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Chapter 9 a substance in a phase must increase as the i mole fraction x increases at constant T
i
Solutions and P [Eq. (4.90)].
Summary
mix G data (as found from vapor-pressure measurements) and statistical-mechanical
arguments show that in solutions where the molecules of different species resemble
one another extremely closely in size, shape, and intermolecular interactions, the
chemical potential of each species is given by m m*(T, P) RT ln x ; such a solu-
i
i
i
tion is called an ideal solution.
9.6 THERMODYNAMIC PROPERTIES OF IDEAL SOLUTIONS
In the last section we started with the molecular definition of an ideal solution and ar-
rived at the thermodynamic definition (9.41). This section uses the chemical potentials
(9.41) to derive thermodynamic properties of ideal solutions. Before doing so, we first
define standard states for ideal-solution components.

Standard States
Standard states were defined for pure substances in Sec. 5.1 and for components of an
ideal gas mixture in Sec. 6.1. The standard state of each component i of an ideal
liquid solution is defined to be pure liquid i at the temperature T and pressure P of the
solution. For solid solutions, we use the pure solids. We have m° m*(T, P), where,
i i
as always, the degree superscript denotes the standard state and the star superscript
indicates a pure substance. The ideal-solution definition (9.41) then is
m m* RT ln x ideal soln. (9.42)*
i
i

i
m° m *1T, P2 ideal soln. (9.43)*
i
i
where m is the chemical potential of component i present with mole fraction x in an
i i
ideal solution at temperature T and pressure P, and m* is the chemical potential of pure
i
i at the temperature and pressure of the solution.
Mixing Quantities
If we know mixing quantities such as G, V, and H, then we know the
mix mix mix
values of G, V, H, etc., for the solution relative to values for the pure components. All
the mixing quantities are readily obtained from the chemical potentials (9.42).
We have G G G* n (m m*) [Eqs. (9.32) and (9.22)]. Equation
mix i i i i
(9.42) gives m m* RT ln x . Therefore
i i i
¢ mix G RT a n ln x ideal soln., const. T, P (9.44)
i
i
i
which is the same as (9.39). Since 0
x
1, we have ln x
0 and G
0, as
i i mix
must be true for an irreversible (spontaneous) process at constant T and P.
From (9.34), V ( G/ P) . But the ideal-solution G in (9.44) does
mix mix T,n i mix
not depend on P. Therefore
¢ V 0 ideal soln., const. T, P (9.45)
mix
There is no volume change on forming an ideal solution from its components at con-
stant T and P, as expected from the molecular definition (Sec. 9.5).
From (9.35), S ( G/ T) . Taking / T of (9.44), we get
mix mix P,n i
¢ mix S R a n ln x ideal soln., const. T, P (9.46)
i
i
i
which is positive. S is the same for ideal solutions as for ideal gases [Eq. (3.33)].
mix