﻿152 
  Messrs. 
  MacGregor 
  and 
  Archibald 
  on 
  tlw 
  

  

  the 
  regions 
  of 
  two 
  or 
  more 
  electrolytes 
  which 
  have 
  common 
  

   ions. 
  

  

  We 
  restrict 
  ourselves 
  here 
  to 
  solutions 
  containing 
  two 
  

   electrolytes 
  with 
  no 
  common 
  ion, 
  which 
  therefore 
  will 
  in 
  

   general 
  contain 
  also 
  two 
  other 
  electrolytes 
  formed 
  from 
  the 
  

   former 
  by 
  double 
  decomposition. 
  We 
  may 
  refer 
  to 
  the 
  

   former 
  electrolytes 
  as 
  1 
  and 
  2, 
  and 
  to 
  the 
  latter 
  as 
  3 
  and 
  4. 
  

   1 
  and 
  2 
  will 
  have 
  no 
  common 
  ion 
  ; 
  nor 
  therefore 
  will 
  3 
  and 
  4. 
  

  

  The 
  application 
  of 
  the 
  law 
  of 
  equilibrium 
  to 
  electrolyte 
  1 
  

   throughout 
  its 
  own 
  region 
  and 
  throughout 
  the 
  regions 
  occu- 
  

   pied 
  by 
  it 
  and 
  3, 
  and 
  by 
  it 
  and 
  4, 
  gives 
  the 
  equations 
  : 
  — 
  

  

  cc 
  l 
  /Y 
  1 
  = 
  a 
  2 
  /Y 
  2 
  = 
  a 
  3 
  /Y 
  5 
  = 
  aJV^ 
  .... 
  (a) 
  

  

  where 
  the 
  a's 
  are 
  the 
  ionization 
  coefficients 
  of 
  the 
  respective 
  

   electrolytes, 
  and 
  the 
  Y's 
  their 
  regional 
  dilutions, 
  i. 
  e., 
  the 
  

   volumes 
  of 
  their 
  regions 
  divided 
  by 
  their 
  content 
  in 
  gramme- 
  

   equivalents 
  of 
  electrolyte. 
  These 
  equations 
  are 
  obtained 
  by 
  

   the 
  same 
  reasoning, 
  mutatis 
  mutandis, 
  as 
  that 
  by 
  which 
  

   Arrhenius* 
  showed 
  that 
  equality 
  in 
  the 
  concentration 
  of 
  ions 
  

   is 
  the 
  condition 
  which 
  must 
  be 
  fulfilled 
  in 
  order 
  that 
  two 
  

   dilute 
  simple 
  solutions 
  of 
  electrolytes 
  with 
  a 
  common 
  ion 
  may 
  

   undergo 
  no 
  change 
  of 
  ionization 
  on 
  being 
  mixed. 
  They 
  need 
  

   not 
  therefore 
  be 
  deduced 
  here. 
  

  

  Another 
  equation 
  may 
  be 
  obtained 
  by 
  applying 
  the 
  law 
  of 
  

   equilibrium, 
  in 
  addition, 
  to 
  each 
  electrolyte 
  throughout 
  the 
  

   whole 
  solution, 
  viz., 
  the 
  equation 
  

  

  where 
  the 
  v's 
  are 
  the 
  volumes 
  of 
  the 
  regions 
  of 
  the 
  respective 
  

   electrolytes. 
  This 
  equation 
  may 
  also 
  be 
  written 
  

  

  N/W^N.V^V, 
  (b) 
  

  

  In 
  addition 
  to 
  these 
  equations 
  obtained 
  from 
  the 
  conditions 
  

   of 
  dissociational 
  equilibrium 
  we 
  have 
  also 
  one 
  expressing 
  the 
  

   equality 
  of 
  the 
  volume 
  v 
  of 
  the 
  solution 
  to 
  the 
  sum 
  of 
  the 
  

   volumes 
  of 
  the 
  regions 
  occupied 
  by 
  the 
  electrolytes, 
  viz., 
  

  

  N 
  1 
  V 
  1 
  +N 
  a 
  V 
  3 
  +N 
  3 
  V 
  3 
  +N 
  4 
  V 
  4 
  =» 
  ( 
  C 
  ) 
  

  

  We 
  have 
  also 
  equations 
  stating 
  that 
  the 
  ionic 
  concentra- 
  

   tions 
  at 
  a 
  given 
  temperature 
  depend 
  only 
  upon 
  the 
  regional 
  

   dilutions, 
  viz., 
  

  

  -i/Vi-ZKVO, 
  « 
  2 
  /v 
  2 
  =/ 
  2 
  (V 
  2 
  ), 
  « 
  3 
  /v 
  3 
  =/ 
  3 
  (V 
  3 
  ), 
  vv 
  4 
  =/ 
  4 
  (V 
  4 
  ), 
  . 
  (d) 
  

  

  where 
  the 
  functions 
  / 
  may 
  be 
  determined 
  for 
  dilute 
  solutions 
  

   by 
  means 
  of 
  conductivity 
  observations 
  on 
  simple 
  solutions 
  

   of 
  the 
  respective 
  electrolytes, 
  provided 
  we 
  may 
  assume 
  that 
  

  

  * 
  Ztschr.f.phys. 
  Che?n. 
  ii. 
  (1888), 
  p. 
  284. 
  

  

  