Thermodynamics and Chemistry

CHAPTER 9 MIXTURES

9.4 LIQUID ANDSOLIDMIXTURES OFNONELECTROLYTES 254

For consistency with Eqs.9.4.27and9.4.28, we can rewrite Eq.9.4.24in the form

B.T; p/Drefx;B.T; p/CRTln

xB

x

(9.4.29)

withx, thestandard mole fraction, given byxD 1.

9.4.6 Solvent behavior in the ideal-dilute solution

We now use the Gibbs–Duhem equation to investigate the behavior of the solvent in an
ideal-dilute solution of one or more nonelectrolyte solutes. The Gibbs–Duhem equation ap-
plied to chemical potentials at constantTandpcan be written

P

ixidiD^0 (Eq.9.2.43).
We use subscript A for the solvent, rewrite the equation asxAdAC

P

i§AxidiD^0 ,
and rearrange to

dAD

1

xA

X

i§A

xidi (9.4.30)

(constantTandp)

This equation shows how changes in the solute chemical potentials, due to a composition
change at constantTandp, affect the chemical potential of the solvent.
In an ideal-dilute solution, the chemical potential of each solute is given byiDrefx;iC
RTlnxiand the differential ofiat constantTandpis

diDRTd lnxiDRTdxi=xi (9.4.31)

(Here the fact has been used thatrefx;iis a constant at a givenTandp.) When we substitute
this expression for diin Eq.9.4.30, we obtain

dAD

RT

xA

X

i§A

dxi (9.4.32)

Now since the sum of all mole fractions is 1 , we have the relation

P

i§AxiD^1 xAwhose
differential is

P

i§AdxiDdxA. Making this substitution in Eq.9.4.32gives us

dAD

RT

xA

dxADRTd lnxA (9.4.33)

(ideal-dilute solution

of nonelectrolytes)

Consider a process in an open system in which we start with a fixed amount of pure
solvent and continuously add the solute or solutes at constantT andp. The solvent mole
fraction decreases from unity to a valuex^0 A, and the solvent chemical potential changes
fromAto^0 A. We assume the solution formed in this process is in the ideal-dilute solution
range, and integrate Eq.9.4.33over the path of the process:

Z (^0) A
A
dADRT
ZxADx (^0) A
xAD 1
d lnxA (9.4.34)
The result is^0 AADRTlnxA^0 , or in general
ADACRTlnxA (9.4.35)