154 PHENOMENA DEPENDENT ON MOLECULAR PATHS 63 



and in the same direction) is thrown into a space which 

 is filled irregularly with molecules at rest, their distribution, 

 however, being such that the density is the same every- 

 where. 



For the solution of the question bound up with this 

 idea, viz. What path will a particle so thrown in probably 

 traverse without a collision ? it is advisable to determine the 

 density of distribution of the particles at rest and express 

 it in terms of their mutual distances apart. If there are 

 N molecules in unit volume, then, considering the volume 

 of this unit to be divided into N equal parts, in fact into N 

 small cubes, we have in each of these small cubes a space 

 which contains on the average only a single molecule. If 

 we denote by the letter X the edge of one of these elemental 

 cubes, which Clausius calls the mean distance between 

 neighbouring molecules, the volume of one of the cubes is 

 X 3 , and the relation 



holds good. 



Since the density p may be expressed in terms of the 

 molecular weight m and the number N by the formula 



(13) 



p = Nm, 



the former formula shows that the density is related to the 

 distance between neighbouring molecules by the equation 



pX 3 = m. 



From the mean distance X between neighbouring mole- 

 cules Clausius deduces the mean probable length of free 

 path by comparison of that mean distance with the smallest 

 possible distance of separation, i.e. with the distance apart 

 of their mean points or centres of gravity at a collision, and 

 of the volume X 3 of the elemental cube with the space which 

 the moving particle must at least have for its motion. 



If, on the collision of two particles, it happens that 

 they come into actual contact, the least possible distance 

 apart of their centres would be the diameter of either, if we 

 could look upon the molecules as being spheres of equal 

 size; if the molecules have any other shape, the calculus 



