13.13: Excess Pressure Inside a Drop and a Bubble
The shape of a small drop of liquid can be considered spherical, neglecting the effect of gravity. This drop can further be considered as two equal hemispherical drops put together due to surface tension. The forces acting on the spherical drop are due to the pressure of the liquid inside the drop, the pressure due to air outside the drop, and the force due to the surface tension acting on the two hemispherical drops.
where γ corresponds to the surface tension. Recalling that the force per unit length is the surface tension gives the force acting on the diameter. The difference between the pressure inside and outside the drop for its given surface area gives the net pressure forces acting on the drop. At equilibrium, the forces inside and outside balance each other, indicating that pressure inside the liquid surface is greater than the external pressure.
In the case of an air bubble inside a liquid, this condition remains unchanged, and the pressure inside the air bubble is greater than the pressure in the surrounding liquid. The pressure difference across this air-liquid interface is proportional to curvature of the interface, which in turn determines the size of the bubble.
A cross-sectional view of a soap bubble shows two surfaces: one in contact with the air inside the bubble and the other in contact with the air outside the bubble. The pressure within the bubble is greater than the air pressure outside the surface. Also, the pressure within the bubble is lower than the air pressure inside the surface by the same quantity. Solving these two conditions gives the Young-Laplace relation for the soap bubble.