ENCYCLOPEDIA OF ENVIRONMENTAL SCIENCE AND ENGINEERING - WATER CHEMISTRYAQUATIC CHEMICAL EQUILIBRIA pot

1256
WATER CHEMISTRY
AQUATIC CHEMICAL EQUILIBRIA
In this section a few example will be given that demonstrate
how elementary principles of physical chemistry can aid in
the recognition of interrelated variables that establish the
composition of natural waters. Natural water systems usu-
ally consist of numerous mineral assemblages and often of
a gas phase in addition to the aqueous phase; they nearly
always include a portion of the biosphere. Hence, natural
aquatic habitats are characterized by a complexity seldom
encountered in the laboratory. In order to distill the perti-
nent variables out of a bewildering number of possible ones,
it is advantageous to compare the real systems with their
idealized counterparts.
Thermodynamic equilibrium concepts represent the
most expedient means of identifying the variables relevant
in determining the mineral relationships and in establishing
chemical boundaries of aquatic environments. Since mini-
mum free energy describes the thermodynamically stable
state of a system, a comparison with the actual free energy
can characterize the direction and extent of processes that
are approaching equilibrium. Discrepancies between equi-
librium calculations and the available data of real systems
give valuable insight into those cases where chemical reac-
tions are not understood sufficiently, where non-equilibrium
conditions prevail, or where the analytical data are not suf-
ficiently accurate or specific.
Alkalinity and Acidity for Aqueous Carbonate
Systems
Alkalinity and acidity are defined, respectively, as the

equivalent sum of the bases that are titratable with strong
acid and the equivalent sum of the acids that are titratable
with strong base; they are therefore capacity factors which
represent, respectively, the acid and base neutralizing
capacities of an aqueous system. Operationally, alkalinity
and acidity are determined by acidimetric and alkalimetric
titrations to appropriate pH end points. These ends points
(equivalence points) occur at the infection points of titra-
tion curves as shown in Figure 1 for the carbonate system.
The atmosphere contains CO
2
at a partial pressure of
3 ϫ 10
Ϫ4
atmosphere, while CO
2
, H
2
CO
3
, HCO
3
Ϫ
and CO
3
2Ϫ

are important solutes in the hydrosphere. Indeed, the carbon-
ate system is responsible for much of the pH regulation in
natural waters.

The following equations define for aqueous carbonate
systems the three relevant capacity factors: Alkalinity (Alk),
Acidity (Acy), and total dissolved carbonate species ( C
T
):
†



Alk HCO CO OH H
[]
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
−
ϭϩ ϩϪ
ϪϪϩ

33
2
2
(1)


Acy H CO HCO H OH
[]
[]
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
ϭϩϩϪ
Ϫϩ Ϫ
2
23 3
* (2)


C
b

ϭϩϩ
ϪϪ
H CO HCO CO
23 3 3
2
*
[]
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
(3)

where [H
2
CO
3
*
] ϭ [CO
2
(aq)] ϩ [H
2
CO
3
].
These equations are of analytical value because they

represent rigorous conceptual definitions of the acid neutral-
izing and the base neutralizing capacities of carbonate sys-
tems. The definitions of alkalinity and acidity algebraically
†
Brackets of the form [ ] refer to concentration, e.g., in moles per
liter.
FIGURE 1 Alkalinity and acidity titration curve for the aque-
ous carbonate system. The conservative quantities alkalinity
and acidity refer to the acid neutralizing and base neutralizing
capacities of a given aqueous system. These parameters can be
determined by titration to appropriate equivalence points with
strong acid and strong base. The equations given below define
the various capacity factors rigorously. Figure from Stumm, W.
and J. Morgan, Aquatic Chemistry, Wiley-Interscience, New
York, 1970, p. 130.
9
7
5
xy
z
pH
[CO
2
–Acy]
[CO
3
2–
–Alk]
[Acy]
Addition of Acid

Addition of Base
[H
+
–Acy]
[Alk]
[OH
–
–
Alk]
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WATER CHEMISTRY 1257
express the net proton deficiency and net proton excess of
the systems with repect to specific proton reference levels
(equivalance points). The definitions can be readily ampli-
fied to account for the presence of buffering components
other than carbonates. For example, in the presence of borate
and ammonia the definition for alkalinity becomes

Alk HCO CO B OH
NH OH H
[]
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦

⎡
⎣
⎤
⎦
[]
⎡
⎣
⎤
⎦
⎡
⎣
⎤
ϭϩ ϩ
ϩϩ Ϫ
ϪϪ Ϫ
Ϫϩ
33
2
4
3
2()
⎦⎦
(4)

Although individual concentrations or activities, such as
[H
2
CO
3
*

] and pH, are dependent on pressure and tempera-
ture, [Alk], [Acy], and C
T
are conservative properties that are
pressure and temperature independent. (Alkalinity, acidity,
and C
T
must be expressed in terms of concentration, e.g., as
molarity, molality, equivalents per liter or parts per million as
CaCO
3
). Note that 1meq/l 5 50 ppm as CaCO
3
.
The use of these conservative parameters facilitates the
calculation of the effects of the addition or removal of acids,
bases, carbon dioxide, bicarbonates, and carbonates to
servative quantities remains constant for particular changes
in the chemical composition. The case of the addition or
removal of dissolved carbon dioxide is of special interest.
Respiratory activities of aquatic biota contribute carbon diox-
ide to the water whereas photosynthetic activities decrease
the concentration of this weak acid. An increase in carbon
dioxide increases both the acidity of the system and C
T
,
the total concentration of dissolved carbonic species, and
it decreases the pH, but it does not affect the alkalinity.
Alternatively, acidity remains unaffected by the addition
or removal of CaCO

3
(s) or Na
2
CO
3
(s). C
T
, on the other
hand, remains unchanged in a closed system upon addition
of strong acid or strong base. For practical purposes, sys-
tems may be considered closed if they are shielded from
the atmosphere and lithosphere or exposed to them only for
short enough periods to preclude significant dissolution of
CO
2
or solid carbonates.
Dissolution of Carbon Dioxide
Though much of the CO
2
which dissolves in solution may ion-
ize to form HCO
3
Ϫ
CO
3
2Ϫ
, depending upon the pH, only a small
fraction (0.3% at 25ЊC) is hydrated as H
2
CO

3
. Hence, the
concentration of the unhydrated dissolved carbon dioxide,
CO
2
(aq), is nearly identical to the analytically determinable
concentration of H
2
CO
3
*
( ϭ [CO
2
(aq)] ϩ [(H
2
CO
3
]).
The equilibrium of a constituent between a gas phase and a
solution phase can be characterized by a mass law relationship.
for the characterization of the CO
2
dissolution equilibrium.
A water that is in equilibrium with the atmosphere (Pco
2
ϭ
10
Ϫ3.5
atm) contains at 25ЊC approximately 0.44 milligram per
liter (10

Ϫ5
M) of CO
2
; K
H
(Henry’s Law constant) at 25ЊC is
10
Ϫ1.5
mole per liter-atm.
Dissolved Carbonate Equilibria
Two systems may be considered: (1) a system closed to
the atmosphere and (2) one that is in equilibrium with the
atmosphere.
Closed Systems In this case H
2
CO
3
*
is considered a non-
volatile acid. The species H
2
CO
3
*
, HCO
3
, CO
3
2Ϫ
and are

interrelated by the equilibria:
†


[][ ][ *]HHCO HCO
ϩϪ
րϭ
323 1
K


[][ ][ ]HCO HCOϩր ϭ
ϪϪ
3
2
32
K

where K
1
and K
2
represent the equilibrium constants (acidity
constants).
The ionization fractions, whose sum equals unity (see
Eq. (3)), can be defined as follows:

a
023
ϭր[*]HCO C

T
(7)


a
13
ϭ
Ϫ
[]HCO C
T
(8)

a
23
2
ϭր
Ϫ
[]CO C
T
(9)
From Eqs. (3) to (9) the ionization fractions can be expressed
in terms of [H
ϩ
] and the equilibrium constants:

a
01 12
21
1ϭϩ ր ϩ ր
ϩϩϪ

([] [])KKKHH (10)

a
112
1
1ϭրϩϩր
ϩϩϪ
([ ] [ ])HHKK (11)


a
2
2
12 2
1
1ϭրϩրϩ
ϩϩϪ
([ ] [ ] ]HHKK K (12)
1 2

HCO
3
Ϫ
and CO
3
2Ϫ
may form complexes with other ions in
the systems (e.g., in sea water, MgCO
3
,

NaCO
3
Ϫ
, CaCO
3
,
MgHCO
3
ϩ
, it is operationally convenient to define a total
concentration of the species to include an unknown number
of these complexes. For example,

[][][ ]
[][ ]
CO CO MgCO
CaCO NaCO
3
2
3
2
3
33
TT
ϪϪ
Ϫ
ϭϩ
ϩϩ ϩ⌳

(13)

