aq lecture
TRANSCRIPT
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THERMODYNAMICS OF AQUEOUS ELECTROLYTES:
IMPLICATIONS TO SOLUBILITY OF METALS IN HYDROTHERMAL
FLUID
Lecture Note
M. K. PANIGRAHI
Department of Geology & Geophysics
I.I.T., Kharagpur
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WB 721302
Email: [email protected]
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Introduction
Fluid-rock interaction is widespread at the lithosphere-hydrosphere interface - on the surface and ocean floor, sometimes penetrating to quite
deeper regions of the crust. There is hardly a rock that we encounter on the earth's surface, that has not interacted with a fluid at some point of time
in its evolution. The constructive and modificative role of fluids in the earth's crust can hardly be overstated. Here, we consider the fluid that
mostly is a water-rich liquid, in the one or two-phase and subcritical or supercritical state and is charged with simple as well as complex and
charged (ions) and neutral species (Fig.1). Rainwater, river water, surface stagnant water (lakes), ground water, hot spring water and of course, sea
water - all fall into this category, characterized by appreciably higher electrical conductivity over that of pure water and appreciably higher total
dissolved solids (TDS). That the hydrosphere runs quite deep is evident from the fluid induced melting and magmatism observed in the
subduction-zone settings and from the hydrothermal activities observed on the ocean
ridges and their continental analogues. The tufa, travertine andsinters deposited on
the mouths of hot springs and fumaroles, deposition of evaporites from present day
and ancient sea water, black/white smokers and sulfide chimneys on several
locations on the ocean ridge crests, widespread regional scale deuteric alterations in
granitic plutons, metasomatism of different kind associated with acid intrusives and
skarns in such environments, extensive quartz veining in rocks of all possible
geologic settings and ages sometimes richly mineralized with base and preciousmetals, zoned alteration pattern with distinctive mineralogy around such veins and
mineralized bodies of rocks - are phenomena resulting from activities of such fluids
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Fig.1 Phase relationship (schematic) in the H2O (+salt) system
showing the boiling curves of pure and salt (NaCl) charged water
and the critical curve. The supercritical region marked by dotted
lines (V) has so far not been understood properly.
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and their interaction with rocks. The naturally occurring aqueous solutions are thus diverse and one could possibly say that they are as diverse as
the rock types and geological settings in the crust.
The foregoing paragraph is enough indication to the importance of the study of aqueous solutions - in fact it is an integral part of geochemistry. It
is just not only for the understanding of the array of phenomena just mentioned, but also for their precise quantification that one needs to have
some insight of the fundamentals of the thermodynamics of aqueous solutions. A detailed treatment of the theory of aqueous electrolytes aspropounded by chemists is beyond the scope of this discussion. However, some fundamental principles based upon solutes and solvent relevant to
geochemistry will be touched upon. The literature on aqueous electrolytes and the methodologies of their treatment is quite vast. Extensive
reviews of the properties of aqueous solutions of a wide spectrum of electrolytes of geochemical relevance, retrieval of standard state
thermodynamic parameters and their extrapolation to higher pressure and temperatures and formulations on activity coefficients in wide
concentration ranges have been done by Helgeson and his coworkers over the last three decades (see Helgeson, Kirkham and Flowers, 1981;
Tanger and Helgeson, 1988; Shock and Helgeson, 1988 and Shock et al., 1989 are some of the important contributions in this regard) and work on
this line is still in progress with shifting of attention to organic species. We shall mostly confine ourselves to fundamental concepts covered in
standard texts (Nordstorm and Munoz(1985), Richardson and McSwain(1990) and Langmuir (1997)) and touch upon more advanced stuff through
example calculations.
Though the present discussion is relevant to hydrothermal environments (metal transport, deposition, alteration and mass-transfer) involving fluids
(of diverse origin) in the broad temperature range of 50 - 5000 C (max up to 5 kb of pressure), the formalism described here also would be useful
to the study of surface and near surface conditions that have tremendous implications to the study of environmental problems. In this discussion
we shall also restrict ourselves to only the inorganic electrolyte species in aqueous solutions and base our discussion entirely on equilibrium
thermodynamics. Irreversible processes of water-rock interaction can also be studied under local and partial equilibrium situations with a bit of
additional information..
