Young–Laplace equation

The Young-Laplace equation. (After Thomas Young and Pierre -Simon Laplace, they independently herleiteten 1805 ) describes the relationship between the surface tension, the pressure and the surface curvature of a liquid

Drops

In a spherical droplet, for example, a small drop of water or a gas bubble in a liquid, there is, due to the surface tension at the interface of liquid / gas, an increased pressure p:

With the spherical radius r. The pressure is thus greater, the smaller the radius of the sphere.

If we reduce the radius as far as to the order of molecular diameters approaching, the surface tension is dependent on the radius:

So that the above simple equation no longer applies.

Any curved surface

If it is not a sphere but with an arbitrarily curved surface, so the equation becomes:

R1 and r2 are the principal curvatures of the circle of curvature.

Soap bubble

For the pressure inside a soap bubble, the pressure is always twice as large because the soap film surfaces two phase gas / liquid has:

• For spherical bubbles:

• For non- spherically symmetric body:

If multiple bubbles are into each other, dividing the sum of the pressures of all bubbles must add each located on the way to the point of consideration.

Derivation

For the surface area A of a sphere

For the volume

With a small change in the radius dr are the changes in the surface

And the volume

The work which is required for changing the surface, so that

The amending of the volume

This gives the formula given above, if the two labor contributions are equated.