Derivation of the Freezing-Point Equation HOWARD M. 'TEETER Bradley Polytechnic Institute, Peoria, Illinois
m H I S simplified derivation of the freezina-point -equation -is resented because it is adequately rigorous, it follows the same logic often used in deriving the corresponding boiling-point eqnation,' and in view of its directness, it can be used in briefer courses in physical chemistry to furnish mathematical justification for an equation which is often presented empiri-
1
a p = aH.aT/T.aV
(1)
where AH, is the molar heat of sublimation and A V is the &ifferencein molar volume of solid and vapor. ~f we assume the molar volume of solid to be negligible, eauation (1) takes the form > , Ap = pAH,aT/RToZ
(2)
ally.^ The following general assumptions will be made: (1) the solution shall be ideal and dilute, so tbat Raoult's Law will apply to the solvent and so tbat the mol fraction of solute, Na, shall equal m/nl, where nl and nz are the number of mols of solvent and solute, respectively; (2) the difference, To - T, between the freezing points of solvent and solution shall be small so that TOT= To2, the vapor pressures of solvent, solution, and solid solvent a t the temperatures To and T will be very nearly identical, and the latent heats of fusion, vaporization, and sublimation may be considered constant. These conditions will be fulfilled if Na 2 0.01. The depression in freezing point, AT, is associated with a depression, Ap, in the vapor pressure of the system.a This decrease in vapor pressure can be evaluated in two ways. If we follow down the vapor pressure curve of the solid solvent from Toto T, we have bv the Clapeyron eauation
where p is the vapor pressure. To evaluate Ap in a second manner we note that it may be obtained in two increments. The first is found by following the vapor pressure curve of supercooled liquid from Toto T. The equation obtained is similar to (2) except tbat now AH", the molar heat of vaporization, appears: Ap, = p AH. AT/RTo2 (3)
Cf. MACDOWGAL, "Physical chemistry," The Macmillan Company, New York City, 1936,,p. 246. a Cf. Bmcnm, "Physical chem~stry,a brief course." PrenticeHall, Inc., New York City, 1940, p. 168. The depressions are very small, in accordance with the assumntions: however. differentials are not used in order t o keen the kathekatics as Ample as possible.
where AHj is the molar heat of fusion. If now we take Na = nz/nl, and n, = w/M, where w and M are the weight of solute added and its molecular weight, respectively, we obtain (6) in its customary form:
The second increment is due to addition of solute and is given by Raoult's Law: Aps = pN*
where Nz is the mol fraction of the added solute. Now since Ap = Ap1 Apz, we have @AH,AT/RTo2= p AH, AT/RTOa+ pN2
(4)
+
(5)
which reduces easily to N2 = AH, AT/RTos
M = wRTo1/n,AH, AT
(6)
(7)
