34 THE THEORY OF IONIZATION OF GASES 



15. Determination of the pressure corresponding to the 

 maximum value of a for a given force. 



Since the values of a/p determined experimentally 

 agree very closely with those given by the formula 

 a NVJ? 

 -=Ne x for the larger values of X/p, the pressure 



for which a is a maximum when X is constant 

 may be obtained by equating to zero the differential 



_NVp 



coefficient of pt~ x with respect to p. The equation 

 in X/p thus obtained is 1^=0. 



Thus for any gas the pressure P, which gives the 



Y 

 greatest conductivity with a force X, is P^^, and 



XT' 



the maximum value of a is NPe~ = ~-. 



The factor by which the original number of ions is 

 multiplied owing to the effects of collisions depends on 



al or ==-==-, where v is the potential difference 



between the plates. Thus in the case in which n Q ions 

 are set free from the negative electrode the number that 

 arrive at the positive electrode has a maximum value 



V 



n e Ve , independent of the distance I between the plates 

 when a given potential difference v is established between 

 them. In order that this equation may hold accurately 

 it is necessary that the negative ions should have the 

 mean velocity corresponding to the force X along the 

 whole distance L This condition will not be established 

 until the ions have travelled a short distance from the 



