140 ON THE DEPRESSION OF THE FREEZING-POINT 
into m ions, then the number of free ions is n ma, and therefore 
the total number of undissociated molecules and free ions in this 
solution is 
(l—a@) n+nma, or mel +a (m—1)). 
On the assumption that a free ion produces the same amount of 
depression of the freezing-point as a molecule, and that in a 
solution the molecules are so far apart that no association of 
molecules occurs, if ¢ is the depression of the freezing-point and 
M the molecular depression, 7. ¢., the depression produced by 
one gramme-molecule or one gramme-ion, we have 
. ft) 
MS (Ee (Gia) eases » > ae eee 
In the case of mixtures of simple solutions, according to the 
above assumption, A the depression of the freezing-point will be 
represented by the expression :— 
A =[M,N, (1+¢,(m,—))4+M,N,(i+2,(m, —1))+ ..]. 
where 1, 2, ete., denote the electrolytes, the m’s the numbers of 
ions into which the molecules of. the respective electrolytes break 
down, the a’s the ionization coefficients in the mixture, the N’s the 
concentrations (in gramme-molecules per litre) of the mixture 
with respect to the respective electrolytes, and the M’s the 
depressions produced by one gramme-molecule or one gramme. 
ion of the undissociated and dissociated portions respectively of 
the electrolytes. The a’s in this. expression are given by the 
method to be tested; the m’s in the case of the electrolytes 
selected can have only one value; and the N’s are of course 
known; but what values the M’s are to be regarded as having 
is doubtful. It was found for simple solutions of the three 
electrolytes employed, that the molecular depressions increased 
as the solutions became more concentrated. This appears to 
indicate that one molecule or one ion, when in the presence 
of a large number of molecules and ions, produces a greater 
depression than when it is in the presence of a smaller 
number. Thus in the case of a solution made by mixing 
simple solutions of different electrolytes, since the number 
