Coulombic interactions between ions in solution are relatively
strong, long-range forces compared to the other types of intermolecular
force in solution. They are thus an important contributor to
the non-ideality of ionic solutions, and in the Debye-Hückel
theory of such solutions, they are taken to dominate the non-ideality
to such an extent that all other contributions may be neglected.
The theory is based around the simple fact that oppositely
charged ions attract one another, whilst like-charged ions repel
As a result, the motion of ions in solution
is not entirely random; there is a slight tendency for ions
of opposite charge to encounter each other more frequently than
ions of the same charge.A time-averaged picture of the solution
shows that near any given ion, there is an excess of counter-ions.
This time-averaged spherical distribution (in which ions of
the same charge as the central ion and counter-ions are both
present, though counter-ions predominate) has a net charge equal
in magnitude but opposite in sign to the central ion, and is
known as its ionic atmosphere.
There is a stabilising Coulombic interaction between the central
ion and its ionic atmosphere, which has the effect of lowering
the energy (and thus the chemical potential) of the central
This model leads to the Debye-Hückel
Limiting Law, which applies only at very low concentrations
of solute (before other contributions to the non-ideality become
important). This law enables calculation of the mean activity
coefficient from basic properties of the solution:
where z+ and z- are the charge number
of respectively the cation and the anion concerned. A
is an empirical parameter, dependant upon the solvent and temperature
(eg, for a solution in water at 298K, A = 0.509).
I is the ionic strength of the
solution, defined as
This means for each type of ion ,i, present in solution, take
the charge number of the ion (positive for cations, negative
for anions), square it, multiply by the molality of the species,
and divide by bº (which has the numerical value 1 mol
kg-1 ). Sum the values for each ion and
divide by two to obtain the ionic strength.
Note in the above expression the charge numbers appear as their
squares, emphasizing the charge on the ions (and hence the Coulombic
interaction between them) as a contributor to non-ideality.
The name 'limiting law' is applied because in the limit of
arbitrarily low molalities (ie as the concentration of the solute
gets closer and closer to zero), all solutions are expected
to behave in a manner consistent with the expression.
However, at more moderate molalities activity coefficients
may differ from the values this law predicts. Experimentally,
it is found that agreement is good up to a molality of approximately
0.001mol kg-1 . At higher solute concentrations,
large deviations from the predictions of the model are observed.