Vol. 6, 1920 
MATHEMATICS: F. L. HITCHCOCK 
193 
concentrations, and also have accurate freezing-point determinations at 
fairly high concentrations, so as to calculate the missing term Pi. It is 
doubtful if any such data of high accuracy now exist over a wide enough 
range for even one substance. If, however, the accurate methods which 
have been used*^ to obtain freezing-point data for very dilute solutions can 
be carried to higher concentrations, and if the accurate data on heat of 
dilution^ which have been obtained at 25° can be carried over a moderate 
temperature range, there seems to be no doubt we should have a theory 
of solution that would be practically complete. A detailed examination 
of existing data in the light of the foregoing theory is now under way, 
but would too greatly lengthen the present paper. A single example must 
here suffice. Taking cane sugar, and, as a first approximation, consider- 
ing only terms in the square of the concentration, i.e., dropping the cube 
and higher powers in the expansion of P{c,T) the coefficient C2 of this 
term was calculated for values of A/", the number of mols sugar per 1000 
grams of water. The data^ of Morse and Fraser gave 
N 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2681 
C2 0.110 0.122 0.098 0.119 0.119 0.118 0.121 0.133 0.115 
while the data^ of Ewan gave 
N 
C2 
Raoult's data^^ gave : 
N 0.8714 1.2908 1.5347 1.6427 2.1108 
C2 0.118 0.104 0.132 0.128 0.132 
N 0.5056 1.0107 
C2 0.117 0.124 
Other data quoted by Landolt and Bernstein gave 
N 0.760 0.4236 0.2840 
C2 0.061 0.094 0.097 
It is evident that the values of C2 from the data of these different ob- 
servers do not agree well. Below about 0.3 normal the solution departs 
so little from the "ideal" that the calculations become meaningless, de- 
pending as they do on differences between ideal and observed numbers. 
11. Application of the foregoing theory to electrolytes. — Let us now sup- 
pose that the only dissolved substance is a single uni-univalent salt, as 
KCl. Calling c the formal concentration per 1000 grams of water we shall 
have 
nio = 55.5, mi = W2 = cy, and W3 = c{\ — 7) 
where 7 is the fraction of salt which is ionized, mi and W2 the masses of 
the ions in mols, and the mols of un-ionized salt. At a given tempera- 
ture P{c,T) becomes a function of c alone and we may write 
P{c) = Ac'^y' + Bc'^yil - 7) + Cc^l - 7)' + • • • (37) 
where A, B, C, ... are constants to be found. It is evident from the form 
of these terms that they will be without influence in very dilute solution, 