The distribution of carbonate species in sea water as a func-
Systems Open to the Atmosphere A very elementary
model showing some of the characteristics of the carbonate
system in natural waters is provided by equilibrating pure
water with a gas phase (e.g., the atmosphere) containing CO
2

at a constant partial pressure. Such a solution will remain in
†
To facilitate calculations the equilibria are written here in terms of
concentration quotients. The activity corrections can be considered
incorporated into the equilibrium “constants” which therefore vary
with the particular solution. Such constants for given media of con-
stant ionic strength, as well as the true thermodynamic constants,
are listed in Tables 2A and 2B.
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Table 1 gives the various expressions and their interrelations
Values for K and K are given in Tables 2A and 2B. Because
aqueous systems. As shown in Figure 2, each of these con-
tion of pH is given in Figure 3.
1258 WATER CHEMISTRY
equilibrium with

p
co
, despite any variation of pH by the addi-
tion of strong base or strong acid. This simple model has its

counterpart in nature when CO
2
reacts with bases of rocks,
for example with clays and silicates.
such a model. A partial pressure of CO
2
equivalent to that
in the atmosphere and equilibrium constants valid at 25ЊC
have been assumed. The equilibrium concentrations of the
individual carbonate species can be expressed as a function
of

and [H
ϩ
]
2
. From Henry’s Law,

[*] ,HCO
CO23
2
ϭ Kp
H
(14)

and Eqs. (5) to (9), one obtains

CKp
TH
ϭ

1
0
2
a
CO
(15)


[]
[]
HCO
H
CO CO3
1
0
1
22
Ϫ
ϩ
ϭϭ
a
a
Kp
K
Kp
HH
(16)
FIGURE 2 Closed system capacity diagram: pH contours for alkalin-
ity versus C
T

(total carbonate carbon). The point defining the solution
composition moves as a vector in the diagram as a result of the addition
(or removal) of CO
2
, NaHCO
3
, and CaCO
3
(Na
2
CO
3
) or C
B
(strong base)
and C (strong acid). (After K.S. Deffeyes, Limnol., Oceanog., 10, 412,
1965.) Figure from Stumm, W. and J. Morgan, Aquatic Chemistry, Wiley-
Interscience, New York, 1970, p. 133.
C
B
C
A
CO
2
NaHCO
3
CaCO
3
Na
2

CO
3
Dilution
1
1
1
1
2
2
0
–0.5
0
1
2
3
11.5
11.4
11.3
11.2
11.1
11.0
10.9
10.8
10.7
10.6
10.5
10.4
10.3
10.2
10.1

10.0
9.9
9.8
9.6
9.7
9.5
9.0
8.5
8.0
7.5
7.0
6.9
6.8
6.7
6.6
6.5
6.4
6.3
6.2
6.1
6.0
5.9
5.8
5.7
5.6
5.5
5.4
5.3
5.2
5.1

5.0
4.5
4.0
3.9
3.8
3.7
3.6
3.5
3.4
Alkalinity (milliequivalents/liter)
C
T
(Total Carbonate carbon; millimoles/liter)
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F igure 4 shows the distribution of the solute species of
TABLE 1
Solubility of gases
Example
a
: CO
2
(g) CO
2
(aq)
Assumptions: Gas behaves ideally; [CO
2
(aq)] ϭ [H2CO
3
*

]
I. Expressions for Solubility Equilibrium
b
(1) Distribution (mass law) constant, K
D
:
K
D
ϭ [CO
2
(aq)]/[CO
2
(g)] (dimensionless) (1)
(2) Henry’s law constant, K
H
:
In (1), [CO
2
(g)] can be expressed by Dalton’s law of partial pressure:
[CO
2
(g)] ϭ p
CO
2
/RT (2)
Combination of (1) and (2) gives
[CO
2
(aq)] ϭ (K
D

/RT)p
CO
2
ϭ K
H
p
CO
2
, (3)
where K
H
ϭ K
D
/RT (mole liter
1
atm
–1
)
(3) Bunsen absorption coefficient, a
B
:
[CO
2
(aq)] ϭ (␣
B
/22.414)p
CO
2
(4)
where 22.414 ϭ RT/p (liter mole

–1
) and
a
B
ϭ K
H
ϫ 22.414 (atm
–1
) (5)
Partial Pressure and Gas Composition
p
CO
2
ϭ x
CO
2
(P
T
– w) (6)
where X
CO
2
ϭ mole fraction or volume fraction in dry gas, P
T
ϭ total pressure and w ϭ water vapor pressure
Values of Henry’s Law Constants at 25ЊC
Gas K
H
(mole liter
–1

atm
–1
)
Carbon Dioxide CO
2
33.8 ϫ 10
–3
Methane CH
4
1.34 ϫ 10
–3
Nitrogen N
2
–
0.642 ϫ 10
–3
Oxygen O
2
1.27 ϫ 10
–3

a
Same types of expressions apply to other gases.

b
The equilibrium constants defined by (1)–(4) are actually constants only if the equilibrium expressions
are formulated in terms of activities and fugacities.
Table from Stumm, W. and J. Morgan, Aquatic Chemistry, Wiley-Interscience, New York, 1970, p. 125.
and

[]
[]
.
CO
H
CO CO3
2
2
0
12
2
22
Ϫ
ϩ
ϭϭ
a
a
KP
KK
Kp
HH
.
(17)
It follows from these equations that in a logarithmic
2 3
*
HCO
3
2
, CO

3
2Ϫ
have slopes of 0, ϩ1, and ϩ2, respectively.
If we equilibrate pure water with CO
2
, the system is
defined by two independent variables, for example, temper-
ature and Pco
2
, In other words, the equilibrium concentra-
tions of all solute components can be calculated by means of
Henry’s Law, the acidity constants and the proton condition
or charge balance if, in addition to temperature, one variable,
such as Pco
2
, [H
2
CO
3
*
] or [H
ϩ
], is known or measured. Use
of the proton condition instead of the charge balance gener-
ally facilitates calculations because species irrelevant to the
calculation need not be considered. The proton condition
merely expressed the equality between the proton excess
and the proton deficiency of the various species with respect
graphic illustration of its use.
Solubility Equilibria

Minerals dissolve in or react with water. Under different
physico-chemical conditions minerals are precipitated and
accumulate on the ocean floor and in the sediments of rivers
and lakes. Dissolution and precipitation reactions impart to
the water and remove from it constituents which modify its
chemical properties.
WATER CHEMISTRY 1259
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to a convenient proton reference level. Figure 4 furnishes a
c oncentration—pH diagram (Figure 4) the lines of H CO ,
1260 WATER CHEMISTRY
TABLE 2B
First acidity constant: H
2
CO
3
ϭ HCO
3
Ϫ
ϩ H
ϩ
K
1
3
23
ϭ
ϩϪ
{}{ }
{}

*
HHCO
HCO
′
K
T
1
3
23
ϭ
ϩϪ
{}[ ]
[]
*
HHCO
HCO
c
T
K
1
3
23
ϭ
ϩϪ
[][ ]
[]
*
HHCO
HCO
Temp., °C

Medium
→ 0 Seawater, 19% Cl
Ϫ
Seawater 1 M NaClO4
Ϫlog K
Ϫlog K
1
Ј
—
Ϫlog
c
K
1
0 6.579
a
6.15
b
——
5 6.517
a
6.11
b
6.01
e
—
10 6.464
a
6.08
b
——

14 — — 6.02
f
—
15 6.419
a
6.05
b
——
20 6.381
a
6.02
b
——
22 — 6.00
c
5.89
c
—
25 6.352
a
6.00
b
, 6.09
d
— 6.04
g
30 6.327
a
5.98
b

——
35 6.309
a
5.97
b
——
40 6.298
a
———
50 6.285
a
———

a
H. S. Harned and R. Davies, Jr., J. Amer. Chem. Soc. , 65, 2030 (1943).

b
After Lyman (1956), quoted in G. Skirrow, Chemical Oceanography, Vol. I, J. P. Riley and G.
Skirrow, Eds., Academic Press, New York, 1965, p. 651.

c
A Distèche and S. Distèche, J. Electrochem. Soc. , 114, 330 (1967).

d
Calculated as log (K
1
/f
HCO3
Ϫ
) as determined by A. Berner, Geochim. Cosmochim. Acta, 29, 947 (1964).

e
D. Dyrssen, and L. G. Sillén, Tellus, 19, 810 (1967).

f
D. Dyrssen, Acta Chem. Scand. , 19, 1265 (1965).

g
M. Frydman, G. N. Nilsson, T. Rengemo, and L. G. Sillén, Acta Chem. Scand. , 12, 878 (1958).
Ref.: Stumm, W. and J. Morgan, Aquatic Chemistry, Wiley-Interscience, New York, 1970, p. 148.
TABLE 2A
Equilibrium constant for CO
2
solubility
Equilibrium: CO
2
(g) ϩ aq ϭ H
2
CO
3

Henry’s law constant: K ϭ [H
2
CO
3
]/p
CO
2
(M.atm
–1

)
Temp., ЊC
→ 0
Medium, 1 M NaClO
4
Seawater, 19% C1
–
–log K –log
c
K –log
cK
0 1.11
a
— 1.19
a
5 1.19
a
— 1.27
a
10 1.27
a
— 1.34
a
15 1.32
a
— 1.41
a
20 1.41
a
— 1.47

a
25 1.47
a
1.51
c
1.53
a
30 1.53
a
— 1.58
a
35 — 1.59
c
—
40 1.64
b
——
50 1.72
b
——

a
Values based on data taken from Bohr and evaluated by K. Buch,
Meeresforschung, 1951.

b
A.J. Ellis, Amer. J. Sci. , 257, 217 (1959).

c
G. Nilsson, T. Rengemo, and L. G. Sillen, Acta Chem. Sand. , 12, 878 (1958).

