QUANTITATIVE RELATIONSHIPS IN HYDROLYSIS 209 



where "k" is a constant and "F" is the concentration of the invertase. 

 As the proportion of sugar to enzyme falls off, however, a point must 

 be reached at which the remnant of sugar is insufficient to bind all of 

 the invertase. At this stage of the reaction, therefore, or if the pro- 

 portion of sugar to enzyme is small to begin with, the reaction-velocity 

 begins to fall off as the reaction proceeds, in accordance with the 

 formula : 



Velocity = kF(a x) 



the numerical value of "k" being different in the two cases because, 

 in the latter case, the "k" includes the equilibrium constant for the 

 reaction : 



Sugar + catalyst *~j sugar-catalyst compound. 



Turning now to the question of the relationship of the Quantity of 

 Substrate hydrolyzed to the Time of Hydrolysis it is at once manifest 

 that if the relationship 



Velocity = kF 



holds good, then, the quantity of sugar decomposed in each unit of 

 time being the same, the quantity "x" decomposed after time "t" 

 must be given by the equation : 



x = kFt 



In the more general case in which the relationship obtains: 



Velocity = kF(a - x) 



the relationship between the mass of substrate transformed and the 

 time occupied in the transformation may be found by a simple opera- 

 tion of the integral calculus to be : 



loge -^- = kFt 



The following results are illustrative of these two types of relation- 

 ship: 

 Invertase on cane-sugar (A. J. Brown) Concentrated Substrate: 



Grams of cane 



sugar per Grams of cane-sugar 



100 c.c. inverted in 60 minutes. 



40.02 . . . . : . . . .V. . 1.076 0.179 



29.96 ........... 1.235 0.206 



19.91 ........... 1.355 0.226 



9.85 ........... 1.355 0.226 



4.89 ........... 1.230 0.206 



Invertase on cane-sugar (A. J. Brown) Dilute Substrate: 



j 



Grams of cane- 10 s a 

 sugar per Grams of cane-sugar 1(PK = -- logio 



100 c c. inverted in 60 minutes. t 



(2.00) ........... (0.308) (132) 



1.00 ........... 0.249 219 



0.50 ........... 0.129 239 



0.25 ........... 0.060 228 



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