155
Ionic Interactions
where S f and A f are constants related to absolute temperature T, and the dielectric constant
D of water (for a 1-1 electrolyte, S f = 0.5116 and A f = 0.3292 at 25°C), I = 1/2 ∑ν i Z i
2 m i is
the molal ionic strength (ν i is the number, Z i is the charge, and m i is the molality of ionic
species[i]), and (a) is the ion size parameter in angstrom units. This equation serves as a
limit in dilute solutions; however, it fails at the high ionic strength of seawater because of
(a) defects in some of the basic assumptions (e.g., treating ions as point charges in a continuous dielectric medium) and (b) deviations that occur because of noncoulombic effects
such as hydration (the Debye–Hückel theory considers only electrical effects).
The classical method of examining the deviations from the Debye–Hückel theory in
concentrated solutions is to use various extended forms involving one or more arbitrary
constants. The difference between this form and the experimental data is attributed to
noncoulombic effects. For example, Guggenheim (1935) used the equation
–log γ ± = 0.551 Z M Z X I 1/2 /(1 + I 1/2 ) + 2ν B MX m
(4.34)
where ν = 2ν M ν X /(ν M + ν X ), I is the molal ionic strength, and m is the molality. By differentiating Equation 4.34 with respect to temperature and pressure, it is possible to examine
the specific interaction model as a function of temperature and pressure.
The most popular method of treating the deviations from the Debye–Hückel theory in
concentrated solutions is the ion- pairing method of Bjerrum. This method assumes that
short- range interactions can be represented by the formation of ion pairs:
M + + A – → MA o
(4.35)
A characteristic association constant is assigned to this formation:
K A = a MA / a M a A = ([MA]/[M+][A–])(γ MA / γ MγA )
(4.36)
where a i , [i], and γ i are, respectively, the activity, molal concentration, and activity coefficient of species i. There are four classes of ion pairs (Figure 4.25):
1. Complexes: when the ions are held in contact by covalent bonds
2. Contact ion pairs: when the ions are in contact and linked electrostatically (with
no covalent bonding)
3. Solvent- shared ion pairs: pairs of ions linked electrostatically, separated by a
single water molecule
4. Solvent- separated ion pairs: pairs of ions linked electrostatically but separated by
more than one water molecule
Bjerrum defined the distance between oppositely charged ions that can be classified as
being associated by q = Z + Z – e 2 /2DkT, where Z i is the charge on the ion i, e is the electrostatic charge, D is the dielectric constant, k is the Boltzmann constant, and T is the absolute
temperature. In this treatment, two ions of opposite charge are considered to form an ion
pair when they are between å, the ion size parameter, and q. This can include ion pairs of
classes 2, 3, and 4. The Bjerrum (1926) theory predicts greater ion pair formation the higher
the valencies and the smaller the dielectric constant of the solvent are, which is in agreement with experimental results.
Ionic Interactions
where S f and A f are constants related to absolute temperature T, and the dielectric constant
D of water (for a 1-1 electrolyte, S f = 0.5116 and A f = 0.3292 at 25°C), I = 1/2 ∑ν i Z i
2 m i is
the molal ionic strength (ν i is the number, Z i is the charge, and m i is the molality of ionic
species[i]), and (a) is the ion size parameter in angstrom units. This equation serves as a
limit in dilute solutions; however, it fails at the high ionic strength of seawater because of
(a) defects in some of the basic assumptions (e.g., treating ions as point charges in a continuous dielectric medium) and (b) deviations that occur because of noncoulombic effects
such as hydration (the Debye–Hückel theory considers only electrical effects).
The classical method of examining the deviations from the Debye–Hückel theory in
concentrated solutions is to use various extended forms involving one or more arbitrary
constants. The difference between this form and the experimental data is attributed to
noncoulombic effects. For example, Guggenheim (1935) used the equation
–log γ ± = 0.551 Z M Z X I 1/2 /(1 + I 1/2 ) + 2ν B MX m
(4.34)
where ν = 2ν M ν X /(ν M + ν X ), I is the molal ionic strength, and m is the molality. By differentiating Equation 4.34 with respect to temperature and pressure, it is possible to examine
the specific interaction model as a function of temperature and pressure.
The most popular method of treating the deviations from the Debye–Hückel theory in
concentrated solutions is the ion- pairing method of Bjerrum. This method assumes that
short- range interactions can be represented by the formation of ion pairs:
M + + A – → MA o
(4.35)
A characteristic association constant is assigned to this formation:
K A = a MA / a M a A = ([MA]/[M+][A–])(γ MA / γ MγA )
(4.36)
where a i , [i], and γ i are, respectively, the activity, molal concentration, and activity coefficient of species i. There are four classes of ion pairs (Figure 4.25):
1. Complexes: when the ions are held in contact by covalent bonds
2. Contact ion pairs: when the ions are in contact and linked electrostatically (with
no covalent bonding)
3. Solvent- shared ion pairs: pairs of ions linked electrostatically, separated by a
single water molecule
4. Solvent- separated ion pairs: pairs of ions linked electrostatically but separated by
more than one water molecule
Bjerrum defined the distance between oppositely charged ions that can be classified as
being associated by q = Z + Z – e 2 /2DkT, where Z i is the charge on the ion i, e is the electrostatic charge, D is the dielectric constant, k is the Boltzmann constant, and T is the absolute
temperature. In this treatment, two ions of opposite charge are considered to form an ion
pair when they are between å, the ion size parameter, and q. This can include ion pairs of
classes 2, 3, and 4. The Bjerrum (1926) theory predicts greater ion pair formation the higher
the valencies and the smaller the dielectric constant of the solvent are, which is in agreement with experimental results.