26* MOLECULAR FREE PATHS 409 



molecules undergo collision from among those that have traversed 

 the path x. The sum of all the paths traversed by these molecules 

 amounts to 



ne- xllX dx, 

 l 



and hence as each particle must certainly collide after traversing 

 some distance between the limits x = and x = oo, the sum of 

 the paths traversed by all the n molecules before collision is 



Jo I 



Thus the mean value of these n free paths is I. 



This mean probable value of the free path is to be understood 

 as corresponding only to particles that move with a certain definite 

 velocity, since we assumed the same motion for all the n particles ; 

 it is therefore denoted by I, so as to be different from the symbol 

 L used in 65. In addition to altering with the speed of the 

 particle, I may in general depend also on position, time, and 

 direction, if the molecular motion of the gas alters with these 

 magnitudes. 



27*. Probability of an Encounter 



Before we determine the value of the free path Z for a particle 

 of a real gas, let us solve, by Clausius' method, a preliminary 

 problem. 



Into a space filled with molecules at rest, of which n are 

 contained in each unit of volume, let a molecule enter with the 

 velocity w. What is the probability that this molecule may in a 

 given interval t, say the unit of time, collide with one of those 

 at rest, the radius of the sphere of action being s ? 



In the time t the molecule traverses the length wt ; its sphere 

 of action therefore moves through the volume 7rs 2 w. Since in this 

 space there are irn^ut molecules at rest, the probable number of 

 encounters which the molecule meets with in the interval t is also 



and the probable number of encounters in unit time is therefore 

 given by the product 