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As has been pointed out by Nordstorm and Munoz(1985), electrolyte theory contrasts with principles of thermodynamics in two ways - it is theory
and it is microscopic. It involved mathematical derivation of thermodynamic functions from basic assumptions about the microscopic behavior of
charged particles. However, the theory has not proceeded enough to formulate a single framework that is adequate for all species under all
conditions. Therefore, empiricism is the rule rather than exception in treatment of aqueous electrolytes as will be discussed later.
The Problem:
Primarily our job is to quantify the process of mineral/rock - fluid interaction - may it be precipitation of minerals/salts from a complex brine like
the seawater or alteration of a mass of rock by a hydrothermal fluid or transport and deposition of metals by it. We have to handle a multiphase
multicomponent system where the compositional variabilities and hence the degrees of freedom are too large. Eventually, we shall have to take
help of equilibrium thermodynamics to compute reaction equilibria considering subsytems within the constraints imposed by the phase rule. We
shall be considering homogeneous equilibrium in the fluid and local or partial equilibrium in irreversible mineral/rock - fluid interaction process to
ascertain the chemical variability of the system. We try to monitor the dynamism of the process by working out the reaction paths through
incorporation of parameters such as 'progress variables' as a measure of change in concentration of species taking part in the reaction. We shall be
discussing these aspects in brief.
In computation of phase equilibria for all possible situations, one must have data on the phase properties i.e. the standard state thermodynamic
parameters such as the molar enthalpy of formation( fH ,0 ), molar entropy( 0S ) and volume( 0V ) as accurate and consistent as possible. In
case of crystalline solids, these have been measured to a fair degree of accuracy and retrieved through appropriate formalism. One must also have
adequate knowledge of the parameters for pressure and temperature dependence of the phase properties for extrapolation to higher temperature and
pressure conditions of geological significance. These parameters are also known to a reasonable degree of certainty. However, the same is not true
for electrolyte species in solutions present either in associated (charged or neutral) or dissociated ionic states. The energetics of such aqueous
solutions are not easy to understand and quantify because it arises out of interaction of atomic size particles (whose time in contact may be on the
order of picoseconds) in microenvironments, hence, their phase properties (and changes in them) are not as directly measurable as those of solids.
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Before going into the details of the formalism of deriving the standard state thermodynamic parameters or seeing how they are measured or
retrieved, it would be worthwhile to digress from here to look at some fundamental aspects of electrolytes in aqueous solutions.
Dissolution of electrolytes in water
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Water has been termed as the universal solvent and also one of the most corrosive substance known. It has got unusual properties compared to
molecules of identical molecular weight (it should have been a gas like NH 3 in normal temperature pressure conditions). Its high dielectric
constant (78.43 at 250C), high heat of fusion and vaporization, high heat capacity, polar nature (water is a dipole with a dipole moment of 1.8 x 10-
18 e.s.u. in the gaseous state compared to that of HCl which is 1.08) makes it almost an anomalous compound and the anomaly is mainly due to the
presence of hydrogen bonding (Fig. 2 and 3) that arises due to the presence of two lone-pair electrons (the 2s 2 and 2pz2) in the oxygen atom. In ice,
all lone pairs are bound to hydrogen atoms of the neighboring water molecules and the lattice consists of a puckered rigid framework of O-H 4
tetrahedra analogous to silicate structure. In liquid water hydrogen bonds are formed with a strength of 1/10 of the O-H molecular bond in the
individual water molecules themselves. Liquid water retains some of the tetrahedral characteristics of ice through lone pair interaction and
hydrogen bonding. Fig.2 depicts the water structure where the electron orbitals just prior to formation of gaseous water showing 2s and 2p lone
pairs and distribution of 2p orbitals is shown in (A), the gaseous water showing near tetrahedral H-O-H angle and lone pair distribution is shown
in (B) and the tetrahedral structure of water in ice showing hydrogen bonding is shown in (C). Fig. 3 depicts the dipolar structure of water where it
can be seen that the positive region is more dispersed than the negative region.
7
he water structure showing the bonding and lone
ctrons
The water dipole showing the dispersion of
e and negative charge
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Fig.4 Dielectric property of water
Fig.4 explains the dielectric property of water because of the high dipole moment.
