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electrolyte solutions, both very large and very small values of g can occur. For exam- Section 10.7

ple, in aqueous solution at 25°C and 1 atm, g [UO (ClO ) ] 1510 at m 5.5 mol/kg The Debye–Hückel Theory of

Electrolyte Solutions
2
i
4 2
and g (CdI ) 0.017 at m 2.5 mol/kg.

2
i
The Practical Osmotic Coefficient. Although the solvent’s chemical potential m A
can be expressed using the mole-fraction scale [Eq. (10.39)], it is customary in work with
electrolyte solutions to express m in terms of the (solvent) practical osmotic coefficient
A
f (phi). The reason for using f instead of the solvent activity coefficient is that, in dilute
electrolyte solutions, the solvent activity coefficient may be very close to 1 even though
the solute activity coefficient deviates substantially from 1 and the solution is far from ide-
ally dilute. It is inconvenient to work with activity coefficients with values like 1.0001. For
details, see Probs. 10.31 to 10.34.
10.7 THE DEBYE–HÜCKEL THEORY
OF ELECTROLYTE SOLUTIONS
In 1923, Debye and Hückel used a highly simplified model of an electrolyte solution
and statistical mechanics to derive theoretical expressions for the ionic activity coeffi-
cients g and g . In their model, the ions are taken to be uniformly charged hard

spheres of diameter a. The difference in size between the positive and negative ions is
ignored, and a is interpreted as the mean ionic diameter. The solvent A is treated as a
structureless medium with dielectric constant e (epsilon r, A). [If F is the force be-
r,A
tween two charges in vacuum and F is the net force between the same charges im-
A
mersed in the dielectric medium A, then F /F 1/e ; see (13.89).]
A r,A
The Debye–Hückel treatment assumes that the solution is very dilute. This re-
striction allows several simplifying mathematical and physical approximations to be
made. At high dilution, the main deviation from ideally dilute behavior comes from
the long-range Coulomb’s law attractions and repulsions between the ions. Debye and
Hückel assumed that all the deviation from ideally dilute behavior is due to interionic
Coulombic forces.
An ion in solution is surrounded by an atmosphere of solvent molecules and other
ions. On the average, each positive ion will have more negative ions than positive ions
in its immediate vicinity. Debye and Hückel used the Boltzmann distribution law of
statistical mechanics (Sec. 21.5) to find the average distribution of charges in the
neighborhood of an ion.
They then calculated the activity coefficients as follows. Let the electrolyte solu-
tion be held at constant T and P. Imagine that we have the magical ability to vary the
charges on the ions in the solution. We first reduce the charges on all the ions to zero;
the Coulombic interactions between the ions disappear, and the solution becomes ide-
ally dilute. We now reversibly increase all the ionic charges from zero to their values
in the actual electrolyte solution. Let w be the electrical work done on the system in
el
this constant-T-and-P charging process. Equation (4.24) shows that for a reversible
constant-T-and-P process, G w ; in this case, w w . Debye and
non-P-V non-P-V el
Hückel calculated w from the electrostatic potential energy of interaction between
el
each ion and the average distribution of charges in its neighborhood during the charg-
ing process. Since the charging process starts with an ideally dilute solution and ends
with the actual electrolyte solution, G is G G id-dil , where G is the actual Gibbs en-
ergy of the solution and G id-dil is the Gibbs energy the solution would have if it were
ideally dilute. Therefore G G id-dil w .
el
G id-dil is known from G id-dil n m id-dil , and G G id-dil is known from calcula-
j j j
tion of w . Therefore G of the solution is known. Taking
G/
n and
G/
n , one gets
el
the ionic chemical potentials m and m , so the activity coefficients g and g in

(10.40) are known. (For a full derivation, see Bockris and Reddy, sec. 3.3.)