Physical Chemistry of Foods

the sphere provides a force that is equal and opposite to the force caused by
the surface tension. Hence, the relationpL¼ 2 g/R.
For a soap bubble of 1 cm diameter and a surface tension of
50 mN?m
^1 ,pLwill be 2(2 6 0.05/0.005)¼40 Pa. The factor 2 before the
bracket is due to a soap film having two surfaces; here, each surface has
(almost) the same radius and the same surface tension. (A ‘‘soap bubble’’ is
in fact a very thin spherical shell of a soap solution.) For a gas bubble of the
same size in the same soap solution,pLwould be 20 Pa. Note thatpLis
greater for a smaller bubble: for one of 1mm diameter,pLwould amount to
2? 105 Pa, or 2 bar. For an emulsion droplet of that size, wheregis smaller,
for instance 10 mN?m
^1 , a Laplace pressure of 0.4 bar results.
The given relation for a sphere is a special case of the generalLaplace
equation

pL¼pconcave 7 pconvex¼g

1

R 1

þ

1

R 2



ð 10 : 7 Þ*

The curvature of a surface can at any place be characterized by two
principal radiiR 1 andR 2 .R 1 is found by constructing a plane surface
through the normal to the surface at the point considered. The curved
surface intersects the plane, resulting in a curve to which a tangent circle is
constructed. The plane then is rotated around the normal until the curvature

FIGURE10.19 Derivation of the Laplace pressure for a sphere of radiusR, where
the sphere’s surface is the phase boundary between two fluids, with interfacial
tensiong;pmeans pressure.