192 
MATHEMATICS: F. L. HITCHCOCK Proc. N. A. S. 
If P(c,T) be dropped, the equation becomes identical (except for slight 
difference of notation) with the melting-point equation established in a 
valuable paper by Washburn^ on the two hypotheses : 
1 °. That the vapor of the solvent acts like a perfect gas. 
2°. That Raoult's law holds good for the lowering of vapor pressure. 
Since our equation (31) has been obtained without any recourse to vapor 
pressure, we may at once follow Washburn's reasoning in inverse order 
and get a certain amount of new light on the two assumptions about vapor 
pressure. For by differentiating (13) with respect to T we have 
I = F'ic)^ (32) 
supposing there is only one dissolved substance. Using F'{c) from 
(21) we have 
Q = (p^ _ (33) 
72 V 1 + c/c'^ 
1 + c/dT 
which aside from slight difference of notation, differs from Washburn's 
differential equation in the presence of P\, which we have seen to be a 
function of c. Taking then the rigorous equation^ 
vT Ldc J T dT 
where p is the vapor pressure, and eliminating dc/dT 
PJ'I _ (35) 
LdcJ-r \ I + c/v 
r V 1 + 
So far we have made no assumptions. If we assume pv = RT we have 
by integrating 
po = p{l + c)eP^ (36) 
a generalization of Raoult's law, to which it reduces if Pi = 0. It is 
noteworthy that Pi is the coefficient of precisely the term in the expansion 
of P{c,T) which does not affect the heat of dilution. In other words the 
departure of the vapor from Raoult's law is dependent on the function 
P{c,T), which affects the freezing-point, minus all the terms which affect 
the heat of dilution. Thus if heat of dilution and freezing-point are known 
we can find the vapor pressure, a fact also pointed out by Washburn. 
Equation (36) shows the precise relation of this fact to the melting-point 
equation provided we assume pv = RT. In any case (35) shows that the 
vapor pressure of a solution can be calculated if we know the coefficient 
of T in the chemical potential of the solvent, and the equation of state 
the vapor. 
It will now be apparent that we can determine empirically the coeffi- 
cients in the expansion of P{c,T) and hence have a complete knowledge of 
the properties of the solution if we can get accurate determinations of 
heat of dilution at various temperature and over a considerable range of 
