Arrhenius equation solid-state reaction kinetics

Galway, A.K. Brown. M.E. A theoretical justification for the application of the Arrhenius equation to kinetics of solid state reactions (mainly ionic crystals). Proc. R. Soc. Lond.. A 1995. 450. 501-512. 

The caveats of Garn, regarding the use of the Arrhenius kinetic equation, incorporating analytical degree of reaction functions encompassed by the Sestak-Berggren general equation in solid state reactions, are well taken, and should be borne in mind by all interested parties. 

One of the main tasks in the studies of solid-state reactions is the determination of the kinetics of the corresponding reactions. The technology of TA is often used to reveal the thermal behavior and thermal character of solid-state reactions, the primary aim is to establish the values of the apparent activation energy E and pre-exponential factor A in the Arrhenius equation, and to choose the most probable mechanism function f(a) of the reaction. The used mathematical apparatus and calculation procedures are quite varied, but they all are related to the mathematical analysis of thermogravimetric curves [6-8]. The analysis of these curves allows determining the mechanism of rate-controlled stage of the conversion, and the values of the kinetic parameters that characterizing it. 

Solid state kinetics were developed from the kinetics of homogeneous systems, i.e. liquids and gases. As it is well known, the Arrhenius equation associates the rate constant of a simple one-step reaction with the temperature through the activation energy (EJ and pre-exponential factor (A). It was assumed that the activation energy (Ea) and frequency factor (A) should remain constant however this does not happen in the actual case. It has been observed in many solid state-reactions that the activation energy may vary as the reaction progresses which were detected by the isoconversional methods. While this variation appears to be contradictory with basic chemical kinetic principles, in reality, it may not be [15]. 

It follows from this discussion that all of the transport properties can be derived in principle from the simple kinetic theory of gases, and their interrelationship through A, and c leads one to expect that they are all characterized by a relatively small temperature coefficient. The simple theory suggests that this should be a dependence on T1/2, but because of intermolecular forces, the experimental results usually indicate a larger temperature dependence even up to r3/2 for the case of molecular inter-diffusion. The Arrhenius 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, then an activation enthalpy of a few kilojoules is observed. It will thus be found that when the kinetics of a gas-solid or liquid reaction depends upon the transport properties of the gas phase, the apparent activation enthalpy will be a few kilojoules only (less than 50 kJ). 

As has been discussed above, we can often use the Arrhenius equation to predict stability in the solid state, even though the kinetics of breakdown are different from those in solution. The exception to this is when equilibrium reactions occur, in which case we can often use the van t Hoff equation (equation 4.57) to predict room-temperamre stability. 

Kinetic parameters. The hterature contains numerous reports of the rate equations identified for particular crystolysis reactions, together with the calculated Arrhenius parameters. However, reproducible values of (Section 4.1.) have been reported by independent researchers for relatively few solid state decompositions. Reversible reactions often yield Arrhenius parameters that are sensitive to reaction conditions and can show compensation effects (Section 4.9.4.). Often the influences of procedural variables have not been carefully identified. Thus, before the magnitudes of apparent activation energies can be compared, attempts have to be made to relate these values to particular reaction steps. 

This situation does not necessarily mean that all kinetic data that have been obtained by the Coats and Redfem and similar methods are incorrect. For example, the calculated activation energy frequently has about the same value regardless of whether the correct rate law has been identified or not. That happens because the rate of the reaction responds to a change in temperature according to the Arrhenius equation. The rate law used to fit the kinetic data does not alter the influence of temperature. Also, many kinetic studies on reactions in the solid state have dealt with series of reactions using similar compounds. As long as a consistent kinetic analysis procedure is used, the trends within the series will usually be vahd. Undoubtedly, however, many studies based on incomplete data analysis procedures have yielded incorrect kinetic parameters and certainly have yielded no reliable information on reaction mechanisms. 

The phenomenon of compensation is not unique to heterogeneous catalysis it is also seen in homogeneous catalysts, in organic reactions where the solvent is varied and in numerous physical processes such as solid-state diffusion, semiconduction (where it is known as the Meyer-Neldel Rule), and thermionic emission (governed by Richardson s equation ). Indeed it appears that kinetic parameters of any activated process, physical or chemical, are quite liable to exhibit compensation it even applies to the mortality rates of bacteria, as these also obey the Arrhenius equation. It connects with parallel effects in thermodynamics, where entropy and enthalpy terms describing the temperature dependence of equilibrium constants also show compensation. This brings us the area of linear free-energy relationships (LFER), discussion of which is fully covered in the literature, but which need not detain us now. 

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