Transition-state theory statistical-mechanical derivation

The derivation of the transition state theory expression for the rate constant requires some ideas from statistical mechanics, so we will develop these in a digression. Consider an assembly of molecules of a given substance at constant temperature T and volume V. The total number N of molecules is distributed among the allowed quantum states of the system, which are determined by T, V, and the molecular structure. Let , be the number of molecules in state i having energy e,- per molecule. Then , is related to e, by Eq. (5-17), which is known as theBoltzmann distribution. 

It is immediately apparent that a theory like transition-state theory is making no pretensions at stating and describing the underlying principles of the behavior of the system. In any serious analysis in terms of the deeper and more fundamental laws of physics (of quantum mechanics, in particular) the further assumptions in its derivation are arbitrary, artificial, and somewhere between wildly simplistic and quite unsound. Nevertheless, the theory is typically introduced via a complex mathematical argument in which it is derived using a series of assumptions and approximations from the supposedly underlying equations of quantum theory and/or statistical mechanics. 

Table 6.6 lists the high pressure limit kinetic parameters for the elementary reaction steps in this complex phenyl + O2 reaction system. These parameters are derived from the canonical transition state theory, the statistical mechanics from the DFT and ab initio data and from evaluation of literature data. Rate constants to all channels illustrated are calculated as function of temperature at different pressure. A reduced mechanism is proposed in Appendix F for the Phenyl + O2 system, for a temperature range of 600K different pressures 0.01 atm, 0.1 atm, latm, and 10 atm. 

An important area of application for QM methods has been determining and describing reaction pathways, energetics, and transition states for reaction processes between small species. QM-derived first and second derivatives of energy calculated at stable and saddle points on PES can be used under statistical mechanics formulations [33, 34] to yield enthalpies and free energies of structures in order to determine their reactivity. Transition state theory and idealized thermodynamic relationships (e.g., AG[Po—>P] = kTln[P/Po]) allow temperature and pressure regimes to be spanned when addressing simple gas phase and gas-surface interactions. 

The rest of this paper will be devoted to the consideration of the second kind of reactions. I shall endeavour rather to emphasise the basic assumptions of the theory than to derive ready formulas. Especially on account of some discrepancies with experiment, I think that it may be useful to see that the transition state method is based, in addition to well-established principles of statistical mechanics, on only three assumptions, two of which arc generally accepted. 

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