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+, |
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.
3.7.3.2 Limitations to the Debye–Hückel approach
None of the assumptions made by Debye and