Activation energy, bimolecular collisions

We are here considering photoactivated energy-rich molecules colliding with relatively cold bath gas molecules (i.e. the system is far from thermal equilibrium). This is the converse of the situation where a gas is heated under bulk conditions and the molecules undergoing dissociation acquire their energy from collisions with the bath gas. In that case, at the high-pressure limit, the rate of dissociation becomes independent of pressure, i.e. dissociation appears to be unimolecular (hence the term) despite the fact that molecules are activated in bimolecular collisions. 

Parameter Calculation and Establishment of Relationships. The use of molecular modeling tools not being evident for nonexperts in the field, alternative tools can be applied for the assessment of values for rate coefficients, preexponential factors, and/or activation energies (22). Collision rate theory provides a simple description of a kinetic process. It counts the number of collisions, Zab, between the reacting species A and B in a bimolecular reaction or between the reacting species and the surface in the case of an adsorption step and applies a reaction probability factor, Prxn, to account for the fact that not every collision leads to a chemical reaction. 

If hvQ is small compared with kT, the partition function becomes kT/hvQ. The function kT jh which pre-multiplies the collision number in the uansition state theoty of the bimolecular collision reaction can therefore be described as resulting from vibration of frequency vq along the transition bond between the A and B atoms, and measures the time between each potential n ansition from reactants to product which will only occur provided that die activation energy, AEq is available. 

An alternative explanation which has been advanced for the first-order reaction is that one molecule is activated during a bimolecular collision, and can retain the activation energy until it finally decomposes some time after the... 

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,... 

This problem was resolved in 1922 when Lindemann and Christiansen proposed their hypothesis of time lags, and this mechanistic framework has been used in all the more sophisticated unimolecular theories. It is also common to the theoretical framework of bimolecular and termolecular reactions. The crucial argument is that molecules which are activated and have acquired the necessary critical minimum energy do not have to react immediately they receive this energy by collision. There is sufficient time after the final activating collision for the molecule to lose its critical energy by being deactivated in another collision, or to react in a unimolecular step. 

When a molecule is supplied with an amount of energy that exceeds some threshold energy, a unimolecular reaction can take place, that is, a dissociation or an isomerization. We distinguish between a true unimolecular reaction that can be initiated by absorption of electromagnetic radiation (photo-activation) and an apparent unimolecular reaction initiated by bimolecular collisions (thermal activation). For the apparent unimolecular reaction, the time scales for the activation and the subsequent reaction are well separated. When such a separation is possible, for true or apparent unimolecular reactions, the reaction is also referred to as an indirect reaction. We will discuss the following. 

For the first time the reaction of CO oxidation was considered in Refs. [137,138], However, the equations for two-site processes differ from corrected Eq. (62b) due to using in these works a different definition for the bimolecular reaction activation energy the term with Ey was absent in the reaction activation energy. Actually calculations were performed with simplified assumption about s = 0 (the collision model). The theoretical curves have given a qualitative agreement with experiment data. 

Even more striking is the regioselective transformation of the highly structured toluene n-complex, with bromine situated specifically over the ortho and para positions to afford the same isomeric (product) mixture of o- and p-bromotoluenes as that obtained in solution [63]. As close as these pre-equilibrium intermediates are structurally akin to the (ordered) transition states for electrophilic bromination, it is important to emphasize that they are formed essentially upon bimolecular collision with no activation energy, and the donor/acceptor binding is in accord with the Mulliken formulation. 

It is easy to understand a bimolecular reaction on the basis of collision theory. Thus, when two molecules A and B collide their relative kinetic energy exceeds the threshold energy, the collision may result in the breaking of bonds and the formation of new bonds. But how can one account for a unimolecular reaction If we assume that in such a reaction (A — P) the molecule A acquires the necessary activation energy by colliding with another molecule, then the reaction should obey second-order kinetics and not the first-order kinetics which is actually observed in several unimolecular gaseous reactions. A satisfactory theory of these reactions was proposed by F.A. Lindemann in 1922. According to him, a unimolecular reaction... 

The preexponential factor, A, has the same dimensions as k, for bimolecular reactions we shall use cm /molecule-sec, and Ea is the activation energy. It has long been recognized that this approach cannot lead to an understanding of reactions at the molecular level, since a conventional experiment determines a rate that is an average over an enormous number of collisions with a wide spectrum of collisional parameters. 

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. 

Figure 2.12 illustrates the potential energy profile for a unimolecular radical decomposition. Bimolecular collisions with a bath gas can activate the molecule so that it contains enough energy to react. However, for reaction to occur this energy must be located in the bond to be broken and hence the dissociation is not instantaneous. During the intramolecular energy redistribution further collisions can occur with the bath gas which will deactivate the molecule. The simple treatment proposed by Linde-... 

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. 

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. 

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