Thermodynamics and Chemistry

CHAPTER 12 EQUILIBRIUM CONDITIONS IN MULTICOMPONENT SYSTEMS

12.5 SOLID–LIQUIDEQUILIBRIA 387

Tf;BD 380 K

Tf;BD 400 K

0

0:2

0:4

0:6

0:8

1:0

300 350 400

T=K

xB

Figure 12.6 Ideal solubility of solid B as a function ofT. The curves are calculated

for two solids having the same molar enthalpy of fusion (Åfus,BHD 20 kJ mol^1 ) and

the values ofTf;Bindicated.

The composition of the liquid mixture in this kind of system is variable, whereas the
composition of the solid compound is fixed. Suppose the components are A and B, present
in the liquid mixture at mole fractionsxAandxB, and the solid compound has the formula
AaBb. We assume that in the liquid phase the compound is completely dissociated with
respect to the components; that is, that no molecules of formula AaBbexist in the liquid.
The reaction equation for the freezing process is

aA(mixt)CbB(mixt)!AaBb(s)

When equilibrium exists between the liquid and solid phases, the temperature is the freezing
pointP Tfof the liquid. At equilibrium, the molar reaction Gibbs energy defined byÅrGD

iiiis zero:

aAbBC(s)D 0 (12.5.12)

HereAandBrefer to chemical potentials in the liquid mixture, and(s) refers to the
solid compound.
How does the freezing point of the liquid mixture vary with composition? We divide
both sides of Eq.12.5.12byTand take differentials:

ad.A=T /bd.B=T /Cdå(s)=T çD 0 (12.5.13)

(phase equilibrium)

The pressure is constant. ThenA=T andB=T are functions ofTandxA, and(s)=T
is a function only ofT. We find expressions for the total differentials of these quantities at
constantpwith the help of Eq.12.1.3on page 366 :

d.A=T /D

HA

T^2

dTC

1

T



@A

@xA



T;p

dxA (12.5.14)

d.B=T /D

HB

T^2

dTC

1

T



@B

@xA



T;p

dxA (12.5.15)