Frequency Factor bimolecular reactions

Glissmann and Schumacher ( ) interpreted their data in terms of a mixed mechanism including a direct bimolecular reaction 2O3 302- In the reinterpretation of these data, Benson and Axworthy (4) decided that there was no evidence for such a reaction. [A reappraisal of the more recent work of Ogg and Sutphen 9) similarly shows that their data do not require the introduction of such a direct bimolecular reaction.] For such a reaction to contribute, let us say 10% to the scheme proposed, k would have to be about 0.2 k (Equation 9) over the pressure and temperature range studied. The frequency factor of Reaction B would be expected to be about 10 to 10 times the frequency of collisions which would be about 2 X 10 liter/mole-sec. If = 0.2 ki is set at 100° C., must be between 18 and 21.5 kcal. per mole, depending on the steric factor used. The observed rate of decomposition of concentrated ozone (I) at low temperatures, where such a reaction has the best chance of being observed, verifies these frequency factors and activation energies as upper and lower limits, respectively, for such a proposed bimolecular path. 

If the concentration unit is mol/L, this must be multiplied by 1000 so that A = 7 X 10 L/mol s. The order of magnitude of the frequency factor for bimolecular reactions is 10 to 10 L/mol sifT 300 K. The frequency factors for reactions involving a light molecule such as H2 are larger about 10 L/mol s. 

The steric factor P is high enough for the bimolecular reaction to occur when two radicals met in the cage. This reaction is limited only by translational diffusion of the reacting radicals and depends on the viscosity of the solvent. The rate constant of such reaction is close to the frequency of encounters of radicals, namely, k = a, kn = 0.25 x 47rrABZ)AB = RT 100007] where spin-statistical factor, rAB is the sum of radii of reactants A and B, and DAB is the sum of their diffusion coefficients. 

Comparison of Rate Constants and Steric Factors of Bimolecular Reactions with Frequency of Reactant Rotation and Orientation in Polymer Matrix (T = 300 K)... 

It is also important to recognize that the A factors for all bimolecular reactions are limited by the collision frequency, Z -b. which is given by... 

Arrhenius recognized that for molecules to react they must attain a certain critical energy, E. On the basis of collision theory, the rate of reaction is equal to the number of collisions per unit time (the frequency factor) multiplied by the fraction of collisions that results in a reaction. This relationship was first developed from the kinetic theory of gases . For a bimolecular reaction, the bimolecular rate constant, k, can be expressed as... 

Elementary reactions are initiated by molecular collisions in the gas phase. Many aspects of these collisions determine the magnitude of the rate constant, including the energy distributions of the collision partners, bond strengths, and internal barriers to reaction. Section 10.1 discusses the distribution of energies in collisions, and derives the molecular collision frequency. Both factors lead to a simple collision-theory expression for the reaction rate constant k, which is derived in Section 10.2. Transition-state theory is derived in Section 10.3. The Lindemann theory of the pressure-dependence observed in unimolecular reactions was introduced in Chapter 9. Section 10.4 extends the treatment of unimolecular reactions to more modem theories that accurately characterize their pressure and temperature dependencies. Analogous pressure effects are seen in a class of bimolecular reactions called chemical activation reactions, which are discussed in Section 10.5. 

A simple way of analyzing the rate constants of chemical reactions is the collision theory of reaction kinetics. The rate constant for a bimolecular reaction is considered to be composed of the product of three terms the frequency of collisions, Z a steric factor, p, to allow for the fraction of the molecules that are in the correct orientation and an activation energy term to allow for the fraction of the molecules that are sufficiently thermally activated to react. That is,... 

The pre-exponential factor of an apparent unimolecular reaction is, roughly, expected to be of the order of a vibrational frequency, i.e., fO13 to 1014 s 1. The pre-exponential factor of a bimolecular reaction is, roughly, related to the collision frequency, i.e., the number of collisions per unit time and per unit volume. 

