Pre-exponential term

Develop (e.g., write) the hyperbolic equation in terms of the dimensionless variables. This breaks the interdependence of exponential and pre-exponential terms. 

Rule 7. Determine the pre-exponential term by setting all variables to center point value, where everything becomes one except the pre-exponential factor, because ... 

The solidity of gel electrolytes results from chain entanglements. At high temperatures they flow like liquids, but on cooling they show a small increase in the shear modulus at temperatures well above T. This is the liquid-to-rubber transition. The values of shear modulus and viscosity for rubbery solids are considerably lower than those for glass forming liquids at an equivalent structural relaxation time. The local or microscopic viscosity relaxation time of the rubbery material, which is reflected in the 7], obeys a VTF equation with a pre-exponential factor equivalent to that for small-molecule liquids. Above the liquid-to-rubber transition, the VTF equation is also obeyed but the pre-exponential term for viscosity is much larger than is typical for small-molecule liquids and is dependent on the polymer molecular weight. 

It is important to distinguish clearly between the surface area of a decomposing solid [i.e. aggregate external boundaries of both reactant and product(s)] measured by adsorption methods and the effective area of the active reaction interface which, in most systems, is an internal structure. The area of the contact zone is of fundamental significance in kinetic studies since its determination would allow the Arrhenius pre-exponential term to be expressed in dimensions of area"1 (as in catalysis). This parameter is, however, inaccessible to direct measurement. Estimates from microscopy cannot identify all those regions which participate in reaction or ascertain the effective roughness factor of observed interfaces. Preferential dissolution of either reactant or product in a suitable solvent prior to area measurement may result in sintering [286]. The problems of identify-... 

Cordes discusses the magnitudes of pre-exponential terms with reference to the partition function for the activated complex in which the following cases are recognized. 

The following assumptions are made (i) the activated complexes are in equilibrium with the reactants, (ii) the energy of a molecule is not altered when an activated complex is substituted for a nearest neighbour, and (iii) the products do not affect the course of reaction, except to define a boundary in surface processes. The various cases can be recognized from the magnitude of the pre-exponential term and calculated values [515] are summarized in Table 7. Low values of A indicate a tight surface complex whereas higher values are associated with a looser or mobile complex. 

The concentration of [El] at equilibrium, which appears in Equation (A1.32) as the pre-exponential term El cq is defined by... 

When small amounts of rhodium were added to palladium, the activation energy E, increased from initial values of 12-14 kcal/mole to a maximum beyond 10% Rh of more than 20 kcal/mole. Beyond 60% Rh, the activation energy again increased rapidly from a minimum to 25 kcal/mole at 97.7% Rh but was 5 kcal/mole less when the reaction was carried out over pure rhodium. Values of the pre-exponential term, A, in the Arrhenius... 

The pre-exponential term includes the surface concentration of organic carbon C0c... 

A naive interpretation of the data on the assumption that the pre-exponential terms Ap and Am of the Arrhenius equations for kp and km are truly independent of temperature shows that Ap 5 x 105 Am, and hence the prevalence of propagation over transfer appears to be due to the great difference between the corresponding pre-exponential factors, reflecting the difference in the corresponding entropies of activation. However, it is very doubtful whether the above-mentioned assumption is even approximately valid under these circumstances. 

How important, though, is nuclear tunnelling for thermal outer-sphere reactions at ordinary temperature If we work in the Golden Rule formalism, an approximate answer was given some time ago. In harmonic approximation, one obtains from consideration of the Laplace transform of the transition probability (neglecting maximization of pre-exponential terms) the following expressions for free energy (AG ) and enthalpy (AH ) of... 

There are two classes of reactions for which Eq. (2.10) is not suitable. Recombination reactions and low activation energy free-radical reactions in which the temperature dependence in the pre-exponential term assumes more importance. In this low-activation, free-radical case the approach known as... 

Note that the term (kBT/h) gives a general order of the pre-exponential term for these dissociation processes. 

