Geology Reference
In-Depth Information
Example 3.2: Calculation of single-ion activity coecients using the
DH and extended DH equations.
If a solution contains 0.01 mol L
1
MgSO
4
, 0.006 mol L
1
Na
2
CO
3
,
and 0.002 mol L
1
CaCl
2
then the ionic strength of solution is
I
¼
0
:
5
ð
0
:
01
ðþ
2
Þ
2
þ
0
:
01
ð
2
Þ
2
þ
0
:
006
2
ðþ
1
Þ
2
þ
0
:
006
ð
2
Þ
2
þ
0
:
002
ðþ
2
Þ
2
þ
0
:
002
2
ð
1
Þ
2
Þ
¼
0
:
5
ð
0
:
04
þ
0
:
04
þ
0
:
012
þ
0
:
024
þ
0
:
008
þ
0
:
004
Þ
¼
0
:
064molL
1
Calculating the single-ion activity coecient for Na
1
using both the
DH and the extended DH equations (use a value of 4.5 for the ion size
parameter, a
i
)
p
0
:
064
Þ¼
0
:
126
DH log g
Na
þ
¼
0
:
5
ðþ
1
Þ
2
g
Na
þ
¼
0
:
747
Ex-DH log g
Na
þ
¼
0
:
5
ðþ
1
Þ
2
p
0
:
064
Þ
=
1
þ
0
:
33
4
:
5
p
0
:
064
ð Þ
¼
0
:
0919 g
Na
þ
¼
0
:
809
At I
¼
0.064 mol L
1
, the single-ion activity coecient calculated by the
DH model is
B
8% lower than that calculated by the extended DH model.
Now consider a solution which has an ionic strength of only 0.001
mol L
1
.
DH log g
Na
þ
¼
0
:
0158 g
Na
þ
¼
0
:
964
Ex
DH log g
Na
þ
¼
0
:
0151 g
Na
þ
¼
0
:
966
Clearly, at lower ionic strength, the single-ion activity coecient is much
closer to unity. Also, the DH and the extended DH models give almost
exactly the same value. This is because the denominator (1
þ
0.33a
i
O
I)of
the extended DH equation approaches a value of unity, i.e. 0.33a
i
O
I
approaches zero, as the ionic strength decreases towards zero. In other
words, the two equations become identical at very low ionic strength.
It should be noted that, for solutions of ionic strength
o
10
1
mol
L
1
, either Equation (3.4) or the Gu¨ ntelberg approximation (3.5), which
incorporates an average value of 3 for a
i
, can be used.
13
The Gu¨ ntelberg
approximation is particularly useful in calculations where a number of
ions are present in solution or when values of the ion size parameter are
poorly defined.