The dielectric constant of water at 250C is 78.47 which is extremely high and is exceeded by any other geologically likely liquid. The solvating
power of water can be easily explained on the basis of the Coulomb's law for electrostatic force between two charged particles.
221
DrqqF= [1]
where D is the dielectric constant of the medium that is equivalent to the ratio of the permitivity of the medium to that of vacuum. This equation
then indicates that higher the dielectric constant of the medium, weaker is the force of attraction and greater is the tendency for ionic solids to
break apart into constituent ions.
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Fig.5 is a schematic presentation of the structure of water around a positively charged ion depicting the hydration shells. This is the Gurney's
cosphere model. (Explain) Since the positive region of the water dipole is more dispersed than the negative region, cations tend to be more highly
hydrated than anions. The hydrated species itself is quite dynamic and all water molecules are constantly moving and exchanging with a half life
as low as pure water structure. Ions of high charge to radius (Z/r) ratio are more hydrated with a predominant Zone-I and are called electrostrictive
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Hydration shells around a positively charged
water.
Fig.5 Hydration shells around a positively
charged ion in water.
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structure making ions (Li+, Na+, H3O+, Ca2+ and similar ions). They tend to decrease solution entropy and increase viscosity and density of the
solution. Larger ions are structure breaking (K+, Rb+, Cs+, Cl-, etc.). Transition metals have hydration shells bound in specific geometries, forming
true molecular entities or complex ions.
As will be discussed later, this solvation of ions contribute significantly to the change in free energy associated with removing an ion of radius ri
and charge Zie from vacuum and place it in a solvent of dielectric constant D (sometimes referred to as )
)11
(2
)( 2=
Dr
eZNG
i
iA[2]
Equation (2) is the Born equation and the terms outside the bracket collectively define the Born coefficient ( ) where NA is the Avagadro's
number, e is the elementary charge, ZI is the charge of the ion and ri also referred to as re,j (effective electrostatic radius) is the radius. Since the
dielectric constant of any medium is greater than vacuum (D>1) the Born free energy will always be negative and free energy of ionic solution
should be more negative for ions of high Z/r ratio in solutions of high dielectric constant. These large free energies explain the stability of ions in
aqueous solutions even without taking the additional stabilizing effect of hydration shells.
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11
Effect of pressure and temperature on dielectric
nt of water. (Boiling curve shown)
Effect of temperature and pressure on
ic constant of water
Effect of temperature on dissociation ofytes (in the higher temperature range)
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increasing pressure tries to restore it. One would therefore expect opposing effects of pressure and temperature on solubilities of electrolytes at
higher temperatures. Fig. 8 shows the effects of increasing pressure and temperature on dissociation constants of HCl and NaCl. The behavior is as
expected from the above discussion. This implies that there will be increasing association of ions with decreasing dielectric constant of water at
higher temperatures at a particular pressure. The upward shift of the curves at higher pressures is to be noted. Fig.9 shows that the log K valuespass through a maxima. This is explainable on the basis of increasing molecular vibration at higher temperatures that facilitate dissociation of ionic
molecules and thus compensates for the decrease in dielectric constant to some extent. These aspects control solubility of metals and formation
constants of complex ions as will be discussed in later sections.
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With this much of background, we shall proceed on to the energetics of solutions in terms of the thermodynamic parameters. As we shall see,
these are mostly dictated by some fundamental parameters of the solvent and aqueous ions in solution -
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Dissociation constants as a
nction of temperature.
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the dielectric constant of the solvent, charge on the ion and effective electrostatic radius of the ion. Before getting into the thermodynamic
quantities associated with solution, it is imperative to brush up our fundamentals on concentration units. Here, the concentration unit is
conventionally expressed as molal(m) i.e. one gram molecular weight in a kg of the solvent as against molar or mole fraction units used in
representing concentration in crystalline solutions. We shall be referring to the thermodynamic quantities in terms of 'partial molal' or 'molal'. It
would also be appropriate to discuss about the standard state in aqueous electrolyte solutions. In crystalline solids and gases, a unit activity
situation is referred to as standard state. In the former, pure phases at temperature and pressure of interest and in the latter, a pure gas at 298.15 K
and 1 bar pressure are taken as the standard state. The standard state is
so chosen that the activity coefficient is unity (ideal case) and mole fraction is also unity (i.e. a
pure phase). If we want to impose the same conditions on aqueous electrolytes using the molal concentration unit, it should be a 1 molal solution
of an electrolyte at unit activity at 298.15 K and 1 bar. This becomes hypothetical because the activity coefficient in that case can never be one.
