Physical Chemistry , 1st ed.

has no variable dictated by the ionic solute except for the charges on the ions!
It is seemingly independent of the identity of the solute.This implies that, for
example, dilute NaCl and dilute KBr solutions have the same properties, since
they are composed of ions having the same charges. However, dilute NaCl and
dilute CaSO 4 would have different properties, despite both being 11 ionic
salts, since the charges on the respective cations and anions are different.
For more precise calculations, the size of the ions involved is a factor also.
Rather than calculating an average activity coefficient , individual ionic
activity coefficients  and  are considered here. A more precise expres-
sion from Debye-Hückel theory for the activity coefficient of an individual
ion is

ln 



1

A



B

z^2





å

I



1

I

/2

1/2 (8.52)

where zis the charge on the ion, å represents the ionic diameter (in units of
meters), and Bis another constant given by the expression

Be

2



N

0 

A

r



k

s

T

olv



1/2

(8.53)

All of the variables were defined above.Istill represents the ionic strength of
the solution, which contains contributions from bothions. Because z^2 is posi-
tive whether zis positive or negative, the negative sign in equation 8.52 ensures
that ln is always negative, so that is always less than 1. Equation 8.52 is
sometimes called the extended Debye-Hückel law.
Equation 8.52 is like equation 8.50 in that the activity coefficient (and there-
fore the activity) is dependent only on properties of the solvent and the charge
and size of the ion, but not the chemical identity of the ion itself. It is there-
fore not uncommon to see tables of data in terms of å and the ionic charge
rather than the individual ions themselves. Table 8.3 is such a table. In using
data from tables like this, you must be extremely careful to make sure the units
work out properly. All units should cancel, leaving a unitless number for the

8.7 Debye-Hückel Theory of Ionic Solutions 231

Table 8.3 Activity coefficients by charge, ionic size, and ionic strength

Ionic strength Ia

å (10^10 m) 0.001 0.005 0.01 0.05 0.10

1-charged ions

9 0.967 0.933 0.914 0.86 0.83

7 0.965 0.930 0.909 0.845 0.81

5 0.964 0.928 0.904 0.83 0.79

3 0.964 0.925 0.899 0.805 0.755

2-charged ions

8 0.872 0.755 0.69 0.52 0.45

6 0.870 0.749 0.675 0.485 0.405

4 0.867 0.740 0.660 0.445 0.355

3-charged ions

6 0.731 0.52 0.415 0.195 0.13

5 0.728 0.51 0.405 0.18 0.115

4 0.725 0.505 0.395 0.16 0.095

Source:a J. A. Dean, ed.,Lange’s Handbook of Chemistry,14th ed., McGraw-Hill, New York, 1992.

Values in this section are for the activity coefficient,.