210 A. J. Lotka — Mode of Growth of Material Aggregates. 



we may express it, the "path" of the change from A to A' 

 might, for instance, assume any such form as the above (fig. 3), 

 for in each case E > E'. 



Evidently, however, the change A A' will not take place 

 spontaneously, except in the case represented by fig. 3a, which 

 is the only one satisfying also condition (2). 



Now, we have no means of ascertaining the character of the 

 "path" of the change A A' by direct observation. The 

 following considerations, however, will show how, from the 

 properties of the aggregate as a whole, we can draw inferences 

 as regards the character of this path for the individual 

 molecules. 



A. Time t' — t required for the change. 



Since k~N molecules pass from A to A' per unit of time, and 

 each takes on an average {t' — t) to complete the change, there 

 will, at a given instant, be kN(t' — t) molecules actually under- 

 going change. The dilution of these in the aggregate wilLbe 



at least -— -, — r. 

 k(t'~t) 



Now it is a matter of universal observation that in all ordi- 

 nary chemical reactions* at any rate, this dilution is so great 

 that the material in the state of transition cannot be detected, 

 no matter what the value of Jc, or the methods^ of analysis, or 

 the general circumstances. In other words, such a system as 

 the one we have been considering, can always be fully 

 described with regard to its composition (within the limits of 

 experimental error) as consisting of a certain quantity of A 

 and a certain quantity of A'. 



We conclude that in all ordinary cases the time t' — t, during 

 which the molecule is in the transitional state, is very small.;}; 



JB. The form of the path A A'. 



In general, the type of the path A A' will, for each mole- 

 cule, depend on its condition at any particular instant, and it 

 may be different for different molecules of the same aggregate. 



(a) Let us first of all suppose that for all molecules for 

 which the condition (1) is satisfied (E > E'), the path is of 

 type (a) ; in this case then the fulfilment of condition (1) will 

 be sufficient to determine the change A A 7 , and it follows that 

 all molecules for which E > E ; will actually be undergoing 



* Evidence of the transitional state in tautomeric change, however, 

 appears to be furnished by the work of Baly and his collaborators. 



■f Ordinary methods of analysis, of course, would not reveal the presence 

 of such transitional bodies. But physical methods should, if they were 

 sufficiently sensitive. In this connection again we must turn to the work of 

 Baly for suggestions. 



X But see remark regarding tautomerism on the next page. 



