GENERAL KINETICS OF MATERIAL TRANSFORMATIONS. 141 



The coefficients a'a, cn-'U, ■ ■ • of the terms of second and higher degree 

 are determined in analogous manner. 



Equations (3), which have so far been introduced merely as a 

 mathematical aiLxiliary, have an obvious physical significance. They 

 represent the conditions for a steady state or an equilibrium. 



Now, if Xi, Xo . . . Xn are entirely independent, it will be seen that 

 (3) completely determines the equilibrium. Physically this means 

 that in such a case, when the values of the parameters P (e. g. volume, 

 temperature) were once fixed, there would then be no further freedom 

 whatever in fixing the equilibrium. This latter would then, for example 

 be wholly independent of the masses of the several components of the 

 system. 



This does not correspond to the actual conditions ordinarily met with 

 in cases of concrete interest. In such cases there are commonly given, 

 in addition to the differential equations (1), which express the general 

 kinetics of the system, a further set of equations of constraint ^ of the 

 form 



^{X„ Xo,...Xn) = (15) 



In physico-chemical systems the equations of constraint are com- 

 monly derived directly from the reaction equations, of which, according 



7 Ordinarily the equations of constraint derived from the reaction equation 

 contain the stoichiometric constants of those equations. However, in the case 

 of certain simultaneous reactions, in which the products of the several reactions 

 are formed in constant proportion, the function $ will contain also certain 

 other constants defining this proportion and not themselves defined by the 

 stoichiometric properties of the system. 



In the case most frequently dealt with in physical chemistry, the case of a 

 system of constant total mass, one of the equations of constraint takes the form 



w A' = const. 



Furthermore, the immutability of the chemical elements (in ordinary reac- 

 tions) furnishes for each element e an equation of constraint. 



S nif = constant 

 where nie denotes the total mass of the element e present iii the system. 



In special cases there may be constraints of entirely different character. 

 Thus, in certain technical processes a stream of gas is passed over a catalytic 

 substance in such manner that an approximately constant mass of the products 

 of reaction issue per unit of time. 



In such case, in addition to equations of type (1) there will be given one or 

 more equations of the form 



-4r = constant = F {d, C2, . . . C„) 

 at 



where C\, C2, . . . C„ are the (constant) concentrations of the substances /Si 



82,- ■ .Sn entering the reaction chamber. 



Cases of this land do not fall within the scope of our present reflection. 



Their theoretical and practical treatment presents, however, no particular 



difficulties. 



