Pressure-Dependent Unimolecular Reactions

In specifying rate constants in a reaction mechanism, it is common to give the forward rate constants parameterized as in Eq. 9.83 for every reaction, and temperature-dependent fits to the thermochemical properties of each species in the mechanism. Reverse rate constants are not given explicitly but are calculated from the equilibrium constant, as outlined above. This approach has at least two advantages. First, if the forward and reverse rate constants for reaction i were both explicitly specified, their ratio (via the expressions above) would implicitly imply the net thermochemistry of the reaction. Care would need to be taken to ensure that the net thermochemistry implied by all reactions in a complicated mechanism were internally self-consistent, which is necessary but by no means ensured. Second, for large reaction sets it is more concise to specify the rate coefficients for only the forward reactions and the temperature-dependent thermodynamic properties of each species, rather than listing rate coefficients for both the forward and reverse reactions. Nonetheless, both approaches to describing the reverse-reaction kinetics are used by practitioners. 

It is important to emphasize that kuni is the observed rate constant (rate coefficient) for a process that is not elementary but is the net result of several contributing reactions. 

The measured rate constant for unimolecular reactions, association reactions, and certain bi-molecular reactions to be considered in the next section can have a complex dependence on total pressure, in addition to the strong temperature dependence of Eq. 9.83. This section introduces the theory of the pressure-dependence of the rate constant kmj the same theory follows to yield the pressure dependence of kassoc. Because kuni and kassoc are related by the equilibrium constant, which is independent of pressure, for a given reaction 

At sufficiently high pressure, kum is typically independent of pressure. The high-pressure limit of the rate constant will be denoted kunji00. Intermolecular collisions of C with other C molecules or with other chemical species present in the gas provide the energy needed to surmount the barrier to reaction, such as the breaking of a bond. The partner in such collisions will be genetically denoted M. 

Energizing collisions are those with sufficient energy that molecule C obtains enough internal energy that it goes on to react. In the limit of sufficiently low pressure, the rate of energizing collisions becomes small relative to the rate of reaction of an energized molecule. As a result, in the low-pressure limit, the rate of reaction becomes bimolecular, that is, proportional to the C-M collision rate. 

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Consider now the Hinshelwood model to describe a pressure-dependent unimolecular reaction ... 

At high temperatures and low pressures, the unimolecular reactions of interest may not be at their high-pressure limits, and observed rates may become influenced by rates of energy transfer. Under these conditions, the rate constant for unimolecular decomposition becomes pressure- (density)-dependent, and the canonical transition state theory would no longer be applicable. We shall discuss energy transfer limitations in detail later. 

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. 

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.)... 

The effective rate law correctly describes the pressure dependence of unimolecular reaction rates at least qualitatively. This is illustrated in figure A3,4,9. In the lunit of high pressures, i.e. large [M], becomes independent of [M] yielding the high-pressure rate constant of an effective first-order rate law. At very low pressures, product fonnation becomes much faster than deactivation. A j now depends linearly on [M]. This corresponds to an effective second-order rate law with the pseudo first-order rate constant Aq ... 

From stochastic molecnlar dynamics calcnlations on the same system, in the viscosity regime covered by the experiment, it appears that intra- and intennolecnlar energy flow occur on comparable time scales, which leads to the conclnsion that cyclohexane isomerization in liquid CS2 is an activated process [99]. Classical molecnlar dynamics calcnlations [104] also reprodnce the observed non-monotonic viscosity dependence of ic. Furthennore, they also yield a solvent contribntion to the free energy of activation for tlie isomerization reaction which in liquid CS, increases by abont 0.4 kJ moC when the solvent density is increased from 1.3 to 1.5 g cm T Tims the molecnlar dynamics calcnlations support the conclnsion that the high-pressure limit of this unimolecular reaction is not attained in liquid solntion at ambient pressure. It has to be remembered, though, that the analysis of the measnred isomerization rates depends critically on the estimated valne of... 

Miller W H 1988 Effect of fluctuations in state-specific unimolecular rate constants on the pressure dependence of the average unimolecular reaction rated. Phys. Chem. 92 4261-3... 

The theory of unimolecular reactions is that the specific rate, k, depends on the pressure as... 

An Arrhenius expression, k = 1012 exp(—24,300/Hr) sec-1, was quoted for the coefficient of the rate-determining step. It is doubtful whether this step is a true unimolecular reaction. The effect of pressures has not been studied. The range of temperatures used appears to be too narrow to discuss the temperature dependence... 

As pointed out before kuni is a pseudo first order rate constant. Since kuni/[M] is independent of [M], kuni/[M] is a second order rate constant at low pressure. It is significant and important for consideration of isotope effects that this second order rate constant for unimolecular reactions depends only on the energy levels of reactant molecules A and excited molecules A, and on the minimum energy Eo required for reaction. It does not depend on the energy levels of the transition state. There will be further discussion of this point in the following section. 

Our interest in thermally activated unimolecular reactions is in the change of kuni with pressure from the high to the zero pressure limit, and in the pressure dependence of the isotope effect over that range. One particularly interesting study carried out by Rabinovitch and Schneider (reading list) focused on the isomerization of methyl isocyanide, CH3NC, to methyl cyanide, CH3CN... 

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. 

Thus transition-state theory provides a relatively straightforward way of estimating Aoo if it is unavailable from experiment. The next section treats the theory of unimolecular reactions, and in particular, their pressure dependence, much more rigorously. 

Lindemann s treatment of unimolecular reactions was introduced in Section 9.4. This early analysis was developed to explain the pressure dependence of the observed unimolecular rate constant fcunj. At sufficiently high pressures, kUni is found to be independent of pressure (although it is typically a very strong function of temperature). However, in the limit of very low total pressure, the unimolecular rate constant is found to depend linearly on the pressure. 

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. 

Perrin s argument that the very nature of a unimolecular reaction demands independence of collisions, and therefore dependence on radiation, is adequately met both by the theory of Lindemann and by that of Christiansen and Kramers. Both these theories have the essential element in common that the distribution of energy among the molecules is not appreciably disturbed by the chemical transformation of the activated molecules thus the rate of reaction is proportional simply to the number of activated molecules and therefore to the total number of molecules, sinc in statistical equilibrium the activated molecules are a constant fraction of the whole. Thus the radiation theory is not necessary to explain the existence of reactions which are unimolecular over a wide range of pressures. 

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