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Chapter 10 It has been found empirically that addition of a term linear in I to the Debye–
m
Nonideal Solutions Hückel equation (10.67) improves agreement with experiment in less dilute solutions.
Davies proposed the following expression containing no adjustable parameters
(Davies, pp. 39–43):

1I >m°2 1>2
m
log g 0.51z 0z 0 c 0.301I >m°2d in H O at 25°C

10
2
m
1 1I >m°2 1>2
m
(10.68)
The Davies equation for log g (or log g ) is obtained by replacement of z z in

10
10

2
2
(10.68) with z (or z ). The Davies modification of the Debye–Hückel equation is typ-

1
ically in error by 1 % at I /m° 0.1. The linear term in (10.68) causes g to go
2

m
through a minimum and then increase as I increases, in agreement with the behavior
m
in Fig. 10.8. As I /m° increases above 0.1, agreement of the Davies equation with
m
experiment decreases; at I /m° 0.5, the error is typically 5 to 10%. It is best to use
m
experimental values of g , especially for ionic strengths above 0.1 mol/kg, but in the

absence of experimental data, the Davies equation can serve to estimate g . The Davies

equation predicts that g will have the same value at a given I for any 1:1 electrolyte.
m

In reality, g values for 1:1 electrolytes are equal only in the limit of high dilution.

In using the Debye–Hückel or the Davies equation in a solution containing sev-
eral electrolytes, note that all the ions in the solution contribute to I in (10.59), but
m
that z and z in (10.67) and (10.68) refer to the ionic charges of the particular elec-

trolyte for which g is being calculated.

Certain information about the Davies equation has been suppressed in this section.
For the full story, see Sec. 10.8.
EXAMPLE 10.3 The Davies equation
Use the Davies equation to estimate g for aqueous CaCl solutions at 25°C with

2
molalities 0.001, 0.01, and 0.1 mol/kg.
1 2 1 2 2 1
We have I © j z m 2 (z m z m ) 2 (4m m ) [Eq. (10.59)].

j
j

2
m

We have m m i and m 2m i , where m is the CaCl stoichiometric molality.

i
2
The ionic strength is I 1 2 (4m 2m ) 3m . The Davies equation (10.68)
i
i
i
m
with z 2, 0z 0 1 , and I 3m becomes

m
i
log g 1.0213m >m°2 1>2 >31 13m >m°2 1>2 4 0.92m >m°
10
i
i
i

Substitution of m /m° 0.001, 0.01, and 0.1, gives g 0.887, 0.722, and

i
0.538, respectively. These calculated values can be compared with the experi-
mental values 0.888, 0.729, 0.517 listed in Sec. 10.6. [For comparison, the
Debye–Hückel equation (10.67) gives 0.885, 0.707, and 0.435.]
Exercise
Use the Davies equation to estimate g at 25°C for (a) 0.001 mol/kg AlCl (aq);
3

(b) 0.001 mol/kg CuSO (aq). [Answers: (a) 0.781; (b) 0.761.]
4
In the 1970s, Meissner and coworkers found that g of a strong electrolyte in

water at 25°C could be rather accurately represented up to ionic strengths of 10 or
20 mol/kg by an empirical equation that contains only one parameter (symbolized by
q) whose value is specific to the electrolyte (Tester and Modell, sec. 12.6). Thus if a
single g value is known at a nondilute molality, activity coefficients can be calculated

over a wide molality range. The Meissner equation (also called the Kusik–Meissner
equation) is given in Prob. 10.40. Meissner and coworkers also developed procedures