Statistical mechanical calculation rates

A quantitative theory of rate processes has been developed on the assumption that the activated state has a characteristic enthalpy, entropy and free energy the concentration of activated molecules may thus be calculated using statistical mechanical methods. Whilst the theory gives a very plausible treatment of very many rate processes, it suffers from the difficulty of calculating the thermodynamic properties of the transition state. 

Although the collision and transition state theories represent two important methods of attacking the theoretical calculation of reaction rates, they are not the only approaches available. Alternative methods include theories based on nonequilibrium statistical mechanics, stochastic theories, and Monte Carlo simulations of chemical dynamics. Consult the texts by Johnson (62), Laidler (60), and Benson (59) and the review by Wayne (63) for a further introduction to the theoretical aspects of reaction kinetics. 

Quantitative estimates of E are obtained the same way as for the collision theory, from measurements, or from quantum mechanical calculations, or by comparison with known systems. Quantitative estimates of the A factor require the use of statistical mechanics, the subject that provides the link between thermodynamic properties, such as heat capacities and entropy, and molecular properties (bond lengths, vibrational frequencies, etc.). The transition state theory was originally formulated using statistical mechanics. The following treatment of this advanced subject indicates how such estimates of rate constants are made. For more detailed discussion, see Steinfeld et al. (1989). 

While details of the solution of the quantum mechanical eigenvalue problem for specific molecules will not be explicitly considered in this book, we will introduce various conventions that are used in making quantum calculations of molecular energy levels. It is important to note that knowledge of energy levels will make it possible to calculate thermal properties of molecules using the methods of statistical mechanics (for examples, see Chapter4). Within approximation procedures to be discussed in later chapters, a similar statement applies to the rates of chemical reactions. 

In transition-state theory, the absolute rate of a reaction is directly proportional to the concentration of the activated complex at a given temperature and pressure. The rate of the reaction is equal to the concentration of the activated complex times the average frequency with which a complex moves across the potential energy surface to the product side. If one assumes that the activated complex is in equilibrium with the unactivated reactants, the calculation of the concentration of this complex is greatly simplified. Except in the cases of extremely fast reactions, this equilibrium can be treated with standard thermodynamics or statistical mechanics . The case of... 

Theoretical rate calculations. Statistical mechanics permits one in principle to compute reaction-rate expressions from first principles if one knows the potential energy surface over which the reaction occurs, and quantum mechanics permits one to calculate this potential energy surface. In Chapter 4 we consider briefly the theory of reaction rates from which reaction rates would be calculated. In practice, these are seldom simple calculations to perform, and one needs to find a colleague who is an accomplished statistical mechanic or quantum mechanic to do these calculations, and even then considerable computer time and costs are usually involved. 

This gives a rate coefticient k = k Kg, where these quantities can be calculated from statistical thermodynamics (which we shall not do here). [It can be shown from statistical mechanics that k = kT/ h, where k is Boltzmann s constant and h is Planck s constant,]... 

The most accurate theories of reaction rates come from statistical mechanics. These theories allow one to write the partition function for molecules and thus to formulate a quantitative description of rates. Rate expressions for many homogeneous elementary reaction steps come from these calculations, which use quantum mechanics to calculate the energy levels of molecules and potential energy surfaces over which molecules travel in the transition between reactants and products. These theories give... 

These theories may have been covered (or at least mentioned) in your physical chemistry courses in statistical mechanics or kinetic theory of gases, but (mercifully) we will not go through them here because they involve a rather complex notation and are not necessary to describe chemical reactors. If you need reaction rate data very badly for some process, you will probably want to fmd the assistance of a chemist or physicist in calculating reaction rates of elementary reaction steps in order to formulate an accurate description of processes. 

The complexity and importance of combustion reactions have resulted in active research in computational chemistry. It is now possible to determine reaction rate coefficients from quantum mechanics and statistical mechanics using the ideas of reaction mechanisms as discussed in Chapter 4. These rate coefficient data are then used in large computer programs that calculate reactor performance in complex chain reaction systems. These computations can sometimes be done more economically than to carry out the relevant experiments. This is especially important for reactions that may be dangerous to carry out experimentally, because no one is hurt if a computer program blows up. On the other hand, errors in calculations can lead to inaccurate predictions, which can also be dangerous. 

By contrast, few such calculations have as yet been made for diffusional problems. Much more significantly, the experimental observables of rate coefficient or survival (recombination) probability can be measured very much less accurately than can energy levels. A detailed comparison of experimental observations and theoretical predictions must be restricted by the experimental accuracy attainable. This very limitation probably explains why no unambiguous experimental assignment of a many-body effect has yet been made in the field of reaction kinetics in solution, even over picosecond timescale. Necessarily, there are good reasons to anticipate their occurrence. At this stage, all that can be done is to estimate the importance of such effects and include them in an analysis of experimental results. Perhaps a comparison of theoretical calculations and Monte Carlo or molecular dynamics simulations would be the best that could be hoped for at this moment (rather like, though less satisfactory than, the current position in the development of statistical mechanical theories of liquids). Nevertheless, there remains a clear need for careful experiments, which may reveal such effects as discussed in the remainder of much of this volume. 

Keizer [455a, 498] has applied non-equilibrium statistical mechanics to the calculation of the reaction rate between two species which can both diffuse with mutual diffusion coefficient ) and encounter distance R. The partially reflecting boundary condition can be incorporated, but in the limit of fast reaction of encounter pairs for identical species... 

Figure 7. Dissociation rates k as extracted from the quantum mechanical calculations (open circles). The statistical rates are represented by the step functions and the filled circles represent the classical rate constants as obtained from elaborate classical trajectory calculations. (Reprinted, with permission of the Royal Society of Chemistry, from Ref. 34.)...

For an outer-sphere reaction, given the translation mobility of the reactants, electron transfer may occur over a range of distances. The problem can be treated in a general way since from statistical mechanics the equilibrium distribution of intemuclear separations can be calculated based on pairwise distribution functions. Integration of the product of the distribution function and ket(r) over all space gives the total rate constant et-32b 48... 

At 315°C. the rate constant ki has a value of 7.0 X 1016 molecules/sec.-cm.2-atm. From the definition of kh this represents the rate of adsorption of methylcyclohexane per cm.2 of bare platinum surface at a methylcyclohexane partial pressure of 1 atm. From kinetic theory and statistical mechanics, one can calculate the number of molecules striking a unit area of surface per unit time with activation energy Ea. This is given by... 

Calculation of the partition functions for reactants is straightforward, but the partition function for the activated complex needs explanation. The activated complex has been shown to have the unique feature of a free translation along the reaction coordinate over the distance occupied by the activated complex. The statistical mechanical quantity for this free translation has already been factorized out from the total partition function for the activated complex in the derivation. This has been done simply because doing so allows cancellation of some awkward terms in the derivation of the rate constant equation. This is why the symbol has appeared along with the symbol f, this latter indicating that the process is one of forming the activated complex, often very loosely termed activation. Q/ is now a partition function per unit volume for the activated complex but with one crucial term missing from it, i.e. the term for the free translation. This is more fully explained in the section below. 

In this connection kinetic models can also be separated into microscopic and macroscopic models. The relations between these models are established through statistical physics equations. Microscopic models utilize the concepts of reaction cross-sections (differential and complete) and microscopic rate constants. An accurate calculation of reaction cross-sections is a problem of statistical mechanics. Macroscopic models utilize macroscopic rates. 

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