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(pure) components at T and P. The star indicates a property of a pure substance or a Section 9.2
collection of pure substances. We have Partial Molar Quantities

V* n V* n V* p n V* a nV* (9.4)
m,r
i
m,i
r
1 m,1
2
m,2
i
After mixing, one finds that the volume V of the solution is not in general equal to the
3
3
unmixed volume; V V*. For example, addition of 50.0 cm of water to 50.0 cm of
3
ethanol at 20°C and 1 atm gives a solution whose volume is only 96.5 cm at 20°C and
1 atm (Fig. 9.1). The difference between V of the solution and V* results from (a) dif-
ferences between intermolecular forces in the solution and those in the pure compo-
nents; (b) differences between the packing of molecules in the solution and the pack-
ing in the pure components, due to differences in sizes and shapes of the molecules
being mixed.
We can write an equation like (9.4) for any extensive property, for example, U, H,
S, G, and C . One finds that each of these properties generally changes on mixing the
P
components at constant T and P.
We want expressions for the volume V of the solution and for its other extensive Figure 9.1
properties. Each such property is a function of the solution’s state, which can be spec-
Volume V of a solution formed by
ified by the variables T, P, n , n , ..., n . Therefore
1 2 r mixing a volume V ethanol of pure
V V1T, P, n , . . . , n 2, U U1T, P, n , . . . , n 2 (9.5) ethanol with a volume
r
1
r
1
3
(100 cm V ethanol ) of pure water
with similar equations for H, S, etc. The total differential of V in (9.5) is at 20°C and 1 atm.
0V 0V 0V 0V
dV a b dT a b dP a b dn p a b dn r
1
0T 0P 0n 0n
P,n i T,n i 1 T,P,n i 1 r T,P,n i r
(9.6)
The subscript n in the first two partial derivatives indicates that all mole numbers are
i
held constant; the subscript n indicates that all mole numbers except n are held
i 1 1
constant. We define the partial molar volume V j of substance j in the solution as
0V
V a b one-phase syst. (9.7)*
j
0n j T,P,n i j
where V is the solution’s volume and where the partial derivative is taken with T, P,

and all mole numbers except n held constant. (The bar in V does not signify an av-
j j
erage value.) Equation (9.6) becomes
0V 0V
dV a b dT a b dP a V dn i (9.8)
i
0T 0P
P,n i T,n i i
Equation (9.8) gives the infinitesimal volume change dV that occurs when the temper-
ature, pressure, and mole numbers of the solution are changed by dT, dP, dn , dn , ....
1 2
From (9.7), a partial molar volume is the ratio of infinitesimal changes in two

extensive properties and so is an intensive property. Like any intensive property, V i
depends on T, P, and the mole fractions in the solution:

V V 1T, P, x , x , p 2 (9.9)
i
1
2
i
From (9.7), if dV is the infinitesimal change in solution volume that occurs when
dn moles of substance j is added to the solution with T, P, and all mole numbers
j
except n held constant, then V equals dV/dn . See Fig. 9.2. V is the rate of change of
j j j j
solution volume with respect to n at constant T and P. The partial molar volume V of
j j
substance j in the solution tells how the solution’s volume V responds to the constant-

T-and-P addition of j to the solution; dV equals V j dn when j is added at constant T
j
and P.