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Fig.10 Standard and reference state in aqueous
electrolytes
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Many electrolytes and minerals reach saturation before 1 molal and no solute can have unit activity at m=1. Therefore a state has to be chosen
where activity coefficient can be one. If we consider an infinitely dilute solution, the interaction between ions can be taken as zero and activity
coefficient can be unity (Henrian solution) and we can take activity = molality at any temperature and pressure. This is defined as the reference
state as depicted in Fig.10 and it is central to the understanding of the thermodynamics of aqueous solutions and all calculations are done on this
basis.
Thermodynamic parameters of aqueous solutions
As has already been mentioned, it is difficult to measure the change in any thermodynamic quantity of an ion on its dissolution. Therefore, what
we measure is basically an apparent quantity. We can theoretically calculate the change in any thermodynamic parameter when an ion is solvated
assuming that there is no change in the partial molal thermodynamic quantities of the solvent, i.e. water. We can express this using the enthalpy
and free energy.
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),( 0,000
rr TPTPf HHHH += [3]
),(0
,000
rr TPTPf GGGG += [4]
The terms on the l.h.s. stand for the conventional apparent standard partial molal quantity (enthalpy and free energy) of formation of an aqueous
species at a given pressure and temperature. The first terms on the r.h.s. of the above two equations are the conventional standard partial molal
enthalpy and free energy of formation of the species from elements at a reference pressure and temperature (1 bar and 298.15 K) and the terms in
parenthesis correspond to the difference in the conventional standard partial molal enthalpy and free energy of formation of an aqueous species at
the temperature and pressure of interest and those at a reference condition.
If we represent a thermodynamic quantity in general by a symbol , then for an electrolyte(k)
absj
jkjk
,0
,
0 = [5]This is the additive relationship among the partial molal properties of aqueous electrolytes that share common ions. In the above equation, j,k is
the stoichiometric coefficient of the ion j in the electrolyte k. The conventional standard partial molal property represented by j0 is
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[6]
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Since H 0,abs is taken as zero for all practical purposes, j0 is equivalent to j0,abs. For electrical neutrality,
0
,
0
j
j
kjk = [7]
Theoretically, it is possible to calculate j0,abs for aqueous ions. However, the equation of state for aqueous ions are restricted to j0 and it does
not impose any restriction on geochemical calculations. Prediction of j0 is sufficient for predicting the standard partial molal properties of
aqueous ions in chemical reactions.
For any aqueous specis in solution, the thermodynamic quantity is a result of an intrinsic and an electrostatic parts that can be written for an
aqueous ion as
abs
je
abs
ji
abs
j
,0
,
,0
,
,0 += [8]
The electrostatic contribution can be thought of as contributed by collapse of water molecules or structural change in water caused by the ion and
solvation, i.e., electrostatic interaction of the ion and the solvent molecule. Thus
abs
js
abs
jc
abs
je
,0
,
,0
,
,0
,+= [9]
The collapse and the intrinsic parts combined can be denoted as 'nonsolvation' contribution to the parameter. That is
abs
ji
abs
jc
abs
jn
,0
,
,0
,
,0
, += [10]
We can write them in the conventional terms as explained before. For the k-th electrolyte
0
,
0
,
0
ksknk+= [11]
and the additivity relationship can be expressed in the conventional form as
0
,,
0
, jn
j
kjkn = [12]
and
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0
,,
0
, js
j
kjks = [13]
Equations (8) to (13) are the basis of the HKF model (Helgeson, Kirkham and Flowers, 1981; Tanger and Helgeson, 1988; Shock and Helgeson,
1988; and Shock et al, 1989)
As mentioned in eq(2), the solvation contribution to the free energy can be expressed by the Born equation. The conventional standard molal
Gibb's free energy of a solute j can be written as
)11(
0
, =
jjsG [14]
j is the Born coefficient of solute j. Similarly, for an electrolyte species k
)11(
0
, =
kks
G [15]
where jj
kjk = , according to the additivity relationship
In HKF model, re,j (effective electrostatic radius) was taken to be independent of pressure and temperature, hence, the terms for electrolytes and
ions were treated as constants.
