196 
MATHEMATICS: F. L. HITCHCOCK 
Proc. N. a. S. 
Lim — = aKo (47) 
dc 
This means that we shall not naturally expect the electrolyte to obey 
the mass law even in very dilute solution. For this would, in strict rigor, 
mean that the graph of K against c should have a horizontal tangent as 
c approaches zero. However, a curve differing little from the horizontal 
would be hard to distinguish experimentally from the other. 
In a highly interesting investigation Washburn has worked out this 
curve for KCl at extreme dilutions, assuming that the graph of K shows 
no marked change in character as c approaches zero. To compare his 
results with the present theory I have assumed that his value of Ao is 
correct and also his value 0.02 for Ko. Determining the function g so 
that the following data (from Washburn and from Adams) should be ex- 
actly satisfied: 
c 0.0004 0.001 0.005 
7 0.98923 0.98163 0.958 
the function g was found to satisfy 
g logio^ = 1075(77 - 47234^(1 - 7) + 240064^272 
Values of 7 calculated from this equation were then compared with 
Washburn's results for round concentrations: 
c 0.00005 0.0001 0.0002 0.0003 0.0007 
7 0.99796 0.99595 0.9931 0.9910 0.98507 (Calc.) 
7 0.99753 0.99529 0.99256 0.99083 0.98511 (Washburn) 
The curve for K plotted from the calculated values of 7 lies very sUghtly 
higher than Washburn's curve at concentrations below 0.0004, but 
approaches 0.02 as c approaches zero. It coincides almost exactly with 
Washburn's curve above 0.0004 differing from his curve in having but 
slight inflection. An inspection of his diagram makes it clear that a 
curve meeting the vertical axis at a small angle with the horizontal, and 
having but little inflection still fits his data very well, and, of course, 
would fit entirely with his supposition that no marked change occurs in 
the character of the curve at infinite dilution. It is highly probable, too, 
that the empirical value of g given above is not the best that can be found. 
Further investigation is now being conducted. 
14. Summary. — The adoption of Gibbs' principle of chemical potentials 
leads to an extention of the ordinary theories of melting-point, heat of 
dilution, vapor pressure, and mass law. In all cases these extensions are 
the result of the presence, in the expression for the chemical potential of 
the solvent, of terms in the square and higher powers of the concentrations. 
In many phenomena these terms are without measurable effect in dilute 
solution. They lead in the case of the mass law to terms in the first 
