172 SURFACES AND MEMBRANES 



Adam [1930] has shown at great length that the surface energy due to 

 the inward pull on the molecules forming the surface is the fundamental 

 property of surfaces. This potential surface energy is of fundamental 

 importance, for a large number of problems relating to the equilibrium 

 of surfaces can be solved without a knowledge of more than the magni- 

 tude of this surface energy. In the solution of such problems a mathe- 

 matical device is used to simplify the analysis. We substitute for the 

 surface energy a hypothetical tension acting in all directions tangent to 

 the surface, and equal to the magnitude of the surface energy per unit 

 area. This hypothetical tension is what is generally understood by the 

 term surface tension. 



The surface energy per unit area is denoted by the number of ergs per 

 square centimeter. This is analogous in dimensions to surface tension 

 expressed in dynes per centimeter. To illustrate its usage by an ele- 

 mentary example, let us analyze the surface energy of a circular surface 

 of radius r having an area irr 2 . Next, let us assume a hypothetical force 

 acting at right angles to the circumference of the surface. This force 

 will act along the radii of the circle tending to shrink the surface. Let its 

 magnitude be T dynes per centimeter of circumference. In order to 

 increase the area, energy must be expended. Let the available energy 

 be sufficient to expand it to an area having a radius (r + dr) cm. The 

 energy used to produce this expansion is dE = 2-n-rTdr, which upon inte- 

 gration becomes E = irr 2 T. Hence the energy per unit area measured 

 in ergs per square centimeter has the same dimensions as T, the so-called 

 surface tension measured in dynes per centimeter. This hypothetical 

 surface tension which is supposed to act in the face of the plane surface is 

 also treated as acting tangentially to any curved surface. To treat it 

 as a vector will often lead to fallacious results. Its use may be illus- 

 trated in calculating the surface energy of a spherical bubble immersed 

 in a liquid, a common biophysical phenomenon. 



The pressure inside a bubble of gas immersed in a liquid must be 

 greater than the external pressure existing at the surface by an amount 

 equal to the internally directed molecular forces per unit area. If T is 

 the hypothetical surface tension in dynes per centimeter at the liquid- 

 gas interface of a bubble of area 4xr 2 , then the surface energy at this 

 interface is E = 4irr 2 T. If the bubble is allowed to expand so that its 

 radius increases to r + dr, then the work done in this expansion is 

 dE = 8irrTdr. If the pressure directed along the radius of the bubble is 

 p dynes per square centimeter, and an increase in volume dV results, 

 then the work done is dE = pdV, or p • 4irr 2 dr = 8irrTdr. Hence 



2T 



V = — 

 r 



