Thermodynamics and Chemistry

CHAPTER 9 MIXTURES

9.2 PARTIALMOLARQUANTITIES 231

bc

0 0:20:40:60:81:0

10

15

20

25

30

35

40

45

xB

.

.V

=n/=

cm

3 mol

^1

(a)

bc

0 0:20:40:60:81:0

 2 :5

 2 :0

1:5

1:0

0:5

0:0

xB

V

.m

mix

/=

cm

3 mol

^1

(b)

bc VB

0 0:20:40:60:81:0

36

37

38

39

40

41

bc

VA

0 0:20:40:60:81:0

14

15

16

17

18

19

20

xB

Vi

=cm

3 mol

^1

(c)

Figure 9.3 Mixtures of water (A) and methanol (B) at 25 C and 1 bar.a

(a) Mean molar volume as a function ofxB. The dashed line is the tangent to the curve

atxBD0:307.

(b) Molar volume of mixing as a function ofxB. The dashed line is the tangent to the

curve atxBD0:307.

(c) Partial molar volumes as functions ofxB. The points atxBD0:307(open circles)

are obtained from the intercepts of the dashed line in either (a) or (b).

aBased on data in Ref. [ 12 ].

obtain 

dVA

dxB



xAC



dVB

dxB



xBD 0 (9.2.17)

and substitute in Eq.9.2.16to obtain

d.V=n/

dxB

DVBVA (9.2.18)

Let the partial molar volumes of the constituents of a binary mixture of arbitrary

compositionxB^0 beVA^0 andVB^0. Equation9.2.15shows that the value ofV=nat the

point on the curve ofV=nversusxBwhere the composition isxB^0 is.VB^0 VA^0 /xB^0 CVA^0.

Equation9.2.18shows that the tangent to the curve at this point has a slope ofVB^0 VA^0.

The equation of the line that passes through this point and has this slope, and thus is

the tangent to the curve at this point, isyD.VB^0 VA^0 /xBCVA^0 , whereyis the vertical

ordinate on the plot of.V=n/versusxB. The line has interceptsyDVA^0 atxBD 0 and

yDVB^0 atxBD 1.

A variant of the method of intercepts is to plot the molar integral volume of mixing
given by

ÅVm(mix)D

ÅV(mix)

n

D

V nAVm;AnBVm;B

n

(9.2.19)