Thermodynamics and Chemistry

CHAPTER 9 MIXTURES

9.4 LIQUID ANDSOLIDMIXTURES OFNONELECTROLYTES 247

Raoult’s law for fugacity can be recast in terms of chemical potential. Section9.2.7
showed that if substanceihas transfer equilibrium between a liquid and a gas phase, its
chemical potentialiis the same in both equilibrated phases. The chemical potential in
the gas phase is given byiDi(g)CRTlnfi=p(Eq.9.3.12). Replacingfibyxifi
according to Raoult’s law, and rearranging, we obtain

iD



i(g)CRTln

fi

p



CRTlnxi (9.4.4)

The expression in brackets is independent of the mixture composition. We replace this
expression by a quantityi, a function ofTandp, and write

iDiCRTlnxi (9.4.5)

Equation9.4.5is an expression for the chemical potential in the liquid phase when Raoult’s
law for fugacity is obeyed. By settingxiequal to 1 , we see thatirepresents the chemical
potential of pure liquidiat the temperature and pressure of the mixture. Because Eq.9.4.5
is valid for any constituent whose fugacity obeys Eq.9.4.3, it is equivalent to Raoult’s law
for fugacity for that constituent.

9.4.2 Ideal mixtures

Depending on the temperature, pressure, and identity of the constituents of a liquid mixture,
Raoult’s law for fugacity may hold for constituentiat all liquid compositions, or over only
a limited composition range whenxiis close to unity.
Anideal liquid mixtureis defined as a liquid mixture in which, at a given temperature
and pressure,eachconstituent obeys Raoult’s law for fugacity (Eq.9.4.3or9.4.5) over the
entire range of composition. Equation9.4.3applies only to a volatile constituent, whereas
Eq.9.4.5applies regardless of whether the constituent is volatile.
Few liquid mixtures are found to approximate the behavior of an ideal liquid mixture.
In order to do so, the constituents must have similar molecular size and structure, and the
pure liquids must be miscible in all proportions. Benzene and toluene, for instance, satisfy
these requirements, and liquid mixtures of benzene and toluene are found to obey Raoult’s
law quite closely. In contrast, water and methanol, although miscible in all proportions,
form liquid mixtures that deviate considerably from Raoult’s law. The most commonly
encountered situation for mixtures of organic liquids is that each constituent deviates from
Raoult’s law behavior by having ahigherfugacity than predicted by Eq.9.4.3—apositive
deviation from Raoult’s law.
Similar statements apply to idealsolidmixtures. In addition, a relation with the same
form as Eq.9.4.5describes the chemical potential of each constituent of an idealgasmix-
ture, as the following derivation shows. In an ideal gas mixture at a givenT andp, the
chemical potential of substanceiis given by Eq.9.3.5:

iDi(g)CRTln

pi

p

Di(g)CRTln

yip

p

(9.4.6)

Hereyiis the mole fraction ofiin the gas. The chemical potential of the pure ideal gas
(yiD 1 ) is

i Di(g)CRTln

p

p

(9.4.7)