P$4(20$ Pts$)$: After a sparkling water bottle is opened, the

carbonated bubbles start to rise and escape out of the liquid

because the liquid cannot hold the excess carbon dioxide in

the solution. At the bottom of the bottle, a $C{O}_2$ bubble has

approximately a diameter of $0\mathrm{.}2$ mm $.$ Estimate the bubble's

diameter when it reaches the surface by considering the $C{O}_2$

bubble as an ideal gas and noting $(PAA{)}_{\text{b}\text{o}\text{t}\text{t}\text{o}\text{m}}=(PAA{)}_{\text{s}\text{u}\text{r}\text{f}\text{a}\text{c}\text{e}}.$

The temperature of the water and bubbles is $10C$ and assume

that the density of the water is the same as that of pure water.