That the bubble has energy may be shown by leaving the end of the tube open. The bubble will contract, forcing the air out, and the current of air blown through the tube may be made to deflect the flame of a candle. If the bubble is in the form of a sphere of radius r this material surface will have an area
S = 4πr². (1)
If T be the energy corresponding to unit of area of the film the surface-energy of the whole bubble will be
ST = 4πr²T. (2)
The increment of this energy corresponding to an increase of the radius from r to r + dr is therefore
TdS = 8πTdr. (3)
Now this increase of energy was obtained by forcing in air at a pressure greater than the atmospheric pressure, and thus increasing the volume of the bubble.
Let Π be the atmospheric pressure and Π + p the pressure of the air within the bubble. The volume of the sphere is
V = 4⁄3 πr3, (4)
and the increment of volume is