Ref.: Stumm, W. and J. Morgan, Aquatic Chemistry, Wiley-Interscience,
New York, 1970, p. 148.
It is difficult to generalize about rates of precipita-
tion and dissolution other than to recognize that they are
usually slower than reactions between dissolved species.
Data concerning most geochemically important solid-solu-
tion reactions are lacking, so that kinetic factors cannot be
assessed easily. Frequently the solid phase initially formed
is metastable with respect to a thermodynamically more
stable solid phase. Relevant examples of such metastabil-
ity are the formation of aragonite under certain conditions
instead of calcite, the more stable form of calcium car-
bonate, and the over-saturation of quartz in most natural
waters. This over-saturation persists due to the extremely
slow establishment of equilibrium between silicic acid and
quartz.
The solubilities of most inorganic salts increase with
increasing temperature. However, a number of compounds
of interest in natural waters (e.g. CaCO
3
, CaSO
4
) decrease in
solubility with increasing temperature. The dependence of
solubility on pressure is very slight but must be considered
for the extreme pressures encountered at ocean depths. For
example, the solubility product of CaCO
3
will increase by
approximately 0.2 logarithmic units for a pressure of 200

atmospheres (ca. 2000 meters).
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0
1
2
3
4
5
67
8
910
11
12
13
13
7
0

–1
–2
–3
–4
–5
–6
–7
0
–1
–2
–3

–4
–5
–6
–7
H
+
P
2
P
1
HC
–
OH
–
C
2–
B
–
HC
–
HB
H
2
C
P
1
P
2
B
–

C
2–
HC
–
HB
H
2
C
seawater
log CONC. (M)
pH
FIGURE 3 Logarithmic concentration—pH equilibrium diagram for seawater as a
closed system. For seawater log B
T
ϭ Ϫ3.37, log C
T
ϭ Ϫ2.62 and the following pK
values: 6.0 for H
2
CO
3
*
, 9.4 for and pK ϭ 13.7. B
T
ϭ total borate boron and C
T
ϭ total
carbonate carbon. Arrows gives [H
ϩ
] for seawater (pH ϭ 8.0) and for two equivalence

points (points of minimum buffer intensity): P
1
, corresponding to a proton reference
level of HB ϩ HC
Ϫ
ϩ H
2
O, and P
2
, corresponding to a proton reference level of HB ϩ
H
2
C ϩ H
2
O (From Dyrssen, D. and L.G. Sillén, Tellus, 19, 110, 1967).
TABLE 2B (continued)
Solid acidity constant: HCO
3
Ϫ
ϭ H
ϩ
ϩ CO
3
2Ϫ
K
2
3
2
3
ϭ

ϩϪ
Ϫ
{}{ }
{}
HCO
HCO
′
K
T
T
2
3
2
3
ϭ
ϩϪ
{}[ ]
[]
HCO
HCO
c
T
K
2
3
2
3
ϭ
ϩϪ
Ϫ

[][ ]
[]
HCO
HCO
Temp., °C
Medium
→ 0
Seawater 0.75 M NaCl 1 M KclO
4
—
Ϫlog K
2
Ϫlog K
2
ЈϪlog K
2
ЈϪlog
c
K
2
5 10.625
a
9.40
b
——
5 10.557
a
9.34
b
——

10 10.490
a
9.28
b
——
15 10.430
a
9.23
b
——
20 10.377
a
9.17
b
——
22 — 9.12
c
9.49
c
—
25 10.329
a
9.10
b
— 9.57
d
30 10.290
a
9.02
b

——
35 10.250
a
8.95
b
——
40 10.220
a
—— —
50 10.172
a
—— —

a
H. S. Harned and S. R. Scholes, J. Amer. Chem. Soc. , 63, 1706 (1941).

b
After Lyman, quoted in G. Skirrow, Chemical Oceanography, Vol. I, J. P. Riley and G. Skirrow,
Eds., Academic Press, New York, p. 651.

c
A. Distèche and S. Distèche, J. Electrochem. Soc. , 114, 330 (1967).

d
M. Frydman, G. N. Nilsson, T. Rengemo, and L. G. Sillén, Acta Chem. Scand. , 12, 878 (1958).
Ref.: Stumm, W. and J. Morgan, Aquatic Chemistry, Wiley-Interscience, New York, 1970, pp. 149
and 150.
WATER CHEMISTRY 1261
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© 2006 by Taylor & Francis Group, LLC

Solubility of Oxides and Hydroxides
If pure solid oxide or hydroxide is in equilibrium with free
ions in solution, for example,
Me(OH)
2
(s) ϭ Me
2 ϩ
ϩ 2OH
Ϫ
(18)
MeO(s) ϩ H
2
O ϭ Me
2 ϩ
ϩ 2OH
Ϫ
(19)
the conventional (concentration) solubility product is given by

c
K
s 0
2233*
[][]( )ϭ
ϩϪ Ϫ
Me OH mole liter ,

(20)

where the subscript “0” refers to solution of the simple, uncom-

plexed forms of the metal ion.
Sometimes it is more appropriate to express the solubility
in terms of reaction with protons, for example,
Me(OH)
2
(s) ϩ 2H
ϩ
ϭ Me
2 ϩ
ϩ 2H
2
O (21)
MeO(s) ϩ 2H
ϩ
ϭ Me
2 ϩ
ϩ H
2
O. (22)
In the general case for a cation of charge z, the solubility
equilibrium for Eqs. (21) and (22) is characterized by

c
c
K
K
z
zz
K
w

z
s
s
0
0
11*()()
[]
[]
[ϭϭ
ϩ
ϩ
ϪϪ
Me
H
mole liter
,
(23)
where K
w
is the ion product of water. This constant and also a
number of solubility equilibrium constants relevant to natu-
Equation (23) can be written in logarithmic form to
express the equilibrium concentration of a cation Me


z

ϩ
as a
function of pH:

log[ ] log
*
Me pH
zc
s
K
ϩ
ϭϪ
0
. (24)
Equation (24) is plotted for a few oxides and hydroxides in
pK
H
2
CO
3
pK
1
pK
2
H
+
H
2
CO
3
C
T
CO

3
OH
–
TRUE
H
2
CO
3
HCO
3
P
log CONCENTRATION (MOLAR)
pH
a
-1
-2
-3
-4
-5
-6
-7
-8
4
5
6
7
8
9
10
11

*
–
-2
FIGURE 4 Logarithmic concentration—pH equilibrium diagram for the aque-
ous carbonate system open to the atmosphere. Water is equilibrated with the at-
mosphere (pCO
2
= 10
Ϫ3.5
atm) and the pH is adjusted with strong base or strong
acid. Eqs. (14), (15), (16), (17) with the constants (25ЊC) pK
H
ϭ 1.5, pK
1
ϭ 6.3,
pK
2
ϭ 10.25, pK(hydration of CO
2
) ϭ Ϫ2.8 have been used. The pure CO
2
solu-
tion is characterized by the proton condition [H
ϩ
] ϭ [HCO
3
Ϫ
] ϩ 2[CO
3
2Ϫ

]+[OH
Ϫ
]
see point P) and the equilibrium concentrations Ϫlog[H
ϩ
] ϭ Ϫlog[HCO
3
Ϫ
] ϭ
5.65; Ϫlog[CO
2
aq] ϭ Ϫlog[H
2
CO
3
] ϭ 5.0; Ϫlog[H
2
CO
3
] Ϸ 7.8; Ϫlog[CO
3
2Ϫ
] ϭ
8.5. Ref.: Stumm, W. and J. Morgan, Aquatic Chemistry, Wiley-Interscience, New
York, 1970, p. 127.
1262 WATER CHEMISTRY
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© 2006 by Taylor & Francis Group, LLC
ral waters are given in Table 3.
Figure 5.

WATER CHEMISTRY 1263
TABLE 3
Equilibrium constants for oxides, hydroxides, carbonates, hydroxide carnonates, sulfates, silicates, and acids
Reaction
Symbol for
equilibrium
constants log K (25ЊC) I
I. OXIDES AND HYDROXIDES
H
2
O(1) ϭ H
ϩ
ϩ OH
Ϫ
K
w
Ϫ14.00 0
Ϫ13.77 1M NaClO
4
(am)Fe(OH)
3
(s) ϭ Fe
3ϩ
ϩ 3OH
Ϫ
K
s0
Ϫ38.7 3M NaClO
4
(am)Fe(OH)

3
(s) ϭ FeOH
2ϩ
ϩ 2OH
Ϫ
K
s1
Ϫ27.5 3M NaClO
4
(am)Fe(OH)
3
(s) ϭ Fe(OH)
2
ϩ
ϩ OH
Ϫ
K
s2
Ϫ16.6 3M NaClO
4
(am)Fe(OH)
3
(s) ϩ OH
Ϫ
ϭ FE(OH)
4
K
s4
Ϫ4.5 3M NaClO
4

2(am)Fe(OH)
3
(s) ϭ Fe
2
(OH)
2
4ϩ
ϩ 4OH
Ϫ
K
s22
Ϫ51.9 3M NaClO
4
(am)Fe(OOH)(s) ϩ 3H
ϩ
ϭ Fe
3ϩ
ϩ 2H
2
O
*
K
s0
3.55 3M NaClO
4
a—FeOOH(s) ϩ 3H
ϩ
ϭ Fe
3ϩ
ϩ 2H

