216 Messrs Jones and Richardson, 



Let us suppose that a substance, whose initial concentration* 

 is G , is subject to any number m of irreversible simultaneous linear 

 decompositions, i.e. decompositions in which the rate of formation 

 of any product is proportional to the concentration of the original 

 substance at the time considered. Let G n be the concentration 

 of the nth product at the time t, then the concentration of 

 the original substance at the same time is C — %G n , X denoting 

 summation over all the m reaction products. The rate of in- 

 crease of G n is thus 



C ^ = k n (G -XG n ), 



where k n is the velocity constant of the reaction which gives rise 

 to the nth product. Since each of the other reactions satisfies 

 a similar equation, we have 



1 a\J n 1 tt^re-i _ _ p -<£p 



fCyi air Hj<ii — ] ajj 



G n Gn-x XG n 



whence V i= V = "- = lF' 



the integration constant vanishing, for 



G n = a n _! = . . . = Ci = when t = 0, 



since none of these products have been formed at the beginning of 

 the reaction. 



From the above we obtain 



G.„ 



dO v 

 ~db 



therefore- log (k n G — G n %lc n ) = — %k n t + A. 



As before, when t = 0, G n — 0, so that the integration constant 

 A = log k n G , and 



log(l-^|) = -2^ (l), 



or Gn = ^(l-e-^) (2), 



with m similar equations for all the m products. This expression 

 gives the concentration G n of the nth. product, at a time t, in terms 

 of the initial concentration of the original substance and of the 



* The concentrations are expressed in gram-molecules per litre. 



^*G n — i Zfon 5 



whence -rr = k n C — G n %k n ; 



