172 Prof. Thomson, The rate of recombination 



two molecules. In order to get numerical results we shall assume 

 that both the charged and the uncharged molecules are conduct- 

 ing spheres of radius a. If c, the distance between the centres 

 of these spheres, is a considerable multiple of a, then the work 

 required to separate the spheres to an infinite distance is approxi- 

 mately -= — — , and if the molecules are to remain paired this 



must be greater than their kinetic energy. We have seen that 

 mV 2 , the kinetic energy of two molecules at 0°C, is equal to e 2 /r 

 when r = 13 x 10~ 6 , hence for union to take place c must be less 



than (| a s r)*. If we put a = 10 -8 cm., c must be less than 2 - 7 x 10 -8 . 

 Thus, if ZV is the number of molecules, n the number of ions per 

 unit volume, the number of complex ions formed per unit time 

 per unit volume is wNn(%*l x 10- 8 ) 2 F. The factor tt(2-7 x lO -8 ) 2 

 is about l/2 - 5 x 10 3 of the factor in the expression for the rate of 

 recombination of oppositely charged ions. The smallness of this 

 factor is, however, far more than compensated by the enormous 

 excess of molecules over ions; the ionization would have to be 

 very intense indeed for the number of ions to amount to 1/1 12 

 of the number of molecules ; thus N will be at least 10 12 times n, 

 and therefore the number of combinations between a charged and 

 an uncharged molecule will in a given time be at least 5 x 10 8 

 times the number of combinations between oppositely charged 

 ions ; hence at the very beginning of its career the charged mole- 

 cule will recombine and form a complex ion ; thus a charge will 

 be carried by the complex ion during practically the whole time 

 of its existence as a free charge. 



Limit to the size of a gaseous ion. After the charged molecule 

 has increased in size by attracting another molecule, the work 

 required to drag an additional uncharged molecule away from it, 

 starting from a given distance, will be less than the work required 

 to drag a molecule from a single charged molecule. The more 

 complex the ion becomes the less is the work required to drag 

 an uncharged molecule away from it ; at a certain stage of 

 complexity the work required to drag a molecule from the ion 

 is less than the kinetic energy possessed by the system in virtue 

 of its temperature, when this stage is reached the ion will cease 

 to attract fresh molecules. 



The work w required to separate an uncharged sphere of radius 

 a from a charged sphere of radius b, the spheres being all but in 

 contact, is (see Maxwell's Electricity and Magnetism, vol. I. p. 275) 

 given by the equation 



1 e 2 1 e- 



w = 



2 a+t _| 27+t (_l^) + f (_^)j 