The 0V (standard molal volume), 0 (compressibility) and0
PC (isobaric heat capacity of solvation) can be derived by taking the appropriate
derivatives of the free enegry expression.
T
k
T
ksks
PQ
P
GV
+=
=
1
10
,,
0[16]
where( )
TT PPQ
=
=
2
1/1[17]
Similarly
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T
k
ksks
PPQN
P
G
+=
=
2
2
2
,02
,0
11
2
[18]
where
=
=2
2
2
2
21
TTPPPT
Q
N
[19]
The isobaric heat capacity term can be written as
PP
k
P
ksksP
TT
TTYTX
T
GTC
+=
=
2
2
2
,02
,,0
11
2
[20]
where
= 2
2
2
2
21
PPTT
X
[21]
and
PTY
=
21 [22]
Similar expressions can be derived for the j-th aqueous ion. The re,j term in the HKF model was expressed as
ZjjxjeZrr += ,, [23]
where rx,j is the crystallographic radius of the j-th ion in . Z is an empirical parameter taken to be 0.94 for cations and 0.0 for anions. While
discussing the HKF model for the standard partial molal quantities, it should be mentioned that it relies heavily on empirical fit to experimental
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evident from the values of0
,
0
,
0
, , kckckc VandC (structural collapse terms to the respective parameters) which increase with decreasing
temperature and approach - at Ts. Tanger and Helgeson(1988) showed from theoretical considerations that solute dependence of k is not
needed and replaced k by (=228K) i.e., a constant irrespective of identity of ions. They proposed revised temperature functions for molar
volume, compressibility and heat capacity as given below.
21
)(
1)(
1)(
0
0
0
=
=
=
TTf
TTf
T
Tf
PC
V
[28]
The advantage this refinement is that the number of solute dependent parameters in the model is decreased. Secondly, the revised function
converts eq(24), (25) and (26) into linear equations with additive coefficients. Thus the additivity relation for nonsolvation contribution (eq. 12)
are exact equalities as the equation of state coefficients of 0n,j for the j-th aqueous ion can be calculated from regressed values of
corresponding coefficients of the k-th electrolyte. On further theoretical grounds, the f(P) term in the nonsolvation contribution was proposed to be
of the form
( ) 21)(
1
)(
PP
Pf
PPf
+
=
+=
[29]
where is a constant characteristic of the solvent H2O (=2600 bar) again on structural grounds.
To account for the discrepancies of predicted (HKF) and experimental values of partial molal volume and heat capacity at pressures corresponding
to the critical pressure of water (the values are too negative at temperatures approaching critical point), correction should be applied to the
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solvation contribution since at high temperatures and pressures (P=Pc,water), the nonsolvation contribution to these values become insignificant.
Therefore revision to eq-23 was proposed as given below(Tanger and Helgeson, 1988).
jZjjxje hZrr ++= ,, [30]
where the additional term hj is a function of temperature and pressure expressed as hj = Djg, where again, Dj is a constant characteristic of the
charge and/or crystallographic radius of the ion(j) and g is function of T and P independent of the identity of the ion. Regression calculation with
NaCl indicated a nature of the function as given below.
( ))4(5.0 2 cbbg = [31]
where
14
1
4
0
,272.3
= =
=
i j
ji
ji Tab and
14
1
4
0
,72.34571.3
= =
=
i j
ji
ji Tac ; is the specific density of water , =N0e
(Avogadro's number x elementary charge) and aI,j are fit parameters. Values of the g-function are tabulated in Tanger and Helgeson(1988) for a
wide range of temperature and pressure.
At this point, we see that we have the required set of equations for expanding the difference term within brackets in eq(3) and (4) to compute the
apparent standard partial quantities using eq(11). For eq(3)
dPETVdTCHH
T
T
P
P
sPTPTP
r r
rr )( 00000, += [32]
where0
sE is the standard partial molal expansivity of an aqueous ion or electrolyte. This can also be expressed as the sum of a solvation and
nonsolvation parts written before for an electrolyte or ionic species.