2
O
*
K
s0
1.6 3M NaClO
4
a—Al(OH)
3
(gibbsite) ϩ 3H
ϩ
ϭ Al
3ϩ
ϩ 3H
2
*
K
s0
8.2 0
g—Al(OH)
3
(bayerite) ϩ 3H
ϩ
ϭ Al
3ϩ
ϩ 3H
2
O
*
K

s0
9.0 0
(am)Al(OH)
3
(s) ϩ 3H
ϩ
ϭ Al
3ϩ
ϩ 3H
2
O
*
K
s0
10.8 0
Al
3ϩ
ϩ 4OH
Ϫ
ϭ Al(OH)
4
Ϫ
K
4
32.5 0
CuO(s) ϩ 2H
ϩ
ϭ Cu
2ϩ
ϩ H

2
O
*
K
s0
7.65 0
Cu
2ϩ
ϩ OH
Ϫ
ϭ CuOH
ϩ
K
1
6.0 (18ЊC) 0
2Cu
2ϩ
ϩ 2OH
Ϫ
ϭ Cu
2
(OH)
2
2ϩ
K
22
17.0 (18ЊC) 0
Cu
2ϩ
ϩ 3OH

Ϫ
ϭ Cu(OH)
3
Ϫ
K
3
15.2 0
Cu
2ϩ
ϩ 4OH
Ϫ
ϭ Cu(OH)
4
2Ϫ
K
4
16.1 0
ZnO(s) ϩ 2H
ϩ
ϭ Zn
2ϩ
ϩ H
2
O
*
K
s0
11.18 0
Zn
2ϩ

ϩ OH
Ϫ
ϭ ZnOH
ϩ
K
1
5.04 0
Zn
2ϩ
ϩ 3OH
Ϫ
ϭ Zn(OH)
3
Ϫ
K
3
13.9 0
Zn
2ϩ
ϩ 4OH
Ϫ
ϭ Zn(OH)
4
2Ϫ
K
4
15.1 0
Cd(OH)
2
(s) ϩ 2H

2ϩ
ϭ Cd
2ϩ
ϩ 2H
2
O
*
K
s0
13.61 0
Cd
2ϩ
ϩ OH
Ϫ
ϭ CdOH
ϩ
K
1
3.8 1M LiClO
4
Mn(OH)
2
(s) ϭ Mn
2ϩ
ϩ 2OH
Ϫ
K
s0
Ϫ12.8 0
Mn(OH)

2
(s) ϩ OH
Ϫ
ϭ Mn(OH)
3
Ϫ
K
s3
Ϫ5.0 0
Fe(OH)
2
(active) ϭ Fe
2ϩ
ϩ 2OH
Ϫ
K
s0
Ϫ14.0 0
Fe(OH)
2
(inactive) ϭ Fe
2ϩ
ϩ 2OH
Ϫ
K
s0
Ϫ14.5 (Ϫ15.1) 0
Fe(OH)
2
(inactive) ϩ OH

Ϫ
ϭ Fe(OH)
3
K
s3
Ϫ5.5 0
Mg(OH)
2
ϭ Mg
2ϩ
ϩ 2OH
Ϫ
K
s0
Ϫ9.2 0
Mg(OH)
2
(brucite) ϭ Mg
2ϩ
ϩ 2OH
Ϫ
K
s0
Ϫ11.6 0
Mg
2ϩ
ϩ OH
Ϫ
ϭ MgOH
ϩ

K
1
2.6 0
Ca(OH)
2
(s) ϭ Ca
2ϩ
ϩ 2OH
Ϫ
K
s0
Ϫ5.43 0
Ca(OH)
2
(s) ϭ CaOH
ϩ
ϩ OH
Ϫ
K
s1
Ϫ4.03 0
Sr(OH)
2
(s) ϭ Sr
2ϩ
ϩ 2OH
Ϫ
K
s0
Ϫ3.51 0

Sr(OH)
2
(s) ϭ SrOH
ϩ
ϩ OH
Ϫ
K
s1
0.82 0
AgOH(s) ϭ Ag
ϩ
ϩ OH
Ϫ
K
s0
Ϫ7.5 0
II. CARBONATES AND HYDROXIDE CARBNONATES
CO
2
(g) ϩ H
2
O ϭ H
ϩ
ϩ HCO
3
Ϫ
K
p1
Ϫ7.82 0
Ϫ7.5 Seawater

5ЊC, 200 atm
seawater
HCO
3
Ϫ
ϭ H
ϩ
ϩ CO
3
2Ϫ
(K
2
)
Ϫ10.33 0
Ϫ9.0 Seawater
(Continued)
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© 2006 by Taylor & Francis Group, LLC
1264 WATER CHEMISTRY
TABLE 3 (continued )
Reaction
Symbol for
equilibrium
constants log K (25ЊC) I
Ϫ9.0 5ЊC, 200 atm
seawater
CaCO
3
(calcite) ϭ Ca
2ϩ

ϩ CO
3
2Ϫ
K
s0
Ϫ8.35 0
Ϫ6.2 Seawater
CaCO
3
(aragonite) ϭ Ca
2ϩ
ϩ CO
3
2Ϫ
K
s0
Ϫ8.22 0
SrCO
3
(s) ϭ Sr
2ϩ
CO
2
2Ϫ
K
s0
Ϫ9.03 0
Ϫ6.8 Seawater
ZnCO
3

(s) ϩ 2H
ϩ
ϭ Zn
2ϩ
ϩ H
2
O ϩ CO
2
(g)
*
K
ps0
7.95 0
Zn(OH)
1.2
(CO
3
)
0.4
(s) ϩ 2H
ϩ
ϭ Zn
2ϩ
ϩ H
2
O ϩ
CO
2
(g)
*

K
0
9.8 0
Cu(OH)(CO
3
)
0.5
(s) ϩ 2H
ϩ
ϭ Cu
2ϩ
ϩ 3/2H
2
O ϩ
1/2CO
2
(g)
*
K
ps0
7.08 0
Cu(OH)
0.67
(CO
3
)
0.67
(s) ϩ 2H
ϩ
ϭ Cu

2ϩ
ϩ
4/3H
2
O ϩ 2/3CO
2
(g)
*
K
ps
7.08 0
MgCO
3
(magnesite) ϭ Mg
2ϩ
ϩ CO
3
2Ϫ
K
s0
Ϫ4.9 0
MgCO
3
(nesquehonite) ϭ Mg
2ϩ
ϩ CO
3
2Ϫ
K
s0

Ϫ5.4 0
Mg
4
(CO
3
)
3
(OH)
2
.3H
2
O(hydromagnesite) ϭ 4Mg
2ϩ
ϩ
3CO
3
2Ϫ
2OH
Ϫ
K
s0
29.5 0
CaMg(CO
3
)
2
(dolomite) ϭ Ca
3ϩ
ϩ Mg
2ϩ

ϩ 2CO
3
2Ϫ
K
s0
Ϫ16.7 0
FeCO
3
(siderite) ϭ Fe
2ϩ
ϩ 2CO
3
2Ϫ
K
s0
Ϫ10.4 (Ϫ10.24) 0
CdCO
3
(s) ϩ 2H
ϩ
ϭ CD
2ϩ
ϩ H
2
O ϩ CO
2
(g)
K
ps0
6.44 1M NaCl

4
MnCO
3
(s) ϭ Mn
2ϩ
2CO
3
2Ϫ
K
s0
Ϫ10.41 0
III. SULFATES, SULFIDES, AND SILICATES
CaSO
4
(s) ϭ Ca
2ϩ
ϩ SO
4
2Ϫ
K
s0
Ϫ4.6 0
H
2
S ϭ H ϩ HS
Ϫ
K
1
Ϫ7.0 0
HS

Ϫ
ϭ H
ϩ
ϩ S
2Ϫ
K
2
Ϫ12.96 0
MnS(green) ϭ Mn
2ϩ
ϩ S
2Ϫ
K
s0
Ϫ12.6 0
MnS(pink) ϭ Mn
2ϩ
ϩ S
2Ϫ
K
s0
Ϫ9.6 0
FeS(s) ϭ Fe
2ϩ
ϩ S
2Ϫ
K
s0
Ϫ17.3 0
SiO

2
(quartz) ϩ 2H
2
ϭ H
4
SiO
4
K
s0
Ϫ3.7
(am)SiO
2
(s) ϩ 2H
2
O ϭ H
4
SiO
4
K
s0
Ϫ2.7 0
H
4
SiO
4
ϭ H
ϩ
ϩ H
3
SiO

4
K
s0
Ϫ9.46 0
IV. ACIDS
NH
4
ϩ
ϭ N
ϩ
ϩ NH
3
(aq)
K
a
Ϫ9.3 0
HOCl ϭ H
ϩ
ϩ OCl
Ϫ
K
a
Ϫ7.53 0
The constants given here are taken from quotations or selections in (a) L. G. Sillén and A. E. Martell, Stability
Constants of Metal Ion Complexes, Special Publ., No. 17, the Chemical Society, London, 1964: (b) W. Feitknecht
and P. Schindler, Solubility Constants of Metal Oxides, Metal Hydroxides and Metal Hydroxide Salts in Aqueous
Solutions, Butterworths, London, 1963; (c) P. Schindler, “Heterogeneous Equilibria Involving Oxides, Hydroxides,
Carbonates and Hydroxide Carbonates”, in Equilibrium Concepts in Natural Water Systems, Advance in Chemistry
Series, No. 67, American Chemical Society, Washington, DC, 1967, p. 196; and (d) J. N. Butler, Ionic Equilibrium,
A Mathematical Approach, Addison-Wesley Publishing, Reading, Mass., 1964. Unless otherwise specified a pressure

of 1 atm is assumed.

