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The Freezing Point Depression Law in Physical chemistry Is It Time for a Change? Hugo F. Franzen The Ames Laboratory-DOE and Iowa State University, Ames, IA 5001 1
The freezing point depression law has been taught for many years as part of both freshman and physical chemistry courses. On the freshman level the law is usually taught phenomenologically, whereas in physical chemistry its oriein in the thermodvnamics of heterogeneous eauilibrium is ;he focus of attention. The teaching of the lawat this level s which are (1) to display the has a number of p u r ~ o s e among thermodynamic&i&n of the law, 121 to provide experience in the determination of molecular weight or freezing point depression, (3) to provide a hasic understanding of eutectic melting and solidification, (4) to provide a basis for a deeper understanding of both Raoult's law behavior (colligative properties) and departures from ideality, and (5) to illustrate the linearization of a more complex functional relationship. The purpose of this article is to suggest a change in physical chemistry courses to a slightly more complicated but significantly more useful generalization of the simple freezing point depression law, one that achieves all of the purposes stated above and a number of additional purposes as well. The time is appropriate for this move for two reasons. The first is that the simple freezing point depression law is currently being taught in freshman chemistry courses making the treatment common to most physical chemistry texts somewhat redundant (not totally so, of course, because the freshman-level treatment generally is not based upon an understandine of calculus or chemical potential). The second is that freezing point depression i s h o longer as important as it once was in the determination of molecular weight. This is where the generalization suggested here, namely treatment of solid-liquid equilibria in which solid-solution is allowed, really shines, for some modern separations and materials handling technologies (zone refining, skull melting) depend for their success upon control of the distribution of impurities between solids and melts. What is suggested here is that the usual two-component, two-phase liquid-solid treatment be generalized to the case of ideally dilute solid-liquid solution equilibrium. This has the advantage of removing the assumption that in bothpractice and principle is false, namely the assumption that there is zero solid solubility; nonetheless, the usual freezing point depression law is recovered when the solubility of solute in the solid is negligibly small. The traditional freezing point depression treatment is based upon the reaction A(pure solid) = A(in liquid solution with B)
A(in solid solution) = A(in liquid solution) B(in solid solution) = B(in liquid solution)
(2)
(3)
be considered instead. Taking the Raoult's law (pure liquid and solid) standard states for A and the Henry's law (B in the hypothetical ideal X B = 1 liquid and solid solutions) standard states for B and applying the Gibhs-Helmholz equation in ideally dilute solution and, as usual in the development of the freezing point depression law, assuming ACpoATto be negligible relative to AH0,yields
and
where XA' and XAa are the mole fractions of A in the liquid and solid phases, Tr is the melting temperature of A, AHof,,8;mis the enthalpy of fusion of pure A, K is the distribution equilibrium constant, XB'IXBS,KO is the limiting value of K a t Tr, and AH'is the standard enthalpy change for reaction 3. For sufficiently dilute solutions
and
where AT = Tr - T, and thus,
Furthermore, taking K = Ko solution
+ AK for sufficiently dilute
and therefore,
(1)
and what is suggested here is that the more realistic net reactions Volume 65
Number 12 December 1988
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Liquid solution
1400 -
-
1300 -
Solid solution
XSi = I
sion law. For K D < 1 (B more soluble in solid than in the liquid) eqs 13 and 14 imply AT< 1 (freezingpoint elevation), and for KO > 1 thev vield deoression of the melting ~ o i n t with solid solution."&r all cases eqs 13 and 14 a r e inear equations appropriate for the analysis of solid-liquid equilibria in sufficiently dilute solutions. The quantity K" can be viewed as a parameter that determines the behavior: KO means zero solid solubility and the usual freezing point depression law, KO > 1means some solid solubility but greater solubility in the liquid and freezing point depression, and KO < 1 means greater solubility in the solid than in the liquid and freezing point elevation. As the solutions become more concentrated the linear laws will no longer apply because the approximations are no longer valid, and because the solutions will in general no longer be ideal (will no longer obey Raoult's and Henry's laws). In the case of higher concentrations of solute the appropriate laws are
-
- XGe
Phase diagram for Oe-Si.
Eliminating Xgl between eq 8 and eq 11 yields
x,=
-
AH0,.i,"AT (1 - KO)RT;~(I AATJ
+
(12)
where A = AHsK~IRT,WP - I), which for sufficiently small AT yields (to first order in AT):
and
These are the generalized limiting laws for solid-liquid twocomponent equilibria. For KO >> 1 (solute much more soluble in liquid than in solid solution) these become Xzs a 0 and X21 AHofusionATRT?, the linearized freezing point depres-
S X the B ~ activity coefficients dewhere K = Y B ~ X B ~ I Y Band pend upon the nature of A and B and their liquid and solid solution interactions. The figure shows the Ge-Si phase diagram.' On the Si-rich end, K o (the limit of XcdIXcd as T goes to 1414 "C, the melting temperature of Si) is greater than unity, and the freezing point is depressed. Furthermore, Si can be purified of a Ge impurity by zone refining. On the Ge-rich end, K" (the limit of X&/Xsis) is less than unity, and the freezing point is elevated. Ge cannot be purified of a Si impurity by zone refining.
'
Binary ANoyPhaseDiagrams; Massaiski. Thaddeus 6.. Ed.; American Society for Metals: Metals Part, OH, 1986: Voi. 2.
Annotated List of Laboratory Experiments Available A revised and updated edition of the Annotated List of Laboratory Experiments in Chemistry from the Journal o f Chemical Education, 1957-1984, prepared by the Division of Chemical Education's Task Force on Project CHEMLAB, is now available. As described in this Journal [1984,61,632] this data base facilitates the use of the more than 2000 laboratory experiments published from 1957 through 1984. Keywords designate principal fields and type and level of experiment (ex., INORG ANAL GAS PREP 2). These are followed by short descriptions of features that are not suggested by the title hut are important in selecting laboratory experiments or projects. The list is complete and includes experiments a t all levels, from some which are useful far elementary school to those intended for advanced or specialized courses. The bound, printed edition, organized by principal fields (ANAL, BIOCHEM, INORG, ORG, PHYS, POLYMER, RADIOCHEM) in a format for easy visual searching, is available from the ACS Education Division, Office of College Chemistry, 1155 Sixteenth St. NW, Washington, DC 20036, for $28. There are no additional shipping costs (within the United States) for prepaid orders. MicroCHEMLAB, computer searchable, is available from Project SERAPHIM (Department of Chemistry, Eastern Michigan University, Ypsilanti, MI 48197) as a set of six disks for $25. It requires an IBM-PC or equivalent with 128K and two double-sided (320K) disk drives to run; a printer is desirable. Project SERAPHIM requires a $2 shipping and handling charge and prepaid orders
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Journal of Chemical Education
