Infections are often accompanied by a biofilm that is very resistant to antibiotics and hard to kill. We can model the situation as a thin film of material atop an inert substrate. The antibiotic diffuses into the biofilm whereupon it reacts to kill the cells. A concentration of \(0.01 \mathrm{moles} / \mathrm{m}^{3}\) of antibiotic is needed to kill the bacteria. Within the biofilm we can model the reaction as first order, \(k^{\prime \prime} c_{a}\). On the surface of the biofilm, the concentration is \(c_{a}=c_{a 0}=\) \(0.1 \mathrm{moles} / \mathrm{m}^{3}\). The diffusivity of antibiotic through the biofilm is \(2.0 \times 10^{-11} \mathrm{~m}^{2} / \mathrm{s}\), the biofilm is \(0.5 \mathrm{~mm}\) thick, and the substrate is impermeable to antibiotic.

a. Derive the differential equation describing the destruction of the antibiotic in the biofilm.

b. What are the boundary conditions for the problem?

c. Solve the differential equation for the concentration profile.

d. Can we kill the bacteria all the way through the biofilm?