Transition state theory equilibrium hypothesis

Transition State Theory [1,4] is the most frequently used theory to calculate rate constants for reactions in the gas phase. The two most basic assumptions of this theory are the separation of the electronic and nuclear motions (stemming from the Bom-Oppenheimer approximation [5]), and that the reactant internal states are in thermal equilibrium with each other (that is, the reactant molecules are distributed among their states in accordance with the Maxwell-Boltzmann distribution). In addition, the fundamental hypothesis [6] of the Transition State Theory is that the net rate of forward reaction at equilibrium is given by the flux of trajectories across a suitable phase space surface (rather a hypersurface) in the product direction. This surface divides reactants from products and it is called the dividing surface. Wigner [6] showed long time ago that for reactants in thermal equilibrium, the Transition State expression gives the exact... 

If only the solvation of the gas-phase stationary points are studied, we are working within the frame of the Conventional Transition State Theory, whose problems when used along with the solvent equilibrium hypothesis have already been explained above. Thus, the set of Monte Carlo solvent configurations generated around the gas-phase transition state structure does not probably contain the real saddle point of the whole system, this way not being a correct representation of the conventional transition state of the chemical reaction in solution. However, in spite of that this elemental treatment... 

This equilibrium hypothesis is, however, not necessarily valid for rapid chemical reactions. This brings us to the second way in which solvents can influence reaction rates, namely through dynamic or frictional effects. For broad-barrier reactions in strongly dipolar, slowly relaxing solvents, non-equilibrium solvation of the activated complex can occur and the solvent reorientation may also influence the reaction rate. In the case of slow solvent relaxation, significant dynamic contributions to the experimentally determined activation parameters, which are completely absent in conventional transition-state theory, can exist. In the extreme case, solvent reorientation becomes rate-limiting and the transition-state theory breaks down. In this situation, rate con-... 

The factor introduces into equation (9) an explicit dependence of m on the concentration of species 1 in the gas adjacent to the interface [see equation (B-78)]. Except for this difference, equation (9) contains the same kinds of parameters as does equation (6), since the coefficient a can be analyzed from the viewpoint of transition-state theory. Although a may depend in general on and the pressure and composition of the gas at the interface, a reasonable hypothesis, which enables us to express a in terms of kinetic parameters already introduced and thermodynamic properties of species 1, is that a is independent of the pressure and composition of the gas [a = a(7])]. Under this condition, at constant 7] the last term in equation (9) is proportional to the concentration j and the first term on the right-hand side of equation (9) is independent of. Therefore, by increasing the concentration (or partial pressure) of species 1 in the gas, the surface equilibrium condition for species 1—m = 0—can be reached. If Pi e(T denotes the equilibrium partial pressure of species 1 at temperature 7], then when m = 0, equation (9) reduces to... 

Bodenstein worked on gas reactions dynamics at the end of 19 Century (Bodenstein, 1899). Reactions in gas phase presents more difficulties and peculiar behaviors respect to liquid ones. Bodenstein accepted the hypothesis of activated species but supposed apparent or false equilibria between them and stable reactants especially for the particular systems he examined. Bodenstein intuited a fully new class of phenomena, what we now call nonequilibrium processes, and initially provoked some interest, but this concept was too early to get a development at the time. Theoretical basis for Transition State Theory, (hereafter called TST), needed a true equilibrium state and this approach become dominant. Other important contributions due to Bodenstein was in clarifying mechanisms of many heterogeneous and catalyzed reactions and the discovery of the mechanism of Chain Reactions around 1920, a field that we will reconsider later analyzing Christiansen work. 

In his original treatment [1] Kresge identified the various with fractionation factors in the transition state. This is a plausible interpretation which we shall use. However, equations 6 and 7 make no reference to transition state theory, and the final result does not depend upon any hypothesis about equilibrium between the transition state and the reactants. 

Mass spectrometry measures the mass of fragment ions formed in a unimolecular dissociation of an energy-rich parent ion. To apply a transition state theory approach to the determination of flie branching into different products requires the assumption that after ionization there is a sufficient delay for the ion to distribute the excess energy over all the available modes. Mass spectrometrists refer to this assumption as the quasi-equilibrium hypothesis, and they are actively concerned about its validity. See Lifshitz (1989), Lorquet (1994, 2000). 

The recrossing question is important because, if none occurs, the number of trajectories traversing the plane per unit time defines the rate of product formation, which is one of the fundamental quantities one wants to get from any kinetic theory. It also turns out that a famous approximation of TST—the supposed thermal equilibrium between reactant and transition state molecules—arises as a direct consequence of the nonrecrossing hypothesis. 

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