Activation derived from Arrhenius plots

In subsequent sections we shall find that the heat of activation, derived from Arrhenius plots, gives pointers to the character of the reaction. For instance, reactions which are controlled by the rate of diffusion have very low energy barriers (30 to 40 kJ mole ), as have the elastic processes in muscle fibres (see Davis Harrington, 1993), while reactions involving protein conformation changes characteristically have values three to four times as large. The major difficulties in the interpretation of many reports... 

Activation-energy differences and A factor ratios derived from Arrhenius plots [51] ... 

The temperature dependency of the overall reaction rates was derived from Arrhenius plots (Figure 2) for which reaction rates measured at comparable conversions and equal partial pressures of CO and H2 were used. The apparent activation energies Eg and the preexponential reaction rates ro are listed in Table III for catalysts A and C-II. The activation energies for the individual compounds obtained for the two catalysts are almost equal considering that the accuracy of Eg is approximately 5 to 10 %. 

The case of m = Q corresponds to classical Arrhenius theory m = 1/2 is derived from the collision theory of bimolecular gas-phase reactions and m = corresponds to activated complex or transition state theory. None of these theories is sufficiently well developed to predict reaction rates from first principles, and it is practically impossible to choose between them based on experimental measurements. The relatively small variation in rate constant due to the pre-exponential temperature dependence T is overwhelmed by the exponential dependence exp(—Tarf/T). For many reactions, a plot of In(fe) versus will be approximately linear, and the slope of this line can be used to calculate E. Plots of rt(k/T" ) versus 7 for the same reactions will also be approximately linear as well, which shows the futility of determining m by this approach. 

R is the gas constant Dq and activation energy Eu are constants derived from an Arrhenius plot for diffusion coefficients applying at different temperatures, and solubility coefficient was obtained from a separate permeation test at TiK. Suitable testing using a specially constmcted permeation cell water-cooled at one end provided good validation data. 

In principle this is derived from an Arrhenius plot of In r+ versus 1/T but such a plot may deviate from a straight line. Hence, the apparent activation energy may only be valid for a limited temperature range. As for the orders of reaction, one should be very careful when interpreting the activation energy since it depends on the experimental conditions. Below is an example where the forward rate depends both on an activated process and equilibrium steps, representing a situation that occurs frequently in catalytic reactions. 

The usual derivation of an activation energy from a set of temperature dependent rates as the slope of an Arrhenius plot gives ... 

An Arrhenius plot of the hydrogen penetration depth is shown in Fig. 7. The activation energy derived from the Arrhenius plot is 0.39 eV. This activation energy is believed to be the energy for breaking an Si—H bond near a B atom. Heating above 40°C appears to be necessary to cause the... 

Different forms of (60) derived for a variety of systems have the exponential temperature dependence in common. In practice 4>a is invariably much larger than kT. The Boltzmann factor therefore increases rapidly with temperature and the remainder of the expression may safely be treated as constant over a small temperature range. The activation energy may be determined experimentally from an Arrhenius plot of In R vs 1/T, which should be a straight line of slope —A/k. 

Thus ((Ax)2) becomes independent of the starting configuration of the cluster. The same equation is obtained if one averages over all configurations in a large number of events. The activation energy of diffusion of the center of mass of the cluster, A cm, as derived from an Arrhenius plot, is related to AE+ and AE by... 

The mechanical properties of Shell Kraton 102 were determined in tensile creep and stress relaxation. Below 15°C the temperature dependence is described by a WLF equation. Here the polystyrene domains act as inert filler. Above 15°C the temperature dependence reflects added contributions from the polystyrene domains. The shift factors, after the WLF contribution, obeyed Arrhenius equations (AHa = 35 and 39 kcal/mole). From plots of the creep data shifted according to the WLF equation, the added compliance could be obtained and its temperature dependence determined independently. It obeyed an Arrhenius equation ( AHa = 37 kcal/mole). Plots of the compliances derived from the relaxation measurements after conversion to creep data gave the same activation energy. Thus, the compliances are additive in determining the mechanical behavior. 

The activation enthalpy is derived from the slope of an Arrhenius plot of log Vmax versus the reciprocal of absolute temperature, and is related to the Arrhenius activation energy... 

The variation of the cathodic peak potential with the scan rate (0.3-0.4 mV precision on each determination, 1 mV reproducibility over the whole set of experiments) allows the determination of the rate constant with a relative error of 3-11%. The results are consistent with those derived from anodic-to-cathodic peak current ratios. Simulation of the whole voltammogram confirms the absence of significant systematic errors that could arise from the assumptions underlying the analysis of kinetic data. Activation parameters derived from weighted regression Arrhenius plots of the data points taken at 5 or 6 tern-... 

Plotting wo as a function of the reciprocal temperature T-1 (Arrhenius representation), one can derive the activation energy from the slope. The temperature dependent pre-exponential factor vq = ,oT2rru is then obtained from the axis cutoff T-1 —> 0 K-1. 

Band-shape analysis can be a powerful mechanistic tool, affording exchange rates, from which activation parameters can be derived via Arrhenius and/or Eyring plots. Thus, possible exchange/reaction mechanisms can be distinguished. Care, however, is needed since ... 

E(j is the activation energy of the deactivation rate and may be determined from an Arrhenius plot for a(T), as shown in Fig. 5. In addition to the a(T) values for the three laboratory runs in Fig. 3, Fig. 5 also contains the a(T) value for Run E-3. Although the data do not fit an Arrhenius model particularly well, the activation energy derived from this figure is about 20 kcal/mole. This value is consistent with the known range of activation energies for sintering, and shows that the rate of catalyst deactivation increases rapidly with temperature. 

In eqn. 2, e a(T)t j a factor that expresses the extent of catalyst deactivation with time, and may be regarded as a time constant for catalyst activity loss. This form of catalyst deactivation factor derives from the assumption that catalyst activity decays exponentially with time, at constant temperature. A and E are, respectively, the preexponential factor and the activation energy for the rate constant. E can be determined from an Arrhenius plot for k1 under conditions where the quantity e a(T)t is essentially constant. Fig. 4 shows the results of experiments that were conducted at the end of the catalyst life tests at 498°K and 538eK shown in Fig. 3. The greater extent of deactivation at 538°K is clearly evident in Fig. 4, since the data at 538°K fall well below those at 498°K. The activation energies derived from the slopes of the two lines on Fig. 4 average t7.9 kcal/mole and agree to within 0.8 kcal/mole. 

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