The Arrhenius law

In discussing reaction-rate constants, we shall focus our attention on a single reaction step, equation (1). For any such process, the temperature T dependence of the specific reaction-rate constant k that appears in equation (4) is given empirically by the Arrhenius expression 

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Mathematically, multiplicities become evident when heat and material balances are combined. Both are functions of temperature, the latter through the rate equation which depends on temperature by way of the Arrhenius law. The curves representing these b ances may intersect in several points. For first order in a CSTR, the material balance in terms of the fraction converted can be written... 

The rate constant k varies with the absolute temperature T of the system according to the Arrhenius law k = k e . ... 

A gas decomposition reaction with stoichiometry 2A —> 2B -i- C follows a second order rate law rj(mol / m s) = kC, where C is the reactant concentration in mol/m. The rate constant k varies with the reaction temperature according to the Arrhenius law ... 

The same experiment is performed at several other temperatures at a single initial pressure of 1.0 atm. The results are shown in Table 3-20. Determine the Arrhenius law parameters (kg and E) for the reaction. 

The reactions were shown, in a representative number of cases, to follow second-order kinetics and to obey the Arrhenius law. The kinetic parameters are, of course, for the entire two-stage process. 

A ten to hundredfold decrease in the velocity of the reaction, seen as a break down of the Arrhenius plot, is observed at a temperature which, for any given pressure, is always higher than that thermodynamically foreseen for the beginning of the a-/3 transition (this discrepancy is smallest at 265 mm Hg pressure). The marked decrease of the rate of reaction is characteristic of the appearance of the /3-hydride phase. The kinetics of reaction on the hydride follows the Arrhenius law but with different values of its parameters than in the case of the a-phase. 

The most general representation of the isokinetic relationship is the plot of logk against the reciprocal temperature. If the Arrhenius law is followed, each... 

Of course, Sqo Sq if the difference is significant, the hypothesis of a common point of intersection is to be rejected. Quite rigourously, the F test must not be used to judge this significance, but a semiquantitative comparison may be sufficient when the estimated experimental error 6 is taken into consideration. We can then decide whether the Arrhenius law holds within experimental error by comparing Soo/(mi-21) with 6 and whether the isokinetic relationship holds by comparing So/(ml — i— 2) with 5. ... 

The idea that /3 continuously shifts with the temperature employed and thus remains experimentally inaccessible would be plausible and could remove many theoretical problems. However, there are few reaction series where the reversal of reactivity has been observed directly. Unambiguous examples are known, particularly in heterogeneous catalysis (4, 5, 189), as in Figure 5, and also from solution kinetics, even when in restricted reaction series (187, 190). There is the principal difficulty that reactions in solution cannot be followed in a sufficiently broad range of temperature, of course. It also seems that near the isokinetic temperature, even the Arrhenius law is fulfilled less accurately, making the determination of difficult. Nevertheless, we probably have to accept that reversal of reactivity is a possible, even though rare, phenomenon. The mechanism of such reaction series may be more complex than anticipated and a straightforward discussion in terms of, say, substituent effects may not be admissible. 

A special case of the isokinetic temperature is still to be mentioned, confined to a single reaction only, not strictly obeying the Arrhenius law (53). Temperature itself thus represents the variable factor, and the relation of AH and AS may be written... 

The kinetic parameters are listed in Table 1. The linearity of lnAr l/r plot is revealed by the correlation coefficient. For all reactions but the deactivation, the rate constants follow the Arrhenius law satisfactorily, implying catalyst deactivation may involve more than one elementary steps. 

The viscosity of a cement affects the pumping properties. The viscosity must be kept low enough to ensure pumpability of the slurry during the entire operation period. In deep wells, because of the increased temperature, the viscosity becomes increasingly lower, which leads to undesirable flow characteristics of the slurry. This effect can be serious, because the viscosity follows the Arrhenius law. Some of the additives used for viscosity control also... 

A different situation arises with a preliminary reduced surface. In this case the measured value of y is within lO - 10 2, and as the temperature increases, the y grows by the Arrhenius Law (Equation) with the activation energy of 5.2 kcal/mole. In addition, there is dependence of y upon the triplet oxygen pressure in the set-up, though the experiment conditions allow us to neglect a priori the impact of homogeneous processes on the spatial distribution of 02( A ) molecules. Prolonged... 

Non-purely thermal effects (other than simple dielectric heating) can be foreseen to have multiple origins. These effects can be rationalized by consideration in terms of the Arrhenius law [19, 20] and can result from modification of each of the terms of this equation. 

Fig. 4.8 Temperature dependence of the dielectric characteristic times obtained for PB for the a-relaxation (empty triangle) for the r -relaxation (empty diamond), and for the contribution of the -relaxation modified by the presence of the a-relaxation (filled diamond). They have been obtained assuming the a- and -processes as statistically independent. The Arrhenius law shows the extrapolation of the temperature behaviour of the -relaxation. The solid line through points shows the temperature behaviour of the time-scale associated to the viscosity. The dotted line corresponds to the temperature dependence of the characteristic timescale for the main peak. (Reprinted with permission from [133]. Copyright 1996 The American Physical Society)...

According to the Arrhenius law, the rate of reaction is correlated linearly to the increase in temperature, with the rate constant k given by... 

If a process obeys the Arrhenius law, then a plot of [logick] versus T should yield a straight line with a slope of -(AE/2.30259R). This indicates that AE represents a barrier to reactivity, perhaps related to organizational events that reflect the conversion of the reactant to an activated species. 

In this region where chemical reaction is rate-controlling, the rate follows the Arrhenius law. At much higher temperatures, because increases much more rapidly with temperature than k, kg k. Then... 

From the Arrhenius law, the value of the frequency factor does not affect the temperature sensitivity. 

Our simulations were performed using the activated dynamics. The probability for removing an atom at a given time step follows the Arrhenius law,... 

One can interpret the log-uniform distribution through the Arrhenius law k — Aexp(—AG/fcT), where AG is the change of the Gibbs free energy inreaction (it includes both energetic and entropic terms AG = AH—TAS, where AH the enthalpy change and AS the entropy change in reaction, T the temperature). The log-uniform distribution of k corresponds to the uniform distribution of AG. 

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