The pre-exponential factor of a bimolecular reaction is related to the reaction cross-section (see Problem 2.3). A relation that is fairly easy to interpret can be obtained within the framework of transition-state theory. Combining Eqs (6.9) and (6.54), we can write the expression for the rate constant in a form that gives the relation to the (hard-sphere) collision frequency ... 

The idea that bimolecular reactions generally, in gas and in solution, should have about the same frequency factors was mentioned in another paper (41) in 1924. For gas reactions it was not at all new at that time, and the step to reactions in solution is not large. 

Another evident mechanism for energy transfer to activated ions may be by bimolecular collisions between water molecules and solvated ion reactants, for which the collision number is n(ri+ r2)2(87tkT/p )l/2> where n is the water molecule concentration, ri and r2 are the radii of the solvated ion and water molecule of reduced mass p. With ri, r2 = 3.4 and 1.4 A, this is 1.5 x 1013 s"1. The Soviet theoreticians believed that the appropriate frequency should be for water dipole librations, which they took to be equal 10n s 1. This in fact corresponds to a frequency much lower than that of the classical continuum in water.78 Under FC conditions, the net rate of formation of activated molecules (the rate of formation minus rate of deactivation) multiplied by the electron transmission coefficient under nonadiabatic transfer conditions, will determine the preexponential factor. If a one-electron redox reaction has an exchange current of 10 3 A/cm2 at 1.0 M concentration, the extreme values of the frequency factors (106 and 4.9 x 103 cm 2 s 1) correspond to activation energies of 62.6 and 49.4 kJ/mole respectively under equilibrium conditions for adiabatic FC electron transfer. 

This could result in an apparently low frequency factor. Thus if fc is in the region of being a bimolecular reaction and has the form At(M), we can compute an upper limit for A from the estimated entropy change in the reaction (Table XIII, 13) and the... 

Because of the limitations imposed by activity coefficients and specific interactions, a precise quantitative check of experimental data against the collision formula presented here is not possible. However, the frequency factors of bimolecular reactions which are diffusion-controlled (i.e., those which occur on nearly every collision) such as free radical recombinations,... 

E is an activation energy, Z a frequency factor, and P a correction or steric factor, intended to allow for unfavourable orientation at the instant of collision. In fact P was chosen simply to get a good fit with experiment and is an unsatisfactory feature of the approach. Nevertheless, most homogeneous bimolecular reactions have rate constants which do conform quite closely to this equation. 

The high pre-exponential factor of Reaction 1 is thus due to a very large entropy of activation for O3. Such large factors are generally not met with in bimolecular reactions. They do occur in this special type of bimolecular reaction, however, which involves energy transfer. Rice 10) has discussed similar effects in the closely related reactions M + X2 M -f 2X, which have even higher frequency factors. 

These generalities are subject to many qualifications—the bimolecular value of A, for example, applies to the interaction of two atoms. For the interaction of more complex structures it may be much smaller. This is sometimes expressed by a steric factor multiplied by the frequency factor. Thus if an atom and polyatomic molecule interact, the value of A will be reduced by a factor of betwen 10" -10" if two polyatomic molecules are involved, the steric factor may be as small as For reactions in solution,... 

Here, k is the electronic transmission coefficient (k = 1 for adiabatic electron transfer) and v x the nuclear frequency factor, whereas is the equilibrium constant for assembly of a precursor state and effectively includes any coulombic work and medium (Debye-Hiickel) terms [4, 5]. Following the approach taken by Stranks [7], the observed volume of activation AV for a simple, adiabatic, outer-sphere, bimolecular electron transfer reaction can be represented as... 

The upper limits are provided by the collision frequency of the radical-molecule pair which is about 1011 3 1/mole-sec at 400°K. This result can also be arrived at from transition state theory by assuming that the centers of the colliding pair lie on a spherical shell 3.5 A in radius and 0.10 A thick. This corresponds to a tight transition state since the small amplitude of motion of 0.10 A is characteristic of bond vibration amplitudes in molecules. The only bimolecular reactions whose A-factors come close to this upper limit are the methathesis reactions of I atoms (27) for which the A-factors equal, or slightly exceed, the collision frequency. 

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