« 
i88 MATHEMATICS: F. L. HITCHCOCK Proc. N. A. S. 
while dnio = — dm, so that (6) simpUfies to 
= -/- rg + /o]<im (8) 
We may perfectly well suppose the total amount of solution to be so great 
that the freezing; out of 1 mol of ice does not affect its composition. The 
expression in brackets is then a constant. Letting Q stand for the latent 
heat per mol we may, therefore, write 
Q = - / - + /o (9) 
dT dT 
6. The form of the melting-point equation. — If we adopt, as a temporary 
notation, 
y =fo ~f (10) 
equation (9) becomes 
-T^ + y = Q (11) 
whose integral is, pressure being regarded as constant, 
^= - J^dT + F{c) + K (12) 
where F{c) is a function of the concentrations of the various solutes, but 
not of the temperature, and K depends on neither concentrations nor tem- 
perature. Since by (5) 3/ = 0 when we have equilibrium, the equation 
of the melting-point curve takes the form 
- J|dr + Fic) + A' = 0 (13) 
7. The chemical potential of the solvent. — To determine the form of the 
function F{c) we now proceed to get an expression for the potential /o 
of the water in the solution. The concentrations may be defined as 
mi m2 .... 
Ci = — , C2 = — , . . . Cn = — (14) 
mo mo mo 
Since the potential fo is a continuous function of the concentration and 
approaches (po the potential of the pure solvent as the solution becomes 
infinitely dilute, we may, as a general concept, think of fo as expanded 
in a power series in the concentrations, the first term being ipo • It is very 
well known, however, that, as a first approximation, the diminution of 
potential when any solute is added to pure water is equal to RTc, that is, 
as a first approximation 
fo = - RT{ci + C2+ ... -i- c„) (15) 
which holds good at infinite dilution. As a second approximation we may 
assume that each of the dissolved substances acts like a perfect gas, which 
leads to the much more accurate equation^ 
fo = <Po - RTln{l + Ci + C2+ ... + Cn) (16) 
