Liquid surfaces and the Laplace-Young equation

Any region of a surface has its geometry specified by the algebraic values of the two principal radii of curvature in that region. Consider a very small element of the free surface of a one-component liquid in equilibrium with its vapour, having principal radii of curvature rj and 2 as shown in Fig. 7.3(a). A sign convention is needed here. For a liquid-vapour interface take the origin of coordinates in the liquid phase and let the positive direction of the normal at the interface be from the liquid phase to the vapour phase. 

A radius of curvature is then considered positive when its centre of curvature is in the liquid phase. In Fig. 7.3(a), both rj and r2 are positive if the surface is concave towards the liquid. This situation will now be analysed. Let the pressure on the 

Ignoring gravitational and other effects, the work that must be done on the liquid to produce the increase in area is given by (Pi P xydr the corresponding increase in the surface free energy is 76a. These two quantities are equal so that  

Equation [7.15] gives the pressure drop that occurs on crossing the liquid-vapour interface in the positive direction, and is known as the Laplace-Young equation. When ri andr2 are both positive Ap is positive, i.e. pi p2, and there is a decrease in pressure on 

When a thin liquid film is considered, two pressure differences occur. These have the same sign and magnitude so that the pressure difference in crossing both surfaces is  

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