The linear and logarithmic rate laws

If the cross-sectional area of the product remains relatively constant but the product layer cracks during growth and so does not impede the access of the gaseous reactants and products to the reacting solid, then a linear rate law is observed 

The logarithmic law is also observed when the oxide film is an electrical insulator such as AI2O3. The transport of electrons through the oxide is mainly due to a space charge which develops between the metal-oxide interface and the oxide-gas interface. The incorporation of oxygen in the surface of the oxide requires the addition of electrons, and if this occurs by a charging process 

This process will clearly slow down as the oxide film thickens and hence 

A further complication is that in these slowly growing oxide films, the spread of the oxide across the metal surface is limited in the early stages by nucleation and growth control. The bare patches of metal between the oxide nuclei will clearly be exposed to a higher oxygen potential and new oxide nuclei will grow at a different initial rate than on the existing nuclei. 

It is clear that the rate of growth of a reaction product depends upon two principal characteristics. The first of these is the thermodynamic properties of the phases which are involved in the reaction since these determine the driving force for the reaction. The second is the transport properties such as atomic and electron diffusion, as well as thermal conduction, all of which determine the mobilities of particles during the reaction within the product phase. 

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