Thermodynamics and Chemistry

CHAPTER 9 MIXTURES

9.2 PARTIALMOLARQUANTITIES 230

Equation9.2.12is theGibbs–Duhem equationfor a binary mixture, applied to partial
molar volumes. (Section9.2.4will give a general version of this equation.) Dividing both
sides of the equation bynACnBgives the equivalent form

xAdVACxBdVBD 0 (9.2.13)

(binary mixture,

constantTandp)

Equation9.2.12shows that changes in the values ofVAandVBare related when the
composition changes at constantTandp. If we rearrange the equation to the form

dVAD

nB

nA

dVB (9.2.14)

(binary mixture,

constantTandp)

we see that a composition change thatincreasesVB(so that dVBis positive) must makeVA
decrease.

9.2.3 Evaluation of partial molar volumes in binary mixtures

The partial molar volumesVAandVBin a binary mixture can be evaluated by themethod
of intercepts. To use this method, we plot experimental values of the quantityV=n(where
nisnACnB) versus the mole fractionxB.V=nis called themean molar volume.
See Fig.9.3(a) on the next page for an example. In this figure, the tangent to the
curve drawn at the point on the curve at the composition of interest (the composition used
as an illustration in Sec.9.2.1) intercepts the vertical line wherexBequals 0 atV=nD
VAD17:7cm^3 mol^1 , and intercepts the vertical line wherexBequals 1 atV=nDVBD
38:8cm^3 mol^1.

To derive this property of a tangent line for the plot ofV=nversusxB, we use Eq.9.2.9

to write

.V=n/D

VAnACVBnB

n

DVAxACVBxB

DVA.1xB/CVBxBD.VBVA/xBCVA (9.2.15)

When we differentiate this expression forV=nwith respect toxB, keeping in mind that

VAandVBare functions ofxB, we obtain

d.V=n/

dxB

D

då.VBVA/xBCVAç

dxB

DVBVAC



dVB

dxB



dVA

dxB



xBC

dVA

dxB

DVBVAC

dV

A

dxB



.1xB/C

dV

B

dxB



xB

DVBVAC



dVA

dxB



xAC



dVB

dxB



xB (9.2.16)

The differentials dVAand dVBare related to one another by the Gibbs–Duhem equation

(Eq.9.2.13):xAdVACxBdVBD 0. We divide both sides of this equation by dxBto