$($b$)[10]$ An exhaust stream from a semiconductor fabrication unit contains $3$ mole $\%$ acetone $($C$\_3$H$\_6$O$)$ and $97$ mole $\%$ air. In order to eliminate any possible environmental pollution, this acetone$-$air stream is to be fed to a mass$-$transfer column in which the acetone will be stripped by a countercurrent, $20^$C falling water stream. The tower is to be operated at a total pressure of $1$atm$.$ If the combined Raoult$-$Dalton equilibrium relation may be used to determine the distribution of acetone between the air and the aqueous phases, determine the following: $($i$)$ The mole fraction of acetone within the aqueous phase which would be in equilibrium with the $3$ mole $\%$ acetone gas mixture. $($ii$)$ The mole fraction of acetone in the gas phase which would be in equilibrium with $20$ ppm acetone in the aqueous phase. Data At $20^$C$,$ the vapour pressure of acetone is $5\mathrm{.}64\backslash$times $10^4$Pa$.$ Atomic masses of oxygen, hydrogen and carbon atoms are $16\mathrm{.}0,1\mathrm{.}0$ and $12\mathrm{.}0,$ respectively.