Reaction of a single grain or massive material

With the assumption of the uniform average surface of A, the speed of growth will be the same one at any point. 

We will indicate by the reactivity of growth and by the space function for a grain /, with 

Ug is the initial amount of A in an i grain Sp indicates at the i time for a nucleus i, bom at the initial time (i.e. in our case for an A grain), the area of the active surface, that is, the area of the internal interface if the growth has internal interface reactions or diffusion as rate-determining steps and the area of the external interface if the growth has external interface reactions (or gas adsorption or desorption) as rate-determining steps. 

From these expressions of the rate, we can, by integration, obtain the fractional extent according to time, indeed 

We can also calculate the rates aeeording to the fraetional extent either by eliminating time between the rate and extent calculated previously, or directly. Illustrating this second method for sphere with inward development and growth with an external interface step as rate determining, we can express 

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