Pressure dependent bimolecular reactions

I have mentioned briefly, at earlier points, the topics of unimolecular reactions induced by intense radiation, of chemical activation, and of pressure-dependent bimolecular reactions and have ignored altogether those of the unimolecular decomposition of ions or of photochemically excited species these are beyond the scope of a book as short as this one, although a few brief remarks about pressure-dependent bimolecular reactions may be helpful. 

Fig. 23. Predicted pressure-dependent bimolecular rate constants for the combination reaction, HO + CIO3 -> HOCIO3.

The Langmuir-Hinshelwood picture is essentially that of Fig. XVIII-14. If the process is unimolecular, the species meanders around on the surface until it receives the activation energy to go over to product(s), which then desorb. If the process is bimolecular, two species diffuse around until a reactive encounter occurs. The reaction will be diffusion controlled if it occurs on every encounter (see Ref. 211) the theory of surface diffusional encounters has been treated (see Ref. 212) the subject may also be approached by means of Monte Carlo/molecular dynamics techniques [213]. In the case of activated bimolecular reactions, however, there will in general be many encounters before the reactive one, and the rate law for the surface reaction is generally written by analogy to the mass action law for solutions. That is, for a bimolecular process, the rate is taken to be proportional to the product of the two surface concentrations. It is interesting, however, that essentially the same rate law is obtained if the adsorption is strictly localized and species react only if they happen to adsorb on adjacent sites (note Ref. 214). (The apparent rate law, that is, the rate law in terms of gas pressures, depends on the form of the adsorption isotherm, as discussed in the next section.)... 

Energy transfer limitations have long been recognized to affect the rates and mechanisms of fission and association reactions (Robinson and Holbrook, 1972 Laidler, 1987). In addition, it is increasingly being recognized that many exothermic bimolecular reactions can exhibit pressure-(density)-dependent rate parameters if they proceed via the formation of a bound intermediate. When energy transfer limitations exist, the rate coefficients exhibit non-Arrhenius temperature dependencies—i.e., the plots of ln(k) as a function of l/T are curved. 

Chain-transfer reactions would be expected to increase in rate with increasing pressure since transfer is a bimolecular reaction with a negative volume of activation. The variation of chain-transfer constants with pressure, however, differ depending on the relative effects of pressure on the propagation and transfer rate constants. For the case where only transfer to chain-transfer agent S is important, Cs varies with pressure according to... 

At first glance, it might appear that the vast majority of the bimolecular reactions with which one deals in the troposphere are simple concerted reactions, that is, during the collision of the reactants there is a reorganization of the atoms, leading directly to the formation of the products. However, it has become increasingly apparent in recent years that some important reactions that appeared to be concerted exhibit characteristics such as pressure dependencies that are not consistent with a direct concerted process. 

As indicated by the involvement of water vapor and an inert third body, this reaction has several channels (see DeMore et al., 1997, for a review). There is both a bimolecular channel, which is pressure independent, and a termolecular channel, which is pressure dependent. In addition, the rate constant increases in the presence of gaseous water, suggesting that the reaction proceeds through a mechanism such as... 

Thus, at 1 atm in air and 298 K, abstraction predominates. The addition channel (45b) would be expected to have a pressure dependence and a negative temperature dependence (see Chapter 5.A.2). Thus is consistent with the observation that the effective overall bimolecular rate constant in 1 atm of air decreases as the temperature increases from 250 to 310 K and that the fraction of the reaction that proceeds via (45a) increases from 0.24 to 0.87 over the same temperature range (e.g., Hynes et al., 1986). 

