Rate of Recombination of Ions in Gases. 147 



For a collision this must be less than s the radius of the sphere 

 of action of the two molecules (supposed forceless). Hence the 

 criterion for a collision is 



P}#\l + 2™l±l£f F(p)dp 



[ m 1 m 2 V 2 J s r r 



Hence the effect of the intermolecular forces is to apparently 

 increase the area of cross section of the sphere of action iu the 



772- ~\~ 7/i-i 



ratio 1 + 2 — - — ^ W where W is the work done by the attraction 

 m x m 2 V 2 



of the particles when they are brought together from an infinite 



distance. 



If we put ??ij = Too = m we can compare this with the result 

 obtained by Mr Sutherland for two particles of equal mass. The 

 correction term in the above formula is exactly twice as great 

 as that found by Mr Sutherland; the discrepancy is due to the 

 different assumptions made, as was pointed out at the beginning 

 of the calculation. 



The formula first found shows that a collision takes place 

 when the straight paths of two molecules whose relative velocity 

 is r approach within a distance b which is less than 



( m 1 m 2 r l J s J 



From this we can find the total number of collisions which take 

 place in the gas by the use of a formula given by Maxwell*. 

 Maxwell shows that, if we have a space containing two gases, of 

 which there are N and N' molecules per c.c. and whose mean 

 velocity squares are a 2 and /3 2 , respectively ; then the number 

 which have a relative velocity between r and r + dr and whose 

 straight paths approach within a distance b, is given by 



-W ,'t^^L* &r*e~£p dr. 

 (a- + /3-)« 



Hence taking molecular attraction into account, the total number 

 of collisions per second is given by 



p4 JkNN' , , S \^ |l + 2 ^ + ™ 2 \ * F(p) dp] r*e-£p dr 

 Jo ^ (a 2 + /3 2 )9( T m^m^Js w p ) 



= 2 JnNN's* (a 2 + /3 2 )* il + T ?-= ^^^ f F( P ) d P \ . 

 { a 2 + {3 2 mjn 2 J s r ) 



If we replace the mean velocity squares by the mean velocities, 



* Scientific Papers, i. 385. 



