Thermodynamics and Chemistry

CHAPTER 11 REACTIONS AND OTHER CHEMICAL PROCESSES

11.1 MIXINGPROCESSES 304

by a superscript “id” as inÅGidm(mix). The general definition of an ideal molar mixing
quantity, analogous to Eq.11.1.5, is

ÅXmid(mix)D

X

i

xi.XiidXi/ (11.1.6)

The chemical potential of constituentiof an ideal mixture is related to the mole fraction
xiby the relation (Eq.9.4.8)
iDiCRTlnxi (11.1.7)

By combining this relation with Eq.11.1.4, we find the molar Gibbs energy of mixing to
form an ideal mixture is given by

ÅGmid(mix)DRT

X

i

xilnxi (11.1.8)

Since each mole fraction is less than one and the logarithm of a fraction is negative, it
follows thatÅGmid(mix) is negative for every composition of the mixture.
We obtain expressions for other molar mixing quantities by substituting formulas for
partial molar quantities of constituents of an ideal mixture derived in Sec.9.4.3into Eq.
11.1.5. FromSiDSiRlnxi(Eq.9.4.9), we obtain

ÅSmid(mix)DR

X

i

xilnxi (11.1.9)

This quantity is positive.

Although the molar entropy of mixing to form anidealmixture is positive, this is not

true for some nonideal mixtures. McGlashan^2 cites thenegativevalueÅSm(mix)D

8:8J K^1 mol^1 for an equimolar mixture of diethylamine and water at 322 K.

FromHiDHi(Eq.9.4.10) andUiDUi(Eq.9.4.12), we have

ÅHmid(mix)D 0 (11.1.10)

and
ÅUmid(mix)D 0 (11.1.11)

Thus, the mixing of liquids that form an ideal mixture is anathermalprocess, one in which
no heat transfer is needed to keep the temperature constant.
FromViDVi(Eq.9.4.11), we get
ÅVmid(mix)D 0 (11.1.12)

showing that the ideal molar volume of mixing is zero. Thus an ideal mixture has the same
volume as the sum of the volumes of the pure components at the sameT andp.^3
Figure11.2on the next page shows howÅGmid(mix),TÅSmid(mix), andÅHmid(mix)
depend on the composition of an ideal mixture formed by mixing two pure substances.
Although it is not obvious in the figure, the curves forÅGidm(mix) andTÅSmid(mix) have
slopes ofC1or1atxAD 0 andxAD 1.

(^2) Ref. [ 113 ], p. 241.
(^3) From the fact mentioned on p. 226 that the volume of a mixture of water and methanol is different from the
sum of the volumes of the pure liquids, we can deduce that this mixture is nonideal, despite the fact that water
and methanol mix in all proportions.