However, the formation of the dimer in the ter-molecular reaction is sufficiently fast under stratospheric conditions that the bimolecular reactions are not important. For example, using the recommended termolecular values (DeMore et al., 1997) for the low-pressure-limiting rate constant of /c,3()0 = 2.2 X 10-32 cm6 molecule-2 s-1 and the high-pressure-limiting rate constant of k3()0 = 3.5 X 10-12 cm3 molecule-1 s-1 with temperature-dependent coefficients n = 3.1 and m = 1.0 (see Chapter 5), the effective rate constant at 25 Torr pressure and 300 K is 1.6 X 10-14 cm3 molecule-1 s-1, equal to the sum of the bimolecular channels (Nickolaisen et al., 1994). At a more typical stratospheric temperature of 220 K and only 1 Torr pressure, the effective second-order rate constant for the termolecular reaction already exceeds that for the sum of the bimolecular channels, 2.4 X 10-15 versus 1.9 X 10-15 cm3 molecule-1 s-1. 

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. 

Transition-state theory is based on the assumption of chemical equilibrium between the reactants and an activated complex, which will only be true in the limit of high pressure. At high pressure there are many collisions available to equilibrate the populations of reactants and the reactive intermediate species, namely, the activated complex. When this assumption is true, CTST uses rigorous statistical thermodynamic expressions derived in Chapter 8 to calculate the rate expression. This theory thus has the correct limiting high-pressure behavior. However, it cannot account for the complex pressure dependence of unimolecular and bimolecular (chemical activation) reactions discussed in Sections 10.4 and 10.5. 

Pressure effects are also seen in a class of bimolecular reactions known as chemical activation reactions, which were introduced in Section 9.5. The treatment in that chapter was analogous to the Lindemann treatment of unimolecular reactions. The formulas derived in Section 9.5 provide a qualitative explanation of chemical activation reactions, and give the proper high- and low-pressure limits. However, that simple treatment neglected many quantum mechanical effects, namely the energy dependence of various excitation/de-excitation steps. 

For the rigid entrance/rigid exit complex-forming bimolecular reaction HO + CO — H + CO2, which passes through HOCO, a separated-step conventional Rice-Ramsperger-Kassel-Marcus (RRKM) treatment extremely well reproduces the experimental temperature and pressure dependences of this four-atom system. 

If the adsorption is small, and the reaction depends upon an essentially bimolecular process among the adsorbed molecules, for example 2A —> A2 or 2A — B + C, then, since the chance that two molecules occupy adjacent positions on the surface depends upon the square of the surface concentration, the rate of reaction is proportional to the square of the gas pressure, and the reaction is kinetically bimolecular. 

In particular, the generally observed dependence of the flame velocity on pressure between p-1/2 and p° corresponds to a mono- or bimolecular reaction and leads to d fp — const or dp = const. We are unaware of any systematic experimental investigations in this direction. 

While we believe the local composition effect discussed above to be an important influence on the measured rates, we could not come to this conclusion without examining the pressure dependence of the bimolecular rate constant for the benzophenone triplet/isopropanol reaction. From Figure 8, where the experimental... 

Figure 1.6 Langmuir-Hinshelwood formalism for a bimolecular reaction dependence of rate on pressure of reactant A (see text).

An example of the somewhat arbitrary choice of structure is set out in Table 1 for the thermal decomposition of CH3NO. It illustrates a further aspect, that the RRKM theory can be applied to analyse the pressure dependence of a pseudo-bimolecular combination reaction, in this case the combination of CH3 with NO [15], viz. 

The studies of bimolecular association reactions are of special interest because they may be expected to show, at sufficiently low concentrations, the same type of dependence of rate on total concentration as is displayed by unimolecular reactions. Indeed the simplest of such systems, the recombination of atoms at normal gas concentrations, never follow simple second-order kinetics but are rather at the extreme end of the concentration-dependent rate law and their kinetics is found experimentally to follow third-order kinetics. From the discussion of the pressure dependence of unimolecular decompositions (Table XI.2) we would expect the region of total concentration dependence to shift to lower and lower concentrations as the number of atoms in the product molecule increases. This is in quali-... 

It is now realized that the need for studies of the pressure dependence of rates of combustion reactions is much greater than had been appreciated. This arises from the fact that many reactions formally written as bimolecular processes occur, unexpectedly, by an addition mechanism with formation of a short-lived adduct. Such reactions will have rate constants that are pressure dependent although not necessarily in a regime relevant to combustion. 

---