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Section 9.4
Determination of
Partial Molar Quantities
Figure 9.8
mix V/n for water–ethanol
solutions at 20°C and 1 atm. The
tangent line is used to find the
partial molar volumes at ethanol
mole fraction equal to 0.5.

EXAMPLE 9.5 Intercept method for V i

Figure 9.8 plots mix V/n against x C 2 H 5 OH for water–ethanol solutions at 20°C and
1 atm. Use this plot to find the partial molar volumes of water (W) and ethanol
(E) in a solution with x 0.5, given that at 20°C and 1 atm, V is 18.05
m
E
3
3
cm /mol for water and 58.4 cm /mol for ethanol.
The tangent line to the curve is drawn at x 0.5. Its intercept at x 0 is

E
E
3
at 1.35 cm /mol, so V 3 W V* 1.35 cm /mol and V W 18.05 cm /mol
3
m,W
3
3
1.35 cm /mol 16.7 cm /mol at x 0.5. The tangent line intersects x 1

E
E

at 0.8 cm 3 /mol, so V V* 0.8 cm 3 /mol and V 57.6 cm 3 /mol at
E
E
m,E
x 0.5.
E
Exercise

Use these results for V E and V W and the equation V n V i to calculate the
i
i
volume of a mixture of 0.50 mol of water and 0.50 mol of ethanol at 20°C and
1 atm. Use V (V V*) V* and Fig. 9.8 to calculate this volume and com-
3
3
pare the results. (Answers: 37.1 cm , 37.1 cm .)
4
5
Exercise

Use Fig. 9.8 to find V E and V W in a solution composed of 3.50 mol of ethanol and
3
3
1.50 mol of water at 20°C and 1 atm. (Answers: 58.0 cm /mol, 16.0 cm /mol.)
5
Drawing tangents at several solution compositions in Fig. 9.8 and using the inter-
cepts to find the partial molar volumes at these compositions, one obtains the results

shown in Fig. 9.9. This figure plots V E and V W against solution composition. Note that

when V E is decreasing, V W is increasing, and vice versa. We will see in Sec. 10.3 that

dV A and dV B must have opposite signs at constant T and P in a two-component solu-

tion. The limiting value of V E at x 1 is the molar volume of pure ethanol.
E
A third way to find partial molar volumes is to fit solution volume data for fixed

n to a polynomial in n . Differentiation then gives V B . See Prob. 9.25.
A
B
Partial Molar Enthalpies, Entropies, and Gibbs Energies

Similar to V n V , the enthalpy H of a solution is given by H n H
i i i i i i
[Eq. (9.26)], where the partial molar enthalpy H i of substance i is H 10H>0n 2
i
i T,P,n j i
[Eq. (9.20)]. The enthalpy of mixing to form the solution from its pure components at