Bimolecular Collision Frequencies

CFIDF end group, no selective reaction would occur on time scales above 10 s. Figure B2.5.18. In contrast to IVR processes, which can be very fast, the miennolecular energy transfer processes, which may reduce intennolecular selectivity, are generally much slower, since they proceed via bimolecular energy exchange, which is limited by the collision frequency (see chapter A3.13). 

Zab bimolecular collision frequency for molecules A and B Oi fraction of catalyst surface covered by species i... 

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... 

Once the collision frequency and density are determined, we focus on example calculations. The capstone for this section is a ballpark calculation of the initial rate of reaction at atmospheric pressure for a gas phase chemical reaction at 2700 K. We present the reaction as described by Levine (//) for a simple, bimolecular reaction step. 

These are easily the largest values ever observed for bimolecular, chemically controlled reactions and imply an enormously loose transition state complex. Since collision frequencies are of the order of 1011 3 liter/mole-sec. we see that we need to account for a positive entropy of activation of the order of 4 Gibbs/mole. 

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. 

This section considers the cross section for reactive collisions ar. Bimolecular reactions will be treated explicitly. The rate (frequency) of collisions depends on the collision cross section. The larger the cross section, the more often molecules run into one another. In a similar way the reactive cross section determines how often molecules run into one another and react. This section introduces the simple line-of-centers model for scaling of the reactive cross section with energy. 

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 ... 

This is the same as Bronsted s theory which was designed particularly for solutions. The concentration of the activated complex can be expressed in terms of the reactants and the equilibrium constant K. Also the heat of the reaction, AH, to give the activated complex, can be calculated approximately from the quantum theory or from the Arrhenius theory. Since AF= —RT In K and AF = AII — TAS, and since K can, in some cases, be calculated from known, fundamental constants, the entropy term remains the only unknown. Rodebush has long pointed out that the unknown quantity 5 in the formula k = se E/RT is related to an entropy term. As a first approximation it has been related to a collision frequency in bimolecular reactions and to a vibration frequency in unimolecu-lar reactions. Combining the two thermodynamic equations23... 

In this last expression, the preexponential factors are all similar in containing a product of two collision frequencies, a steric factor, and a mean lifetime. The latter may be approximated in a number of ways, each of which yields about 10 sec. Since bimolecular collision frequencies are about 10 liters/mole-sec, this would make Z V about 10 liters /mole -sec. The collision theory thus leads to a frequency of termolecular collisions of about 10 liters /mole -sec, which as we shall see from Table XII.9, is about the order of magnitude observed for the fastest reactions. 

Equation (5-6) gives the number of bimolecular collisions per unit time and volume, but not all of these collisions lead to reaction, and so we write rate = collision frequency X fraction of collisions having energy equal to or greater than that required for reaction, or... 

Observations of first-order kinetics for various gas decompositions, A —> P. seemed strange to early kineticists because the mechanism of activation of the reactant was unclear. If such a reaction were controlled by collisional activation it should be bimolecular. since the pair-collision frequency in a unit volume of pure A is proportional to [A]. Careful measurements for such reactions indeed showed a decrease in the apparent rate coefficient 7 ./[A] with decreasing pressure, suggesting that collisional activation was beginning to limit the reaction rate. 

The mechanism of quenching had previously been established by observing the formation of free radical ions using flash photolysis.345 Rehm and Weller proposed the empirical Equation 5.5 to fit the data, where AetG° is the free energy of photoinduced electron transfer in the contact pair (Equation 5.1), AG is the free energy of activation that accounts for the structural and solvent reorganization required for the transfer of an electron, kd and k d are the rate constants for the formation and separation of the encounter complex, respectively, Kd = kd/k d is the equilibrium constant of complex formation and Z is the bimolecular collision frequency in an encounter complex, Z 1011 s 346 A value of kd/(ZKd) = 0.25 was used. 

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. 

In bimolecular elementary reactions the frequency factor is limited by the fact that the reacting molecules must collide for reaction to take place. The maximum rate of reaction cannot therefore be greater than the frequency of collisions between molecules. This can be calculated from the kinetic theory of gases to be no higher than about 10u cc/mol/sec for the collision of two atoms at room temperature, and less for the collisions of complex molecules. Collision frequencies are weakly dependent on temperature and can be taken to be constant for all practical purposes. Thus the maximum frequency factor can be taken as 1014 cc/mol/sec. 

Once again, any decrease from the maximum collision frequency can be ascribed to a negative entropy of activation, this time stemming from the requirement that complex molecules must collide in a specific orientation for a given reaction to take place. Bimolecular frequency factors cannot be estimated from other data except by comparison to the frequency factors of similar reactions. 

Solvent effects enter through the potential of mean force and the activation energy they may cancel or nearly cancel in the expression for kj (cf. Northrup and Hynes ). The collision frequency per unit density of B, ab(8 b /Mab) 1 expression for k° for a bimolecular reaction, takes the place of the frequency Wq in (3.23) for an isomerization reaction. This analysis shows that the transition state expression for the rate coefficient appears in this theory as a singular contribution to the rate kernel for the hard-sphere model of the reaction. 

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