Arrhenius equation solid state

Following the general trend of looldng for a molecular description of the properties of matter, self-diffusion in liquids has become a key quantity for interpretation and modeling of transport in liquids [5]. Self-diffusion coefficients can be combined with other data, such as viscosities, electrical conductivities, densities, etc., in order to evaluate and improve solvodynamic models such as the Stokes-Einstein type [6-9]. From temperature-dependent measurements, activation energies can be calculated by the Arrhenius or the Vogel-Tamman-Fulcher equation (VTF), in order to evaluate models that treat the diffusion process similarly to diffusion in the solid state with jump or hole models [1, 2, 7]. 

APPLICATION OF THE ARRHENIUS EQUATION TO SOLID STATE REACTIONS... 

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). 

Figure 6 shows the calculated tc values plotted vs. inverse absolute temperature. In the temperature range below —15° C, it is seen that the rc values are not dependent on Wc but dependent on temperature. The rc value increases from ca, 3 x 10-8 sec at —15°C to ca. 3 x 10 7 sec at — 60°C. This shows that the bound water in the system is in the state between viscous liquid and non-rigid solid in this temperature range. As seen from the figure, the In rc vs. temperature-1 (A-1) plots are apparently linear. The temperature dependence of rc may be expressed with considerable accuracy by the Arrhenius equation in the form (12)... 

Figure 5.23 illustrates the applicabilily of this equation to the solid state powder reaction between Si02 and BaCOa giving BaSiOa plus CO2 (gas). In Figure 5.23(a) the linear time dependence of [1 - (1 — is plotted for several temperatures. The slopes equal to 2KJR are plotted as a function of in Figure 5.23(6). Figure 5.23(c) shows the Arrhenius expression,... 

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. 

Gam [5,27] believes that no discrete activated state is generated in decompositions of solids. This conclusion is based on the observed wide variations in magnitudes of calculated activation energies. Vibrational interactions transfer energy rapidly within a crystalline solid so that no substantial difference from the average energy can be achieved or sustained. The statistical distributions upon which the Arrhenius equation is based (including the Boltzmaim function) are thus not apphcable to solids and consequently this use of the Arrhenius equation lacks a theoretical foundation. 

Activation of interface levels by the most energetic electron or phonon quanta according to distributions expressed by equations (4.4) or (4.5) provides a theoretical explanation of the fit of the Arrhenius equation to solid state decompositions proceeding at an interface [41]. 

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. 

A theoretical explanation for the applicability of the Arrhenius equation to solid state reactions has recently been provided [35] (see also Chapter 4.). Determinations of the available energy levels associated with the reaction zone and their occupancy, perhaps by spectroscopic methods, may help in establishing the steps which control bond redistributions under these conditions. 

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. 

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. 

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