$(8)3.$ Unsteady Difiusion into a Semi$-$inffinite Medium

Liquid benzene, $C_6{H}_6,$ a common industrial solvent with a density of $0\mathrm{.}88\frac{g}{c}{m}^3,$ leaked from a storage tank and seeped into the ground below. As the density of benzene is less than the density of water, the liquid benzene formed a light nonaqueous$-$phase liquid layer on top of the water$-$saturated soil, as shown belowThere is no groundwater flow through the water$-$saturated soil layer: it is essentially stagnant. We are interested in the transport of benzene into the water$-$saturated soil so that we may identify how much soil must be dug up and treated. This system can be approximated as a semi$-$infinite diffusion medium. Liquid benzene is sparingly soluble in water, and its solubility limit is $24mo\frac{l}{m^3}$ at $293K.$ The effective diffusion coefficient of benzene in the water$-$saturated soil is $1{10}^{-9}\frac{m^2}{s}$ at $293K.$ Initially, there is no benzene dissolved in the water$-$safurated soil. The pure liquid benzene resting on top of the soil provides a continual source due to the low solubility of benzene in water.

A$.$ How long will it take for the concentration of benzene in water at a depth of $1m$ to reach the toxicity limit of $1\frac{g}{m^3}?$

B$.$ Find the concentration of benzene in the water in the soil after $1\mathrm{.}5$ years at depths of $0\mathrm{.}25m,0\mathrm{.}5m,0\mathrm{.}75m,$ and $1m.$ Draw$-$a graph of the concentration profile from $z=0m$ to $z=1m$ and use your graph to estimate the amount of benzene dissolved in $1$ cubic meter of surface soil after $1\mathrm{.}5$ years.

Please do the work by hand with pen and paper for easier visualization.