The Debye-Hückel model
This model is based on electrostatic theory,
and it only considers long-range interactions between ions:
| |
log γi = |
−Aγ Zi2
√I /
(1 + åBγ √I )
|
where I is the ionic strength,
γi is the
activity coefficient of an
ion “i” with electric charge Zi and
Aγ and Bγ are solvent parameters
which are temperature- and pressure-dependent (see Table;
Aγ = 0.509 (mol/kg)−½ for water at 25°C).
The ion-size parameter å may be different for each electrolyte,
but according to theory the same value of å must be used for all
ions in mixtures of salts. A value of
å = 0.4569 × 10−9 m,
corresponds to åBγ = 1.5 (mol/kg)−½ at 25°C.
The Debye-Hückel equation is only useful in very dilute solutions.
For most real applications other activity coefficient models
that include short-range ion interactions must be used.
To be precise, the above equation must also include the term:
where the sum of molalities includes all dissolved species, both charged and neutral.
This term is needed to convert log γi from the
rational (mole fraction) scale to the practical (molality) scale. But because
the Debye-Hückel equation is only valid at very dilute solutions, the sum
of molalities approaches zero, so that this term becomes negligible and it is generally disregarded.
The activity of water (H2O) is given by:
| |
log aH2O = |
− Φ ∑mk /
(ln(10) 55.508)
|
where “k” are all solute species in the aqeuous solution, and “Φ”
is the osmotic coefficient, which for the Debye-Hückel model it is given by:
| |
Φ = |
1 −
(2 AΦ I3/2
(1/∑mk)
σ(1 + åBγ √I
))
|
where AΦ = (1/3) ln(10) Aγ, and
| |
σ(x) = |
(3/x3) ((1+x) − 1/(1+x) − 2 ln(1+x))
|
Models for activity coefficients
