292 Mr. S. R. Milner on the Heats of 



The actual calculation is more complicated than in the 

 above simplified case in at least two respects. In the first 

 place, instead of the liquid having an indefinitely thin 

 surface, there must certainly in reality be a stratum of finite 

 thickness in which the density varies continuously from that 

 of the liquid to that of the vapour. Also, in dealing with 

 liquids it is necessary to apply the corrections which van der 

 Waals has indicated for the effect of the actual size of the 

 molecules in altering their mean free paths. 



With these modifications, consider therefore the region of 

 varying density to be divided into infinitely thin layers of 

 thickness dz, in passing upward through the boundary 

 between two of which a molecule receives a sudden increase 

 d(j) in its mechanical potential, by doing work against the 

 downward force. Putting X the number of molecules per 

 c.c. of the liquid in the lower layer, we have 



TSdf^TpdLi, ...... (1) 



where dhi is the element of internal latent heat per gram 

 between the two layers, and p the density of the liquid at this 

 point. 



If there were no collisions among the molecules, the 

 minimum velocity, v v which a molecule must have in order 

 to pass upwards through the boundary plane of the layers, 

 would be given by the condition that its kinetic energy 

 perpendicular to the plane must be equal to dcf>, or 



^mv± cos 2 6 = d<p. 



in which m is the mass, and the angle the direction of 

 motion makes with the perpendicular to the plane. 



As a matter of fact, however, collisions take place; and 

 after a collision the molecule struck moves on from a position 

 in advance of the centre of the striking molecule by an 

 amount which (at constant temperature) is on the average a 

 constant fraction of the distance between the centres of the 

 molecules at impact. If s be this distance, and I the mean 

 free path, then as van der Waals has shown *, 



where v is the volume of a gram of the liquid, and b a 



* Continuity of State : Phys. Soc. Translation, p. 374. I is the mean 

 free path calculated by neglecting the efiect of the extension of the 

 molecules in the direction of their relative motion. 



