32 The Law of Distribution of Velocities [CH. n 



The average duration of a free path is accordingly 



1 /km 



2^V27r 



The distance described per unit time by the v molecules occupying unit 

 volume 



+ 00 



h ^7>7^ 



r e~ hmc * cdudvdw 



-/// 



/h a m* r 



\/ ~^~ e 



V 7T 3 Jo 



= 4nrv \/ | e- hr 2 c 3 dc 

 2v 



This distance is the aggregate of all the free paths, of which the number is 

 given by expression (47). By division we find as the length of the mean ~J h 

 free path 



1 7071.. 



V27TZ/0- 2 



If the average is taken in any other way, the result is of course different. 

 We might for instance average over all the free paths which are being 

 described at a particular instant of time. 



Tait* takes a particular instant of time, and defines the mean free path 

 as the average of the distances described by each molecule between this 

 instant and the instant of its next collision. He calculates as the value of 

 the mean free path defined in this way, 



(50), 



7TVO- 



the factor '677... arising from an integral of which the value cannot be 

 calculated in finite terms. We shall return later to the actual calculations 

 by which this result is obtained. 



From the results obtained in this section the numerical values given in 

 8 can be calculated without trouble. 



Apparent Irreversibility of Motion. 



31. When a gas is not in a steady state, it follows from 23 (p. 21), 

 that dH/dt must be negative. Former writers have interpreted this to mean 

 that H will continually decrease, until it reaches a minimum value, and will 

 then retain that value for ever after. A motion of this kind would, however, 

 be dynamically irreversible, and therefore inconsistent with the dynamical 

 equations of motion from which it ought to have been deduced. As will 

 appear from the next chapter, the truth is that we have at this point reached 

 the limit within which the assumption of molecular chaos leads to accurate 

 results. The motion is, in point of fact, strictly reversible, and the apparent 

 irreversibility is merely an illusion introduced by the imperfections of the 



statistical method. 



* Royal Soc. Edin. Trans, xxxiu., p. 74 (1886). 



