We wish to design a system too slowly release an insecticide into the air within a warehouse. The insecticide (A) is provided as an aqueous solution with 5% molar of A. We will encapsulate the solution in the core of a small hollow spheres so that the insecticide can diffuse slowly through a polymeric membrane that will conform the outer shell of the sphere (see figure). The company that fabricates the capsules informs us that they can produce capsules with radius between 0.5 and 2mm, and membrane thickness between 0.01 and 0.03mm. The solubility of insecticide in the membrane (which is impermeable to water), is given by c_A,m = 0.001c_A where c_A is the concentration of insecticide in the solution in contact with the membrane. The diffusivity of A in the polymeric membrane is D_AB = 3 x 10^-7 cm^2/s at 25C.

(a) Show that a mole balance of A in the spherical capsule can be written as: d(Vc_A)/dt = -N_A*A where c_A is the concentration of insecticide in the solution contained in the capsule (assumed to be uniform), N_A is the flux of A through the polymeric membrane, and V and A are the volume and surface area off the sphere, respectively.

(b) Specify a capsule design (radius and membrane thickness) such that half of the insecticide initially contained in the capsule is released in 2 days. State all assumptions and approximations.