Temperature dependence activation theory

Molecular theories of kinetics predict that rate coefficients can be expressed in terms of the product of a pre-exponential factor and an exponent, as does the Arrhenius equation, but in contrast to the Arrhenius equation, both the pre-exponential factor and the activation energy are usually predicted to be temperature dependent. Although theories do not predict a simple form for the temperature dependence of the pre-exponential factor, experimental data can seldom discern... 

A plot of In k vs. pressure is shown in Fig. 15, and confirms that the rate of thermal decomposition of a RDX increases with increasing pressure in the range of concern here. Within our experimental error, the slopes of these lines are independent of temperature. From activation theory, the slopes of these lines permit us to calculate a volume of activation of -5.6 cm mole" for the decomposition reaction. Reactions with positive pressure dependence of the rate constant indicate a bimolecular-type reaction mechanism, where two or more molecular entities combine to form an activated complex whose combined molar volume is less than the separate entities. In this case the volume decrease is 5.6 cm /mole. 

It follows from this discussion that all of the transport properties can be derived in principle from the simple kinetic dreoty of gases, and their interrelationship tlu ough k and c leads one to expect that they are all characterized by a relatively small temperature coefficient. The simple theory suggests tlrat this should be a dependence on 7 /, but because of intermolecular forces, the experimental results usually indicate a larger temperature dependence even up to for the case of molecular inter-diffusion. The Anhenius equation which would involve an enthalpy of activation does not apply because no activated state is involved in the transport processes. If, however, the temperature dependence of these processes is fitted to such an expression as an algebraic approximation, tlren an activation enthalpy of a few kilojoules is observed. It will thus be found that when tire kinetics of a gas-solid or liquid reaction depends upon the transport properties of the gas phase, the apparent activation entlralpy will be a few kilojoules only (less than 50 kJ). 

A more interesting possibility, one that has attracted much attention, is that the activation parameters may be temperature dependent. In Chapter 5 we saw that theoiy predicts that the preexponential factor contains the quantity T", where n = 5 according to collision theory, and n = 1 according to the transition state theory. In view of the uncertainty associated with estimation of the preexponential factor, it is not possible to distinguish between these theories on the basis of the observed temperature dependence, yet we have the possibility of a source of curvature. Nevertheless, the exponential term in the Arrhenius equation dominates the temperature behavior. From Eq. (6-4), we may examine this in terms either of or A//. By analogy with equilibrium thermodynamics, we write... 

If a data set containing k T) pairs is fitted to this equation, the values of these two parameters are obtained. They are A, the pre-exponential factor (less desirably called the frequency factor), and Ea, the Arrhenius activation energy or sometimes simply the activation energy. Both A and Ea are usually assumed to be temperature-independent in most instances, this approximation proves to be a very good one, at least over a modest temperature range. The second equation used to express the temperature dependence of a rate constant results from transition state theory (TST). Its form is... 

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. 

In experimental practice, we usually ignore the temperature dependence of the prefactor and extract the activation energy by making an Arrhenius plot, as discussed in Chapter 2. The consequence of collision theory, however, is that a curved plot, rather than a straight line, will result if the activation energy is of the same order of k T. 

Various statistical treatments of reaction kinetics provide a physical picture for the underlying molecular basis for Arrhenius temperature dependence. One of the most common approaches is Eyring transition state theory, which postulates a thermal equilibrium between reactants and the transition state. Applying statistical mechanical methods to this equilibrium and to the inherent rate of activated molecules transiting the barrier leads to the Eyring equation (Eq. 10.3), where k is the Boltzmann constant, h is the Planck s constant, and AG is the relative free energy of the transition state [note Eq. (10.3) ignores a transmission factor, which is normally 1, in the preexponential term]. 

In the foregoing derivations we have assumed that the true pH value would be invariant with temperature, which in fact is incorrect (cf., eqn. 2.58 of the Debye-Hiickel theory of the ion activity coefficient). Therefore, this contribution of the solution to the temperature dependence has still to be taken into account. Doing so by differentiating ET with respect to T at a variable pH we obtain in AE/dT the additional term (2.3026RT/F) dpH/dT, which if P (cf., eqn. 2.98) is neglected and when AE/dT = 0 for the whole system yields... 

In summary, to apply the Marcus theory of electron transfer, it is necessary to see if the temperature dependence of the electron transfer rate constant can be described by a function of the Arrhenius form. When this is valid, one can then determine the activation energy AEa only under this condition can we use AEa to determine if the parabolic dependence on AG/ is valid and if the reaction coordinate is defined. 

Coming back to aromatic anion radicals, a more accurate comparison between the experimental reaction kinetics and the predictions of the dissociative electron transfer theory revealed that the agreement is good when steric hindrance is maximal (tertiary carbon acceptors) and that the reaction is faster and faster than predicted as steric hindrance decreases, as discussed in detail in Section 3.2.2 (see, particularly, Figure 3.1). These results were interpreted as indicating an increase in the ET character of the reaction as steric hindrance increases. Similar conclusions were drawn from the temperature dependence of the kinetics, showing that the entropy of activation increases with steric hindrance, paralleling the increase in the ET character of the reaction. 

The idea that an activated complex or transition state controls the progress of a chemical reaction between the reactant state and the product state goes back to the study of the inversion of sucrose by S. Arrhenius, who found that the temperature dependence of the rate of reaction could be expressed as k = A exp (—AE /RT), a form now referred to as the Arrhenius equation. In the Arrhenius equation k is the forward rate constant, AE is an energy parameter, and A is a constant specific to the particular reaction under study. Arrhenius postulated thermal equilibrium between inert and active molecules and reasoned that only active molecules (i.e. those of energy Eo + AE ) could react. For the full development of the theory which is only sketched here, the reader is referred to the classic work by Glasstone, Laidler and Eyring cited at the end of this chapter. It was Eyring who carried out many of the... 

Occasionally, the rates of bimolecular reactions are observed to exhibit negative temperature dependencies, i.e., their rates decrease with increasing temperature. This counterintuitive situation can be explained via the transition state theory for reactions with no activation energy harriers that is, preexponential terms can exhibit negative temperature dependencies for polyatomic reactions as a consequence of partition function considerations (see, for example, Table 5.2 in Moore and Pearson, 1981). However, another plausible explanation involves the formation of a bound intermediate complex (Fontijn and Zellner, 1983 Mozurkewich and Benson, 1984). To... 

Effect of Solvent on Arrhenius Plots. If water is a substrate, then the presence of an organic solvent, which may disrupt the structure and/or orientation of water, may alter the Arrhenius plot. For example, a linear plot is seen with fumarate hydratase in the presence of 10% methanol. However, the plot is biphasic in the presence of 10% ethanol . See Boltzmann Distribution Collision Theory Temperature Dependency, Transition-State Theory Energy of Activation On... 

---