Solvation equilibrium

The treatment of equilibrium solvation effects in condensed-phase kmetics on the basis of TST has a long history and the literature on this topic is extensive. As the basic ideas can be found m most physical chemistry textbooks and excellent reviews and monographs on more advanced aspects are available (see, for example, the recent review article by Tnihlar et al [6] and references therein), the following presentation will be brief and far from providing a complete picture. 

Onsager s reaction field model in its original fonn offers a description of major aspects of equilibrium solvation effects on reaction rates in solution that includes the basic physical ideas, but the inlierent simplifications seriously limit its practical use for quantitative predictions. It smce has been extended along several lines, some of which are briefly sunnnarized in the next section. 

Van der Zwan G and Hynes J T 1984 A simple dipole isomerization model for non-equilibrium solvation dynamics in reactions in polar solvents Chem. Phys. 90 21-35... 

There are two major approaches to including nonequilibrium effects in reaction rate calculations. The first approach treats the inability of the solvent to maintain its equilibrium solvation as the system moves along the reaction coordinate as a frictional drag on the reacting solute system.97, 100 The second approach adds one or more collective solvent coordinate to the nuclear coordinates of the solute.101 107 When these solvent coordinates are... 

Even at this level of dynamical theory, one is not restricted to considering equilibrium solvation of the gas-phase saddle point or of configurations along the gas-phase reaction path [109, 338-344], and to the extent that the solvent is allowed to affect the choice of the reaction path itself, dynamic (i.e., nonequilibrium) solvation effects begin to appear in the theory. 

One should take careful note of the fact that in the nonadiabatic solvation, or frozen solvent" limit, it is the absence of solvation dynamics that is important. But is just this lack that is responsible for the deviation from equilibrium solvation, which instead assumes the dynamics are effective in always maintaining equilibrium. 

We consider the reactive solute system with coordinate x and its associated mass p, in the neighborhood of the barrier top, located at x=xi=0, and in the presence of the solvent. We characterize the latter by the single coordinate. v, with an associated mass ps. If the solvent were equilibrated to x in the barrier passage, so that there is equilibrium solvation and s = seq(x), the potential for x is just -1/2 pcc X2, where (, , is the equilibrium barrier frequency [cf. (2.2)]. To this potential we add a locally harmonic restoring potential for the solvent coordinate to account for deviations from this equilibrium state of affairs ... 

But the entire conception here is that of equilibrium solvation of the transition state by the Debye ionic atmosphere, and closer inspection [51] indicates that this assumption can hardly be justified indeed, time scale considerations reveal that it will nearly always be violated. The characteristic time for the system to cross the reaction barrier is cot, 0.1 ps say. On the other hand, the time required for equilibration of the atmosphere is something like the time for an ion to diffuse over the atmosphere dimension, the Debye length K- this time is = 1 ns for a salt concentration C= 0.1M and only drops to lOps for C 1M. Thus the ionic atmosphere is perforce out of equilibrium during the barrier passage, and in analogy with ionic transport problems, there should be an ionic atmosphere friction operative on the reaction coordinate which can influence the reaction rate. 

These aspects were examined in a study [51] which employed a generalized Debye-Falkenhagen description for the ionic atmosphere dynamical friction and GH theory for the rate. It was found that, while indeed the atmosphere is almost never equilibrated during the barrier passage and to a large extent is frozen on this time scale, the atmosphere frictional derivations from the equilibrium solvation TST result... 

In words, s describes the interaction of the solute charge distribution component p, with the arbitrary solvent orientational polarization mediated by the cavity surface. The arbitrary weights p,, previously defined by (2.11), enter accordingly the definition of the solvent coordinates, and reduce, in the equilibrium solvation regime, to the weights tv,, such that the solvent coordinates are no longer arbitrary, but instead depend on the solute nuclear geometry and assume the form se<> = lor. weq. In equilibrium, the solvent coordinates are correlated to the actual electronic structure of the solute, while out of equilibrium they are not. 

The coefficients c, in solution correspond to the global minimum of the free energy (note that this is not the equilibrium solvation condition), and satisfy the system of equations... 

Although in principle one could choose a set of arbitrary values for the solvent coordinates sm, solve the eigenvalue equation (2.23), and compute the free energy (2.12), in practice a preliminary aquaintance with the equilibrium solvation picture for the target reaction system serves as a computationally convenient doorway for the calculations in the nonequilibrium solvation regime. We show this below in the section dedicated to an illustration of the method for a two state case reported in BH-II. 

Most continuum models are properly referred to as equilibrium solvation models. This appellation emphasizes that the design of the model is predicated on equilibrium properties of the solvent, such as the bulk dielectric constant, for instance. The amount of time required for a solvent to equilibrate to the sudden introduction of a solute (i.e., the solvent relaxation time) varies from one solvent to another, but typically is in the range of molecular vibrational and rotational timescales, which is to say on the order of picoseconds. 

In Section 11.4.6, the limitations of continuum models in their ability to treat non-equilibrium solvation, at least in their simplest incarnations, were noted and discussed. In principle, exphcit solvent models might be expected to be more appropriate for the study of chemical processes characterized by non-equilibrium solvation. In practice, however, the situation is not much better for the explicit models than for the implicit. 

With Monte Carlo methods, the adoption of the Metropolis sampling scheme intrinsically assumes equilibrium Boltzmann statistics, so special modifications are required to extend MC methods to non-equilibrium solvation as well. Fortunately, for a wide variety of processes, ignoring non-equilibrium solvation effects seems to introduce errors no larger than those already inherent from other approximations in the model, and thus both implicit and explicit models remain useful tools for studying chemical reactivity. 

At the next level of approximation, we continue to imagine the solvent to be fully equilibrated to tlie reacting system at every point, but instead of working with the solvated MEP from the gas-phase surface, we find tlie equilibrium solvation patli (ESP) which is the MEP on the fully solvated surface (see Figure 11.1). While both die gas-phase and solvated surfaces are defined entirely in terms of solute coordinates, tlie I iSP may be quite different from the gas-phase MEP because solvation effects may push the patli in directions orthogonal to the gas-phase reaction coordinate (see Figure 11.5). With die ESP in hand, TST (or VTST) analysis may be carried out in the usual way lo obtain a condensed-phase rate constant. 

Note that the region where solvent is least well equilibrated to the solute is expected to be in the vicinity of the activated complex, since it has so short a lifetime. Since non-equilibrium solvation is less favorable than equilibrium solvation, the non-equilibrium free energy of the activated complex is higher than the equilibrium free energy, and the non-equilibrium lag in solvent response thus slows the reaction. This effect is sometimes referred to as solvent friction and can be accounted for by inclusion in the transmission factor a. 

Fig. 13. The interaction energy of Fe(III) and H formed in non-equilibrium solvation states upon photolysis of Fe(II) vs. the distance between these particles [38]. The energy of the initial state Fe(IT).lq + H30 1 is taken to be zero. R0 is the radius of the Fe(II)aq ion.

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