Arrhenius plots/behavior

Exploration of the region 0 < T < requires numerical calculations using eqs. (2.5)-(2.7). Since the change in /cq is small compared to that in the leading exponential term [cf. (2.14) and (2.18)], the Arrhenius plot k(P) is often drawn simply by setting ko = coo/ln (fig. 5). Typical behavior of the prefactor k and activation energy E versus temperature is presented in fig. 6. The narrow intermediate region between the Arrhenius behavior and the low-temperature limit has width... 

An Arrhenius plot of the rate constant, consisting of the three domains above, is schematically shown in fig. 45. Although the two-dimensional instanton at Tci < < for this particular model has not been calculated, having established the behavior of fc(r) at 7 > Tci and 7 <7 2, one is able to suggest a small apparent activation energy (shown by the dashed line) in this intermediate region. This consideration can be extended to more complex PES having a number of equivalent transition states, such as those of porphyrines. 

Usually the Arrhenius plot of In k vs. IIT is linear, or at any rate there is usually no sound basis for coneluding that it is not linear. This behavior is consistent with the conclusion that the activation parameters are constants, independent of temperature, over the experimental temperature range. For some reactions, however, definite curvature is detectable in Arrhenius plots. There seem to be three possible reasons for this curvature. 

The time required to produce a 50% reduction in properties is selected as an arbitrary failure point. These times can be gathered and used to make a linear Arrhenius plot of log time versus the reciprocal of the absolute exposure temperature. An Arrhenius relationship is a rate equation followed by many chemical reactions. A linear Arrhenius plot is extrapolated from this equation to predict the temperature at which failure is to be expected at an arbitrary time that depends on the plastic s heat-aging behavior, which... 

FIGURE 13.23 An Arrhenius plot is a graph of In A against 11T. If, as here, the line is straight, then the reaction is said to show Arrhenius behavior in the temperature range studied. This plot has been constructed from the data in Example 13.8. 

Evidence on this question may be taken by the behavior of the electrical conductivity CT as a function of temperature. A thermally activated process T dependence on log(CT), Arrhenius plot) is expected if doping takes place, whereas j -i/4 dependence, characteristic of a variable range hopping at the Fermi level is expected for a nondoping situation. 

Figure 6.6 Arrhenius plots in crystals (a) almost pure crystals with low impurity concentrations (b) crystals with low-temperature defect clusters and (c) the ionic conductivity of Ce02 doped with 10 mol % Nd203, showing defect cluster behavior. [Part (c) adapted from data in I. E. L. Stephens and J. A. Kilner, Solid State Ionics, Y77, 669-676 (2006).]...

Because JPS is limited by reaction kinetics and mass transport a dependency on the HF concentration cHf and the absolute temperature Tcan be expected. An exponential dependence of JPS on cHf has been measured in aqueous HF (1% to 10%) using the peak of the reverse scan of the voltammograms of (100) p-type electrodes. If the results are plotted versus 1/7) a typical Arrhenius-type behavior... 

Fig. 4.10 Critical current density JPS of (100) oriented silicon electrodes for different HF concentrations plotted versus the inverse absolute temperature 1/71 Arrhenius-type behavior, with an activation energy of 0.345 eV, is observed.

The behavior described above is exemplified in figure 10.11, where Arrhenius plots of solid/liquid conventional partition coefficients for transition elements reveal more or less linear trends, with some dispersion of points ascribable to compositional effects. 

A reaction for which the mechanism changes in a temperature-dependent manner may also exhibit anti-Arrhenius behavior. See Arrhenius Equation Arrhenius Plot, Nonlinear... 

ANTI-ARRHENIUS BEHAVIOR ARRHENIUS EQUATION ARRHENIUS PLOT, NONLINEAR Antibiosis,... 

Even more unusual behavior is obseived for the temperature dependence of the rate constant. Figure 6.11 shows these data in Arrhenius form for the reactions of toluene and 1,2,3-trimethylbenzene. At the higher temperatures, the Arrhenius plot is linear with a normal activation energy (i.e., the rate constant increases with increasing temperature). However, as the temperature is lowered, there is a sharp discontinuity in the plot and at lower temperatures the temperature dependence is reversed i.e., the rate constants decrease with increasing temperature. 

An example of this approach was presented earlier in Figure 3.34, which contains Arrhenius plots (rate vs. l/T cf. Section 3.0.2) at different total pressures. Figure 3.34 clearly shows the two types of deposition rate behavior. At low temperatures (higher 1/r) the reaction kinetics are slow compared to mass transport, and the deposition rate is low. At higher temperatures (lower HT) chemical kinetic processes are rapid compared to mass transport, resulting in a distinct change in slope and a higher deposition rate. 

The isomerizations of n-butenes and n-pentenes over a purified Na-Y-zeolite are first-order reactions in conversion as well as time. Arrhenius plots for the absolute values of the rate constants are linear (Figure 2). Similar plots for the ratio of rate constants (Figure 1), however, are linear at low temperatures but in all cases except one became curved at higher temperatures. This problem has been investigated before (4), and it was concluded that there were no diffusion limitations involved. The curvature could be the result of redistribution of the Ca2+ ions between the Si and Sn positions, or it could be caused by an increase in the number of de-cationated sites by hydrolysis (6). In any case the process appears to be reversible, and it is affected by the nature of the olefin involved. In view of this, the following discussion concerning the mechanism is limited to the low temperature region where the behavior is completely consistent with the Arrhenius law. 

The expected Arrhenius plot for cation self-diffusion in KC1 doped with Ca++ is shown in Fig. 8.13. The two-part curve reflects the intrinsic behavior at high temperatures and extrinsic behavior at low temperatures. 

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