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Pressure Inside a Bubble

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Anatomy of a Soap Bubble

When you blow a soap bubble, you are creating a pressurized bubble of air which is contained within a thin, elastic surface of liquid. Most liquids cannot maintain a stable surface tension to create a bubble, which is why soap is generally used in the process ... it stabilizes the surface tension through something called the Marangoni effect.

When the bubble is blown, the surface film tends to contract. This causes the pressure inside the bubble to increase. The size of the bubble stabilizes at a size where the gas inside the bubble won't contract any further, at least without popping the bubble.

In fact, there are two liquid-gas interfaces on a soap bubble - the one on the inside of the bubble and the one on the outside of the bubble. In between the two surfaces is a thin film of liquid.

The spherical shape of a soap bubble is caused by the minimization of the surface area - for a given volume, a sphere is always the form which has the least surface area.

Pressure Inside a Soap Bubble

To consider the pressure inside the soap bubble, we consider the radius R of the bubble and also the surface tension, gamma, of the liquid (soap in this case - about 25 dyn/cm).

We begin by assuming no external pressure (which is, of course, not true, but we'll take care of that in a bit). You then consider a cross-section through the center of the bubble.

Along this cross section, ignoring the very slight difference in inner and outer radius, we know the circumference will be 2pi R. Each inner and outer surface will have a pressure of gamma along the entire length, so the total. The total force from the surface tension (from both the inner and outer film) is, therefore, 2gamma (2pi R).

Inside the bubble, however, we have a pressure p which is acting over the entire cross-section pi R2, resulting in a total force of p(pi R2).

Since the bubble is stable, the sum of these forces must be zero so we get:

2gamma (2pi R) = p(pi R2)

or

p = 4 gamma / R

Obviously, this was a simplified analysis where the pressure outside the bubble was 0, but this is easily expanded to obtain the difference between the interior pressure p and the exterior pressure pe:
p - pe = 4 gamma / R

Pressure in a Liquid Drop

Analyzing a drop of liquid, as opposed to a soap bubble, is simpler. Instead of two surfaces, there is only the exterior surface to consider, so a factor of 2 drops out of the earlier equation (remember where we doubled the surface tension to account for two surfaces?) to yield:
p - pe = 2 gamma / R
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