﻿tlie 
  Ions 
  of 
  Gases. 
  355 
  

  

  equal 
  to 
  that 
  of 
  CO2, 
  a 
  difference 
  ^Yhicll 
  could 
  be 
  approxi- 
  

   mately 
  allowed 
  for 
  by 
  a 
  short 
  calculation, 
  but 
  it 
  is 
  hardly 
  

   worth 
  making, 
  as 
  the 
  point 
  under 
  consideration 
  is 
  well 
  enough 
  

   presented 
  by 
  comparing 
  the 
  mean 
  of 
  0"131 
  and 
  0'136 
  with 
  

   O"024'5. 
  Thus 
  we 
  find 
  the 
  resistance 
  to 
  the 
  diffusion 
  of 
  the 
  

   ion 
  5*-i 
  times 
  as 
  great 
  as 
  the 
  corresponding 
  resistance 
  to 
  that 
  

   of 
  the 
  molecule. 
  This 
  result 
  is 
  in 
  general 
  agreement 
  with 
  

   the 
  demonstration 
  given 
  that 
  the 
  ne«' 
  viscosity 
  6 
  increases 
  

   the 
  resistance 
  of 
  ordinary 
  gaseous 
  viscosity 
  to 
  %'& 
  times 
  its 
  

   amount. 
  The 
  hypothesis 
  of 
  molecular 
  clusters 
  is 
  unnecessary 
  

   to 
  explain 
  the 
  slow 
  diffusion 
  of 
  ions 
  in 
  gases. 
  

  

  We 
  must 
  also 
  discuss 
  another 
  quantity 
  closely 
  related 
  to 
  

   a/, 
  namely 
  a, 
  the 
  coefficient 
  of 
  recombination 
  of 
  ions 
  in 
  gases 
  

   according 
  to 
  the 
  very 
  simple 
  formula 
  introduced 
  by 
  J. 
  J. 
  

   Thomson 
  : 
  

  

  dE,ldt=-ci^^^ 
  (15) 
  

  

  There 
  are 
  certain 
  fundamental 
  objections 
  to 
  the 
  excessive 
  

   simplicity 
  of 
  this 
  formula 
  which 
  makes 
  the 
  recombination 
  

   take 
  place 
  according 
  to 
  the 
  chemical 
  law 
  of 
  mass 
  action, 
  

   notwithstanding 
  the 
  large 
  range 
  of 
  the 
  electric 
  forces 
  of 
  

   attraction 
  and 
  repulsion 
  amongst 
  ions. 
  Langevin 
  has 
  given 
  

   {Ann. 
  de 
  CJi. 
  et 
  de 
  Ph. 
  [7] 
  xxviii. 
  1903, 
  pp. 
  289 
  & 
  433) 
  the 
  

   now 
  generally 
  accepted 
  proof 
  that 
  this 
  simplicity 
  is 
  justified. 
  

   The 
  essence 
  of 
  his 
  argument 
  is 
  this 
  : 
  that 
  the 
  electric 
  force 
  

   at 
  the 
  surface 
  of 
  a 
  sphere 
  surrounding 
  an 
  ion 
  being 
  ^/r^, 
  

   and 
  the 
  number 
  of 
  opposite 
  ions 
  that 
  can 
  be 
  drawn 
  across 
  the 
  

   surface 
  of 
  the 
  sphere 
  ibeing 
  proportional 
  to 
  ^Trr^Ni^i, 
  the 
  

   number 
  entering 
  the 
  sphere 
  in 
  unit 
  time 
  is 
  proportional 
  to 
  

   ^ireSiUi, 
  and 
  is 
  thus 
  independent 
  of 
  the 
  radius 
  of 
  the 
  sphere 
  

   and 
  of 
  the 
  range 
  of 
  the 
  electric 
  forces 
  of 
  the 
  ions. 
  If 
  we 
  

   take 
  account 
  of 
  the 
  mobiHties 
  of 
  both 
  sorts 
  of 
  ions 
  we 
  get 
  

   Langevin's 
  formula 
  

  

  a 
  = 
  47r6(z^i 
  + 
  U2), 
  (16) 
  

  

  •where 
  e 
  is 
  that 
  fraction 
  of 
  the 
  ions 
  drawn 
  into 
  a 
  sphere 
  which 
  

   combine 
  to 
  form 
  neutral 
  molecules. 
  Langevin 
  has 
  inves- 
  

   tigated 
  values 
  of 
  e 
  experimentally, 
  and 
  Richardson 
  (Phil. 
  

   Mag. 
  [6] 
  X. 
  1905, 
  p. 
  242) 
  has 
  applied 
  considerations 
  of 
  

   probability 
  to 
  the 
  calculation 
  of 
  e. 
  But 
  it 
  seems 
  to 
  me 
  that, 
  

   though 
  Langevin's 
  theory 
  gives 
  as 
  a 
  first 
  approximation 
  the 
  

   justification 
  of 
  J. 
  J. 
  Thomson's 
  equation 
  (15), 
  a 
  second 
  

   approximation 
  is 
  necessary 
  to 
  bring 
  the 
  theory 
  of 
  ions 
  in 
  gases 
  

   into 
  agreement 
  with 
  experimental 
  facts 
  and 
  the 
  kinetic 
  theory 
  

   of 
  gases. 
  Experimental 
  proof 
  of 
  the 
  insufficiency 
  of 
  (15) 
  

   has 
  been 
  given 
  by 
  Barus 
  [Ann. 
  der 
  Fliys. 
  xxiv. 
  1907, 
  p. 
  225), 
  

   using 
  J. 
  J. 
  Thomson's 
  condensation 
  method 
  of 
  counting 
  N^. 
  

  

  