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Prove that if two bubbles of radii r(1) ...

Prove that if two bubbles of radii `r_(1)` and `r_(2)` coalesce isothermally in vacuum then the radius of new bubble will be `r=sqrt(r_(1)^(2)+r_(2)^(2))`

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When two bubbles coalesce then number of molecules of air will remains constant and temperature also constant
so `n_(1)+n_(2)=nimpliesP_(1)V_(1)+P_(2)V_(2)=Pvimplies(4T)/(r_(1))((4)/(3)pir_(1)^(3))+(4T)/(r_(2))((4)/(3)pir_(2)^(3))=(4T)/(r)((4)/(3)pir^(3))impliesr=sqrt(r_(1)^(2)+r_(2)^(2))`
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