Excess of Pressure -- Drops and Bubbles

Because a liquid's free surface behaves like a stretched elastic membrane, always trying to contract to the smallest possible area, a curved liquid surface always has a slightly higher pressure on its concave (inner) side than on its convex (outer) side -- the surface tension acting all around the curved boundary produces a net inward force, which must be balanced by a correspondingly higher pressure pushing outward from inside. This pressure difference is called the excess of pressure, $\Delta P$.

A spherical liquid drop, or an air bubble inside a liquid. Both of these have only a single free (curved) surface -- for a liquid drop in air, the single surface is the outer boundary between the liquid and the surrounding air; for an air bubble inside a liquid, the single surface is the boundary between the air inside the bubble and the liquid surrounding it. For a sphere of radius $r$ with one such surface, of surface tension $T$, the excess of pressure on the inside over the outside is:

A soap bubble in air. A soap bubble, unlike a simple liquid drop, is a thin spherical film of soap solution with air both inside and outside it -- so it has two free surfaces under tension, an inner surface (film-to-inside-air) and an outer surface (film-to-outside-air), both contributing to holding the bubble's shape and both adding to the excess pressure. Since each surface individually contributes an excess pressure of $2T/r$, and the two surfaces act together (both surfaces of the very thin soap film have essentially the same radius $r$), the total excess of pressure inside a soap bubble is twice that of a single-surface drop of the same radius and the same surface tension:

$\Delta P = \frac{4T}{r}$ …