Thermodynamics and Chemistry

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 dxAtodxB.) 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)CXz(aq)

HereCandare the numbers of cations and anions in the formula unit of the salt, and
zCandzare the charge numbers of these ions. The solution in equilibrium with the