THE ACTION OF HEAT, ALKALIES AND ACIDS 261 



were determined. The principal experimental observations connected 

 with the inversion of cane sugar are given below : — 



1. Rate of Inversion. — When all other conditions are unchanged, the 

 rate of inversion is proportional to the active mass, i.e., when the temperature 

 and the concentration of the acid are unchanged, a 20 per cent, solution of 

 cane sugar inverts twice as fast as a 10 per cent, solution. Developed 

 mathematically, this statement becomes reduced to the follo\^dng form : — 



In a sugar solution let there be a parts of sugar present ; in a small interval 

 of time, t, let x parts be inverted. There are then present a — x parts of cane 

 sugar. Since the rate of change is proportional to the active mass, 



d X 



^— = k [a — x) where A is a constant. 



it t 



WTience, by integration, log = k t 



or — log = k 



t ^ a— X 



The constant k gives a means of comparing the strength of difTerent acids, 

 or, under the ionic h>-pothesis, the degree of dissociation. This law was 

 found experimentally by Wilhelmy^^ in 1850, and developed on a -priori 

 reasoning by Guldberg and Waage ^^ in 1867. It forms a typical instance 

 of the universal law that rate of chemical change is proportional to the 

 active mass. 



As definitely appUed to a sugar solution in acid medium, let the total 

 change in polarization due to inversion be a ; then a is proportional to the 

 amount of sugar originally present. Let the fall in polarization, i.e., the 

 algebraical difference between the initial reading and the reading after any 

 time interval, t, be x ; then x is proportional to the amount of sugar inverted. 

 The calculation of the constant will then appear as in the following example. 



Initial reading, 40° ; reading after complete inversion, — 12° ; total 

 change = a = 52° ; reading after 60 minutes, 30° ; proportionate amount of 

 sugar inverted = a; = 40 — 30=10. Then 



I ^2 



Constant = ^- log — = • 001546. 



60 ^ 52 — 10 ^^ 



2. Influence of Acid. — The constant k was determined by Ostwald^^ in 

 1884 for a large number of acids ; some values as found by him are given 

 below. These are referred to half normal strength, to 25' C. temperature, the 

 time being expressed in minutes, and the logarithms being common ones. 



Acid. Constant. Acid. Constant. 



Hydrobromic ... o- 002187 Sulphurous ... ... 0-0006630 



Hydrochloric ... 0'002438 ' Oxalic... ... ... 0-0004000 



Nitric ... ... 0-002187 ' Phosphoric ... ... 0-0001357 



Sulphuric ... 0-001172 Acetic... ... ... 0-0000088 



3. Effect of Concentration of Acid. — Within comparatively narrow limits 

 the rate of inversion is nearly directly proportional to the concentration of the 

 acid. With the stronger acids, however, the rate of inversion decreases more 

 rapidly than does the decrease in concentration ; with weaker acids, the re- 

 verse holds. 



