Arrhenius Temperature Dependence of the Rate Constant

The rate constant normally depends on the absolute temperature, and the functional form of this relationship was first proposed by Arrhenius in 1889 (see Rule III in Chapter 1) to be  

Suggest an experimental approach to obtain these rate constant data and calculate the activation energy and pre-exponential factor. (Adapted from C. G. Hill, An Introduction to Chemical Engineering Kinetics Reactor Design, Wiley, New York, 1977.) 

Note that the rate constants are for a first-order reaction. The material balance for a closed system at constant temperature is  

If the pressure rise in the closed system is monitored as a function of time, it is clear from the above expression how the rate constant can be obtained at each temperature. 

In order to determine the pre-exponential factor and the activation energy, the In k is plotted against T as shown below  

---

The exponential terms in these two equations correspond to the Arrhenius temperature dependence of the rate constant. 

Quantum effects can also be included artificially by a tunneling correction in the expression for the rate eonstant of the process, proportional to the diffusion coefficient D [33]. In this way, the diffusion coefficient is obtained as a modification of the Arrhenius temperature dependence of the rate constant ... 

The simplest expression for the temperature dependence of the rate constant k is given by the Arrhenius equation... 

The temperature dependence of A predicted by Eq. (5-11) makes a very weak contribution to the temperature dependence of the rate constant, which is dominated by the exponential term. It is, therefore, not feasible to establish, on the basis of temperature studies of the rate constant, whether the predicted dependence of A is observed experimentally. Uncertainties in estimates of A tend to be quite large because this parameter is, in effect, determined by a long extrapolation of the Arrhenius plot to 1/T = 0. 

Arrhenius acid Species that, upon addition to water, increases [H+], 86 Arrhenius base Species that, upon addition to water, increases [OH-], 86 Arrhenius equation Equation that expresses the temperature dependence of the rate constant In k2/ki = a(l/Ti — 1 IT2)IR, 302-305... 

The temperature dependence of a rate is often described by the temperature dependence of the rate constant, k. This dependence is often represented by the Arrhenius equation, /c = Aexp(- a/i T). For some reactions, the temperature relationship is instead written fc = AT" exp(- a/RT). The A term is the frequency factor for the reaction, which reflects the number of effective collisions producing a reaction. a is known as the activation energy for the reaction, and is a measure of the amount of energy input required to start a reaction (see also Benson, 1960 Moore and Pearson, 1981). 

Comparison of this equation with the Arrhenius form of the reaction rate constant reveals a slight difference in the temperature dependences of the rate constant, and this fact must be explained if one is to have faith in the consistency of the collision theory. Taking the derivative of the natural logarithm of the rate constant in equation 4.3.7 with respect to temperature, one finds that... 

The second method of detecting tunnelling relies on the fact that the primary hydrogen KIE shows an anomalous temperature dependence when significant tunnelling takes place. In the absence of tunnelling, the temperature dependence of the rate constant should follow the Arrhenius equation (42)... 

The third-order rate expression (Equation 9) is applicable over the temperature range 121° to 187°C. The Arrhenius relationship describing the temperature dependence of the rate constant k3 (Figure 7) is... 

A plot of In k against the reciprocal of the absolute temperature (an Arrhenius plot) will produce a straight line having a slope of —EJR. The frequency factor can be obtained from the vertical intercept. In A. The Arrhenius relationship has been demonstrated to be valid in a large number of cases (for example, colchicine-induced GTPase activity of tubulin or the binding of A-acetyl-phenylalanyl-tRNA to ribosomes ). In practice, the Arrhenius equation is only a good approximation of the temperature dependence of the rate constant, a point which will be addressed below. 

To a first approximation over the relatively small temperature range encountered in the troposphere, A is found to be independent of temperature for many reactions, so that a plot of In k versus T l gives a straight line of slope —E.d/R and intercept equal to In A. However, the Arrhenius expression for the temperature dependence of the rate constant is empirically based. As the temperature range over which experiments could be carried out was extended, nonlinear Arrhenius plots of In k against T 1 were observed for... 

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. 

The temperature dependence of the rate constant for the step A -> B leads to the term /(0) in the dimensionless mass- and heat-balance eqns (4.24) and (4.25). The exact representation of an Arrhenius rate law is f(9) — exp[0/(l + y0)], where y is a dimensionless measure of the activation energy RTa/E. As mentioned before, y will typically be a small quantity, perhaps about 0.02. Provided the dimensionless temperature rise 9 remains of order unity (9 < 10, say) then the term y9 may be neglected in the denominator of the exponent as a first simplification. 

