Space functions in isotropic growth

We consider the two directions of development, with the two modes located at the interface as the rate-determining step. The formed solid B is a part of the sphere. 

Inward development with the rate-determining step at the internal 

However, integrating [9.5] between 0 and ri and moments rand t (f is constant under our experimental conditions), it becomes  

The volume of half-sphere CKC at time t, for the fractional extent a of the reaction, is  

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