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Two soap bubbles of radii a and b in air coalesse together under isothermal conditions forming a bubble

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Two soap bubbles of radii a and b in air coalesse together under isothermal conditions forming a bubble of radius c. P is the atmospheric pressure, show that

S is surface tension of soap solution.

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Let P1 , v1 denote total pressure and volume of air in bubble of radius a. P2 , v2 and P3 , v3 are corresponding quantities for bubble of radius b and c respecitvely. Then

Under isothermal conditinos

Rearranging term we have

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