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less than 0.1, where z is the average number of nearest neighbors for solute i. For ap- Section 9.8
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proximately spherical solute and solvent molecules of similar size, z is roughly 10. Thermodynamic Properties
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of Ideally Dilute Solutions
For solutes with large molecules (for example, polymers), z can be much larger. A
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polymer solution becomes ideally dilute at a much lower solute mole fraction than a
nonpolymer solution, since much higher dilutions are required to ensure that a poly-
mer solute molecule is, to a high probability, surrounded only by solvent molecules.
Ideal solutions and ideally dilute solutions are different and must not be confused
with each other. Unfortunately, people sometimes use the term “ideal solution” when
what is meant is an ideally dilute solution.
At the high dilutions for which (9.55) applies, the mole fraction x is proportional
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to the molar concentration c and to the molality m to a high degree of approximation
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(Prob. 9.8). Therefore (9.55) can be written as m RT ln c h (T, P) or as m
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RT ln m k (T, P), where h and k are functions related to f . Therefore, molalities or
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molar concentrations can be used instead of mole fractions in dealing with solutes in
ideally dilute solutions.
Summary
diln G data (as found from vapor-pressure measurements) and statistical-mechanical
arguments show that in the limit of high dilution of a solution (x close to 1), the solute
A
chemical potentials are given by m f (T, P) RT ln x and the solvent chemical
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potential is m m*(T, P) RT ln x . This is an ideally dilute solution.
A
A
A
9.8 THERMODYNAMIC PROPERTIES OF
IDEALLY DILUTE SOLUTIONS
Before deriving thermodynamic properties of ideally dilute solutions from the chem-
ical potentials (9.55) and (9.56), we define the standard states for components of ide-
ally dilute solutions.
Standard States
The standard state of the solvent A in an ideally dilute solution is defined to be pure
A at the temperature T and pressure P of the solution. Therefore the solvent standard-
state chemical potential is m° m*(T, P), and (9.56) can be written as m m°
A
A
A
A
RT ln x for the solvent.
A
Now consider the solutes. From (9.55), we have m f (T, P) RT ln x . The stan-
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dard state of solute i is defined so as to make its standard-state chemical potential m° i
equal to f (T, P); m° f (T, P). This definition of m° gives
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m m° RT ln x solute in ideally dil. soln. (9.57)
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What choice of solute standard state is implied by taking m° equal to f (T, P)? When
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x becomes 1 in (9.57), the log term vanishes and the equation gives m (at x 1) as
i i i
equal to m°. It might therefore be thought that the standard state of solute i is pure i at
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the temperature and pressure of the solution. This supposition is wrong. The ideally
dilute solution relation (9.57) is valid only for high dilution (where x is much less
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than 1), and we cannot legitimately take the limit of this relation as x goes to 1.
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However, one could imagine a hypothetical case in which m m° RT ln x holds
i i i
for all values of x . In this hypothetical case, m would become equal to m° in the limit
i i i
x → 1. The choice of solute standard state uses this hypothetical situation. The standard
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state for solute i in an ideally dilute solution is defined to be the fictitious state at the
temperature and pressure of the solution that arises by supposing that m m°
i i
RT ln x holds for all values of x and setting x 1. This hypothetical state is an extrapo-
i i i
lation of the properties of solute i in the very dilute solution to the limit x → 1.
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