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, $c_{AR},$ 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 $f=r{C}_A,$ rewrite the model differential equation and obtain the concentration profile in the aggregate.