus, step (i) is usually not rate determining step in reactions in solutions. When the reaction takes place in solution, it is step (ii) that determines the rate of a bimolecular reaction. This conclusion is supported by the fact that the rates of these reactions do not depend upon the viscosity of the solvent. The rate should be effected by the solvent if diffusion of reactant is the rate determining step. 

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