in the Analogy between Solutions and Gases. 95 



We can follow this relation still further, and, assuming 

 Avogadro's law for solutions, deduce the above normal mole- 

 cular depression of the freezing-point from other data. It 

 bears a close relation to the heat of fusion of the solvent, as is 

 shown by applying the second law of thermodynamics to a 

 reversible cycle. Imagine a very dilute solution containing 

 P per cent, of a dissolved substance, which has produced a 

 depression of freezing-point, A ; the solution itself freezes at 

 T, and its heat of fusion is W per kilogram. 



1. By use of piston and cylinder with semipermeable 

 walls, the solution is deprived at temperature T of that amount 

 of the solvent in which a kilogram-molecule of the dissolved 

 substance existed ; the amount of the solution is moreover so 

 great that no appreciable change of concentration occurs, 

 hence the work expended on it is 2T. 



2. This quantity of solvent, — p— , is then allowed to freeze 



at T by withdrawing p calories; the solution and the 



solvent, now solid, are cooled through A degrees, and the 



solvent is now allowed to melt in contact with the solution, 



thereby absorbing the above quantity of heat. Finally, the 



temperature is raised A degrees. 



^ . ... .,, 100MW . . . , 

 During this reversible process p calories rise from 



A to T, corresponding to an expenditure of energy equal to 



100MWA T ., . . - MA , 



prp . In this expression, however, -p- is the molecular 



depression of the freezing-point, which we may represent by 



• a r 100W* , .. 



the letter t ; the work done is therefore — ~; — , and it was 



shown above to be equal to 2T. Hence the equation 



100W* T 2 



i^- =2T ; whence t = 0'02 ±- 



This theoretical deduction receives ample confirmation from 

 experimental data. The following table exhibits the molecular 

 depression of freezing-point experimentally determined by 

 Eaoult (Annalesde Chimie et de Physique, [5] xxviii.; [6] xi.), 

 along with the values calculated by means of the above 

 formula : — 



