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Both solvent–solute and solute–solute interactions in electrolytes give rise to excess free energies and nonideal behavior. By developing a model to account for these two kinds of interactions, we can develop an equation that will predict the activity of ions in electrolyte solution.

In an electrolyte solution, each ion will exert an electrostatic force on every other ion. These forces will decrease with the increase in square of distance between ions. The forces between ions will be reduced by the presence of water molecules, due to its dielectric nature. As total solute concentration increases, the mean distance between ions will decrease. Thus, we can expect that activity will depend on the total ionic concentration in the solution. The extent of electrostatic interaction will also obviously depend on the charge of the ions involved: the force between Ca²⁺ and Mg²⁺ ions will be greater at the same distance than between Na⁺ and K⁺ ions.

Figure 3.14 An ion surrounded by a cloud of oppositely charged ions, as assumed in Debye–Hückel theory.

In the Debye–Hückel theory (Figure 3.14), a given ion is considered to be surrounded by an atmosphere or cloud of oppositely charged ions (this atmosphere is distinct from, and unrelated to, the solvation shell). If it were not for the thermal motion of the ions, the structure would be analogous to that of a crystal lattice, though considerably looser. Thermal motion, however, tends to destroy this structure. The density of charge in this ion atmosphere increases with the square root of the ionic concentrations but increases with the square of the charges on those ions. The dielectric effect of intervening water molecules will tend to reduce the interaction between ions. Debye–Hückel theory also assumes the following:

 All electrolytes are completely dissociated into ions.

 The ions are spherically symmetrical charges (hard spheres).

 The solvent is structureless; the sole property is its permittivity.

 The thermal energy of ions exceeds the electrostatic interaction energy.

With these assumptions, Debye and Hückel (1923) used the Poisson–Boltzmann equation, which describes the electrostatic interaction energy between an ion and a cloud of opposite charges, to derive the following relationship (see Morel and Hering, 1993, for the full derivation):

(3.74)

I is ionic strength, in units of molality or molarity, calculated as:

(3.75)

where m is the concentration and z the ionic charge. The parameter å is known as the hydrated ionic radius, or effective radius (significantly larger than the radius of the same ion in a crystal). A and B constants are known as solvent parameters and are functions of T and P. Equation 3.74 is known as the Debye–Hückel extended law; we will refer to it simply as the Debye–Hückel equation. Table 3.2a summarizes the Debye–Hückel solvent parameters over a range of temperatures and Table 3.2b gives values of å for various ions (and see Example 3.3).

For very dilute solutions, the denominator of eqn. 3.74 approaches 1 (because I approaches 0), hence eqn. 3.74 becomes:

(3.76)

Table 3.2a Debye–Hückel solvent parameters.

T°C	A	B (108 cm)
0	0.4911	0.3244
25	0.5092	0.3283
50	0.5336	0.3325
75	0.5639	0.3371
100	0.5998	0.3422
125	0.6416	0.3476
150	0.6898	0.3533
175	0.7454	0.3592
200	0.8099	0.3655
225	0.8860	0.3721
250	0.9785	0.3792
275	1.0960	0.3871
300	1.2555	0.3965

From Helgeson and Kirkham (1974).

Table 3.2b Debye–Hückel effective radii.

Ion	å (10–8 cm)
Rb+, Cs+, , Ag+	2.5
K+, Cl–, Br–, I–,	3
OH–, F–, HS–, , ,	3.5
Na+, , , , ,	4.0–4.5
Pb2+, ,	4.5
Sr2+, Ba2+, Cd2+, Hg2+, S2–	5
Li+, Ca2+, Cu2+, Zn2+, Sn2+, Mn2+, Fe2+, Ni2+	6
Mg2+, Be2+	8
H+, Al3+, trivalent rare earths	9
Th4+, Zr4+, Ce4+	11

From Garrels and Christ (1982).

This equation is known as the Debye–Hückel limiting law (so-called because it applies in the limit of very dilute concentrations).

Davies (1938, 1962) introduced an empirical modification of the Debye–Hückel equation. The Davies equation is:

(3.77)

where A is the same as in the Debye–Hückel equation and b is an empirically determined parameter with a value of around 0.3. It is instructive to see how the activity coefficient of Ca²⁺ would vary according to Debye–Hückel and Davies equations if we vary the ionic strength of the solution. This variation is shown in Figure 3.15. The Davies equation predicts that activity coefficients begin to increase above ionic strengths of about 0.5 m. For reasons discussed below and in greater detail in Chapter 4, activity coefficients do actually increase at higher ionic strengths. On the whole, the Davies equation is slightly more accurate for many solutions at ionic strengths of 0.1−1 m. Because of this, as well as its simplicity, the Davies equation is widely used.