CONTRIBUTION TO THE GENERAL KINETICS OF 

 MATERIAL TRANSFORMATIONS. 



By Alfred J. Lotka. 



Received November 28, 1919. Presented February 11, 1920. 



F. JiJTTNER ^ has given a general discussion of the differential equa- 

 tions of chemical dynamics, in so far as they apply to a single chemical 

 reaction taking place in accordance with the law of mass action. 



A certain interest attaches to a still more general treatment wliich 

 should make us independent of the law of mass action, and which 

 should cover the wider field of concurrent reactions of various types. 

 Such a treatment is therefore given below. 



The systems with which we have to deal comprise a number of 

 components Si S2. . .S„, whose mass at any instant may be denoted 

 b}' A'l X2. . Xn. Observational data furnish directly or indirectly a 

 system of differential equations^ 



dX, 



dt 

 dX. 



di 



dXn 



dt 



— Fi(Xi, Xi, . . .Xn] Pi, P2, ■ ■ ■ Pj) 

 = F^iXi, X2,. . .Xn', Pi, Pi,- ■ -Pi) 



= Fn(Xi, X2,. . -Xn', Pi, P2, ■ -Pi) 



(1) 



1 Zeitschr. f. phys. Chemie, 1911, v. 77, p. 735. See also R. Marcelin, 

 Ann. de Phys. 1895, v. 3, p. 120. 



2 The system of equations (1) although very general, does not, however, 

 cover cases in which geometric factors (e. g. diffusion effects) play a dominant 

 role. Such cases are therefore excluded from our present considerations. 

 Certain reactions in heterogeneous systems, nevertheless, do fall within the 

 scope of this discussion, namely those the velocity of which is small as compared 

 with the rate of diffusion, so that the course of the reaction is determined prac- 

 tically by the reaction velocities alone, irrespective of diffusion velocity. 



A class of reactions which, perhaps, may not always fall witliin the scope of 

 equations of the type (1) and (2) are so-called induced, sympathetic or coupled 



reactions. In such cases it may be that the velocities -^ can not be repre- 

 sented as functions of the Z's alone, but that the velocities -j- themselves 



must appear explicitly in these functions. 



