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2
2
Chapter 9 where z/( x y) z/( y x) [Eq. (1.36)] was used. Use of (9.20) and (9.22) gives
Solutions
0m i 0G i
a b a b S i (9.30)
0T 0T
P,n j P,n j
which corresponds to (9.29) with extensive variables replaced by partial molar quan-
tities. Similarly, partial differentiation with respect to n of ( G/ P) T,n j V leads to
i
0m i 0G i
a b a b V i (9.31)
0P 0P
T,n j T,n j
The subscript n in (9.31) indicates that all mole numbers are held constant.
j
Importance of the Chemical Potentials
The chemical potentials are the key properties in chemical thermodynamics. The ’s
m
i
determine reaction equilibrium and phase equilibrium [Eqs. (4.88) and (4.98)].
Moreover, all other partial molar properties and all thermodynamic properties of the
solution can be found from the m ’s if we know the chemical potentials as functions of
i
T, P, and composition. The partial derivatives of m i with respect to T and P give S i

and V i [Eqs. (9.30) and (9.31)]. The use of m i H TS i [Eq. (9.28)] then gives H .
i
i

The use of U H PV i (Prob. 9.19) and C P,i 10H >0T 2 P,n j gives U i and C . Once
i
i
i
P,i

we know the partial molar quantities , S ,m i i V i , etc., we get the solution properties as

G © i n G i , S © i n S , V © i n V , etc. [Eq. (9.26)]. Note that knowing V as a
i
i i
i
i
function of T, P, and composition means we know the equation of state of the solution.

EXAMPLE 9.4 Use of M to get V i
i

Find V i for a component of an ideal gas mixture starting from m . i
The chemical potential of a component of an ideal gas mixture is [Eq. (6.4)]
m m°1T2 RT ln 1P >P°2 m°1T2 RT ln 1x P>P°2
i
i
i
i
i

Use of V 10m >0P2 [Eq. (9.31)] gives, in agreement with (9.11),
i i T,n j
0ln1x P>P°2 RT
i
V RT a b
i
0P P
T,n j
Exercise

Use the result V i RT/P to verify the relation V n V i [Eq. (9.16)] for an
i
i
ideal gas mixture.
Summary
The partial molar volume V i of component i in a solution of volume V is defined as

V 10V>0n 2 . The solution’s volume is given by V © i n V i . Similar equa-
i
i

i T,P,n j i
G
tions hold for other extensive properties (U, H, S, G, etc.). Relations between , ,
H
i
i

S i , and V i were found; these resemble corresponding relations between G, H, S, and V.
All thermodynamic properties of a solution can be obtained if the chemical potentials

m G i are known as functions of T, P, and composition.
i
9.3 MIXING QUANTITIES
Similar to defining V V V* at constant T and P [Eq. (9.17)], one defines other
mix
mixing quantities for a solution. For example,
¢ mix H H H*, ¢ mix S S S*, ¢ mix G G G*