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P
T
T
P
T
TP
P
sk
P
nk
sknkk
TP
PYN
TPaa
T
P
PY
TQN
T
T
V
T
VEEE
+
++=
+
=
+
=+=
1111
111
2
43
0
,
0
,0
,
0
,
0
[33]
Hence
( )( )
rrrr
rr
rr TPrTP
TP
TP
P
r
r
r
r
r
rf
YTT
TTY
P
PaPPa
T
T
P
Pa
PPaTT
cTTcHH
+
+
++
+
+
++
+
+
+=
1111
11
ln2
ln
)(11
)(
4322
121
00
[34]
Eq(4) can be expressed as
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dPVdTT
CTdTCTTSGG
P
P
T
T
T
P
T
T
PrTPTPTP
rrr
rrrr ++= 00
0000
,)( [35]
Now we can write the expression for standard partial molal Gibb's free energy change as
( )( )
( )( )
rrrr
rr
rr
rr
TPrTP
TP
TP
r
r
r
r
r
r
r
r
r
rTPf
YT
P
PaPPa
TP
Pa
PPaTT
TTTT
TTc
TTT
T
TcTTSGG
+
+++
+
+++
+
+
+=
11
11
ln1
ln
)(ln11
ln)(
432
122
1
000
[36]
Now we have the required equation of state for apparent standard partial molal free energy of the aqueous species in solution. The standard partial
molal free energy and enthalpy of formation and partial molal entropy at 298.15K and 1 bar, the empirical parameters c1, c2 and a1 to a4 and j are
tabulated for about sixty odd aqueous charged (both simple and complex ions) and a number of neutral complex species in Shock and
Helgeson(1988), Shock et al.(1989) according to the modifications to HKF model. There is extensive tabulation of data on dielectric constant of
water at different pressures,rrTP
Y , values of the 'g'function (g(P,T)) for the extrapolation of the Born coefficient ( ) to high P and T in
Tanger and Helgeson(1988). One can take the values directly from the tables while calculating the free energy change of aqueous species at
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isothermal isobaric conditions or incorporate the equations into the equation of state (eq.33). This will permit us to evaluate the free energy change
of reactions involving aqueous species and minerals (sulfide, oxide, silicate) with data on these minerals available in the literature.
From the well known relationship
KRTGG TPrTPr ln0
,,,, += [37]
and imposing equilibrium of the participating mineral and aqueous phases, Gr,P,T = 0, hence G0r,P,T = -RT lnK, we can calculate the equilibrium
constant of any reaction.
Example-1
Suppose that we have an assemblage of magnetite, chalcopyrite and pyrite and we want to show the stabilities of the minerals and also characterize
the physicochemical environment of the fluid in terms of compositional variables in the Cu-Fe-S-O-H-S system. We can write simple hydrolysis-
sulfidation reactions involving the minerals and a fluid phase.
Fe3O4 + 4H+ + 2 H2S = FeS2 + 2 Fe
2+ + 4H2O (R1)
Fe3O4 + 2 Cu+ + 4H2S = 2 CuFeS2 + Fe2+ + 4 H2O (R2)
FeS2 + 2 Cu+ + Fe2+ + 2 H2S = 2 CuFeS2 + 4 H
+ (R3)
As we see, the reactions are written so that charge balance is maintained. As is clear from the reaction, stabilities of the phases are controlled by
pH, activities of Fe2+, Cu+ and fugacity of H2S at any given temperature and pressure. Here we have four release parameters other than P and T. If
we take activity ratios Cu+ and H+ (aCu+/aH+) and Fe2+ and H+ (aFe
2+/a2H+) we can decrease the degrees of freedom and the release parameters to three.
So, to depict the phase relations, we have to fix any one of the three. If we fix the fugacity of H2S, we can plot stability fields in the log (aFe2+/a2H+)
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vs log (aCu+/aH+) space. This might give us some idea as to the nature of variation of the activities of the meatal ions, that may also help us in
calculation of solubility of the metals in the fluid in the given condition once we are in a position to calculate the activity coefficients accurately
(discussed later).
Assuming that we have calculated the G0P,T of the reaction using eq(36) and have calculated the log K values of the reaction using eq(37), we
can expand the K of the three reactions as follows.
