173
Ionic Interactions
ln γ M = Equation 4.86 + Σ c m c (2θm c + Σ a m a Ψm ca ) + ΣΣm a m a′ Ψ aa M
(4.82)
ln γ X = Equation 4.87 + Σ a m a (2θ Xa + Σ c m c Ψ cc X) + ΣΣm cc Ψ cc X
(4.83)
The values of θ ij are related to the interaction of like- charged ions (e.g., Na + – K + ), and the
values of Ψ ijk are related to the triplet interactions (e.g., Na + – K + – Cl – ). If the values of B MX ,
B’ MX , and C MX are calculated at a given ionic strength (as well as fγ and E), the calculation
of γ is straightforward. This can be demonstrated by giving the equations for M and X in a
simple seawater media made up of Na + , Mg 2+ , Cl – , and SO 4
2– . The equations are
ln
Z f
m B
EC
m
B
EC
M
M
Cl
MCl
MCl
SO
MSO
M
γ
γ
=
+
+
+
+
2
2
2
4
4
(
)
(
S SO
Na
Cl
M NaCl
m NaCl
Mg
Cl
M
m m Z B
Z C
m m Z
4
2 1
)
(
)
(
+
+
+
2 2 1
2 1
2
2
4
4
B
Z C
m m
Z B
Z
MgCl
m MgCl
Na
SO
M NaSO
+
+
+
)
(
M M Na SO
Mg
SO
M MgSO
m MgSO
C
m m
Z B
Z C
2
4
4
4
4
2 1
)
(
)
+
+
(4.84)
ln
Z f
m B
EC
m B
EC
X
X
Na
NaX
NaX
Mg
MgX
MgX
γ
γ
=
+
+
+
+
2
2
2
(
)
(
)
)
(
)
(
+
+
+
m m Z B
Z C
m m Z B
Na
Cl
M NaCl
X NaCl
Mg
Cl
X Mg
2 1
2
C Cl
X MgCl
Na
SO
X NaSO
X NaSO
Z C
m m
Z B
Z C
2
2
4
4
1
2 1
+
+
+
)
(
4 4
4
4
4
4
2
)
(
)
+
+
+
Z C
m m
Z B
Z C
X NaSO
Mg
SO
X MgSO
X MgSO
(4.85)
It first should be noticed that the double sum terms, m Na m Cl , and so on, are the same in
both equations and are only functions of the media. It is thus possible to calculate and sum
all the B 1 and C terms into
Z M
2 ΣΣ + Z M Σ
(4.86)
where
ΣΣ = ′ + ′
+ ′
+ ′
B
B
B
B
NaCl
MgCl
NaSO
MgSo
2
4
4
(4.87)
Σ =
+
+
+
C
C
C
C
NaCl
MgCl
NaSO
MgSo
2
4
4
(4.88)
For seawater, Σ = 0.03503 and ΣΣ = 0.00027, which can be compared to Σ = 0.03234 and
ΣΣ = 0.00033 in NaCl and Σ = 0.03543 and ΣΣ = 0.00026 in NaCl + MgSO 4 . Since the values
of Σ and ΣΣ do not vary much, they are not strong functions of the media composition. If
the Pitzer B MX and C MX are known for Na + and Cl – salts, it is possible to make a reliable
estimate of γ T for many ions for seawater media. If the appropriate terms are available for
Mg + and SO 4
2– salts, the estimates are more reliable. Estimates for some simple ions are
given in Table 4.13.
To make reliable estimates for some of the anions, it is necessary to consider the strong
interactions that occur between Mg 2+ and Ca 2+ . For H + , it is also necessary to consider the
strong interactions with SO 4
2− . The values for H + , F – , OH – , HCO 3
– , B(OH) 4
– , and CO 3
2– in seawater in Table 4.13 have been corrected for these effects. At present, reliable Pitzer parameters and the necessary ion- pairing constants are available for all the major components
of seawater at 25°C. More recent work has extended the parameters to 25°C for Na + , Mg 2+ ,
Ca 2+ , Cl – , and SO 4
2− . Future work will include reliable data for more ions over a wide range
of temperatures.
