A new plastic bottle is being considered for carbonated beverages $($that contain dissolved carbon dioxide$).$ We want to calculate the leakage rate of $C{O}_2.$ Approximate the bottle as a liquid$-$filled cylindrical shell with inner diameter $8\mathrm{.}5cm,$ height $17\mathrm{.}5cm$ and wall thickness $0\mathrm{.}75mm.$ The concentration of $C{O}_2$ in the liquid $($beverage$)$ is $0\mathrm{.}25mo\frac{l}{?}$ liter, and that in the ambient air is $2\mathrm{.}0{10}^{-5}mo\frac{l}{?}$ liter liter $($:$=0\mathrm{.}001m^3\}.$ The diffusivity of $C{O}_2$ in the plastic is $1{10}^{-9}c\frac{m^2}{?}s.$ The concentration of $C{O}_2$ in the plastic in equilibrium with the liquid is determined by

$S_{PL}=\frac{C_{C{O}_2}(plastic)}{C_{C{O}_2}(liquid)}=1\mathrm{.}5$

and the $C{O}_2$ concentration in equilibrium with air obeys

$S_{PA}=\frac{C_{C{O}_2}(plastic)}{C_{C{O}_2}(air)}=3$

where $S_{PL}$ and $S_{PA}$ are solubilities.

$($a$)(20$ pts$)$ Calculate the diffusion rate of carbon dioxide through the plastic in moles$/$second$.$ Assume that the $C{O}_2$ concentration is uniform in both the liquid and in the air; that is$,$ the plastic layer is the only diffusion resistance.

$($b$)(5$ pts$)$ Sketch the concentration profile of $C{O}_2,$ showing the concentrations in the liquid, plastic and air. A qualitative sketch is sufficient $-$ you don't need to calculate the profile.