224 
Transactions of the Royal Society of South Africa. 
/8, . . . the numbers of molecules of each which take part in the reaction ; x 
is the molecular amount transformed in time t, and k is the velocity constant 
of the reaction. 
When investigating the influence of relative initial concentration on the 
course of a reaction, or when comparing the velocities of different reactions, 
it is advantageous to work in terms of a " modular time "—that is, the time 
which elapses before a specified fraction, \, of the total transformation has 
taken place.* 
Of the initial concentrations concerned, there must be at least one which 
is not greater than any of the others. Let one such be a. The reaction will 
be complete when x = a. Refer all concentrations to a as unit, and write 
X = \a so that X is the fraction transformed at any stage ; and, for the 
ratios b/a, c/a, . . . write no, . . . Equation (5) then becomes in modular 
form 
= /.a--^+^+v+ - (1-X)« {n,-X)^ {no-X)y ... . (6) 
The integration of equation (6) can always be effected by elementary 
methods,t but if the indices a, /5, . . . are greater than 3 the process becomes 
laborious. Indeed, the chemist apparently prefers to choose, when possible, 
%p ^2' • • . = 1, so as to obtain a more tractable equation. That procedure, 
however, has the disadvantage of obscuring the effects of the influence of 
the relative initial concentrations of the reactants upon the course of the 
reaction. It may, perhaps, be of utility to show how the integrals of some 
of the commoner types of reaction may be obtained immediately as special 
cases of a general formula. 
If there are^ reactants in all, and if only one molecule of each reactant 
is concerned, equation (6) becomes 
71 ^7 7 ^ = Jcav^d^ : . (7) 
where a is the lowest initial concentration involved. The result of inte- 
grating (7) in the usual way may be written 
2 
[0(r . 
_ = kaP-^^ . . (8) 
II (nj-'^^d 
Now 
i=p-i 1 
^ II O^j—ni) 
(9) 
*Cf. Todd, Phil. Mag. xxxv, p. 281 (1918). 
t See, e. g., Todhunter, Integral Calculus, Cli. II. 
