236 



Free Path Phenomena 



[CH. xi 



287. It is clear from the form of this expression that free paths which 

 are many times greater than the mean free path will be extremely rare. 

 Since B c is the reciprocal of X c (equation (538)), the probability that a molecule 

 moving with velocity c shall describe a path greater than n times \ c is 

 f(n\ c } or, by equation (548), is e~ n . Thus only one in 148 describes a path as 

 great as 5X C , only one in 22027 a path as great as 10X C , only one in 9 x 10 43 

 a path as great as 100\ c , and so on. 



This result applies of course only to molecules moving with a given 

 velocity c. 



At any given instant the fraction of the whole number of molecules 

 which have described a distance greater than I since their last collision, is 



/7T 3 r 



V AWJo 



(550). 



This function is not easy to calculate in any way. As the result of a 

 rough calculation by quadrature, I have found that through the range of 

 values for I in which its value is appreciable, it does not ever differ by more 

 than about 1 per cent, from e~ l '' a ^, the value for molecules moving with 

 velocity 1/V/m. 



Persistence of Velocities after Collision. 



288. We have next to consider what is the average effect of a collision, 

 as regards reversal of path. We shall find that on the average a collision does 

 not reverse the velocity in the original direction of motion, or even reduce it 

 to rest, but that there is a tendency for the original velocity to persist after 

 collision. Obviously it is of importance to form a numerical estimate of the 

 extent to which this occurs. 



289. Let us begin by considering two equal molecules colliding with 

 velocities a, b, the magnitude of these velocities but not their directions 

 being known. In fig. 17 let OP and OQ represent these velocities, and 

 let R be the middle point of PQ, Then we can resolve the motion of the 



FIG. 17. 



