SURFACES OF DISCONTINUITY 515 



This expression is of course essentially negative by the con- 

 vention stated above, which means that the numerical value 

 of (2) is the amount by which the energy of these molecules is 

 less than what it would be were they all widely separated from 

 one another at the same temperature, i.e., in the gaseous state. 

 If we now wish to obtain the potential energy of all the mole- 

 cules in the body of liquid, we must not merely multiply the 

 expression (2) by the volume. To do so would be to overlook 

 the vital point that if a molecule lies in the layer of depth h at 

 the surface, part of the sphere of molecular action lies outside 

 the Uquid and the expression (1) is not correct for the potential 

 energy of this molecule. For such a molecule the contribution 

 to expression (2) is numericalUj smaller since n is zero* for certain 

 elements of the spherical volume of radius h surrounding it; 

 but as </)(r) is negative for the values of r considered, the con- 

 tribution of this molecule to the total potential energy is 

 greater than for a molecule in the interior of the liquid. In 

 short, if a body of liquid is divided into two portions which are 

 then separated from one another against mutual attraction we 

 know that the potential energy of the whole is increased. This 

 increase is made up of the larger contributions of those mole- 

 cules which lie near to the two new surfaces created by the 

 division. This increase can be calculated in terms of 0(r) and 

 we can thus obtain an expression which represents the "surface 

 energy," meaning by that the extra energy associated with the 

 molecules in the surface layer of thickness h over and above 

 that which would be associated with them if they were all in the 

 interior of the liquid mass. This is not the place to enter into 

 the analytical details, but it can be shown that the whole 

 potential energy of the body of liquid can be written in the form 



pV + <tA, 



where V and A are the volume and superficial area of the mass ; 



* Actually it is the concentration of the vapor or gas phase, rather 

 than zero. 