a
Most of the symbols used for the equilibrium constants are those given in Stability Constants of Metal-Ion Complexes
Table from Stumm, W. and J. Morgan, Aquatic Chemistry, Wiley-Interscience, New York, 1970, pp. 168 and 169.
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WATER CHEMISTRY 1265
The relations in Figure 5 do not fully describe the solu-
bility of the corresponding oxides and hydroxides, since
in addition to free metal ions, the solution may contain
hydrolyzed species (hydroxo complexes) of the form

. The
solubility of the metal oxide or hydroxide is therefore
expressed more rigorously as

Me Me Me OH
T
z
n
zn
n
ϭϩ
ϩϪ
[][()].
1
∑
(25)

for ferric hydroxide, zinc oxide, and cupric oxide.
Solubility of Carbonates
The maximum soluble metal ion concentration is a function
of pH and concentration of total dissolved carbonate species.
Calculation of the equilibrium solubility of the metal ion for a
given carbonate for a water of a specific analytic composition
discloses whether the water is over-saturated or undersaturated
with respect to the solid metal carbonate. In the case of calcite

[]
[]
Ca
CO
2
0
3
2
0
2
ϩ
Ϫ
ϭϭ
KK
C
ss
T
a
.
(26)
Since a

2
is known as a function of pH, Eq. (26) gives the
equilibrium saturation value of Ca
2 ϩ
as a function of C
T

and pH. An analogous equation can be written for any
metallic cation in equilibrium with its solid metallic car-
bonate. These equations are amenable to simple graphical
representation in a log concentration versus pH diagram as
†
Note that this solubility product is expressed for activities, as
represented by {}.
1
2
3
4
5
6
2
4
68
10
12
Fe
3+
Al
3+
Cu

2+
CuO(s)
Cu
2+
Zn
2+
Fe
2+
Cd
2+
Mg
2+
Ag
+
CO
2+
–log [Me
z+
]
pH
FIGURE 5 Solubility of oxides and hydroxides: free metal ion concentration
in equilibrium with solid oxides ore hydroxides. As shown explicitly by the equi-
librium curve for copper, free metal ions are constrained to concentrations to the
left of (below) the respective curves. Precipitation of the solid hydroxides and
oxides commences at the saturation concentrations represented by the curves. The
formation of hydroxo metal complexes must be considered for the evaluation of
complete solubility of the oxides or hydroxides. Ref.: Stumm, W. and J. Morgan,
Aquatic Chemistry, Wiley-Interscience, New York, 1970, p. 171.
Control of Solubility
Solubility calculations, such as those exemplified above, give

thermodynamically meaningful conclusions, under the speci-
fied conditions (e.g., concentrations, pH, temperature and pres-
sure), only if the solutes are in equilibrium with that solid phase
for which the equilibrium relationship has been formulated.
For a given set of conditions the solubility is controlled by the
solid giving the smallest concentration of solute. For example,
within the pH range of carbonate bearing natural waters, the
stable solid phases regulating the solubility of Fe(II), Cu(II),
and Zn(II) are, respectively, FeCO
3
(siderite), CuO (tenorite)
and Zn (OH) (CO
3
) (hydrozincite).
Unfortunately, it has not yet been possible to determine
precise solubility data for some solids important in the reg-
ulation of natural waters. Among these are many clays and
dolomite (CaMg(CO
3
)
2
), a mixed carbonate which con-
stitutes a large fraction of the total quantity of carbonate
rocks. The conditions under which dolomite is formed in
nature are not well understood and attempts to precipitate
it in the laboratory from solutions under atmospheric con-
ditions have been unsuccessful. These difficulties in ascer-
taining equilibrium have resulted in a diversity of published
figures for its solubility product, ({Ca
2ϩ

}{Mg
2ϩ
}{Co
3
2Ϫ
}2}
†
,
ranging from 10
Ϫ16.5
to 10
Ϫ19.5
(25ЊC).
The Activity of the Solid Phase
In a solid-solution equilibrium, the pure solid phase is defined
as a reference state and its activity is, because of its constancy,
C023_002_r03.indd 1265C023_002_r03.indd 1265 11/18/2005 1:32:09 PM11/18/2005 1:32:09 PM
© 2006 by Taylor & Francis Group, LLC
Plots of this equation as a function of pH are given in Figure 6
illustrated in Figure 7.
1266 WATER CHEMISTRY
set equal to unity. This means that the activity of the solid phase
is implicitly contained in the solubility equilibrium constant.
Therefore, experimental differences from precisely known
equilibrium constants can be used to deduce information about
the constitution of the solid phase.
There are various factors that affect the activity of the
solid phase: (1) the lattice energy, (2) the degree of hydra-
tion, (3) solid solution formation, (4) the free energy of the
surface and (5) the presence of constituents affecting the

purity of the solid.
The solids occurring in nature are seldom pure sub-
stances. For example, isomorphous replacement by a foreign
constituent in the crystalline lattice is an important factor by
which the activity of the solid phase may be decreased.
Redox Equilibria and Electron Activity
There is a conceptual analogy between acid-base and
oxidation-reduction reactions. In a similar way that acids
and bases have been interpreted as proton donors and proton
acceptors, reductants and oxidants are defined as electron
donors and electron acceptors. Since free electrons do not
exist in solution, every oxidation is accompanied by a reduc-
tion and vice versa. An oxidant is thus a substance which
causes oxidation to occur, while itself becoming reduced.
The oxidation states of the reactants and products change
as a result of the electron transfer which mechanistically may
occur as a transfer of a group that carries one or more elec-
trons. Since electrons are transferred in every redox reac-
tion they can be treated conceptually like any other discrete
reacting species, namely, as free electrons. The following
redox reaction is illustrative:
O
2
ϩ 4H
ϩ
ϩ 4e ϭ 2H
2
O reduction
4Fe
2ϩ

ϭ 4Fe
3ϩ
4e oxidation
O
2
ϩ 4Fe
2ϩ
ϩ 4H
ϩ
ϭ 4Fe
3ϩ
ϩ 2H
2
O redox
reaction (27)
2
4
6
8
10
6
8
10
12
pH
Zn
+2
Fe
+2
S

R
+2
CO
+2
Ks
0
C
T
1
2
1
2
(Ks
0
)
Mg
+2
CO
3
–log CONC. (M)
–2
FIGURE 7 Solubility of carbonates in solutions of con-
stant total dissolved carbonate carbon. The maximum soluble
metal ion concentration ([Me
2ϩ
] ϭ k
s0
/[(CO
3
2ϩ

] ϭ K
s0
/a
2
C
T
)
is shown as a function of pH for C
T
ϭ 10
Ϫ2.5
M. The acid-
ity constants for H
2
CO
3
*
are indicated on the horizontal axis.
Dashed portions of the curves indicated conditions under
which MeCO
3
(s) is not thermodynamically stable due to
the formation of the more stable solid hydroxide or oxide.
Ref.: Stumm, W. and J. Morgan, Aquatic Chemistry, Wiley-
Interscience, New York, 1970, p. 179.
FIGURE 6 Solubility of amorphous Fe(OH)
3
, ZnO, and CuO. The
equilibrium solubility curves for the hydroxo metal complexes and
for the free metal ion have been combined to yield the composite

curve bordered by the cross hatching. The constituent curves were
nuclear complexes, for example Fe
2
(OH)
2
2Ϫ
, CU
2
(OH)
2
2ϩ
, has been
ignored. Such complexes do not change the solubility characteristics
markedly for the solids considered. Also ignored is complexing with
other ligands such as NH
3
. Ref.: Stumm, W. and J. Morgan, Aquatic
Chemistry, Wiley-Interscience, New York, 1970, p. 173.
– log CONC.
– log CONC.
ZnO(s)
CuO(s)
CuOH
+
Cu
+2
ZnOH
+
ZnOH
3

–
am-Fe(OH)
3
(s)
*
8
8
8
6
6
6
4
4
2
2
Zn(OH)
4
–2
7911
pH
Zn
+2
10 12
pH
Cu(OH)
4
–2
Cu(OH)
3
–

b
a
(M)
(M)
Fe
+3
FeOH
+2

pH
Fe(OH)
2
2
4
6 8 10 12
Fe(OH)
4
2
4
6
8
c
–
+
C023_002_r03.indd 1266C023_002_r03.indd 1266 11/18/2005 1:32:09 PM11/18/2005 1:32:09 PM
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constructed from the data in Table 3. The possible occurrence of poly-
WATER CHEMISTRY 1267
This treatment of electrons is no different from that of
many other species such as hydrogen ions and silver ions,

which do not actually exist in aqueous solution as free and
unhydrated species but which are expressed as H
ϩ
and Ag
ϩ

in reactions. For example, the H
ϩ
really takes the form of
hydrated protons (H
9
O
4
ϩ
) or hydrogen complexes (acids
such as HCl).
Redox Intensity
By introducing the definition pH ϭ Ϫlog[H
ϩ
], which under
idealized conditions is formulated as
pH ϭ Ϫlog{H
ϩ
}, (28)
Sørenson (1909) established a convenient intensity parame-
ter that measures the relative tendency of a solution to donate
or transfer protons. In an acid solution this tendency is high
and in an alkaline solution it is low.
Similarly, Jørgensen (1945) has established an equally
convenient redox intensity parameter,

p ␧ ϭ Ϫlog{e}, (29)
that measures the relative tendency of a solution to donate
or transfer electrons. In a highly reducing solution the ten-
dency to donate electrons, that is, the hypothetical “electron
pressure” or electron activity, is relatively large. Just as the
activity of hypothetical hydrogen ions is very low at high
pH, the activity of hypothetical electron is very low at high
p ␧. Thus a high p ␧d indicates a relatively high tendency for
oxidation. In equilibrium equations H
ϩ
and e are treated in
an analogous way. Thus oxidation or reduction equilibrium
constants can be defined and treated similarly to acidity con-
stants as shown by the following equations:
For protolysis:
HA ϭ H
ϩ
ϩ A
Ϫ
(30)
{H
ϩ
}{A
Ϫ
}/{HA} ϭ K
HA
, or
pH ϭ Ϫlog K
HA
ϩ log({A

