$14-66$ You probably have noticed that balloons inflated with helium gas rise in the air the first day during a party but they fall down the next day and act like ordinary balloons filled with air. This is because the helium in the balloon slowly leaks out through the wall while air leaks in by diffusion.

Consider a balloon that is made of $0\mathrm{.}2-$mm$-$thick soft rubber and has a diameter of $15$ cm when inflated. The pressure and temperature inside the balloon are initially $120$ kPa and $25C.$ The permeability of rubber to helium, oxygen, and nitrogen at $25C$ are $9\mathrm{.}4{10}^{-13},7\mathrm{.}05{10}^{-13},$ and $2\mathrm{.}6{10}^{-13}$kmo$\frac{l}{m}*s\frac{*}{b}ar,$ respectively. Determine the initial rates of diffusion of helium, oxygen, and nitrogen through the balloon wall and the mass fraction of helium that escapes the balloon during the first $5$ h $,$ assuming the helium pressure inside the balloon remains nearly constant. Assume air to be $21$ percent oxygen and $79$ percent nitrogen by mole numbers, and take the room conditions to be $100$ kPa and $25C.$

Assumptions

Boundary Cond

Procedure using integrals and Fick law

Complete answeres with units