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molecules. Because the B and C molecules have no differences in intermolecular in- Section 9.5
teractions or sizes and shapes, the B and C molecules have no preference as to their Ideal Solutions
locations and will be distributed at random in the container. We want the probability
ratio p /p for the condition of random spatial distribution in state 2, with each mole-
1
2
cule having no preference as to which molecules are its neighbors.
We could use probability theory to calculate p and p , but this is unnecessary be-
2
1
cause we previously dealt with the same situation of the random distribution of two
species in a container. When two ideal gases mix at constant T and P, there is random
distribution of the B and C molecules. For both an ideal gas mixture and an ideal so-
lution, the probability that any given molecule is in the left part of the mixture equals
V*/(V* V*) V*/V, where V* and V* are the unmixed volumes of B and C and V
B
C
B
C
B
B
is the mixture’s volume. Therefore p and p will be the same for an ideal solution as
1
2
for an ideal gas mixture, and mix S, which equals k ln (p /p ), will be the same for ideal
1
2
solutions and ideal gas mixtures.
For ideal gases, Eq. (3.32) gives
¢ mix S n R ln 1V*>V2 n R ln 1V*>V2
B
C
B
C
and this equation gives S for ideal solutions. Since B and C molecules have the
mix
same size and the same intermolecular forces, B and C have equal molar volumes:
V* V* . Substitution of V* n V* , V* n V* n V* , and V V* V*
m,B m,C B B m,B C C m,C C m,B B C
(n n )V* into the above S equation gives S n R ln x n R ln x
B C m,B mix mix B B C C
for an ideal solution.
Substitution of H 0 and S n R ln x n R ln x into G
mix mix B B C C mix
H T S then gives the experimentally observed G equation (9.39) for
mix mix mix
ideal solutions.
It might seem puzzling that an equation like (9.39) that applies to ideal liquid mix-
tures and solid mixtures would contain the gas constant R. However, R is a far more
fundamental constant than simply the zero-pressure limit of PV/nT of a gas. R (in the
form R/N k) occurs in the fundamental equation (3.52) for entropy and occurs in
A
other fundamental equations of statistical mechanics (Chapter 21).
As noted in Sec. 9.2, the chemical potentials m in the solution are the key ther-
i
modynamic properties, so we now derive them from G of Eq. (9.39). We have
mix
G G G* n m n m*[Eq. (9.32)]. For an ideal solution, G
mix i i i i i i mix
RT n ln x [Eq. (9.39)]. Equating these G expressions, we get
i i i mix
a n m a n 1m* RT ln x 2 (9.40)
i
i
i
i
i
i i
where the sum identities (1.50) were used. The only way this last equation can hold
for all n values is if (see Prob. 9.63 for a rigorous derivation)
i
m m*1T, P2 RT ln x ideal soln. (9.41)
i
i
i
where (since G is at constant T and P), m*(T, P) is the chemical potential of pure
mix i
substance i at the temperature T and pressure P of the solution.
We shall adopt (9.41) as the thermodynamic definition of an ideal solution. A
solution is ideal if the chemical potential of every component in the solution obeys
(9.41) for all solution compositions and for a range of T and P.
Just as the ideal-gas law PV nRT is approached in the limit as the gas density
goes to zero, the ideal-solution law (9.41) is approached in the limit as the solution
components resemble one another more and more closely, without, however, becom-
ing identical. Figure 9.15
Figure 9.15 plots m versus x at fixed T and P for an ideal solution, where m
i i i The chemical potential m of a
m* RT ln x . As x → 0, m → q. As x increases, m increases, reaching the i
i i i i i i component of an ideal solution
chemical potential m* of pure i in the limit x 1. Recall the general result that m of plotted versus x at fixed T and P.
i i i i