A gas bubble of radius r in a liquid breaks into eight equal-radius bubbles. The total interfacial-area/gas-volume ratio changes from 3/r to 3n/r. Find n.
Volume conservation gives $8r_{new}^3=r^3$, so each new radius is r/2. For any collection of equal spheres, total area divided by total volume is 3/radius. The new ratio is $3/(r/2)=6/r=3(2)/r$, hence n = 2.
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