Ionic Interactions
ln γ M = Equation 4.86 + Σ c m c (2θm c + Σ a m a Ψm ca ) + ΣΣm a m a′ Ψ aa M
(4.82)
ln γ X = Equation 4.87 + Σ a m a (2θ Xa + Σ c m c Ψ cc X) + ΣΣm cc Ψ cc X
(4.83)
The values of θ ij are related to the interaction of like- charged ions (e.g., Na + – K + ), and the
values of Ψ ijk are related to the triplet interactions (e.g., Na + – K + – Cl – ). If the values of B MX ,
B’ MX , and C MX are calculated at a given ionic strength (as well as fγ and E), the calculation
of γ is straightforward. This can be demonstrated by giving the equations for M and X in a
simple seawater media made up of Na + , Mg 2+ , Cl – , and SO 4
2– . The equations are
ln
Z f
m B
EC
m
B
EC
M
M
Cl
MCl
MCl
SO
MSO
M
γ
γ
=
+
+
+
+
2
2
2
4
4
(
)
(
S SO
Na
Cl
M NaCl
m NaCl
Mg
Cl
M
m m Z B
Z C
m m Z
4
2 1
)
(
)
(
+
+
+
2 2 1
2 1
2
2
4
4
B
Z C
m m
Z B
Z
MgCl
m MgCl
Na
SO
M NaSO
+
+
+
)
(
M M Na SO
Mg
SO
M MgSO
m MgSO
C
m m
Z B
Z C
2
4
4
4
4
2 1
)
(
)
+
+
(4.84)
ln
Z f
m B
EC
m B
EC
X
X
Na
NaX
NaX
Mg
MgX
MgX
γ
γ
=
+
+
+
+
2
2
2
(
)
(
)
)
(
)
(
+
+
+
m m Z B
Z C
m m Z B
Na
Cl
M NaCl
X NaCl
Mg
Cl
X Mg
2 1
2
C Cl
X MgCl
Na
SO
X NaSO
X NaSO
Z C
m m
Z B
Z C
2
2
4
4
1
2 1
+
+
+
)
(
4 4
4
4
4
4
2
)
(
)
+
+
+
Z C
m m
Z B
Z C
X NaSO
Mg
SO
X MgSO
X MgSO
(4.85)
It first should be noticed that the double sum terms, m Na m Cl , and so on, are the same in
both equations and are only functions of the media. It is thus possible to calculate and sum
all the B 1 and C terms into
Z M
2 ΣΣ + Z M Σ
(4.86)
where
ΣΣ = ′ + ′
+ ′
+ ′
B
B
B
B
NaCl
MgCl
NaSO
MgSo
2
4
4
(4.87)
Σ =
+
+
+
C
C
C
C
NaCl
MgCl
NaSO
MgSo
2
4
4
(4.88)
For seawater, Σ = 0.03503 and ΣΣ = 0.00027, which can be compared to Σ = 0.03234 and
ΣΣ = 0.00033 in NaCl and Σ = 0.03543 and ΣΣ = 0.00026 in NaCl + MgSO 4 . Since the values
of Σ and ΣΣ do not vary much, they are not strong functions of the media composition. If
the Pitzer B MX and C MX are known for Na + and Cl – salts, it is possible to make a reliable
estimate of γ T for many ions for seawater media. If the appropriate terms are available for
Mg + and SO 4
2– salts, the estimates are more reliable. Estimates for some simple ions are
given in Table 4.13.
To make reliable estimates for some of the anions, it is necessary to consider the strong
interactions that occur between Mg 2+ and Ca 2+ . For H + , it is also necessary to consider the
strong interactions with SO 4
2− . The values for H + , F – , OH – , HCO 3
– , B(OH) 4
– , and CO 3
2– in seawater in Table 4.13 have been corrected for these effects. At present, reliable Pitzer parameters and the necessary ion- pairing constants are available for all the major components
of seawater at 25°C. More recent work has extended the parameters to 25°C for Na + , Mg 2+ ,
Ca 2+ , Cl – , and SO 4
2− . Future work will include reliable data for more ions over a wide range
of temperatures.