Specific rate function

Despite the obvious correspondence between scaled elasticities and saturation parameters, significant differences arise in the interpretation of these quantities. Within MCA, the elasticities are derived from specific rate functions and measure the local sensitivity with respect to substrate concentrations [96], Within the approach considered here, the saturation parameters, hence the scaled elasticities, are bona fide parameters of the system without recourse to any specific functional form of the rate equations. Likewise, SKM makes no distinction between scaled elasticities and the kinetic exponents within the power-law formalism. In fact, the power-law formalism can be regarded as the simplest possible way to specify a set of explicit nonlinear functions that is consistent with a given Jacobian. Nonetheless, SKM seeks to provide an evaluation of parametric representation directly, without going the loop way via auxiliary ad hoc functions. 

The specific rate function k(E) as an inverse Laplace transform... 

Fig. 5.4. Specific rate functions, d,=k(E), for the two molecules of Fig. 5.3, as a function of energy above threshold. Note the minor irregularities at low energies, due to the changeover from direct count to approximate methods for calculating the state densities.

A molecular dynamic approach to specific rate functions... 

In the preceding chapter, we discovered that the fall-off behaviour for a large class of thermal unimolecular reactions could be reproduced more-or-less to perfection if the specific rate function was assumed to have the form given in equation (4.9). At first sight, this may seem to be a rather artificial form to choose but, in fact, as we will now see, when one attempts a state-to-state synthesis, a rather similar result ensues. 

The reason for this is twofold, for not only is the number of initial states minimised, but the choice of rotational state for the product N2 molecule drops out also. It has been pointed out, rightly, [80.L1] that this is a considerable assumption reaction is considered to take place as a non-adiabatic transition between two electronic states of the N2O molecule, and although the ground state is linear, the other one is not consequently, the bending motions should play a part in the reaction process. Thus, whilst one may regard the numerical results of the simpler treatment with some circumspection, it remains an ideal vehicle for illustrating the state-to-state synthesis of specific rate functions. 

Now the specific rate function k E) is not what is plotted in Figure 6.1 k(E) is the mean rate constant to be applied to all reactant molecules having an internal energy in the range < ,+, regardless of the... 

Fig. 6.5. Comparison of the synthetic specific rate function (dashed line) for the thermal isomerisation of methyl isocyanide with that derived from the inverse Laplace transform (solid line).

Without wanting to appear repetitive, it is clear from the build-up of the specific rate function shown in Section 6.2 that the external rotational states must be included. 

For your chosen reaction, examine the effects of varying, one at a time, the rotational constants, selected vibrational frequencies and their respective anharmonicities upon the density of states p( ), the specific rate function k E), and the rate constant also, examine the effect of omitting the external rotational constants altogether. 

Compare the shape of the fall-off for a given reaction which is assumed to be either strict Arrhenius in form (i.e. the standard calculation), or modified Arrhenius in form you will need to use equation (4.12) for the specific rate function in this case. 

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