Ϫ
}/{HA}). (31)
For the oxidation of Fe
2 ϩ
to Fe
3 ϩ
:
Fe
2 ϩ
ϭ Fe
3 ϩ
ϩ e (32)
{e}{Fe
3 ϩ
}/{Fe
2 ϩ
} ϭ K
ox
, or
p ␧ ϭ Ϫlog K
ox
ϩ log({Fe
3 ϩ
/Fe
2 ϩ
}). (33)
As seen from Eq. (33) p ␧ increases with the ratio of the activ-
ities (or concentrations) of oxidized to reduced species.
†

†
The sign convention adopted here is that recommended by IUPAC
(International Union of Pure and Applied Chemistry).
From the following general reduction reaction:
aA ϩ bB ϩ ne ϭ cC ϩ dD
the generalized expression for any redox couple is given by

ppεεϭϪϪϩϩ
ϭ
⌬
0
1
23
a
n
b
n
c
n
d
n
n
G
RT
pppp
ABCD
. Ј
(34)
where p X ϭ Ϫlog[ X ] and

p
n
K
n
K
n
G
nRT
εϭϪ ϭ ϭ
⌬111
23
log log
.
ox red

K
ox
and K
red
are the equilibrium constants for the oxidation
and reduction reactions; n is the number of electrons trans-
ferred in the reaction. ⌬G represents the free energy change
for the reduction. Thus it is seen that p ␧ is a measure of the
electron free energy level per mole of electrons.
Equation (34) permits the expression of the redox
intensity by p␧ for any redox couple for which the equi-
librium constant is known. Numerical illustrations of the
calculation of p ␧ values (25ЊC) are given for the following
equilibrium systems in which the ionic strength, I,
‡

is
assumed to approach O:
a) An acid solution 10
Ϫ5
M in Fe
3 ϩ
and 10
Ϫ3
M in
Fe
2 ϩ
.
b) A natural water at pH ϭ 7.5 in equilibrium with
the atmosphere (Po
2
ϭ 0.21 atm.).
c) A natural water at pH ϭ 8 containing 10
Ϫ5
M
Mn
2 ϩ
in equilibrium with g ϪMnO
2
(s).
“Stability Constant of Metal-Ion Complexes” gives the
following equilibrium constants ( K
red
):
a) Fe
3 ϩ

ϩ e ϭ Fe
2 ϩ
;

KKϭϭ
ϩ
ϩ
{}
{}{}
;log .
Fe
Fe e
2
3
12 53

b) 1/2 O
2
( g ) ϩ 2H
ϩ
ϩ 2e;
H
2
O(1)
‡
Ionic strength, I, is a measure of the interionic effect resulting pri-
marily from electrical attraction and repulsions between the various
ions; it is defined by the equation t = 1/2 ⌺
i
C

i
Z
i
2
. The summation is
carried out for all types of ions, cations and anions, in the solution.
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1268 WATER CHEMISTRY

K
P
Kϭϭ
ϩ
1
41 55
2
12 2 2
oH e
/
{}{}
;log .

c) g ϪMnO
2
(s) ϩ 4H
ϩ
ϩ 2e
ϭ Mn
2

ϩ 2H
2
O(1);

KKϭϭ
ϩ
ϩ
{}
{}{}
;log . .
Mn
He
2
42
40 84

For the conditions stipulated the following p ␧ values are
obtained:
a) p
Fe
Fe
 ϭϩ ϭ
ϩ
ϩ
12 53 10 53
3
2
.log
{}
{}

.
b) poH ϭϩ ϭ
ϩ
20 78 1 2 13 11
2
12 2
. / log( { } ) .
/
P

c)
p
H
Mn
 ϭϩր ϭ
ϩ
ϩ
20 42 1 2 6 92
4
2
.log
{}
{}
.
Equilibrium Distribution in the Sulphur System
Figure 8 shows the p ␧ dependence of a 10
Ϫ4
M SO
4
2Ϫ

ϪHS
Ϫ
system at pH ϭ 10 and 25ЊC. The reaction is

SO H e HS H O
4
2
2
98 41
Ϫϩ Ϫ
ϩϩϭϩ () (35)

and the redox equilibrium equation is

p
SO H
HS
εϭր ϩր
Ϫϩ
Ϫ
18 18
4
2
log log
[][]
[]
K
(36)
where log K (for the reduction reaction) is 10

34
. Hence,

ppHSOHS ϭϪ ϩր Ϫր
ϪϪ
425 1125 18 1
4
2
. . log[ ] log[ ]

or, for pH ϭ 10,

pSOHS ϭϪ ϩ ր Ϫ ր
ϪϪ
718 18
4
2
log[ ] log[ ].

HS
Ϫ
is the predominant S(ϪII) species at pH ϭ 10. Figure 8
shows that the lines for [SO
4
2Ϫ
] and [HS
Ϫ
] intersect at p ␧ ϭ Ϫ7.
The asymptotes for [SO
4

2Ϫ
] have slopes of ϩ8 and 0, whereas
those for [HS
Ϫ
] have slopes of 0 and Ϫ8.
Lines for the equilibrium partial pressure of O
2
and H
2
are
also given in the diagram. As the diagram shows, rather high
relative electron activities are necessary to reduce SO
4
2Ϫ
.

At
the pH value selected, the reduction takes place at p ␧ values
slightly less negative than for the reduction of water. Thus in
the presence of oxygen and at pH ϭ 10, only sulfate can exist;
its reduction is possible only at p ␧ values less than Ϫ6.
Equilibrium Constants for Redox Reactions
Equilibrium constants for some redox processes pertinent in
sive reference source for such constants is Stability Constants
for Metal-Ion Complexes, L. G. Sillén and A. E. Martell, The
Chemical Society, London (1964) and its Supplement (1971).
Significantly, the first section of this reference deals with the
electron as a ligand, similarly to its treatment above. Another
compilation, somewhat outdated though still very useful, is
Oxidation Potentials, 2nd ed., W. M. Latimer, Prentice-Hall,

(1952). This treatise lists redox potentials rather than equi-
librium constants, but, as shown in the next section, the latter
are readily obtained from the former.
The Determination of p␧ and Redox Potential
As with pH, p ␧ can be measured with a potentiometer using
an indicator electrode (e.g., a platinum or gold electrode)
and a reference electrode. The result is read as a potential
difference in volts. When a reversible hydrogen electrode, at
which the electrode reaction is H
2
ϭ 2H
ϩ
ϩ 2e, is used as
the reference, the resulting potential difference is termed the
redox potential, E
H
, where the suffix H refers to the hydro-
gen electrode as the reference. Usually another reference
electrode is used, e.g., a calomel electrode, but the addition
of a constant factor (i.e., the potential difference between
the calomel electrode and the hydrogen electrode) to the
p⑀
–12 –8
–4
0
4
812
HS
–
–4

–8
–12
–16
HS
–
10
–92
10
–76
10
–60
10
–44
10
–28
10
–28
10
–12
10
+4
Po
2
PO
2
PH
2
pH
2
10

–44
10
–36
10
–20
10
–12
10
–4
10
+4
SO
4
–2
SO
4
–2
pH=10
log CONC. (M or atm.)
FIGURE 8 Equilibrium distribution of sulfur compounds as a func-
tion of p
␧ at pH ϭ 10 and 25ЊC. Total concentration is 10
Ϫ4
M. The
dotted curve shows that solid sulphur cannot exist thermodynami-
cally at pH ϭ 10, since its activity never becomes unity. Figure from
Stumm, W. and J. Morgan, Aquatic Chemistry, Wiley-Interscience,
New York, 1970, p. 311.
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aquatic conditions are listed in Table 4. A quite comprehen-
WATER CHEMISTRY 1269
†
With dilute solutions it is convenient to express concentrations on
a minus log concentration scale, e.g., pNa
ϩ
ϭ Ϫlog[Na
ϩ
].
observed voltage gives E
H
. p ␧ is then readily computed by
dividing E
H
by a constant factor according to Eq. (27):

pε
E
RT F
H
(. )
,
23 ր

where
F ϭ the Faraday constant (96,500 coulombs per
equivalent),
R ϭ the gas constant (8.314 volt-coulombs per
degree—equivalent),
and

T ϭ the absolute temperature in degrees Kelvin.
At 25ЊC, (2.3 ␻T / F ) ϭ 0.059 volt.
Both conceptually and computationally, it is more conve-
nient to use the dimensionless p ␧ rather than the more directly
measured E
H
, as a measure of the redox intensity. Every ten-
fold change in the activity ratio of Eq. (34) causes a change in
p ␧ of one unit divided by the number of electrons transferred
in the redox reaction. The fact that one electron can reduce one
proton is another reason for expressing the intensity parameter
for oxidation in a form equivalent to that used for acidity.
Incidentally, pH is also not measured directly. It is deter-
mined by measuring the potential (in volts) of an indicator
electrode (e.g., a glass electrode) with respect to a reference
electrode. If the hydrogen electrode is used or referred to as
the reference electrode, the resulting potential difference is
called the acidity potential. The pH is calculated by dividing
the acidity potential by 2.3 RT / F.
The Peters-Nernst Equation
Since much literature still describes the electron activity in
volts rather than on a scale

