Exponential distribution of activation

It is seen from these figures that as the concentration of the additive increases, the value of E and Kq increase at first, simultaneously reach a maximum, and then drop again. The range of values covered by E and K0 are quite extensive and these effects, as pointed out by Cremer, cannot be explained by an exponential distribution of active centers on a surface. 

This concept was first mathematically developed by Clark and Bailey according to an exponential distribution of active centres with respect to adsorption energy. The Langmuir-Hinshelwood mechanism for adsorption and reaction was found to fit experimental results (ethylene polymerization over chromium oxide-silica-alumina catalyst) more closely than the Rideal mechanism. 

J. R. Macdonald, "Frequency Response of Unified Dielectric and Conductive Systems Involving an Exponential Distribution of Activation Energies," Journal of Applied Physics, 58 (1985) 1955-1970. 

The important point to be made here is that the assumption of (1) an exponential distribution of activation energies and (2) an exponential form for Tlead directly to CPE behavior. The exponential distribution of activation energies has been further discussed by Macdonald [1963]. See also Section 2.2.3.5. 

The physical origin of a power law distribution function for waiting times might arise from an exponential distribution of activation energies. Suppose, the distribution function of activation energies was of the form... 

In a full data analysis, one would first detennine the most appropriate model and then use it to carry out fits for each different temperature available. Here, only partial results for fits of the present T = 225 K data with a few other models will be discussed. First, Sp values for CNLS proportional weighting fits with the CKOEL, OMF KIEL, and DSD EDAEEL models were all close to 0.007 excellent fits. Here, the EDAE model involves an exponential distribution of activation energies fitted at the complex dielectric level and assuming dielectric-system dispersion. Since the fits were all comparable, selection of a best model must depend on other criteria. 

In the complex plane, this element appears as a straight line inclined at the angle a ir/2 to the real axis. Macdonald has shown that, for physical situations in which a relaxation time description is appropriate, CPE behavior may arise from an exponential distribution of activation energies for the relaxation process (30). For porous electrodes, such a description... 

Since dead sites have zero activity, the overall activity of the catalyst is a(n) = a (n), where a is the vector giving the activity levels of the active states. In order to see how overall activity a changes over time, first consider what happens if one starts with a "quasi-steady-state" distribution of active states, i.e., let v(0) = Cj, the eigenvector of Pn corresponding to the dominant eigenvalue Aj of Pn. In this case Piiei=AiCj, so s(n) = Pn"ei = Ai"ei = Aj"s(0). Thus the relative proportions of sites in active states remain unchanged over time there is simply an overall exponential decrease in the total population of active sites. Similarly, in this quasi-steady-state case we have a(n) = a s(n) = A "a ei = Ai"a(0) i.e., the overall activity decreases exponentially. The decay constant Aj is very close to 1 since the columns of Pu all have a sum very close to 1. In fact, if the columns of Pu all have identical sums P, then Aj = P this corresponds to the situation where the probability of sudden death is the same from each active state, namely b = 1-P. 

It is reasonable to assume that the solids flow is fully backmixed with an exponential distribution of residence times. Based on this assumption and by writing an expression for the average activity of the leaving catalyst stream (which contains particles of all ages with their corresponding activities), the following equations are derived ... 

Molecular tunnelling processes have been detected in the recombination of HbCO after flash photolysis at low temperature ( < 10 K) and attempts to analyse the data using non-adiabatic molecular group transfer theory have met with reasonable success. At higher temperatures, (< 20 K) a non-exponential Arrhenius pathway is detected suggesting a distribution of activation enthalpies depend-... 

Table 2 shows the calculation of top event probability for the FT given in Figure 7. The failure rates and repair rates have been generated randomly for exponential distribution of failure and repair activities. The data set used for ( 3, 3), ( 3, 3), ( 3, 3)] is [(0.16, 0.69), (0.11),... 

Seconday Current Distribution. When activation overvoltage alone is superimposed on the primary current distribution, the effect of secondary current distribution occurs. High overpotentials would be required for the primary current distribution to be achieved at the edge of the electrode. Because the electrode is essentially unipotential, this requires a redistribution of electrolyte potential. This, ia turn, redistributes the current. Therefore, the result of the influence of the activation overvoltage is that the primary current distribution tends to be evened out. The activation overpotential is exponential with current density. Thus the overall cell voltages are not ohmic, especially at low currents. 

Sub-cellular fractionation of five strains reveal the same numbers of bands. The distribution of PG activity in sub-cellular organelles was broadly similar in these five strains. PG activity was detected in low-density vesicles, vacuoles and ER fractions in samples harvested during the early exponential phase of growth. However, PG levels were always lower (at least 1.5 fold) than those found in wild type. Cells of the mutants harvested during stationary phase of growth showed that 84% of total intracellular PG activity was located in the vesicle fraction. No intracellular PG activity was found in stationary phase wild type cells. 

Having hypothetically assumed that rates Vn( ) and V21 ( ) of an active center transition through the interface do not depend on length of the growing terminal block of a macroradical, one will find the distribution of blocks for length (Eq. 75) to be exponential. In this unreal case, the solution of Eqs. 73 and 74 will formally reduce to the solutions of the traditional equations of radical copolymerization [76] for the concentrations Ra(l) of radicals with... 

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