202 THE HYDROLYZING ENZYMES 



The rationale of this law, which is known as Wilhelmy's Law, may 

 be made clear in the following way: In every instant innumerable 

 collisions are taking place between sugar-molecules and water-molecules. 

 Only a small proportion of these collisions are effective in accomplishing 

 the breaking up of a disaccharide molecule. The proportion of effective 

 collisions is, however, the same no matter how many sugar-molecules 

 may chance to be present. We may picture to ourselves, without 

 seeking to employ the analogy too literally, the effective collisions as 

 "head-on" collisions, the ineffective collisions being "glancing." 

 Each sugar-molecule is independent of all the rest, and its chance of 

 achieving an effective collision with a water-molecule is the same as 

 that of all the rest. Suppose every thousandth collision is effective, 

 i. e., one tenth of a per cent, of the total collisions per second. If the 

 solution of sugar be 2-molecular, the total number of collisions per 

 cubic centimeter per second will be twice as great as when the solution 

 is 1 -molecular, because there are twice as many molecules of sugar in a 

 given space. In each solution the percentage of effective collisions is 

 the same. Out of one thousand collisions in a 2-molecular solution one 

 will be effective, and out of a thousand collisions in a 1-molecular solu- 

 tion one will also be effective. But as there are twice as many collisions 

 per second in the former as there are in the latter solution, there will 

 also be twice as many effective collisions per second, and the amount of 

 sugar transformed in a second, i. e., the Velocity of Hydrolysis in the 

 2-molecular solution must be twice as great as it is in the 1-molecular 

 solution. 



This very simple relationship may also be expressed in an algebraical 

 formula : 



Velocity of hydrolysis = k(a x) 



where (a x) is the mass of unaltered sugar at any instant, "a" being 

 the initial amount and "x" the quantity which has already undergone 

 hydrolysis at the moment of observation. The constant "k" expresses 

 the constant ratio which, as we have seen, subsists between the mass 

 of unhydrolyzed sugar which is present and the velocity with which 

 hydrolyzed sugar is making its appearance. It is, in fact, the velocity 

 of hydrolysis when (a x)=l, that is, when the mass of unconverted 

 sugar is unity, one gram-molecule, or one gram, or whatever mass 

 we may arbitrarily choose as a unit, provided we measure all the 

 quantities in the equation in the terms of the same unit. Also, and 

 this is very important to notice, the constant "k" is a direct measure 

 of the percentage of effective collisions between sugar-molecules a ad 

 water-molecules, for if the percentage of effective collisions be doubled 

 by any means then the velocity of hydrolysis must obviously be 

 doubled also. In the equation : 



Velocity = k(a x) 