It is important to consider the definition of q. Defined as the amount of heat evolved by chemical reaction in a unit volume per unit time, q is the product of the terms involving the energy content of the fuel and its rate of reaction. The rate of the reaction can be written as Ze E,RT. Recall that Z in this example is different from the normal Arrhenius pre-exponential term in that it contains the concentration terms and therefore can be dependent on the mixture composition and the pressure. Thus,... 

SI symbol for absorbance (unitless) SI symbol for Helmholtz energy ( J ) SI symbol for the pre-exponential term in Arrhenius equation (mol W)"... 

Deactivation Energy, Kcal/mole Fraction of ZSM-5 in inventory Catalyst Inventory, tons Deactivation Rate Constant Pre-Exponential Term of Kp Water Partial Pressure Effect Makeup Rate of Base Catalyst, tons/min Makeup Rate of ZSM-5 Catalyst, tons/min Motor Clear Octane Number Base Motor Octane Number Deactivation Order... 

Although this form differs from the Arrhenius equation in that the pre-exponential term depends slightly on T, because the exponential dependence usually dominates, the weak dependence of the pre-exponential term on T may be regarded as negligible and the whole term A T regarded as a constant A. Hence, it is possible to roughly derive the Arrhenius relation from the collision theory. 

It should be pointed out here that large decreases in the pre-exponential term of the Fowler-Nordheim equation occur on oxygen adsorption, so that the slope of a log Z/F vs. 1/V plot should be used for determination of X- These decreases were also found by Muller (1) for oxygen and by the writer for hydrogen on tungsten. They may correspond, at least in part, to a decrease in effective emitting area, both micro- and macroscopically. 

Table I. Diffusion Coefficients as a Function of a Pre-exponential Term and an Activation Energy...

The expressions for ket in equations (29) and (30) are the products of two factors (1) an exponential term which, assuming a common force constant, gives the fractional population of reactants at temperature T with vibrational distributions at the intersection regions for each of the trapping vibrations, and (2) a pre-exponential term ve, the electron transfer frequency ve is defined in equation (31). 

Returning to equation (38), in the limit that ve vn, Ke = 1 and vet = vn. Electron transfer reactions that fall into this domain where the probability of electron transfer is unity in the intersection region have been called adiabatic by Marcus. Reactions for which Kei < 1, have been called non-adiabatic . In the limit that ve 2vn and e = vjvn, the pre-exponential term for electron transfer is given by vet = ve. This was the limit assumed in the quantum mechanical treatment using time dependent perturbation theory. 

The difference between the two results is in the pre-exponential term. In the quantum mechanical treatments either vet= vaKe or vet = 0.5(tu + reyl depending on the model adopted. In the Marcus equation the pre-exponential term is ZKee p( w lRT) and the time dependence is introduced through the collision frequency, Z. In any case, in the non-adiabatic limit, ve < v , and kobs is given by equation (41). In the adiabatic limit, ve > vn and kobs is given by equation (42), with vn in the range 10n —1013 s-1. 

The initial rise section in Figure 7.7, which follows the quadratic law (/—r2) given by the pre-exponential term in Eq. (7.13), corresponds to the formation of the first mono-layer. Figure 7.7 also shows the theoretical i-t transients for the formation of successive layers under conditions of progressive nucleation. The theoretical current-time transient for the three-dimensional nucleation is shown in Figure 7.8. The difference between the 2D and 3D nucleation (Fig. 7.7 and 7.8) is in the absence of damped oscillations in the latter case. A comparison between the theoretical and the experimental transients for the 2D polynuclear multilayer growth is shown in Figure 7.9. 

The pre-exponential term would have a relatively small effect on the temperature dependence of the rate. According to Equation 14, the pre-exponential term would cause a slight increase of the rate with temperature. 

Time-resolved emission spectra were reconstructed from a set of multifrequency phase and modulation traces acquired across the emission spectrum (37). The multifrequency phase and modulation data were modeled with the help of a commercially available global analysis software package (Globals Unlimited). The model which offered the best fits to the data with the least number of fitting parameters was a series of bi-exponential decays in which the individual fluorescence lifetimes were linked across the emission spectrum and the pre-exponential terms were allowed to vary. 

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