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 P_a.

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 m_i is the mass, P_i is the pressure, and V_i = (4/3) πR³_i 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.