Consider a blister package of tablets that is stored in a bathroom medicine cabinet. The package is composed of a flat, aluminum lidding sheet and a formed polymer sheet. The polymer sheet includes toughs in a frustum shape, as shown schematically below. The polymer sheet is $L=40m$ thick. The top and bottom of each trough are of diameters $D_1=4mm$ and $D_2=5mm,$ respectively. The depth is $h=3mm.$ Assume the aluminum sheet is impermeable to water vapor and the polymer sheet has a binary diffusion coefficient for water vapor is $D_{AB}=8{10}^{-14}\frac{m^2}{s}.$ The temperature is $295K$ and the relative humidity outside and inside the package is ${}_1=0\mathrm{.}9$ and ${}_2=0\mathrm{.}1,$ respectively.

a$)$ What are the partial pressures of water vapor inside and outside the package?

b$)$ What are the molar concentrations of water vapor at the outer and inner surfaces of the polymer trough?

c$)$ If solubility of water vapor in the polymer material is $1\mathrm{.}3{10}^{-2}$kmo$\frac{l}{m^3}\frac{*}{b}ar,$ determine the molar transfer rate at which water vapor is transferred through the polymer through wall to the tablet?

d$)$ Considering a different polymer material selected for the trough sheet, which leads to the diffusivity of water vapor is reduced by $15\%.$ Determine the total water vapor mass transfer rate.