CHAPTER 12 EQUILIBRIUM CONDITIONS IN MULTICOMPONENT SYSTEMS
12.5 SOLID–LIQUIDEQUILIBRIA 389
bc
0 0:2 0:4 0:6 0:8 1:0
650
700
750
800
850
900
Mg mole fraction Zn Zn
Tf
=K
Figure 12.7 Solid curve: freezing-point curve of a liquid melt of Zn and Mg that
solidifies to the solid compound Zn 2 Mg.a The curve maximum (open circle) is at
the compound compositionx^00 ZnD2=3and the solid compound melting pointTf^00 D
861 K. Dashed curve: calculated using Eq.12.5.23withÅfusHD15:8kJ mol ^1.
aRef. [ 50 ], p. 603.
By making the approximations thatÅsolHis independent ofT andxA, and is equal to
ÅfusH, we can separate the variables and integrate as follows:
ZTf 00
Tf^0
dTf
Tf^2
D
R
ÅfusH
(^) Z
x^00 A
x^0 A
a
xA
dxAC
Zx (^00) B
x^0 B
b
xB
dxB
!
(12.5.22)
(The second integral on the right side comes from changing dxAto dxB.) The result of
the integration is
1
Tf^0
D
1
Tf^00
C
R
ÅfusH
aln
xA^00
xA^0
Cbln
x^00 B
x^0 B
(12.5.23)
(ideal liquid mixture in
equilibrium with solid
compound,ÅsolHDÅfusH)
LetTf^0 be the freezing point of a liquid mixture of compositionxA^0 andxB^0 D 1 xA^0 , and
letTf^00 be the melting point of the solid compound of compositionxA^00 Da=.aCb/andxB^00 D
b=.aCb/. Figure12.7shows an example of a molten metal mixture that solidifies to an
alloy of fixed composition. The freezing-point curve of this system is closely approximated
by Eq.12.5.23.
12.5.5 Solubility of a solid electrolyte
Consider an equilibrium between a crystalline salt (or other kind of ionic solid) and a solu-
tion containing the solvated ions:
MCX (s)ïCMzC(aq)C Xz (aq)
HereCand are the numbers of cations and anions in the formula unit of the salt, and
zCandz are the charge numbers of these ions. The solution in equilibrium with the