set up in Molecules hy Collisions. 283 



To take a second example, ^vhich is of interest as beino- a 

 case ot' failure of the oeneral proof of § 6, suppose that 



Y = e-<^'' 



/Y.Ar 



«/> V K, 



In this case the '* elasticity " is represented in the enercry 

 of vibration by a factor e~P--'^ . The appropriate unit of time 

 is K~i, and if p is, in these units, represented by 200, 

 jtr/2«:= 20.000, and the energy gained by the vibrations at 

 collision i? reduced by the " elasticity ^' to e~-^^^^^ of what it 

 would otherwise have been. 



§ 8. These two instances have been selected at random, 

 but it is obvious enough, from the general theorems of 

 §§5 and 6, that any other form for V would give a very 

 similar result. The smaller of the two factors introduced by 

 the " elasticity " of the molecule has been ^-^^^. This means 

 that if the molecules were all moving with average velocity 

 the number of collisions required to dissipate a given fraction 

 of the energy would be increased by the "elasticity" in a 

 ratio of about e"^^^ : 1. In other words, the " elasticity '' 

 could easily make the difference between dissipation of 

 energy in a fraction of a second and dissipation in billions of 

 years. 



§ 9. It has, however, been seen (§ 5) that appreciable 

 vibrations will be set up by a collision in which the relative 

 velocity is comparable with 2 X 10' cms. per second. AYe 

 must therefore examine the frequency of sach collisions in 

 gases at normal temperatures. 



The number of collisions in which the molecules have 

 velocities between c and c + clc, c' and c' + dc' inclined at an 

 angle between <j) and ^ + <^(^ is proportional to * 



e-^><c2+c^^ c"c''g sin (^ d<j) dc dc', ... (15) 



where g is the relative velocity, given by 



g'' = c- + c''' — 2cd co^<p (16) 



Let us suppose that of the velocities c, c^ the latter is the 

 greater, and write 



u = c^—c, v = c' + c. 

 From equation (16) it follows that g must lie between ic 

 and V. By diflPerentiation, keeping ?«, v and therefore also 

 c, c' constant, we have gdg = cc' >\m ^ d<^. Expression (15) is 

 now proportional to 



^_p;„(„2+,2) [^' — li^yf dg du dv. 

 * Boltzmann, Gastheorie, i. p. 64, equation 61. 



