Physical Chemistry Third Edition

4.6 Euler’s Theorem and the Gibbs–Duhem Relation 193

Experimental

curve

Tangent line

Value of V 2 at x 15 0.500

0 0.2

0

0.4

x 1

0.6 0.8 1.0

Tangent point

atx 15 0.500 5 x ́ 1

10

Vm

/cm

3 mol

21

20

30

40

50

60

Value of

V 1 at

x 15 0.500

_

_

Figure 4.2 The Mean Molar Volume of an Ethanol-Water Solution as a Function of

Mole Fraction of Ethanol.

EXAMPLE4.24

From the intercepts in Figure 4.3, determine the partial molar volumes of ethanol (substance 1)

and water (substance 2) if the mole fraction of ethanol is equal to 0.500. The molar volumes

of the pure substances areVm,1∗  58 .4cm^3 mol−^1 ,Vm,2∗  18 .02 cm^3 mol−^1.

Solution

The tangent line is drawn in the figure.

right interceptV 1 −Vm,1∗ − 0 .83 cm^3 mol−^1

left interceptV 2 −Vm,2∗ − 1 .31 cm^3 mol−^1

V 1  58 .4cm^3 mol−^1 − 0 .83 cm^3 mol−^1  57 .6cm^3 mol−^1

V 2  18 .02 cm^3 mol−^1 − 1 .31 cm^3 mol−^1  16 .7cm^3 mol−^1

If the method is to be applied to the partial molar Gibbs energy, the partial molar

energy, the partial molar enthalpy, or the partial molar entropy, the second version of

the method must be used since the actual values ofG,U,H, andSare not defined

(they can have any constants added to their values without physical effect).