4.6 Euler’s Theorem and the Gibbs–Duhem Relation
193
Experimental
curve
Tangent line
Value of V 2 at x 1 5 0.500
0
0.2
0
0.4
x 1
0.6
0.8
1.0
Tangent point
at x 1 5 0.500 5 x´ 1
10
V
m / cm 3
mol 21
20
30
40
50
60
Value of
V 1 at
x 1 5 0.500
_
_
Figure 4.2 The Mean Molar Volume of an Ethanol-Water Solution as a Function of
Mole Fraction of Ethanol.
E X A M P L E 4.24
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 are V ∗
m,1 58.4 cm 3 mol −1 , V ∗
m,2 18.02 cm 3 mol −1 .
Solution
The tangent line is drawn in the figure.
right intercept V 1 − V ∗
m,1 −0.83 cm 3 mol −1
left intercept V 2 − V ∗
m,2 −1.31 cm 3 mol −1
V 1 58.4 cm 3 mol −1 − 0.83 cm 3 mol −1 57.6 cm 3 mol −1
V 2 18.02 cm 3 mol −1 − 1.31 cm 3 mol −1 16.7 cm 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 of G, U, H, and S are not defined
(they can have any constants added to their values without physical effect).