Normal free-energy region

For many reasons it is convenient to distinguish between two cases. The first is called the normal free-energy region (or the normal Marcus region) and is defined by —A < AG12 < A (Fig. 1) with the reaction rate expressed as follows ... 

Figure 1. Profile of free-energy surfaces for the reactants and products as a function of the standard free-energy change for the reaction. (A) Normal free-energy region. (B) Activationless region.

This relation has been used to predict and interpret both self-exchange and crossreaction rates (or even "12), depending on which of the quantities have been measured experimentally. Alternatively, one could study a series of closely related electron-transfer reactions (to maintain a nearly constant X12) as a function of AG 2 a plot of In ki2 vs. In A 12 is predicted to be linear, with slope 0.5 and intercept 0.5 In ( 11 22)- The Marcus prediction (for the normal free-energy region) amounts to a linear free-energy relation (LFER) for outer-sphere electron transfer. 

The rate constant drops with increasing aG° in the inverted free energy region, in contrast to the rate constant increase in the normal region, aG° high-frequency nuclear modes is that the rate constant falls off more slowly with increasing aG° as excited vibrational states of the high-frequency mode(s) open new decay... 

For each window, p( ) is estimated by using the exact analog of (3.14). However, reconstruction of the full probability distribution directly is not possible because the total normalization constant is not known. Instead, we exploit the fact that p( ) (or, equivalently, the free energy) is a continuous function of . If consecutive windows overlap one can build the complete probability distribution by matching p( ) in the overlapping regions, as illustrated in Fig. 3.1. How to do this in a systematic fashion will be discussed later in this section. 

This expression may be interpreted as a ratio of two partition functions. In the denominator we have the partition function Z / of all trajectories starting in region with endpoint anywhere the integral in the numerator is the partition function Zai t) of all trajectories starting in. c/ and ending in 38 [this is the normalizing factor of (7.11)]. We can then view the ratio of partition functions as the exponential of the free energy difference between these two ensembles of trajectories... 

Figure 4. Schematic diagram to show the reorganization energy X for nonisotopic reactions for harmonic free energy profiles. This figure shows a normal region activation barrier when-AG° < an activationless situation when -AC =. l.and an inverted region activation barrier when-AG° > A for the harmonic potential inii andGfin represent the initial (reactant) and the final (product) system free energy, respectively.

It should be mentioned that most authorities (17) consider the solid solubility of silicon in nickel to be several per cent in the temperature region of this study. The present sample contained only 0.3 per cent Si. This would indicate that a temperature-dependent fraction of the total finds it more economical, from the free energy standpoint, to occur as a surface phase. It may be that certain types of catalyst poisoning consist of the formation of surface phases of this kind on normally active regions of the catalyst. 

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