Thermodynamics and Chemistry

CHAPTER 12 EQUILIBRIUM CONDITIONS IN MULTICOMPONENT SYSTEMS

12.5 SOLID–LIQUIDEQUILIBRIA 388

då(s)=T çD

Hm(s)

T^2

dT (12.5.16)

When we substitute these expressions in Eq.12.5.13and solve for dT=dxA, settingTequal
toTf, we obtain

dTf

dxA

D

Tf

aHACbHBHm(s)

"

a



@A

@xA



T;p

Cb



@B

@xA



T;p

(12.5.17)

The quantityaHACbHBHm(s) in the denominator on the right side of Eq.12.5.17
isÅsolH, the molar differential enthalpy of solution of the solid compound in the liquid
mixture. The two partial derivatives on the right side are related through the Gibbs–Duhem
equationxAdACxBdB D 0 (Eq.9.2.27on page 233 ), which applies to changes at
constantTandp. We rearrange the Gibbs–Duhem equation to dBD.xA=xB/dAand
divide by dxA: 
@B
@xA



T;p

D

xA

xB



@A

@xA



T;p

(12.5.18)

Making this substitution in Eq.12.5.17, we obtain the equation

dTf

dxA

D

xATf

ÅsolH



a

xA

b

xB



@A

@xA



T;p

(12.5.19)

which can also be written in the slightly rearranged form

dTf

dxA

D

bTf

ÅsolH



a

b

xA

1 xA



@A

@xA



T;p

(12.5.20)

Suppose we heat a sample of the solid compound to its melting point to form a liquid
mixture of the same composition as the solid. The molar enthalpy change of the fusion
process is the molar enthalpy of fusion of the solid compound,ÅfusH, apositivequantity.
When the liquid has the same composition as the solid, the dissolution and fusion processes
are identical; under these conditions,ÅsolHis equal toÅfusHand is positive.
Equation12.5.20shows that the slope of the freezing-point curve,TfversusxA, is zero
whenxA=.1xA/is equal toa=b, orxADa=.aCb/; that is, when the liquid and solid have
the same composition. Because.@A=@xA/T;pis positive, andÅsolHat this composition
is also positive, we see from the equation that the slope decreases asxAincreases. Thus,
the freezing-point curve has a maximum at the mixture composition that is the same as
the composition of the solid compound. This conclusion applies when both components of
the liquid mixture are nonelectrolytes, and also when one component is an electrolyte that
dissociates into ions.
Now let us assume the liquid mixture is an ideal liquid mixture of nonelectrolytes in
whichAobeys Raoult’s law for fugacity,ADACRTlnxA. The partial derivative
.@A=@xA/T;pthen equalsRT=xA, and Eq.12.5.19becomes

dTf

dxA

D

RTf^2

ÅsolH



a

xA

b

xB



(12.5.21)