†

similar to that for other reagents,
the Peter-Nernst equation is still of utility. It is easily
expressed by substituting Eq. (37) into Eq. (34):

EE

RT
nF reduced
G
nF
HH
ϭϩ
ϭ
Ϫ⌬
0
23.
log
{}
{}
,
oxidized
(38)

where

ERTF
G
nF
H
0
0
23ϭրϭ
Ϫ⌬
(. ) ; .pε
(39)

Measurement of E
H

It is essential to distinguish between the concept of a poten-
tial and the measurement of a potential. Redox or electrode
potentials (as listed in “Stability Constants of Metal-Ion
Complexes” and other references) have been derived from
equilibrium data, thermodynamic data, the chemical behav-
ior of a redox couple with respect to known oxidizing and
reducing agents, and from direct measurements of electro-
chemical cells.
Direct measurement of E
H
for natural water environments
involves complex theoretical and practical problems in spite
of the apparent simplicity of the electrochemical technique.
For example, the E
H
of aerobic (dissolved oxygen contain-
ing) waters, measured with a platinum or gold electrode,
does not agree with that predicted by Eq. (38). Even when
reproducible results are obtained, they often do not represent
reversible Nernst potentials. Among the considerations hin-
dering the direct measurement of E
H
are the rates of electron
exchange at certain electrodes and the occurrence of mixed
potentials. A mixed potential results when the rate of oxida-
tion of one redox couple is compensated by the rate of reduc-
tion of a different couple during the measurement. Although

aqueous systems containing oxygen or similar oxidizing
agents will usually give positive E
H
values and anaerobic
systems will usually give negative ones, detailed quantita-
tive interpretation with respect to concentrations of redox
species is generally unwarranted.
Since natural waters are normally in a dynamic rather
than an equilibrium condition, even the concept of a single
oxidation–reduction potential characteristic of the aqueous
system as a whole cannot be justified. At best, measurement
can give rise to an E
H
value applicable to a particular redox
reaction or to redox species in partial chemical equilibrium
and even then only if these redox agents are electrochemi-
cally reversible at the electrode surface at a rate that is rapid
compared with the electron drain or supply by the measuring
electrode system.
p␧–pH Diagrams
A p ␧ –pH stability field diagram shows in a panoramic way
which species predominate at equilibrium under any condi-
tion of p ␧ (or E
H
) and pH. The primary value of a p ␧ –pH dia-
gram is its simultaneous representation of the consequences
of the equilibrium constants of many reactions for any com-
the various species pertinent to the chlorine system. These
diagrams are readily constructed from thermodynamic data
Log Concentration—p␧ Diagrams

The predominant redox species are depicted as a function
of p ␧ or redox potential in p ␧ -log concentration diagrams.
Upper or lower bounds of p ␧ values for the occurrence of
specific redox reactions are immediately evident from these
double log-arithmetic diagrams. Such diagrams, constructed
for a few elements in the biochemical cycle.
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such as those listed in Table 4.
bination of p ␧ and pH. Figure 9 shows stability fields for
from the data in Table 4 with pH ϭ 7, are shown in Figure 10
1270 WATER CHEMISTRY
The boundary conditions for the stability of water
(Figure 10(d)) are given at high p ␧ values by the oxidation
of water to oxygen:
1/2 H
2
O ϭ 1/4 O
2
ϩ H
ϩ
ϩ e (40)
and at low p ␧ values by the reduction of water to hydrogen
H
2
O ϩ e ϭ 1/2 H
2
ϩ OH
Ϫ
. (41)

Water in equilibrium with the atmosphere ( P o
2
ϭ 0.21 atm.)
at pH ϭ 7.0 (25ЊC) has a p␧ ϭ 13.6 ( E
H
ϭ 0.8 volt.)
Figure 10(a) gives the relationships among several oxi-
dation states of nitrogen as a function of p ␧. For most of
the aqueous range of p ␧, N
2
gas is the most stable species.
However, at large negative p ␧ values ammonia becomes
predominant and for p ␧ greater than ϩ12 nitrate dominates
pH ϭ 7. The fact that the nitrogen gas of the atmosphere has
not been converted largely into nitrate under the prevailing
aerobic conditions at the land and water surfaces indicates a
lack of efficient biological mediation.
FIGURE 10 Equilibrium concentrations of
biochemically important redox components
as a function of p
␧ at a pH of 7.0. These equi-
librium diagrams have been constructed from
for the following concentrations representa-
tive of natural water systems: pH ϭ 7.0; C
T

(total carbonate carbon) ϭ 10
Ϫ3
M; [H
2

(aq)]
ϭ [H
2
S(aq)] ϩ [HS
Ϫ
] ϩ [SO
4
2Ϫ
] ϭ 10
Ϫ5
M;
[NO
3Ϫ
] ϩ [NO
2Ϫ
] ϩ NO
4ϩ
] ϭ 10
Ϫ3
M; p
N2
2
ϭ
0.78 atm. and thus [N
2
(aq)] ϭ 0.5 ϫ 10
Ϫ3
M.
Figure from Stumm, W. and J. Morgan,
Aquatic Chemistry, Wiley-Interscience, New

York, 1970, p. 331.
1
2
3
4
+25
+20
+15
+10
+5
0
0510
pH
0
0
5
+0.5
+10
10
15
20
25
+15
p⑀
p⑀
H
2
H
2
O

H
2
O
O
2
HOCl
OCl
–
Cl
–
Cl
2
(aq)
E
H
(v)
FIGURE 9 Stability field diagram for the chlorine
system: p
␧ versus pH. The curves labelled 1,2,3
and 4 represent equilibria derived from Eqs. (25)
a
value is 7.53 at 25ЊC and is represented by the verti-
cal line between HOCl and OCl
Ϫ
stability fields. In
dilute solutions Cl
2
(aq) exists only at low pH. Cl
2
,

OCl
Ϫ
, and HOCl are all unstable or metastable in
water since they are all slowly reduced by water as
shown by the position of their stability fields, with
respect to the H
2
O–O
2
equilibrium curve. Figure
from Stumm, W. and J. Morgan, Aquatic Chemistry,
Wiley-Interscience, New York, 1970, p. 320.
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and (26) of Table 4 and pK for HOCl: This latter
equilibrium constants listed in Table 3 and 4
WATER CHEMISTRY 1271
TABLE 4
Equilibrium constants of redox processes pertinent in aquatic conditions (25ЊC)
Reaction
pε°(ϭ Ϫlog K) pε°(W)
a
(1) 1/4 O
2
(g) ϩ H
ϩ
(W) ϩ e ϭ 1/2 H
2
O ϩ20.75 ϩ13.75
(2) 1/5 NO

3
Ϫ
ϩ 6/5H
ϩ
(W) ϩ e ϭ 1/10 N
2
(g) ϩ 3/5 H
2
O ϩ21.05 ϩ12.65
(3) 1/2 MnO
2
(S) ϩ 1/2HCO
3
Ϫ
ϩ 3/2 H
ϩ
(W) ϩ e
ϭ 1/2MnCO
3
(s) ϩ 3/8 H
2
O
ϩ20.46 ϩ9.96
ϩ8.46
(3a) 1/2 MnO
2
(s) ϩ 2H
ϩ
(W) ϩ e ϭ 1/2 Mn
2ϩ

ϩ H
2
O ϩ20.42 ϩ6.42
(4) 1/2 NO
3
Ϫ
ϩ H
ϩ
(W) ϩ e ϭ 1/2 NO
2
Ϫ
ϩ 1/2 H
2
O ϩ14.15 ϩ7.15
(5) 1/8 NO
2
ϩ 5/4 H
ϩ
(W) ϩ e ϭ 1/8 NH
4
ϩ
ϩ 3/8 H
2
O ϩ14.90 ϩ6.15
(6) 1/6 NO
2
Ϫ
ϩ 4/3 H
ϩ
(W) ϩ e ϭ 1/6 NH

4
ϩ
ϩ 1/3 H
2
O ϩ15.14 ϩ5.82
(7) 1/2 CH
3
OH ϩ H
ϩ
(W) ϩ e ϭ 1/2 CH
4
(g) ϩ
1/2 H
2
O
ϩ9.88 ϩ2.88
(8) 1/4 CH
2
O ϩ H
ϩ
(W) ϩ e ϭ 1/4 CH
4
(g) ϩ 1/4 H
2
O ϩ6.94 Ϫ0.06
(9) FeOOH(s)ϩ HCO
3
Ϫ
ϩ 2H
ϩ

(W) ϩ e ϭ FeCO
3
(s) ϩ
2H
2
O
ϩ15.33 ϩ1.33
Ϫ1.67
b
(9a) Fe
3ϩ
ϩ e ϭ Fe
2ϩ
ϩ12.53 ϩ 12.53
(10) 1/2 CH
2
O ϩ H
ϩ
(W) ϩ 3 ϭ 1/2 CH
3
OH ϩ3.99 Ϫ3.01
(11) 1/6 SO
4
2Ϫ
ϩ 4/3 H
ϩ
(W) ϩ e ϭ 1/6 S(s) ϩ 2/3 H
2
O ϩ6.03 Ϫ3.30
(12) 1/8 SO

