Smaller Bubble, Higher Pressure: Use 1/R Not R
The extra pressure inside a curved liquid surface is inversely proportional to its radius. This single fact tells you which of two connected bubbles collapses, what the radius of their shared wall is, and how bubble sizes compare.
Motion and force questions become simpler when direction, starting condition, time and the acting forces are marked first.
Excess pressure formulas
A liquid drop has one surface and its excess pressure is 2 T / R. A soap bubble has two surfaces, so its excess pressure is 4 T / R. An air bubble inside a liquid has excess pressure 2 T / R. In every case the excess pressure is proportional to 1 / R, so if the excess pressure of bubble A is twice that of B, then R_A = R_B / 2 and the volumes (proportional to R³) are in the ratio 1 : 8.
Two bubbles in contact
When two soap bubbles of radii r1 and r2 touch over a common wall, the pressure difference across the wall is the difference of the two excess pressures: 4 T / r_wall = 4 T / r1 - 4 T / r2, so 1 / r_wall = 1 / r1 - 1 / r2 and r_wall = r1 r2 / (r2 - r1).
the wall curves towards the larger bubble because the smaller bubble has the higher pressure.
Which way does air flow?
If two bubbles are connected through a valve or tube, air flows from the bubble with higher excess pressure (smaller radius of curvature) to the one with lower excess pressure, so the smaller bubble shrinks and the larger one grows. A hemispherical bubble of radius r has a smaller radius of curvature than a sub-hemispherical (shallower) bubble on the same tube, so air flows from the hemispherical bubble into the shallower one and the hemispherical bubble gets smaller.
OlympIQ exam lens
- ✓Drop: 2T/R; air bubble in liquid: 2T/R; soap bubble: 4T/R; always proportional to 1/R
- ✓Equal tension and excess pressure ratio fix the radius ratio; volumes go as R³
- ✓Wall between two touching bubbles: 1/r = 1/r1 - 1/r2
- ✓Air flows from the smaller (higher-pressure) bubble to the larger one
Do not assume the bigger bubble has the bigger pressure. The excess pressure is larger for the smaller bubble, so when two bubbles are connected, the small one empties into the large one.
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