Problem $3$:

A cylindrical rubber plug of thickness $20mm$ and cross$-$sectional area of $300m{m}^2$ is used to cap a $1\mathrm{.}0$ liter vessel that contains pure $C{O}_2($you can picture it like a cork in a wine bottle$).$ Initially, the gas pressure in the vessel is $5$ bar, but over time $C{O}_2$ diffuses slowly through the rubber plug. The vessel walls are impermeable to $C{O}_2,$ and the vessel is kept in a laboratory at $25C.$

The solubility and diffusivity of $C{O}_2$ in rubber are $40mo\frac{l}{m^3\frac{*}{b}ar}$ and $0\mathrm{.}11*{10}^{-9}\frac{m^2}{s},$ respectively, at $25C.$

A$)$ Derive an equation for the molar flux in the rubber plug.

B$)$ Calculate the rate of mass loss of $C{O}_2$ from the vessel under the conditions described above.

C$)$ Derive an expression for the $C{O}_2$ concentration distribution in the rubber plug and sketch the concentration profile in a graph.

D$)$ Derive an equation for the $C{O}_2$ pressure inside the vessel as a function of time.

E$)$ Calculate how long it takes before the pressure in the vessel drops by $1\%.$