Global rate definition

The precise experimental definition of a delay depends on the reactor being used and on the properties it is possible to measure. Thus, for a batch reactor, the recording of the pressure increase as a function of time allows the time for which the global rate d(Ap)/dt is at a maximum, to be determined. The intersection of the tangent to the curve Ap(t) at this point with the horizontal Ap = 0 gives the value of the delay. 

The concept of effectiveness developed separately for external or internal transport resistances can be extended to an overall effectiveness factor for treating the general diffusion-reaction problem where both external and internal concentration and temperature gradients exist The overall effectiveness factor, D, is defined for relating the actual global rate to the intrinsic rate, that is, -Ra)p to (-Ra)6- To stun up the definitions for y, 7], and D,... 

By definition, the global rate is simply the intrinsic rate multiplied by the effectiveness factor. In order to obtain an expression for the effectiveness factor, conservation equations for the diffusion and reaction taking place in a pellet need to be solved. Take as an example the simple case of a first-order, isothermal reaction occurring in a thin slab-like pellet as shown in Figure 4.1. If only bulk diffusive flux is considered, with constant effective diffusivity i> , a steady state mass balance in one dimension gives ... 

Thus, the intrinsic rate is reduced by a factor of l/ when diffusion limitations are severe. Properties of the effectiveness factor emerge from Eqs. 4.5 and 4.6 the effectiveness factor approaches unity when is small (diffusion-free reaction) and the product of 17 and approaches unity when is large (diffusion-limited reaction). It is clear from the definition of the Thiele modulus that a higher intrinsic rate of reaction, a larger pellet, or a lower effective diffusivity results in a higher Thiele modulus and thus a relatively lower global rate. 

This technique was employed to study the binding dynamics of Pyronine Y (31) and B (32) with /)-CD/ s The theoretical background for this particular system has been discussed with the description of the technique above. Separate analysis of the individual correlation curves obtained was difficult since the diffusion time for the complex could not be determined directly because, even at the highest concentration of CD employed, about 20% of the guest molecules were still free in solution. The curves were therefore analyzed using global analysis to obtain the dissociation rate constant for the 1 1 complex (Table 12). The association rate constant was then calculated from the definition of the equilibrium constant. 

Ncff is just the inverse of the rate fluctuation. This is the traditional definition of the number of effective decay, or reaction, channels in the random matrix approach to the statistics of decay rates. This approach has been used both in nuclear (16) and chemical (17) physics. Comparing this result with the RRKM prediction, one can see that Mc(t(E) replaces N (E). One can use either a vibrationally adiabatic tunneling model (17) or a model of hopping between two electronic surfaces in the Condon approximation (40) to show that, when a global random matrix model is used for the Hamiltonian, Neff = N in the classical limit. 

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