Thermodynamics and Chemistry

CHAPTER 9 MIXTURES

9.2 PARTIALMOLARQUANTITIES 233

the molar volume, so we can simplify the notation by usingVAandVBinstead. Hereafter,
this book will denote molar quantities of pure substances by such symbols asVA,HB, and
Si.
The relations derived above for the volume of a binary mixture may be generalized for
any extensive propertyXof a mixture of any number of constituents. The partial molar
quantity of speciesi, defined by

Xi defD



@X

@ni



T;p;nj§i

(9.2.24)

is an intensive property that depends onT,p, and the composition of the mixture. The
additivity rule for propertyXis

XD

X

i

niXi (9.2.25)

(mixture)

and the Gibbs–Duhem equation applied toXcan be written in the equivalent forms
X

i

nidXiD 0 (9.2.26)

(constantTandp)

and
X

i

xidXiD 0 (9.2.27)

(constantTandp)

These relations can be applied to a mixture in which each speciesiis a nonelectrolyte sub-
stance, an electrolyte substance that is dissociated into ions, or an individual ionic species.
In Eq.9.2.27, the mole fractionximust be based on the different species considered to
be present in the mixture. For example, an aqueous solution of NaCl could be treated as
a mixture of components A=H 2 O and B=NaCl, withxBequal tonB=.nACnB/; or the
constituents could be taken as H 2 O, NaC, and Cl, in which case the mole fraction of NaC
would bexCDnC=.nACnCCn/.
A general method to evaluate the partial molar quantitiesXAandXBin a binary mixture
is based on the variant of the method of intercepts described in Sec.9.2.3. The molar mixing
quantityÅX(mix)=nis plotted versusxB, whereÅX(mix) is.XnAXAnBXB/. On this
plot, the tangent to the curve at the composition of interest has intercepts equal toXAXA
atxBD 0 andXBXBatxBD 1.
We can obtain experimental values of such partial molar quantities of an uncharged
species asVi,Cp;i, andSi. It is not possible, however, to evaluate the partial molar quanti-
tiesUi,Hi,Ai, andGibecause these quantities involve the internal energy brought into the
system by the species, and we cannot evaluate the absolute value of internal energy (Sec.
2.6.2). For example, while we can evaluate the differenceHiHifrom calorimetric mea-
surements of enthalpies of mixing, we cannot evaluate the partial molar enthalpyHiitself.
We can, however, include such quantities asHiin useful theoretical relations.

As mentioned on page 226 , a partial molar quantity of achargedspecies is something

else we cannot evaluate. It is possible, however, to obtain values relative to a reference