Oil spills on top of a muddy marsh of depth L. At the bottom of this marsh (z = L), biocatalyst has been added, which degrades the oil and maintains a constant concentration C_ACAT << C_AS. At the top (z = 0), the immiscible and less dense oil layer forms a thick layer, and the limited solubility of oil in the marsh maintains C_AS at z = 0. Assume C_AS is relatively small compared to the concentration of water in the marsh, so that the oil in the marsh can be viewed as very dilute. Because of severe density gradients in the muddy marsh, the diffusivity of oil is not constant but instead varies linearly according to: D_oil-in-marsh = D₀ - 0.5*D₀/L * z.

1. Start with a mole balance for the transport of oil in the marsh and derive an expression for the concentration of oil in the marsh as a function of z.
2. Using your answer for part a, derive an expression for the flux of oil at z = 0.
3. Now using your answer for part b, derive an expression for the time it will take to deplete the entire oil layer, if the initial thickness of the oil layer is W and the molar density of the oil layer at the top is fixed at C_AN.