The Contracting Area Rate Law

Consider a particle of a solid that is shaped as a cylindrical rod of radius r and length h as follows. 

For simpUcity, we will suppose that the cylinder is very long compared to the radius, which means that h r. Therefore, the area of the ends wiU be considered to be insignificant compared to the area of the curved surface, and we will assume that there is no significant reaction on the ends. Consequendy, we are assuming that the length remains constant during the reaction. Since the amount of material in the cylinder is represented by the volume and the reaction occurs on the surface, we can write 

The assumption that the area of the ends of the cyHnder is insignificant means that the first term on the right-hand side of this equation can be ignored and that the area can be represented as lirrh. 

Substituting for S in the rate expression shown in Eq. (7.35) followed by substitution for r gives 

We can represent the quantity 2ko TThy as k and integrate from Vo at t = 0 to a volume of V at some later time, t. 

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