﻿Ionic 
  Velocities 
  , 
  and 
  Atomic 
  Sizes. 
  173 
  

  

  These 
  tables 
  and 
  their 
  graphs 
  bring 
  out 
  more 
  clearly 
  the 
  

   important 
  fact 
  that 
  xi 
  or 
  \r)/\ 
  rj 
  passes 
  through 
  a 
  minimum. 
  

   The 
  curves 
  are 
  parabolic, 
  those 
  for 
  BaCl 
  2 
  , 
  (NH 
  4 
  ) 
  2 
  S0 
  4 
  , 
  and 
  

   Z11SO4 
  having 
  axes 
  inclined 
  to 
  the 
  axis 
  of 
  xi. 
  The 
  sinuosities 
  

   of 
  the 
  CaCl 
  2 
  curve 
  are 
  smoothed 
  out 
  in 
  the 
  dotted 
  curve. 
  

   The 
  equations 
  for 
  the 
  BaCl 
  2 
  and 
  Z11SO4 
  graphs 
  are 
  respectively 
  

  

  (•969m 
  1 
  / 
  3 
  -i--350) 
  2 
  = 
  7-18/ 
  + 
  -824?>2 
  1 
  /3-5-35 
  . 
  (12) 
  

  

  (•969m 
  1 
  /3_/_-j 
  : 
  42) 
  2 
  = 
  3-51i 
  + 
  -403m 
  1 
  ' 
  3 
  -l-433 
  . 
  (13) 
  

  

  with 
  which 
  the 
  values 
  marked 
  xi 
  calc. 
  in 
  Table 
  II. 
  have 
  

   been 
  found. 
  For 
  reasons 
  which 
  are 
  obvious 
  after 
  what 
  has 
  

   been 
  said 
  in 
  discussing 
  the 
  NaCl 
  graph, 
  little 
  importance 
  

   attaches 
  to 
  these 
  equations. 
  The 
  main 
  point 
  about 
  the 
  graphs 
  

   is 
  that 
  they 
  make 
  it 
  plain 
  that 
  xi 
  attains 
  a 
  minimum 
  value 
  at 
  

   a 
  certain 
  strength 
  of 
  the 
  solution. 
  In 
  interpreting 
  the 
  form 
  

   of 
  these 
  equations 
  we 
  must 
  first 
  consider 
  how 
  m 
  l 
  3 
  and 
  m 
  2/3 
  

   come 
  in. 
  On 
  general 
  principles 
  it 
  is 
  clear 
  that 
  they 
  do 
  not 
  

   enter 
  because 
  of 
  a 
  direct 
  dependence 
  of 
  ionization 
  on 
  the 
  

   distance 
  apart 
  of 
  the 
  molecules 
  of 
  solute, 
  which 
  is 
  proportional 
  

   to 
  m 
  -1 
  ' 
  3 
  . 
  It 
  is 
  possible 
  that 
  the 
  m 
  1 
  ^ 
  3 
  enters 
  on 
  account 
  of 
  the 
  

   following 
  train 
  of 
  circumstances. 
  The 
  solute 
  molecules 
  in 
  a 
  

   watery 
  solution, 
  such 
  as 
  that 
  of 
  NaCl, 
  probably 
  change 
  a 
  

   certain 
  amount 
  of 
  (H 
  2 
  0) 
  3 
  into 
  (H 
  2 
  0) 
  2 
  , 
  as 
  is 
  shown 
  by 
  the 
  

   occurrence 
  of 
  shrinking 
  on 
  solution, 
  and 
  they 
  also 
  probably 
  

   dissociate 
  into.#0[> 
  2 
  (#H) 
  2 
  t> 
  a 
  number 
  of 
  trihydrol 
  molecules. 
  

   Let 
  y 
  be 
  the 
  average 
  part 
  of 
  each 
  second 
  for 
  which 
  £Ot) 
  2 
  (#H) 
  2 
  |? 
  

   is 
  separate, 
  1 
  — 
  ?/ 
  the 
  part 
  for 
  which 
  it 
  is 
  combined 
  with 
  others, 
  

   then 
  the 
  actions 
  producing 
  fresh 
  stions 
  do 
  so 
  at 
  a 
  rate 
  

   c(i—y)m, 
  while 
  those 
  forming 
  trihydrol 
  out 
  of 
  the 
  stions 
  do 
  so 
  

   at 
  a 
  rate 
  c 
  f 
  y^m 
  d 
  , 
  and 
  for 
  equilibrium 
  c(l— 
  ■y)m 
  = 
  c'i/ 
  ? 
  'm' 
  d 
  . 
  and 
  

   when 
  y 
  is 
  small, 
  2/ 
  3 
  x 
  m~ 
  2 
  and 
  y 
  varies 
  as 
  m~ 
  23 
  . 
  

  

  Now 
  the 
  rate 
  at 
  which 
  the 
  stions 
  #Ot> 
  2 
  (#H) 
  2 
  |? 
  form 
  the 
  

   labile 
  compound 
  Clt>#0|? 
  2 
  (#H) 
  2 
  t?#Na 
  will 
  be 
  proportional 
  both 
  

   to 
  y 
  and 
  to 
  m, 
  that 
  is 
  to 
  say, 
  it 
  varies 
  as 
  m 
  1/3 
  , 
  and 
  therefore 
  

   the 
  index 
  1/3 
  enters 
  because 
  of 
  the 
  3 
  in 
  the 
  formula 
  for 
  

   trihydrol. 
  Our 
  equations 
  connecting 
  xi 
  and 
  m 
  ls 
  are 
  there- 
  

   fore 
  equations 
  of 
  chemical 
  equilibrium, 
  expressing 
  that 
  a 
  rate 
  

   of 
  combination 
  denoted 
  by 
  a 
  square 
  is 
  proportional 
  to 
  a 
  rate 
  

   of 
  dissociation 
  denoted 
  by 
  a 
  linear, 
  whence 
  the 
  parabolic 
  

   graphs.* 
  It 
  would 
  not 
  be 
  profitable 
  to 
  follow 
  this 
  train 
  of 
  

   thought 
  further, 
  until 
  the 
  dielectric 
  capacities 
  of 
  solutions 
  

   have 
  been 
  quantitatively 
  studied. 
  The 
  complete 
  theory 
  of 
  

   the 
  ionization 
  of 
  binary 
  electrolytes 
  in 
  aqueous 
  solutions 
  is 
  

   more 
  complicated 
  even 
  than 
  it 
  has 
  been 
  hitherto 
  supposed 
  

   to 
  be. 
  

  

  