Vol. 6, 1920 
MATHEMATICS: F. L. HITCHCOCK 
187 
^dnto+^ dm, + ^Jtdm,+ ... +^dm, = 0 (2) 
dwo c>mi dm2 dntn 
which is one of the requirements that the free energy be a minimum. 
The partial derivatives d<^/dmo, etc., were called by Gibbs the chemical 
potentials of the respective masses. I shall denote these chemical poten- 
tials by the symbols /o, /i, etc., that is 
dip , bip dip 
Jo = - — , /I = — , - ■ - Jyt - r — {o) 
onto OMi ontn 
so that the general condition of equilibrium may be written 
fodnio + fidmi + f2dm2 + . . . + /ndm^ = 0 (4) 
Equation (4) may be said to be the kernel of the Gibbs theory. 
4. Application to the melting-point curve of ice in contact with an aqueous 
solution. — Let our chemical system now consist of nio mols of water in 
which are dissolved mi, m2, . . .mn mols of various solutes in contact with 
a mass of ice m (without subscript). If any ice melts an equal amount of 
water is formed, that is dmo = — dm. The masses of the solutes are un- 
changed, that is dmi = 0, dm2 = 0, etc. Hence, equation (4) becomes 
for this case 
fodmo — fdmo = 0 
that is 
/o - / = 0. (5) 
When ice is in equilibrium with an aqueous solution the chemical potential 
of the ice is equal to the chemical potential of the solvent. 
It is important to notice that fo is not the potential of the solvent in the 
pure state, but its potential as it actually exists in the solution ; it is a func- 
tion of the concentrations of the various dissolved substances as well as 
of temperature and pressure. On the other hand /, the chemical poten- 
tial of the ice, is a function of temperature and pressure only. It is also 
important to notice that one mol of ice or water here means merely one 
formula-weight, and assumes nothing as to actual molecular weight. 
Thus the chemical potentials are the free energies per formula-weight 
of their respective masses. 
5. The latent heat of melting ice in contact with the solution. — When a 
little water freezes out of the solution the heat emitted is determined by 
equation (1). The only variables are Wo and m. We may, therefore, 
write 
dQ = Afr^ _ Am„ + Afr^ _ Xm (6) 
dmo\ 57 / dw\ 57 / 
by the ordinary formula for a total differential. But we also have, iden- 
tically, 
d dip _ b dip _ bfo . d dip df 
