Transition State Theory for Reactions in Solution

The effect of the solvent on the rate constant is considered in terms of non-ideality, charge on reactants, relative permittivity and change in solvation pattern of the solvent. Because of the difficulty of assessing partition functions in solution, the thermodynamic formulation is used. A simplified version is given here. 

The basic equation of transition state theory in this form is 

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For these reasons we cannot use (7(R) as a rigid support for dynamical studies of trajectories of representative points. G(R) has to be modified, at every point of each trajectory, and these modifications depend on the nature of the system, on the microscopic properties of the solution, and on the dynamical parameters of the trajectories themselves. This rather formidable task may be simplified in severai ways we consider it convenient to treat this problem in a separate Section. It is sufficient to add here that one possible way is the introduction into G (R) of some extra coordinates, which reflect the effects of these retarding forces. These coordinates, collectively called solvent coordinates (nothing to do with the coordinates of the extra solvent molecules added to the solute ) are here indicated by S, and define a hypersurface of greater dimensionality, G(R S). To show how this approach of expanding the coordinate space may be successfully exploited, we refer here to the proposals made by Truhlar et al. (1993). Their formulation, that just lets these solvent coordinates partecipate in the reaction path, allows to extend the algorithms and concepts of the above mentioned variational transition state theory to molecules in solution. 

The thermodynamic formulation of the transition state theory is useful in considerations of reactions in solution when one is examining a particular class of reactions and wants to extrapolate kinetic data obtained for one reactant system to a second system in which the same function groups are thought to participate (see Section 7.4). For further discussion of the predictive applications of this approach and its limitations, consult the books by Benson (59) and Laidler (60). Laidler s kinetics text (61) and the classic by Glasstone, Laidler, and Eyring (54) contain additional useful background material. 

Another term used to describe rate processes is molecu-larity, which can be defined as an integer indicating the molecular stoichiometry of an elementary reaction, which is a one-step reaction. Collision theory treats mo-lecularity in terms of the number of molecules (or atoms, if one or more of the reacting entities are single atoms) involved in a simple collisional process that ultimately leads to product formation. Transition-state theory considers molecularity as the number of molecules (or entities) that are used to form the activated complex. For reactions in solution, solvent molecules are counted in the molecularity, only if they enter into the overall process and not when they merely exert an environmental or solvent effect. 

In the context of transition-state theory, for ions and molecules in the liquid to attach to the crystal, they must first become an activated complex. The same is true for detachment (Figure 4-5). For clarity of discussion, use calcite (Cc), assumed to be pure CaCOs, growth from an aqueous solution as an example. The reaction is... 

A common approach for the study of activated barrier crossing reactions is the transition state theory (TST), in which the transfer rate over the activation barrier V is given by (0)R/2jt)e where 0)r (the oscillation frequency of the reaction coordinate at the reactant well) is an attempt frequency to overcome the activation barrier. For reactions in solution a multi-dimensional version of TST is used, in which the transfer rate is given by... 

The force controls the remarkably persistent coherence in products, a feature that was unexpected, especially in view of the fact that all trajectory calculations are normally averaged (by Monte Carlo methods) without such coherences. Only recently has theory addressed this point and emphasized the importance of the transverse force, that is, the degree of anharmonicity perpendicular to the reaction coordinate. The same type of coherence along the reaction coordinate, first observed in 1987 by our group, was found for reactions in solutions, in clusters, and in solids, offering a new opportunity for examining solvent effects on reaction dynamics in the transition-state region. 

Although possessing certain inherent limitations (Benson, 1960a), transition state theory seems adequate to permit the quantitative computation of kinetic parameters from first principles. As we have seen, however, practical application of the theory is impeded by incomplete information about the molecular properties of the activated complex and, for reactions in solution, the lack of a quantitative description of molecular interactions in condensed phases. It would be highly useful, therefore, to have some other basis on which to assess... 

The most traditional theory for chemical reaction rates is the transition state theory (TST) established in 1940 s. It has recently been disclosed, however, that the TST caimot be applied to varieties of solution reactions. Examples can be found in biological enzymatic reactions, electron or proton transfer reactions atom-group transfer reactions, and isomerization reactions. Smdy of solution reactions is one of the most traditional as well as the most fundamental subjects in chemistry. The situation mentioned above means, nevertheless, that we have not yet established a general expression on rates of solution reactions. Accordingly, many discussions have been stimulated for investigating the unknown general expression. ... 

Transition state theory was also developed as a means of rationalizing rate constants for gas phase reactions and their temperature dependence. It is most directly applied to bimolecular reactions and is based on three fundamental postulates for reactions in solution ... 

The classical, Marcus/Hush, limit corresponds to Equation (1) with /Cei= 1 and = (FC)r=o-This condition is achieved if either (i) the structural differences between the reactants and products do not implicate high-frequency vibrational modes or (ii) the exchange of energy (heat) between the high-frequency vibrational modes and the solvent is fast on the time scale for electron transfer. Statement (ii) is equivalent to the equilibrium assumption of transition-state theory. That this assumption is not always correct for reactions in solution has been demonstrated in ultrafast kinetic studies of reactions that vary ... 

The above outline of the transition-state theory has been primarily concerned with bimolecular collision processes in perfect gas reaction. for which (15 26) and (15 27) are correct thermodynamic expressions for the assumed equilibrium. For reactions in solution these should... 

Harvey JN (2010) Ab initio transition state theory for polar reactions in solution. Faraday Discuss 145 487-505... 

Because the rates of reaction are much less sensitive to the effect of pressure than of temperature, studies of the effects of external pressure for reactions in solution are more difficult to carry out, and need pressures of several thousand atmospheres (kilobars). However, they do yield important information on the transition state, which helps understanding mechanisms. The theory of pressure effects was initially formulated by van t Hoff in terms of the effect of pressure on equilibrium constants... 

Transition state theory is presented with an emphasis on solution reactions and the Marcus approach. Indeed, to allow for this, I have largely eliminated the small amount of material on gas-phase reactions that appeared in the First Edition. Several treatments have been expanded, including linear free-energy relations, NMR line broadening, and pulse radiolytic and flash photolytic methods for picosecond and femtosecond transients. 

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