S 67 MOLECULAR FREE PATHS 161 



path we were calculating was in motion, all the others being 

 at rest. All the particles, however, are in motion in the 

 actual case. It is easy to see that this general motion must 

 increase the probability of a collision of one particle with 

 the others ; for the particle can also be struck by another 

 that moves from the side and with which it would not 

 come into contact as a result of its own motion. By the 

 general motion, therefore, of all the particles the probability 

 of a collision is increased, and thus the mean value of the 

 molecular free path is diminished. 



Clausius has calculated this shortening of the path for 

 the case in which all the particles move with equal speeds 

 but in all possible directions in space. With this supposi- 

 tion we find the number of collisions increased in the ratio 

 of 4 : 3, and, therefore, the free path shortened in the ratio 

 3 : 4. We obtain then, as is proved in 28*, the value 



L = fX 3 / 2 



for the mean free path of a particle in the uniformly moving 

 medium, and this differs only by the factor f from the 

 former value. From this equation also we can deduce a pro- 

 portion like the former and of similarly simple meaning, viz. 



L : ^ = X 3 : TTS*. 



68. Molecular Free Path with an Unequal 

 Distribution of Speeds 



But these calculations do not correspond exactly to the 

 real state of things, since the underlying assumption as to 

 the way in which the speeds are distributed among the 

 molecules cannot possibly be right. The supposition that 

 all the particles of a gaseous medium are to have equal 

 speeds gives no real picture of the motion which exists in a 

 gas that is in equilibrium under a pressure which is every- 

 where the same and at a temperature which is everywhere 

 the same. The true law according to which the molecules 

 arrange their speeds is, as we know ( 24), that discovered 

 by Maxwell. 



M 



