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Chapter 9 Ideal Gas Mixtures
Solutions We have thought in terms of liquid and solid ideal solutions in this section. However,
it is clear that an ideal gas mixture meets the molecular definition of an ideal solution,
since mixing ideal gases produces no energetic or structural changes. Moreover, one
can show (Prob. 9.45) that the chemical potentials in an ideal gas mixture can be put
in the form (9.41) defining an ideal solution. An ideal gas mixture is an ideal solution.

9.7 IDEALLY DILUTE SOLUTIONS

An ideal solution occurs in the limit where the molecules of the different species re-
semble one another very closely. A different kind of limit is where the solvent mole
fraction approaches 1, so that all solutes are present in very low concentrations. Such
a solution is called an ideally dilute (or ideal-dilute) solution. In an ideally dilute so-
lution, solute molecules interact essentially only with solvent molecules, because of
the high dilution of the solutes.
Consider such a very dilute solution of nonelectrolytes. (In electrolyte solutions, the
strong interionic forces give substantial solute–solute interactions even at very high di-
lutions; hence the ideally dilute solution model is not useful for electrolyte solutions.
Also, each electrolyte gives two or more ions in solution, and so the chemical potential
m of an electrolyte solute differs in form from m of a nonelectrolyte, even in the limit
i i
of infinite dilution. Electrolyte solutions are treated in Chapter 10.) We shall use A to
denote the solvent and i to signify any one of the solutes. The condition of high dilution
is that the solvent mole fraction x is very close to 1. For such a very dilute solution,
A
solute molecules are generally surrounded by only solvent molecules, so that all solute
Figure 9.19 molecules are in an essentially uniform environment; see Fig. 9.19.
To arrive at a thermodynamic definition of an ideally dilute solution, one uses
In an ideally dilute solution, solute vapor-pressure data for highly dilute nonelectrolyte solutions to arrive at an equation
molecules (shaded) interact only
with solvent molecules. for diln G, the Gibbs energy change that occurs when an ideally dilute solution is di-
luted by the addition of a certain amount of the solvent A. One then derives the chem-
ical potentials m and m in the ideally dilute solution from the G equation in the
i A diln
same way that the chemical potentials (9.41) in an ideal solution were derived from
the G equation (9.39). The details of the derivation of m and m from G are
mix i A diln
given in Prob. 9.47. One finds
m RT ln x f 1T, P2 solute in ideally dil. soln. (9.55)
i
i
i
m m*1T, P2 RT ln x solvent in ideally dil. soln. (9.56)
A
A
A
where R is the gas constant, f (T, P) is some function of T and P, m*(T, P)
i A
G* (T, P) is the chemical potential of pure liquid solvent A at the T and P of the so-
m,A
lution, and x and x are the mole fractions of solute i and solvent A in the solution.
i A
Statistical-mechanical derivations of (9.55) and (9.56) are given in E. A. Guggenheim,
Mixtures, Oxford, 1952, sec. 5.04; A. J. Staverman, Rec. Trav. Chim., 60, 76 (1941).
The laws of thermodynamics are general and cannot supply us with the explicit forms
of equations of state or chemical potentials for specific systems. Such information
must be obtained by appeal to molecular (statistical-mechanical) arguments or to ex-
perimental data (as in the use of PV nRT for low-density gases).
We adopt as the thermodynamic definition: An ideally dilute solution is one in
which the solute and solvent chemical potentials are given by (9.55) and (9.56) for a
range of composition with x close to 1 and for a range of T and P.
A
As a real solution becomes more dilute, the chemical potentials approach (9.55)
and (9.56) more closely. Just how dilute a solution must be in order to be considered
ideally dilute depends on how accurately one wants to represent the solution’s ther-
modynamic properties. A rough rule for nonelectrolyte solutions is that z x should be
i i