Young-Laplace Equation Calculator
Connect a small bubble to a large one and the small one empties — the intuition is exactly backwards.
The formula
dP = 2 gamma / r for one surface, 4 gamma / r for a bubble
Small means high pressure
The pressure jump across a curved surface goes as one over the radius, so the smaller the droplet the harder it squeezes. A one-millimetre water droplet holds about 146 Pa above its surroundings; a one-micrometre droplet holds 146,000. This is why fine mists evaporate so much faster than the same water in a puddle, and why the alveoli in your lungs need a surfactant to keep the smallest ones from collapsing into the larger.
A bubble has two surfaces
A droplet of liquid in air has one interface. A soap bubble has an inner and an outer surface with a thin film between, so the pressure difference is twice as large for the same radius. Getting this factor wrong is the standard error, and it is a factor of two rather than something subtle.
Why two bubbles do not equalise
Connect a small bubble to a large one and the small one empties into the large one, because its pressure is higher. The intuition that they would even out is exactly backwards, and the demonstration is a staple of first-year lectures for that reason.
Where these models stop
Michaelis-Menten assumes a single substrate and a steady state; Langmuir assumes one layer on identical sites with no interaction between them; Stokes-Einstein assumes a hard sphere much larger than the solvent molecules around it. Each is an idealisation that happens to describe real systems well over a useful range, and each fails predictably outside it. Knowing which assumption a number rests on is usually more valuable than the number.