The Contracting Sphere Rate Law

We have described some of the difficulties that accompany kinetic interpretations ofsohd state reactions. In spite of this, a number of models can be described that essentially represent trial efforts for interpreting experimental results. We can expect, however, that detailed knowledge of a particular reaction is Hkely to be specific to that reaction. Having described the parabolic and first-order rate laws, we now consider some geometrical models. In the first of these models, we will derive the rate law for the reaction of spherical particles. 

Suppose that a reaction involving a spherical solid particle of radius r takes place on the surface of the particle. Actually, it does not matter whether it is a single particle or a collection of particles of spherical shape. For this particle, the volume is given by 

If the reaction takes place on the surface of the sphere, the rate will be proportional to the surface area, S. However, if we assume a uniform density, the quantity of material present is proportional to V. As a result, the volume of the particle is decreasing with time according to 

Another way to interpret the contracting sphere rate law is to substitute for V in terms of the radius of the particle. When this is done, we obtain 

This expression is equivalent to saying that the radius of the reacting particle is decreasing at a constant rate. 

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