Activation energy distribution function

Calculations have been performed for the three reaction mechanisms suggested. The specific temperature-independent rate constants are given in the following table. The parameters Em and exchange data. The ratio iV./iV, was 10-4. 

Let there be the ensemble of particles reacting between themselves under the first-order kinetic law but with differing activation energies of transformation. What will be the shape of the activation energy distribution function of particles if it is known that the distribution function depends on the transformation rate constant according to a hyperbolic law n(k) l/k (kmm< k < Armax) ... 

Accordingly, the catalytic activity in a given catalytic reaction depends on only four factors. Two of them are specific for the system as a whole the activation energy and the reaction order. The latter may be reduced to the heat of adsorption, as b0 is a nearly universal constant. The other two factors are, at least in first approximation, properties of the catalyst its surface area F and its energy distribution function. Future work will have to answer the question of which parameters control, qualitatively and quantitatively, these four factors. 

Z Partition function for all active degrees of freedom of the molecule /(e) The energy distribution function for the system of interest K(e) The thermal Boltzman distribution function... 

Sections II-E,4 and 5 contain a general formulation of the isotope effect for any experimental energy distribution function. The distribution may be one typical of a conventional thermal system, or it may be a special function, characteristic of the particular activation technique. The simplest of the latter type, and one which allows the clearest demonstration of the nature of the energy dependence of the kinetic isotope effect, is a monoenergetic excitation function or 5-function. Then the intermolecular isotope effect is described by eq. (10), where the subscripts H and D stand for any two isotopically labeled species, and hydrogen and deuterium are the important cases,... 

Thermally Activated Systems. The equilibrium (high pressure) kinetic isotope effect in thermal activation systems is the one conventionally measured and the theoretical basis for this limiting case has been well formulated.3 In the low-pressure non-equilibrium regime, very large inverse statistical-weight secondary isotope effects can occur. 20 b These effects are dependent on the ambient temperature and the thermal energy distribution function the latter is considered in Sec. III-E, and discussion of these effects is postponed until Sec. III-E,4. 

In order actually to evaluate eqs. (19)-(22), the energy distribution function must be known or calculable. This function depends upon the technique used to produce the excited species and its evaluation will be considered here for chemical and thermal activation direct photoactivation, if it occurs, requires a special case in each instance and a general form cannot be given. 

Distribution Functions and Hydrogen-Deuterium Isotope Effects in Nonthermal Activation Systems. In Sec. II-D, hydrogen-deuterium isotopic rate ratios for monoenergetic systems were discussed. In practice, the measured effects are ratios modified by the energy distribution functions and should be compared to kan/kaD rather than to k,n/ktn. A s appropriate for the system under investigation, one of eqs. (19)-(22) is written for each of the isotopic species and a ratio, kttn/kaD, is thus constructed for comparison of isotope effects. These need not be listed in detail. It should be noted that the distribution function for the normal and isotopically substituted systems will usually be somewhat different (Fig. SB). 

Distribution Functions and Hydrogen-Deuterium Isotope Effects in Thermal Activation Systems. The isotope effects in thermal activation systems are determined to a large degree by the energy distribution function and kt and /(e) must be simultaneously considered. The equilibrium high-pressure effect is considerably different from the non-equilibrium low-pressure case, and they are discussed separately. 

All models, however, have the same deficiency, that is, the concentrations in the solid and solution can be measured experimentally however, the surface activity coefficients, the surface electric work, as well as the energy distribution function can only be estimated. It means that the models are adapted to the experimental data and the best-fitted model is used, and therefore the selected model has no thermodynamically significant meaning (Cemik et al. 1995). In this chapter, these problems will be illustrated and discussed. 

The nature of the different groups of active sites, for the catalytic oxidation of CO, has also been investigated from the experimentally determined energy distribution functions. The existence of three groups of active sites is observed as shown in Figs. 3 and 4, - which is also expected from thermal desorption spectroscopy (TDS) for the adsorption of CO over group VIII noble metals. 

Fig. 6. Tuned compositional kinetic model, (a) Activation energy distribution as a function ot the total potential, (b) molar composition of the fluid as defined for each E. ...

The pore size distribution fimctions obtained by using NLDFT method [171] for the carbons studied are shown in Fig. 13. The energy distribution functions for these samples are presented in Fig. 14. The pore size distributions indicate quite significant differences in the porosity of the carbons studied in the range of micropores and mesopores. While active... 

Tables 3 and 4 present the ranges of Ei value changes for individual systems. The shape of the curves of the thermodesorption energy distribution functions presented in Figure 12 are similar to Gaussian curves with one distinct peak indicating the existence of one main active centre on the surface of the adsorbents. The increase of desorption energy dmax for polar liquids (in kJ/mol 32.29< dmax<84.59 for water and 24.44< dmax<60.06 for n-butanol) is evidence of an increase of the adsorbent-adsorbate interaction forces. Broadening of bands on the desorption energy distribution curves indicates the increase of energetic heterogeneity of the studied materials.

Fig. 4 Comparative presentation of the energy distribution function

We are using three different symbols for the energy distribution function /(E) denotes the Boltzmann (equilibrium) distribution function, 3(E) refers to an unspecified distribution function, and /)(E) describes the overall energy distribution of reactants in chemically activated systems. 

With thermal activation molecular excitation due to energy exchange with the activating particles depends on temperature and this specifies the molecular energy distribution function. 

The non-equilibrium energy distribution function of molecules AB depends much more on the activation mechanism than the fall-off curve. The qualitative behaviour of the population ratio X(E)/X (E) for these two cases is shown in Fig. 25. For the strong-collision mechanism the depletion of population is seen to occur only for energy levels above Ea whereas for the stepladder mechanism it is below this level. 

Qa and Xv.p are the heat of adsorption and the latent heat of vaporization, and and E,i are the activation energies of transport at the first layer and at the second and above layers. E,i is determined from temperature dependency of liquid viscosity of adsorbate and Em is considered to be proportional to as discussed in 4.3.3. Further, this model was extended to the ease of heterogeneous surface, where energy distribution function is involved. Comparison of the model and the data on vycor glass is shown in Fig. 4.7. 

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