Internal relaxation rate rates

We have assumed so far that the electron has sufficient time to relax within the substructure of Sy and therefore reach the lowest state Sy (0) before any decay to the ground state occurs. This is entirely justified if the total decay rate from Sy is much smaller than the typical internal relaxation rate in Sy,... 

Fig. 5.1. Comparison of the theoretical fall-off curve for the thermal isomerisation of cyclopropane at 765 K with the experimental results. The theoretical curve is calculated from the parameters listed in [78.Y2], except that the infinite pressure rate constant is taken to be 3.57 x 10 s rather than 3.41 x lO s as recommended by Falconer, Hunter Trotman-Dickenson [61.F1], see Footnote 4 the internal relaxation rate constant is r, = 5.50x10 Torr s. The experimental data are those of Chambers Kistiakowsky [34.C] (diamonds). Corner Pease [45.C] (crosses), and of Pritchard, Sowden Trotman-Dickenson [53.P2], also Appendix 2 (circles) the dotted line shows the high pressure limit of [61.F1],...

Table 5.1. Apparent internal relaxation rate constants for strong collision reactions...

So what happens if we change our consideration to a molecule of different complexity In practice, there are many variables which complicate the analysis, for not only will the and d change, but /<, and Aoo will also be different. Let us imagine a hypothetical molecule C3D3 which possesses the same internal relaxation rate constant as does cyclopropane, and which reacts to form some product with the same values of and of A. We will also assume that it has the same two moments of inertia as does cyclopropane, so that the only thing different about it is its vibrational frequencies it has 12 normal modes of vibration instead of 21, and for the purposes of this illustration, I have simply made an arbitrary deletion of nine of the original modes of the cyclopropane molecule. 

Fig. 5.10. Comparison of fall-off curves for two of the four reaction products in the thermal isomerisation of monofluorocyclopropane, in the strong collision approximation. The upper theoretical curve corresponds to the rate of formation of (rans-1-fluoropropene, and the lower one to that of 2-fluoropropene. The points are the experimental results of Casas, Kerr Trotman-Dickenson [64.C] see Footnote 14 also the position of these curves is determined by an assumed internal relaxation rate constant r,= 3x 10 Torr s .

Having dispensed with the notion that the observed internal relaxation rate g discriminates between strong and weak collision regimes, we are left with only one criterion for diagnosing a weak collision reaction, the occurrence of a fall-off curve which departs from that predicted by the strong collision expression, equation (5.14). We have seen two examples, that of methyl isocyanide in the previous chapter, and that of nitrous... 

However, when we go on to calculate the internal relaxation rate constant, we find a different value for this quantity from each reaction. 

Show that the magnitude of the feature in Figure 7.3 becomes larger if the internal relaxation rate n is made ten times larger, and that it almost disappears if it is made ten times smaller. 

This simple relaxation theory becomes invalid, however, if motional anisotropy, or internal motions, or both, are involved. Then, the rotational correlation-time in Eq. 30 is an effective correlation-time, containing contributions from reorientation about the principal axes of the rotational-diffusion tensor. In order to separate these contributions, a physical model to describe the manner by which a molecule tumbles is required. Complete expressions for intramolecular, dipolar relaxation-rates for the three classes of spherical, axially symmetric, and asymmetric top molecules have been evaluated by Werbelow and Grant, in order to incorporate into the relaxation theory the appropriate rotational-diffusion model developed by Woess-ner. Methyl internal motion has been treated in a few instances, by using the equations of Woessner and coworkers to describe internal rotation superimposed on the overall, molecular tumbling. Nevertheless, if motional anisotropy is present, it is wiser not to attempt a quantitative determination of interproton distances from measured, proton relaxation-rates, although semiquantitative conclusions are probably justified by neglecting motional anisotropy, as will be seen in the following Section. 

For a rigidly held, three-spin system, or when existing internal motion is very slow compared to the overall molecular tumbling, all relaxation methods appear to be adequate for structure determination, provided that the following assumptions are valid (a) relaxation occurs mainly through intramolecular, dipolar interactions between protons (b) the motion is isotropic and (c) differences in the relaxation rates between lines of a multiplet are negligibly small, that is, spins are weakly coupled. This simple case is demonstrated in Table V, which gives the calculated interproton distances for the bicycloheptanol derivative (52) of which H-1, -2, and -3 represent a typical example of a weakly coupled, isolated three-spin... 

In the theory of liquids, such a phenomenon is well known under the term of de Gennes narrowing [152]. There the peak in S(Q) renormalizes the relaxation rate for density fluctuations Deff = D/S(Q). While in the liquid this is an effect involving different independent particles, here it occurs for the internal density fluctuations of one entity. 

A comparison with Burchard s first cumulant calculations shows qualitative agreement, in particular with respect to the position of the minimum. Quantitatively, however, important differences are obvious. Both the sharpness as well as the amplitude of the phenomenon are underestimated. These deviations may originate from an overestimation of the hydrodynamic interaction between segments. Since a star of high f internally compromises a semi-dilute solution, the back-flow field of solvent molecules will be partly screened [40,117]. Thus, the effects of hydrodynamic interaction, which in general eases the renormalization effects owing to S(Q) [152], are expected to be weaker than assumed in the cumulant calculations and thus the minimum should be more pronounced than calculated. Furthermore, since for Gaussian chains the relaxation rate decreases... 

---