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Analogous to (9.16) and (9.19), the Gibbs energy G of a solution is Section 9.2
Partial Molar Quantities

G a n G a n m one-phase syst. (9.23)
i
i
i
i
i i
Equations like (9.23) and (9.19) show the key role of partial molar properties in solu-
tion thermodynamics. Each extensive property of a solution is expressible in terms of
partial molar quantities.
If Y is any extensive property of a solution, the corresponding partial molar prop-
erty of component i of the solution is defined by

Y 10Y>0n 2 (9.24)*
i
i T,P,n j i
Partial molar quantities are the ratio of two infinitesimal extensive quantities and so
are intensive properties. Analogous to (9.8), dY is
0Y 0Y
dY a b dT a b dP a Y dn i (9.25)
i
0T 0P i
P,n i T,n i
The same reasoning that led to (9.16) gives for the Y value of the solution

Y a n Y one-phase syst. (9.26)*
i i
i

Equation (9.26) suggests that we view n Y i as the contribution of solution
i
component i to the extensive property Y of the phase. However, such a view is over-

simplified. The partial molar quantity Y i is a function of T, P, and the solution mole

fractions. Because of intermolecular interactions, Y i is a property of the solution as a
whole, and not a property of component i alone.
As noted at the end of Sec. 7.1, for a system in equilibrium, the equation dG
SdT VdP m dn is valid whether the sum is taken over all species actually
i
i
i

present or over only the independent components. Similarly, the relations G © i n G i

i
and Y © i n Y i [Eqs. (9.23) and (9.26)] are valid if the sum is taken over all chemi-
i
cal species, using the actual number of moles of each species present, or over only the
independent components, using the apparent numbers of moles present and ignoring
chemical reactions. The proof is essentially the same as given in Prob. 7.70.
Relations between Partial Molar Quantities
For most of the thermodynamic relations between extensive properties of a homoge-
neous system, there are corresponding relations with the extensive variables replaced
by partial molar quantities. For example, G, H, and S of a solution satisfy
G H TS (9.27)
If we differentiate (9.27) partially with respect to n at constant T, P, and n j i and use
i

the definitions (9.20) to (9.22) of , , and S ,GH i i i we get
10G>0n 2 10H>0n 2 T10S>0n 2
i T,P,n j i i T,P,n j i i T,P,n j i

m G H TS i (9.28)
i
i
i
which corresponds to (9.27).
Another example is the first equation of (4.70):
0G
a b S (9.29)
0T
P,n j
Partial differentiation of (9.29) with respect to n gives
i
0S 0 0G 0 0G
a b a a b b a a b b
0n i T,P,n j i 0n i 0T P,n j T,P,n j i 0T 0n i T,P,n j i P,n j