The temperature dependence of the rate constants of radical addition (k ) is described by the Arrhenius equation (Section 10.2). At a given temperature, rate variations due to the effects of radical and substrate substituents are due to differences in the Arrhenius parameters, the frequency factor, A , and activation energy for addition, . For polyatomic radicals, A values span a narrow range of one to two orders of magnitude [6.5 < log (A /dm3 mol-1 s-1) < 8.5] [2], which implies that large variations in fcj are mainly due to variations in the activation energies, E. This is illustrated by the rate constants and Arrhenius parameters for the addition to ethene of methyl and halogen-substituted methyl radicals shown in Table 10.1. 

The temperature dependence of the rate constant is usually taken as a generalized Arrhenius law... 

The amounts of the reacted CO molecules and formed C02 molecules were monitored by volumometry. They proved to be close to each other. The SG => SC was monitored optically (cf Sections 9.4 and 9.6) by recording changes in the band intensities of these centers at 5.65 and 5.3 eV, respectively (in this case, the sample was cooled to 300 K). The rate constants for the reactions were derived from the kinetic curves of the CO molecules consumed at various temperatures. The temperature dependence of the rate constant for this reaction is shown in Figure 7.3a, in the Arrhenius coordinates. The activation energy for the process was found to be 20.7kcal/mol. Similar method was used to measure the rate constant for the reverse reaction ... 

The value of n is determined by the properties of the nucleus. The Arrhenius activation energy was determined from the temperature dependence of the rate constant k. The results are shown in Table 7.18. In any case there is still a question as to whether the kinetics were studied for truly anhydrous salts. 

The Arrhenius equation describes temperature dependence of the rate constant of a chemical reaction ... 

Head-to-tail dimerization of 1,1-diphenylsilene (19a), produced by laser flash photolysis of 1,1-diphenylsilacyclobutane (17a), yields the 1,3-disilacyclobutane 2761,62 with a rate constant fcdim = (1-3 0.3) x 1010 M 1 s 1 in hexane solution at 25 °C (equation 17)46. This value is within a factor of two of the diffusional rate constant in hexane at this temperature, indicating that dimerization of this silene is faster than reaction with even the most potent of nucleophilic trapping reagents (see Table 3). More recently, the temperature dependence of the rate constant for dimerization of 19a has been studied63. The results of these experiments are shown in Figure 1, and lead to Arrhenius activation parameters of a = -4 2 Id moD1 and log(A/M 1 s"1) = 9.2 0.4. 

The temperature dependence of the rate constant k is normally expressed by an Arrhenius law with the intrinsic activation energy E. In contrast, the temperature dependence of the effective diffusivity De is much weaker. Normally, De is obtained from... 

For an elementary reaction the temperature dependency of the rate constant is given by the Arrhenius equation, Equation (6), which accounts only for elementary reactions. It is important to note that this equation gives the dependency of the rate constant k on the temperature, not the dependency of r. The preexponential factor P also shows a dependency on the temperature, but its dependency is weak compared to the exponential dependency of k ... 

It means that in reality the macroradicals are concentrated in a thin layer near the surface of polymer particles. Fig. 8 shows the temperature dependence of the rate constants of oxidation reactions for the three polymers investigated (curves b, c, d). One can see that in the temperature range investigated this dependence is in agreement with the Arrhenius equation. Let us examine the initial sections of oxidation curves. Analysis of the curves shows that they can be represented as a superposition of two different exponents corresponding to two different rate constants of radical oxidation. The temperature dependence of rate constants determined from the initial sections of oxidation curves of polymethyl-metacrylate is shown in Fig. 8 (curve a). Following fact is of interest ... 

Violation of the Arrhenius law was observed for the first time in the reaction of electrochemical hydrogen evolution, where the temperature dependence of the rate constant of H-ion discharge with formation of the hydrogen atom adsorbed at the metal,... 

The crucible contents of the TGA were modelled as a dynamic system with distributed parameters assuming plate geometry. The radius of the sample was taken as the characteristic length for the heat transport. The overall rate of reaction was approximated by an irreversible reaction of T order biomass -> solids + volatiles, assuming the validity of the Arrhenius expression for the temperature dependency of the rate constant. 

---