Thermodynamics and Chemistry

CHAPTER 9 MIXTURES

9.2 PARTIALMOLARQUANTITIES 232

versusxB, as illustrated in Fig.9.3(b). ÅV(mix) is the integral volume of mixing—the
volume change at constantTandpwhen solvent and solute are mixed to form a mixture of
volumeVand total amountn(see Sec.11.1.1). The tangent to the curve at the composition
of interest has interceptsVAVm;AatxBD 0 andVBVm;BatxBD 1.

To see this, we write

ÅVm(mix)D.V=n/xAVm;AxBVm;B

D.V=n/.1xB/Vm;AxBVm;B (9.2.20)

We make the substitution.V=n/D.VBVA/xBCVAfrom Eq.9.2.15and rearrange:

ÅVm(mix)D



VBVm;B

VAVm;A



xBC

VAVm;A

(9.2.21)

Differentiation with respect toxByields

dÅVm(mix)

dxB

D

VBVm;B

VAVm;A

C



dVB

dxB



dVA

dxB



xBC

dVA

dxB

D

VBVm;B

VAVm;A

C



dVA

dxB



.1xB/C



dVB

dxB



xB

D

VBVm;B

VAVm;A

C



dVA

dxB



xAC



dVB

dxB



xB

(9.2.22)

With a substitution from Eq.9.2.17, this becomes

dÅVm(mix)

dxB

D

VBVm;B

VAVm;A

(9.2.23)

Equations9.2.21and9.2.23are analogous to Eqs.9.2.15and9.2.18, withV=nre-

placed byÅVm(mix),VAby.VAVm;A/, andVBby.VBVm;B/. Using the same

reasoning as for a plot ofV=nversusxB, we find the intercepts of the tangent to a

point on the curve ofÅVm(mix) versusxBare atVAVm;AandVBVm;B.

Figure9.3shows smoothed experimental data for water–methanol mixtures plotted in
both kinds of graphs, and the resulting partial molar volumes as functions of composition.
Note in Fig.9.3(c) how theVAcurve mirrors theVBcurve asxBvaries, as predicted by the
Gibbs–Duhem equation. The minimum inVBatxB0:09is mirrored by a maximum inVA
in agreement with Eq.9.2.14; the maximum is much attenuated becausenB=nAis much
less than unity.

Macroscopic measurements are unable to provide unambiguous information about mo-

lecular structure. Nevertheless, it is interesting to speculate on the implications of the

minimum observed for the partial molar volume of methanol. One interpretation is that

in a mostly aqueous environment, there is association of methanol molecules, perhaps

involving the formation of dimers.

9.2.4 General relations

The discussion above of partial molar volumes used the notationVm;AandVm;Bfor the
molar volumes of pure A and B. The partial molar volume of a pure substance is the same as