Linear free energy relationships in solid state diffusion processes
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Linear free energy relationships in solid state diffusion processes Rajat Kapoor and S. T. Oyamaa) Department of Chemical Engineering, Virginia Polytechnic, Blacksburg, Virginia 24061-0211 (Received 3 April 1996; accepted 25 October 1996)
This paper presents a new form of linear free energy (LFE) relationship for diffusive mass transport in oxides and other binary compounds. The relationship applies to a family of related compounds. For a given substance, i, solid-state diffusivity is related to the equilibrium constant Ki or the free energy of transformation, DGi0 , via a transfer coefficient g, through the expression ln Di g ln Ki 1 constant s2gDGi0yRTp 1 constantd. The system investigated here is the series of suboxide intermediates of vanadium pentoxide formed during temperature-programmed synthesis of vanadium nitride. The value of g for this series is 0.27. The diffusivity values are determined by fitting a mathematical model to the experimental data. Diffusivity data are presented graphically in contour diagrams which correlate pre-exponential values, activation energies, particle sizes, and heating rates used in the temperature-programmed syntheses. An Evans–Polanyi linear relation, DEi aDsDHi0 d, relating activation energy, Ei , to enthalpy change of transformation, DHi0 , via a transfer coefficient a 0.53, is also shown to exist for the above system. The discrepancy between a and g is resolved by using the Horiuti concept of the stoichiometric number of the rate-determining step.
I. INTRODUCTION
Relationships linking kinetics and thermodynamics are of considerable interest because they provide insights into the mechanism of the kinetic processes taking place, and the possibility of predicting kinetic information from thermodynamic data. Linear free energy (LFE) relationships correlating rate constants with equilibrium constants or free energies for series of closely related reactions are examples of such links between kinetics and thermodynamics. These are not a part of formal thermodynamics and have been termed extrathermodynamic relationships.1 The relations are empirical, and various forms have been suggested so far. (a) The Br¨onsted equation, ln ki a ln Ki 1 constant, relates the rate constant, ki , and the equilibrium constant, Ki , for a family of reactions through a socalled transfer coefficient, a, with values lying between 0 and 1.2 Numerous examples of this relation have been tabulated for liquid phase acid or base catalyzed organic reactions.1 (b) The Hammett equation, lnskiyk0 d rs, relates ratios of rate constants or equilibrium constants skiyk0 d for families of reactions to two parameters, one characteristic of the reaction, r, and the other of the reactant, s. This relation was originally proposed to systematize the effect of substituents in meta- and a)
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http://journals.cambridge.org
J. Mater. Res., Vol. 12, No. 2, Feb 1997
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parasubstituted benzene
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