30 The Law of Distribution of Velocities [CH. n 



into the hydrodynamical equations, these equations will lead to a solution 

 in which the ultimate state is one in which there is no mass motion in the 

 gas. On the hydrodynamical view, the energy of the original motion has 

 been "dissipated" by viscosity. On the Kinetic Theory view, this energy has 

 been converted into molecular motion. In fact the Kinetic Theory enables 

 us to trace as molecular motion, energy which other theories are content to 

 regard as lost. 



Number of Collisions, Mean Free Path, etc. 



30. We shall now use the results which have been obtained, to calculate 

 the total number of collisions per unit volume of gas. Since the number of 

 collisions is not affected by the mass-motion of the gas, we shall take this 

 mass-motion to be zero. 



In expression (4) we found the number of collisions of class a occurring 

 per unit time to be 



v 2 f(u, v, w)f(u', v, w') F<7 2 cos ddu dvdw du'dv'dw' do) ...... (41), 



and the problem of determining the total number of collisions amounts to 

 integrating this expression over all values of the variables when f(u, v, w) 

 has the form appropriate to the steady state, i.e., when 



(42). 



In expression (41), Fis the relative velocity and 6 is the angle between this 

 velocity and that of the line of centres. If <j> is the azimuth of the line of 

 centres referred to any definite plane through the direction of the relative 

 velocity, we may, in expression (41), replace dm by sin.dddd<f>. Since 

 collisions can occur for all values of $ and for all values of 6 from to ?r/2, 

 we must integrate expression (41) from < = to ^> = 2?r and from 6 = to 

 6 = 7T/2. Performing the integrations, and substituting for f(u, v, w) from 

 equation (42), we obtain 



h s/ m 3 \ 



- - e~ hm < C2+C ' 2 ' ) Va-^du dvdw du'dv'dw' ............ (43), 



IT ) 



IT 



as the total number of collisions in which the molecules before collision have 

 velocities lying within the usual limits du dvdw du'dv'dw'. 



Let us now suppose that the variables are transformed to new variables, 

 given by 



u = i ( u + u ')> etc -> a = u' u, etc., 



so that u, v, w are the components of the velocity of the centre of gravity of 

 the two molecules, and a, /3, 7 are the components of the velocity of the 

 second molecule relatively to the first. We have 



d (u. ) , -| 



d(u,u') -I, 1 



