Consider a homogeneous spherical aggregate of bacteria of radius R$.$ Species A diffuses into the bacteria and undergoes an irreversible $1$st$-$order reaction. The concentration of species A at the surface of the bacteria, cAR, is known.

a$)$ Develop a model equation describing the consumption of species A through the bacteria aggregation, and determine the boundary conditions to your model.

b$)$ Introducing a transformation f$=$ r CA$,$ rewrite the model differential equation and obtain the concentration profile in the aggregate.