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  XXXI. 
  General 
  Curves 
  for 
  the 
  Velocity 
  of 
  Complete 
  Homo- 
  

   geneous 
  Reactions 
  between 
  Two 
  Substances 
  at 
  Constant 
  

   Volume. 
  By 
  George 
  W. 
  Todd, 
  D.Sc.(Birm.\ 
  B.A. 
  

   (Camb.)*. 
  

  

  [Plate 
  IX.] 
  

  

  WHEN 
  m 
  molecules 
  of 
  a 
  substance 
  A 
  react 
  with 
  n 
  

   molecules 
  of 
  a 
  substance 
  B 
  to 
  give 
  one 
  or 
  more 
  

   resultants, 
  there 
  being 
  no 
  back 
  reaction, 
  the 
  velocity 
  of 
  the 
  

   reaction 
  is 
  given 
  by 
  

  

  where 
  x 
  is 
  the 
  change 
  in 
  the 
  concentration 
  C 
  in 
  time 
  t 
  and 
  

   k 
  is 
  the 
  velocity 
  constant. 
  If 
  k 
  is 
  known, 
  the 
  changes 
  in 
  

   concentration 
  for 
  various 
  initial 
  concentrations 
  of 
  the 
  reacting 
  

   substances 
  can 
  be 
  worked 
  out 
  by 
  integrating 
  the 
  above 
  

   equation, 
  but 
  the 
  integration 
  often 
  absorbs 
  valuable 
  time. 
  

   By 
  choosing 
  suitable 
  quantities 
  it 
  is 
  possible 
  to 
  plot 
  curves 
  

   which 
  will 
  apply 
  geuerally 
  to 
  all 
  reactions 
  of 
  a 
  similar 
  type. 
  

   The 
  author 
  has 
  worked 
  out 
  some 
  of 
  these, 
  and 
  puts 
  them 
  on 
  

   record 
  hoping 
  that 
  they 
  may 
  save 
  much 
  time 
  and 
  labour. 
  

  

  Bi-molecular 
  Reaction. 
  

  

  If 
  the 
  reaction 
  is 
  bi-molecular 
  of 
  the 
  type 
  A 
  + 
  B->1 
  or 
  

   more 
  resultants, 
  the 
  reaction 
  velocity 
  is 
  given 
  by 
  

  

  -£=k(a-x)(b-x), 
  

  

  where 
  a, 
  b 
  are 
  the 
  initial 
  concentrations 
  of 
  A, 
  B 
  respectively. 
  

   The 
  equation 
  may 
  be 
  written 
  

  

  at 
  \ 
  a 
  J 
  \a 
  a 
  J 
  

  

  Putting 
  -=X, 
  where 
  X 
  = 
  fraction 
  changed, 
  

  

  g=K(l-X)(p-X), 
  

  

  where 
  K 
  = 
  &a 
  and 
  /?=-. 
  

  

  Take 
  (i.) 
  initial 
  concentrations 
  equal, 
  i.e. 
  a 
  = 
  b 
  or 
  p=l 
  y 
  

   then 
  

  

  X 
  

  

  X 
  

  

  f* 
  dX 
  _ 
  X 
  

  

  * 
  Communicated 
  by 
  the 
  Author. 
  

  

  