4
2Ϫ
ϩ 5/4Hϩ (W) ϩ 3 ϭ 1/8 H
2
S(g) ϩ
1/2 H
2
O
ϩ5.75 Ϫ3.50
(13) 1/8 SO
4
2Ϫ
ϩ 9/8 H
ϩ
(W) ϩ e ϭ 1/8 HS
Ϫ
ϩ 1/2 H
2
O ϩ4.13 Ϫ3.75
(14) 1/2 S(s) ϩ H
ϩ
(W) ϩ e ϭ 1/2 H
2
S(g) ϩ2.89 Ϫ4.11
(15) 1/8 CO
2
(g) ϩ H
ϩ
(W) ϩ e ϭ 1/8 CH
4

(g) ϩ 1/4 H
2
O ϩ2.87 Ϫ4.13
(16) 1/6 N
2
(g) ϩ 4/3 H
ϩ
(W) ϩ e ϭ 1/3 NH
4
ϩ
ϩ4.68 Ϫ4.68
(17) 1/2 (NADP
ϩ
) 1/2 H
ϩ
(W) ϩ e ϭ1/2 (NADPH) Ϫ2.0 Ϫ5.5
c
(18) H
ϩ
(W) ϩ e ϭ 1/2 H
2
(g) 0.0 Ϫ7.00
(19) Oxidized ferredoxin ϩ e ϭ reduced ferredoxin Ϫ7.1 Ϫ7.1
d
(20) 1/4 CO
2
(g) ϩ H
ϩ
(W) ϩ e ϭ 1/24 (glucose) ϩ
1/4 H

2
O
Ϫ0.20 Ϫ7.20
e
(21) 1/2 HCOO
Ϫ
ϩ 3/2H
ϩ
(W) ϩ e ϭ 1/2 CH
2
O ϩ
1/2 H
2
O
ϩ2.82 Ϫ7.68
(22) 1/4 CO
2
(g) ϩ H
ϩ
(W) ϩ e ϭ 1/4 CH
2
O ϩ 1/4 HO
2
Ϫ1.20 Ϫ8.20
(22a) 1/4 HCO
3
Ϫ
ϩ 5/4 H
ϩ
(W) ϩ e ϭ 1/4 CH

2
O ϩ
1/2 H
2
O
ϩ0.76 Ϫ7.99
Ϫ8.74
b
(23) 1/2 CO
2
(g) ϩ 1/2 H
ϩ
(W) ϩ e ϭ 1/2 HCOO
Ϫ
Ϫ4.83 Ϫ8.73
(24) 1/2 C1
2
(aq) ϩ e ϭ C1
Ϫ
ϩ23.6 ϩ23.6
(25) HOCl ϩ H
ϩ
(W) ϩ e ϭ 1/2 C1
2
(aq) ϩ H
2
O ϩ26.9 ϩ19.9
a
(W) signifies that the pH ϭ 7.
b

These data correspond to (HCO
3
Ϫ
) ϭ 10
Ϫ3
M rather than unity and so are not exactly pε(W); they represent
typical aquatic data conditions more nearly than pε(W) values do.
c
M. Calvin and J.A. Bassham, The Photosynthesis of Carbon Compounds, Benjamin, New York, 1962.
d
D. I. Arnon, Science, 149, 1460 (1965).
e
A. L. Lehninger, Bioenergetics, Benjamin, New York, 1965:
Table from Stumm, W. and J. Morgan, Aquatic Chemistry, Wiley-Interscience, New York, 1970, p. 318.
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© 2006 by Taylor & Francis Group, LLC
1272 WATER CHEMISTRY
Because of these peculiarities in equilibria with N
2
between NO
3
Ϫ
, and NO
2
Ϫ
by assuming the transition between
bound nitrogen and N
2
to be hindered. This diagram shows that
the shifts in relative predominance of the three species, NH

4
ϩ
,
NO
2
Ϫ
and NH
3
Ϫ
occur within a rather narrow p ␧ range. That
each of these species has a dominant zone within this range
would seem to be a contributing factor to the highly mobile
characteristics of the fixed species of the nitrogen cycle. Note
that the relevant species of N (ϪIII) at pH ϭ 7 is NH
4
ϩ
rather
than NH
3
since the ammonia system p K
a
ϭ 9.3 .
The reduction of SO
4
2Ϫ
to H
2
S or HS
Ϫ
provides a good

example of the application of equilibrium concepts to aquatic
relationships. Figure 10(d) shows that significant reduction
of SO
4
2Ϫ
to H
2
S at pH ϭ 7 requires p␧ Ͻ Ϫ3. The biological
enzymes that mediate this reduction must therefore operate
at or below this p␧. Because the system is dynamic rather
than static only an upper bound can be set in this way, for
the excess driving force in terms of p␧ at the mediation site
cannot be determined by equilibrium computations. Since,
however, many biologically mediated reactions seem to oper-
ate with relatively high efficiency in utilizing free energy, it
O
2
H
2
O
NO
3
NO
3
–
–
MnO
2
Mn(I)
NH

4
+
FeOOH
CH
2
O
Fe(II)
CH
3
OH
SO
4
2–
HS
–
CO
2
CH
4
H
2
O
H
2
8
–5
0
5
p⑀
10

15
900
600
EH
(mv)
25°
300
0
–300
a
Electron Transfer Capacity “Titration Curve”
(REDOX INTENSITY VS REDOX CAPACITANCE)
REDUCTION BY ORGANICS
OXIDATION BY OXYGEN
N
2
meq / liter
5
10 15
FIGURE 11 Electron transfer capacity “Titration Curve.” In a system with
excess organic material, the redox intensity falls as the electron acceptors are suc-
cessively reduced. This diagram was constructed from the redox intensities and
reactions indicated on the curve proceed sequentially during the stagnation peri-
od in the deeper waters of a polluted lake, vertically downward or temporally in
sediments, sequentially after starting an anaerobic digester and chronologically in
ground water contaminated with organic nutrients.
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© 2006 by Taylor & Francis Group, LLC
initial concentrations (expressed as electron equivalents) listed in Table 3. The
, Figure

10(b) has been constructed for the metastable equilibria
WATER CHEMISTRY 1273
appears likely that the operating p p␧ value does not differ
greatly from the equilibrium value. The diagram also indi-
cates the possibility of formation of elemental sulfur within
a narrow p␧ range at neutral and low pH.
The major feature of the carbon system, shown in
C (ϪIV): i.e., the reduction of carbon dioxide bicarbonates
and carbonates to methane and the reverse oxidation reac-
tions. Formation of solid carbon is thermodynamically pos-
sible close to p ␧ ϭ Ϫ4 but its inclusion does not change
other relationships. Other carbon compounds exist under
equilibrium conditions only at very small concentrations
(Ͻ10
Ϫ9
M). The existence of myriad synthetic and bio-
chemical organic compounds at ambient p␧ levels is due to
their exceedingly slow rates of equilibration to CO
2
or CH
3

or, in the case of the complicated organic structures in living
systems, to the constant input of energy.
Microbial Mediation and Free Energy of
Redox Reactions
To survive and hence reproduce, microorganisms must not
only capture a significant fraction of the thermodynamically
available energy but must also acquire this energy at a rate
compatible with the maintenance of life. Thus the salient

capability is that of power production per unit biomass and
therefore kinetics (or the rate of movement toward equilib-
rium) should be considered as well as thermodynamics. To
provide for such energy production microorganisms have
developed highly efficient and specific biological com-
pounds (enzymes) which catalyze energy yielding reac-
tions and cell constituent producing processes.
Organisms do not oxidize organic substrates or reduce
O
2
or SO
4
2Ϫ

; they only mediate those reactions which are
thermodynamically possible, or more specifically, the elec-
tron transfer occurring in these reactions. The p ␧ range in
which certain oxidation or reduction reactions are possible
may be estimated by calculating the equilibrium concen-
trations of the relevant species as a function of p␧. Since,
for example, SO
4
2Ϫ
can be reduced only below a given p␧
or redox potential, an equilibrium model can characterize
the p␧ ranges in which reduction of sulfate is possible and
is not possible. Such models are graphically presented in
a parameter that characterizes the ecological milieu in a
restrictive fashion.
tions of the listed half reactions to give complete redox

reactions. Those that are thermodynamically possible are
always accompanied by a decrease in free energy. The free
energy change of the complete redox reaction, ⌺⌬ G , is
easily calculated from the p ␧ values through rearrange-
ment of Eq. (34):

⌬GRT
nn
ϭϪ
ϫ
23.
[( ) ( ) ].
∑
∑
pp
for reduction for oxidation
εε
(42)

If Ϫ⌬ G is negative, the reaction can occur. Combinations that
All of these reactions are mediated by microorganisms.
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© 2006 by Taylor & Francis Group, LLC
lead to energetically possible reactions are given in Table 3.
The data in Table 4 permit the calculation of combina-
Figure 10(e), is simply the interconvertibility of C(IV) to

bial mediation. These diagrams manifest the use of p␧ as
Figure 11, where all the reactions are amenable to micro-
1274 WATER CHEMISTRY
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WERNER STUMM (DECEASED)
Swiss Federal Institute of Technology, Zürich
MARTIN FORSBERG
STEVEN GHERINI

Harvard University
WASTES OF INDUSTRY: see INDUSTRIAL WASTE MANAGEMENT
WATER—DESALTING: see DESALINATION

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© 2006 by Taylor & Francis Group, LLC