GENERAL KINETICS OF MATERIAL TRANSFORMATIONS. 147 



dY 



or, shifting the origin after the manner of (5), (6), 



f=/(.) (27) 



and hence, by the same procedure as in the case of (7) 



^=n!/-{-bi/+... (28) 



The sohition (8) in this case takes the form 



y = czi c^' + an ^''^^ + «m c'^' + ... (29) 



where 



X = a (30) 



dy 



Now if the equihbrium at ?/ = is stable, it is evident that we must 

 have 



(32) 



Hence we conchide that 



1. The series (29) converges for large values of t. 



2. The variable y ultimately approaches zero asymptotically, 

 since the coefficient a, from the nature of things, can not be complex. 



A proof of this asymptotic approach to equilibrium has been given 

 by Jiittner, for the case of a single chemical reaction following the law 

 of mass action. The proof given above is independent of the law of 

 mass action and applies to all cases satisfying only the broad condition 

 that the equilibrium corresponding to y = is stable. 



Special Cases (continued). 



2. If n — V} > 1 the transformation can not be regarded as one 

 single reaction. The state of the system at any instant, for given 

 values of the parameters P. and A requires for its definition {n — m) 



