Distribution of activation energies

Step 4 of the thermal treatment process (see Fig. 2) involves desorption, pyrolysis, and char formation. Much Hterature exists on the pyrolysis of coal (qv) and on different pyrolysis models for coal. These models are useful starting points for describing pyrolysis in kilns. For example, the devolatilization of coal is frequently modeled as competing chemical reactions (24). Another approach for modeling devolatilization uses a set of independent, first-order parallel reactions represented by a Gaussian distribution of activation energies (25). 

In a further development of the continuous chain model it has been shown that the viscoelastic and plastic behaviour, as manifested by the yielding phenomenon, creep and stress relaxation, can be satisfactorily described by the Eyring reduced time (ERT) model [10]. Creep in polymer fibres is brought about by the time-dependent shear deformation, resulting in a mutual displacement of adjacent chains [7-10]. As will be shown in Sect. 4, this process can be described by activated shear transitions with a distribution of activation energies. The ERT model will be used to derive the relationship that describes the strength of a polymer fibre as a function of the time and the temperature. 

At higher temperatures, a pressure-dependent process occurs from - 100 to 60 °C, which is reversible. Such a broad temperature range indicates a site heterogeneity and a broad distribution of activation energies. By considering a thermally activated motion, an estimate of the mean activation energy leads to 38 kj mol 1 and for the activation volume to 25 cm3 mol 1 for the two types of polycarbonates. 

The investigations of PMMA at two temperatures (- 40 and 60 °C) by multidimensional solid-state 13C and 2H NMR (Sect. 8.1.4) have led to quite a precise description of the ester group motions and the associated main-chain motions. However, it has not been possible to get information on the origin of the observed distribution of activation energies, nor on the extent of cooperativity along the main chain required by the 7r-flip of the asymmetric ester group. 

In this model,110 it was assumed that all C 2H bonds perform thermally activated rotational jumps within energy landscapes on the surface of a cone. Specifically, six basins were supposed to be separated by six energy barriers at positions 0, 60,..., 300° around the axis of the cone. For each cone, the barriers were drawn anew from the distribution of activation energies determined for TOL in DS.12,19 Further, it was assumed that all positions on the surface of the cone, except for the barriers, have the same energy, i.e., a random-barrier model was considered. The thermally activated jumps lead to a random new position in one of the two neighboring basins. This means that several back-and-forth jumps occur over relatively low energy barriers until relatively high barriers are crossed. In other words, many... 

Fig. 8. The initial distribution of activation energies in the Szilard-Chalmers reaction in potassium chromate jh (Fo) against Eo in electron volts.

J. R. Macdonald, "Generalizations of Universal Dielectric Response and a General Distribution-of-Activation-Energies Model for Dielectric and Conducting Systems," Journal of Applied Physics, 58 (1985) 1971-1978. 

Comparison of Reaction Rate Constants of Various Oil Shale Samples. Based on Anthony and Howard s assumption of a Gaussian distribution of activation energies and on the Arrhenius equation, the kinetic parameters and the reaction rate constant for a specific temperature were calculated according to the following equation ... 

The three semicircles are flattened considerably, indicating a spread of rate constants attributed to surface heterogeneity, and this effect was modelled by assuming a plausible Gaussian distribution of activation energies with a standard deviation of the order of ksT for the electron injection steps. The rate constants obtained from the fit... 

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