408 CHEMICAL DYNAMICS 



constant proportion /3 to x^. Introducing this factor into equa- 

 tion (vi) we obtain : 



dx F 



-JT = r~i 7 \ 3~i ' k{a - x). (vii) 



dt 1 -^ a{a — X) — fix^ 



Integrating, we obtain: 



(1 - ^a"") log —^— + (a + /3a) X + ^a;2 = kFt. (viii) 



In the derivation of Henri's equation it is assumed either that 

 the ferment is unaltered during the processes which accomplish 

 the introduction of the elements of water into the protein mole- 

 cule or else that the equilibrium in equation (D) lies so far to the 

 right that for all practical purposes the ferment is present wholly 

 in the hydrated, proteolytically active form. Equation (viii) 

 is Henri's equation generalized to the extent of including any 

 species of equilibrium in equation (D). 

 Equation (viii) may be written: 



^""^a-x^ \- &a? ^ 2{\- ^a') (1 - M ^ ^ 



from which it is evident that it contains three mutually inde- 

 pendent constants. This fact deprives it of any great utility for 

 the purpose of comparing numerical data with the theoretical 

 deductions, since an equation containing three mutually inde- 

 pendent constants may be made to fit with tolerable exactitude 

 any continuous curve by an appropriate choice of constants. 

 In the absence of very numerous and very exact data, a numerical 

 comparison between theory and experiment involving this 

 equation, therefore, would be devoid of utility. It is of interest 

 to observe, however, that all the relationships between x, F and 

 t found by the various observers, whose investigations we have 

 cited above and in the previous chapter, are especial cases of the 

 general relation which is expressed in equation (ix). Thus if 

 j8 be small, that is, if the shift in the equilibrium between the 

 hydrated and unhydrated forms of the enzyme due to unit mass 

 of the products of the hydrolysis is very small, or if the equi- 

 librium in equation (D) lies far to the right, then we regain 

 Henri's equation. If not only jS but a is small, i.e., if the part 

 played by the protein and its products in determining the equi- 

 librium between the different forms of ferment is small, then we 



