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Date: 30-8-2016
1016
Date: 9-8-2016
3794
Date: 21-8-2016
875
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Soap Bubble Coalescence
Two soap bubbles B1 and B2 of radii R1 and R2 become one bubble B0 of radius R0. Find the surface tension coefficient for the soap solution. The ambient pressure is Pa.
SOLUTION
Assume that, during the coalescence, the total mass of air inside the bubbles and the temperature do not change. So,
(1)
where m0, m1, m2 are the masses of air inside bubbles B0, B1, B2 respectively. By the ideal gas law,
(2)
where mi is the mass, Pi is the pressure, and Vi = (4/3) πR3i is the volume in the ith bubble, and μ is the molar mass of the trapped air. The equilibrium condition for a bubble is
(3)
The coefficient 2 in front of the second term results from the presence of two surfaces of the soap film enclosing the air. From (2) and (3) we obtain
(4)
Substituting (4) into (1), we obtain
(5)
and so
(6)
Note that if a is very small the volume of the new bubble is close to the sum of the original volumes, whereas if it is very large the surface area of the new bubble is roughly the sum of the original surface areas.
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