You are designing a novel treatment device that will be used to extract trichloroethylene $($TCE$)$ from a wet sediment slurry. The sediment is $50\%$ soil $($particle density $=2\mathrm{.}0$ g$/$cm$3)$ and $50\%$ water by volume. Assume a linear partitioning model for soil$/$water equilibrium with a Kd value of $20$ L$/$kg$.$ The device works by creating a completely mixed slurry of water and soil and exposing it to a membrane on the other side of which is an organic solvent $($in a flow through chamber$)$ with a high capacity for TCE $($the concentration in the solvent is always zero$).$ The initial concentration on the soil is $10$ mol TCE$/$kg dry soil. The pilot device uses a $10$ L sediment slurry chamber, and a $0\mathrm{.}5$ m$2$ membrane that is $2$ mm thick. The diffusivity of TCE through the membrane is $0\mathrm{.}013$ cm$2/$s$.$ The system is designed to run for $10$ minutes per batch. $($a$)$ What is the initial concentration in the water fraction of the sediment, and $($b$)$ what is my expected sediment concentration after $10$ minutes? $($Hints: this is a mass balance and the flux of TCE will change as the concentration drops and TCE is in both the soil and water phase in the sediment$)$

Answers

a$.0\mathrm{.}5$ mol$/$L

b$.3\mathrm{.}86$ mol$/$kg