Soap Bubbles in Equilibrium

Two soap bubbles of radius R₀ are connected by a thin “straw” of negligible volume compared to the volume of the bubbles (see Figure 1.1). The ambient pressure is P_a, the temperature is τ, and the surface tension coefficient is σ.

a) Is this system in stable equilibrium? What is the final state?

b) Calculate the entropy change between the final-state configuration and the configuration in Figure 1.1. Assume 2σ/R₀ << P_a.

Figure 1.1

SOLUTION

a) The equilibrium is unstable. It is obvious from purely mechanical considerations that if the radius of one bubble decreases and the other increases, the pressure in the first bubble (which is inversely proportional to R₀) will increase and that in the second bubble will decrease, leading to further changes in respective radii until the system becomes one bubble with radius R₁ (see Figure 1.2). The same result can be obtained by considering the free energy of the system.

Figure 1.2

b) The free energy of the bubble consists of two parts: a volume part, which is just the free energy of a gas, and a surface part, which is associated with the surface tension:

(1)

The Gibbs free energy

(2)

The entropy change

where G₁ is the potential of the system with one bubble and G₀ is the potential of the initial configuration. We then find

(3)

where we used the fact that the number of particles did not change and q is the heat necessary to produce a unit area of the film:

So

(4)

We can eliminate R₁ from the final result by using the following equations:

(5)

(6)

(7)

where the last equation represents the ideal gas law at constant temperature. This yields the equation

(8)

Solving this cubic equation in the small σ limit gives

(9)

Substituting (9) into (4) yields (in the same approximation)

(10)