Removal of an Assumption in Deriving the Phase Change Formula AT= * ~ . m
Frederic E. Schubert Suffolk County Community College Selden, New York 11784
The freezing point depression and boiling point elevation of a dilute solution can be approximated using formulas of the form AT=K.rn (1) where AT is the change in freezing or boiling point and m is the solution molality. The constant, K, isspecific to the phase chance and solvent A. P& of the uncertainty involved in use of this formula arises from mathematical aooroximations made in its derivation. Atrention is focused h& on the most seriousof thesc, namely that the mole fraction of solure B isequivalent to the solutesolvent mole ratio; that is X 6 - - nB (2)
-
nA
I t is shown that this assumption, generally employed in complete derivations of (1)in physical chemistry texts (1,2) and also in less rigorous developments in college chemistry texts (3-61,is unnecessary. The following equation serves to initiate the discussion ( I , 9).
-1,
AH -InXA = - R
(~-2 1
(3)
In eqn. (3), X A is the mole fraction of solvent A, AH is the enthalpy change for the transition for pure A a t the normal freezing or boiling point To,R is the gas constant, and T the phase transition temperature of the mixture. The formula may he rewritten -In (1 -Xd =-AH AT RTTn with X g the mole fraction of solute B. The logarithmic term can he expanded -In (1 - XB) = XB 112XB2 + 113XB3.. . (5) which for small XB is well approximated by the first-order term. Noting this and also that T To eqn. (4) may then be written AHAT xn=(6) RTo2 Equation (6) can be modified further, since if X e is small, eqn. (2) holds. Then
+
(7)
There is another more direct method of arriving at eqn. (7). I t involves the introduction of the logarithmic expansion and the conversion from mole fraction to mole ratio in one step. First -lnXA=ln
but so that
(-)
=I~(I+!%)
l n u + Y ) = Y - I I Z P + I Iw... -In XA= 3 2 - 112 (:)'+ nA
113 (%)'+. . . nA
(8) (9)
(10)
Using only the first term in the expansion allows one to arrive at eqn. (7) without employing eqn. (2). It is clear that when higher-order terms in the logarithmic expansion are small, the errors in eqns. (6)and (7) are about equal in magnitude and opposite in sign. Conversion of mole ratio to molality brings eqn. (7) to the familiar form of eqn. (1). The removal of the assumption of eqn. (2) implies that in studying near ideal systems, for example in introductory freezing point depression labs, it is reasonable to use solutions with concentrations in the neighborhood of 1 molal while still maintaining acceptable accuracy. Work of C. R. Bury and H. 0.Jenkins presented in Glasstone (7) indicates that a 1.2 molal solution of C C 4 in benzene leads to an error of only about 6% in the use of eqn. (1). For this near-ideal system about 5%of the error is due to the mathematical approximations discussed here. In conclusion then. the attractivelv simple eqn. (1) is based on a firmer mathematical foundation than had been supoosed. It is in fact essentiallv as accurate as eqn. (6) which %her treatments suggest is mire precise. Literature Cited (11 Moore. W. J.. "Phpieal Chemistry." 4th Ed.. Prenfiee-Hall Inc.. New J e w , 1912, p. 249.
121 Barrow. G.. "Physical Chemistry," 3rd Ed.. McGraw~Hillhe.. New Jersey. 1973, p. 603. (31 Monimcr. C. E., "Chemistry A ConeeptualAppmach."3rd Ed.. D. Van Nostrand Ca., New York, 1915,~. 284. (41 B r w i e , P.,Arents. J., Maidich. H.. and Turk. A., ''Fundamentals of Chemistry." 3rd Ed., Academic Pres he., New York, 1975, pg. 184. (5) SienLo. M. J.. and Plane, R. A,. "Chemicsl Principles and Properties." 2nd Ed., McGrsa-Hill Inc., NevYork. 1914. p 234. (61 Mahsn, B. H., "Uniwrsity Chemistry." 3rd Ed.. Addison-Waslev Publiahinec~..Inc.. Ma~sachusetts,1976, p. 154. (71 Glasstone, S., "Physical Chemistry." 2nd Ed.. D. Van Nmtrand Co., lnc.. New York, 1946. p. 645.
Volume 56, Number 4, April 1979 1 259
