Transformation from the Liquid to the Gaseous State. 235 



of the surface film separating the liquid and vapour. It is 

 clear, therefore, that if «r remains sensibly constant, p must 

 increase as r increases, or in other words, the external pressure 

 and the volume must increase simultaneously, if equilibrium 

 is to be maintained. 



The saturated vapour-pressure «r, however, is not quite 

 constant, but varies at constant temperature with the curva- 

 ture of the film with which it is in contact, and if ot be 

 taken to represent the normal saturated vapour-pressure, that 

 is the pressure of a saturated vapour in contact with a plane 

 surface of its own liquid, then the saturated vapour-pressure 

 in contact with a concave spherical surface, of radius r, is 

 easily shown to be 



r ' >i-/V 



where p ± is the density of the liquid, and p 2 the density of the 

 saturated vapour. Hence the relation (1) connecting p and 

 r becomes 



r r P1-P2 



In this equation all the quantities other than p and r may 

 be taken as remaining constant during an isothermal trans- 

 formation, and consequently, within certain limits, the volume 

 and external pressure of the mass should increase together. 



This equation, however, cannot be expected to hold in the 

 extreme case in which the bubble is so small that the mass 

 within it ceases to possess distinctly the properties of a 

 vapour, or in the other extreme case, in which the bubbles 

 become so large and numerous that the remaining liquid, by 

 reason of being drawn out into thin films, or otherwise, ceases 

 to behave as a liquid in regard to the transmission of hydro- 

 static pressure &c. Within certain limits, however, equation 

 (2) gives the relation between the external pressure and the 

 volume of the mass. 



Thus, in the case of a single bubble, if the whole mass be 

 taken as unity, and the mass of vapour within the bubble be 

 m, then the mass of the liquid portion will be 1 — m, and the 

 whole volume will be 



P2 Pi 



But, if the radius of the bubble be r, we have 



4 

 3 



m=«7rp 2 r 3 ; (4) 



