# The Young-Laplace Equation

The shape of a liquid drop is governed by what is known as the Young-LaPlace equation. It was derived more or less simultaneously by Thomas Young (1804) and Simon Pierre de Laplace (1805). Read here how they did it.

## I. Introduction

The shape of liquid drop is governed by what is known as the Young-LaPlace equation. It was derived more or less simultaneously by Thomas Young (1804) and Simon Pierre de Laplace (1805). A short derivation of this equation is presented here.

Consider a small section of a curved surface with carthesian dimensions x and y. On the normal n two arcs can be constructed that support the lengths of x and y:

When the radii are enlarged by a small amount dR the work dW performed equals:

Where ? is the surface energy. The work is done against a pressure. According to thermodynamics this work can also be written as:

So we obtain for P:

From the first equation it can simply be shown that:

Insertion of the results from (5) into eq (4 yields the Young-LaPlace equation:

An expression for R1 for the cylindrical symmetrical can be deduced as follows:

Consider a contour f(z) with derivative f’(z) equaling n1. The length of the segment dL over distance dz equals:

This is equated to α.R1 . The angle α is derived from the difference in derivatives in z and z+dz:

Upon completion we obtain for R1:

The second radius R2 is derived as follows: take a small segment from the circumference of the profile. From the length of this segment dC we can see that:

From the figure we learn that:

From which it can readily be deduced that:

As a final result we obtain or the Young-Laplace equation:

C. van Veen