Truncated parabola

To examine the shape that this equation enables us to predict for log k or AG as a function of AG, we substitute the parameter for a specific case. The value of kfc will be taken as 7.4 x 109 L mol-1 s l, that being the value in water at 298 K. Values of k calculated from Eq. (10-66) are shown in Fig. 10-10 as a function of AG. Values of AG are also depicted. The value A = 80 kJ mor1 was used and Z was taken from TST as 6.21 x 1012 s l at 298 K. The effect of introducing the diffusion-controlled limit is that the plot is shaped like a truncated parabola. This figure was drawn with K = k /k-Ac = 0.2 L mol-1. The left side of each of the diagrams shows the inverted region where k decreases and AG increases as AG becomes more negative. 

The integrals extend from — so and +So lying at the edges of the barrier. For the truncated parabola the solution reduces to Equation 6.8. Note that the tunneling corrections introduced in Sections 6.3.1 and 6.3.1.1 can be used either with TST or with VTST. 

Analytical expressions for the contributions of tunneling to the transmission coefficient have been obtained for some model barriers, V(x), notably the Eckart potential, for which the solution is exact [46], and the truncated parabola, for which an approximate, but accurate, solution has been found [47]. These references should be consulted for details of the calculations and tabulations of tunneling corrections. A hrst approximation to tunneling through a barrier of arbitrary shape is given by the following equation [48] ... 

For the present purposes the Bell truncated-parabola treatment will be adopted when a specific barrier is needed. The conclusion should not be drawn that the barrier is a truncated parabola with such and such dimensions E and a, but rather that the data resemble those calculated for a hypothetical one-dimensional barrier of these dimensions. The statement in this form recognizes the artificiality and inadequacy of the model, yet allows a level of quantitative treatment... 

In many of these cases equation 1 has been used to fit the experimental data, with and a used as adjustable parameters, and with nP = 2m . These calculations must be taken with a grain of salt since not only is the truncated parabola a rather unrealistic barrier, but the assumption about the mass is an extreme one. We shall return later to this problem. Nevertheless the values of and a are qualitatively reasonable, and support the contention that the low values of APjA demonstrate tunnelling. 

Actually the parabolas are truncated at the diffusion-controlled limit because of considerations we met in Chapter 9. We can develop this again here in an abbreviated fashion by writing a two-step scheme, the first being entirely diffusion and the second intramolecular electron transfer ... 

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