Vol. 6, 1920 MATHEMATICS: F. L. HITCHCOCK 
195 
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G{c) = k (41) 
where G{c) is a function of concentration, which, if expanded in series, 
begins with terms of the first degree in the masses of the dissolved sub- 
stances. The coefficients in the expansion are functions of temperature 
(and pressure), and k is also a function of temperature. If we omit 
P{c,T) in /o we must also omit G{c) in this extended mass law (41). It 
is noteworthy that in the logarithmic term we have rigorously Xi, X2, x?^, 
not nti/nio, etc., although in solutions not very concentrated it does not 
matter much. 
13. Discussion of the generalised mass law. — It is shown in the fore- 
going theory that if the. function P{c,T) can be accurately determined we 
can at once calculate the function G{c) and hence predict to what extent 
ionization departs from the mass law. This could be done from accurate 
measurement of heat of dilution, checked as to terms in T by freezing- 
point data. Conversely, if ionization be known, we can find G{c) and 
hence P{c) for that temperature. Many other relations can be brought 
to bear particularly those connecting specific heats with change in equi- 
librium under varying temperature. 
Furthermore equation (41) can be employed to study the theoretical 
form of the curve obtained bj plotting the "constant" K against the 
formal concentration c. By putting Wi = W2 = cy and = c{l — 7) 
we may write (41) as 
K = = Koc' (42) 
1 — 7 
where g is a function of concentrations which may be expanded 
g = acy a'c{l - 7) + etc. (43) 
By differentiating with respect to c we have 
^' = K.e^ 'L' (44) 
dc dc 
while dg/dc found from (43) is 
* = a7 + a\\ - 7) + (a - a')c^ (45) 
dc dc 
and the value of dy/dc from (42) is 
dy _ [acy + a'c{l - y) - l]y{l — 7) 
dc c[2 - 7 + cy{l - y){a' - a)] 
Now as c approaches zero, 7 approaches unity. By inspection of (46) 
we see that cdy/dc approaches zero, for the denominator approaches 
unity on cancelling c. Hence from (45) we see that dg/dc approaches 
the constant a. Hence by (44) we have 
(46) 
