Oxygen diffuses through the wall of drug containers and oxidizes many drugs rendering them inactive. It is the oxygen diffusion/reaction scenario that limits the shelf-life of many pharmaceutical products. To limit the oxidation of drugs, oxygen scavengers, like sodium bisulfite \(\left(\mathrm{NaHSO}_{3}\right)\) are often added. The reaction to remove oxygen is:

\[2\mathrm{HSO}_{3}^{-}+\mathrm{O}_{2}\rightarrow\mathrm{SO}_{4}^{2-}+\mathrm{H}_{2} \mathrm{O}\]

Consider a liquid drug stored in a cylindrical polyethylene container. The container is \(15 \mathrm{~cm}\) high and has an inner diameter of \(6 \mathrm{~cm}\). The initial concentration of \(\mathrm{NaHSO}_{3}\) in the drug formulation is \(1 \mathrm{~g} /\) liter. How thick must the walls of the container be to ensure that \(90 \%\) of the \(\mathrm{NaHSO}_{3}\) remains after 1 year? Neglect diffusion through the top and bottom of the container and assume that the \(\mathrm{NaHSO}_{3}\) reacts instantly with the \(\mathrm{O}_{2}\) so that the \(\mathrm{O}_{2}\) concentration in the drug is always 0 . The effective diffusivity of \(\mathrm{O}_{2}\) through polyethylene is \(9 \times 10^{-13} \mathrm{~m}^{2} / \mathrm{s}\) and you may assume the partition coefficient is \(1, P=1 \mathrm{~atm}, T=20^{\circ} \mathrm{C}\), and the process operates at steady-state.