( )( ) ( )
=
+
+
24
2
1
2
2
SHH
FeR
fa
aK [38]
which can be rearranged so that two variables are on the either side as
( ) 2/)log2(log/log2
2 1
2
SHRHFefKaa +=++ [39]
Similarly for R2 and R3 we can write
( ) ( )( ) ( ) ( )
=
++
++
422
2
2
2
2
SHHCu
HFeR
faa
aaK [40]
leading to ( ) SHHCuRHFe faaKaa 22 log4)/log(2log/log 22 ++= ++++ [41]
and
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( )( ) ( ) ( )
=
++
+
22
4
3
22 SHFeCu
HR
faa
aK [42]
leading to ( ) SHHCuRHFe faaKaa 22 log2)/log(2log/log 32 = ++++ [43]
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We can plot these reactions at isothermal-isobaric and iso-fH2S conditions to depict the phase assemblage as shown in Fig.11. These kind of phase
diagrams where the effect of the activity of any one species (such as pH in the present case) are called abridged activity diagrams that are
commonly used in geochemistry (silicate alterations etc.). We can work out the phase relationship and can have the physicochemical constraints in
the fluid known in terms of activities of ionic species in any mineral-fluid interaction process. We can have several such abridged activity,
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activity-activity (analogous to - diagrams), activity-fugacity (fO2 - pH) diagrams for depicting phase relations at any pressure and
temperature of interest. The purpose behind this discussion is to show that when we are interested in only deciphering the phase relations like this,
we are not concerned about calculation of activities of the individual ionic species. However, when we are assigned with the job of calculating
solubilities of metals in hydrothermal fluids or computation of mass transfer between minerals and fluid or working out the sequence of
precipitation of minerals/salts from complex brines like seawater, which we do either by considering equilibrium of the fluid with minerals of
interest or considering homogeneous equilibria in the fluid, formation constants of complex forming reactions, mass and charge balance equations,we do need to calculate activities of individual aqueous species and in turn their activity coefficients. In these situations, we are far removed from
ideal behavior of aqueous species because of dominance of short range solute-solute and solute-solvent interactions. Therefore, before going to the
formalism of dealing with such processes, let us have some idea about activity coefficients of aqueous species in solutions of our interest.
Activity coefficients of aqueous electrolytes
Activity coefficient of species in aqueous solutions has been an outstanding problem in solution chemistry. As mentioned in the introduction, the
theory is yet to develop to the required level and the data generated so far in this regard are inadequate and will remain so for quite sometime. We
shall review the fundamental aspects in brief here.
Definition of activity coefficient: Usually denoted by as the ratio of the activity of the electrolyte or solute species and its molaility, referred to
as practical activity coefficient by Nordstorm and Munoz(1985). It is a dimensionless quantity, hence, activity of aqueous species has the
dimension of concentration (molal), whereas it is dimensionless in crystalline solutions by convention. In the reference state discussed earlier, lim
33
Fig. 11 Abridged activity diagram showingstability of minerals in Cu-Fe-S-
O-H sysyem
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(ai/mI) =1. For a strong electrolyte, where there is complete dissociation of the species ABaqAaq+ + Baq- for which we can write AB,aq = A,aq+
+ B,aq- because chemical potentials are additive. Assuming ideal condition ( A,aq = B,aq =1, we can write
aqBaqAaqBaqAaqABmmRT ++++=
,
0
,
0
,, ln [44]
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For a uni-univalent electrolyte (1:1), aqBaqAaqAB mmm ,,, == , hence eq (44) reduces to2
,
0
,, ln aqABaqABaqAB mRT+= , which indicates, for
an 1:1 electrolyte a m2 or a/m constant. For any salt dissolving into + positive and - negative ions for a total of = + + - ions (e.g.
for MgCl2, = 3), the relationship becomes a m
. Fig.12 depicts the relationship for three electrolyte species of different stoichiometries.
Though we write in terms of chemical potentials of individual ions as shown already, these potentials can not be experimentally measured
35
Fig.12 Plots showing the
proportionality relationship
in NaCl, CaCl2 and AlCl3
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individually. Each ion is related to a stoichiometric amount of an oppositely charged ion by the principle of electrical neutrality. An expression
that recognizes the existence of separate ions but allows to make real measurement of solution activities is needed. This has given rise to the
concept of 'mean ion chemical potential' and mean ion activities and activity coefficients. The mean value of the ionic chemical potential for an
electrolyte of composition +
BA that dissociates into +AZ+ + -BZ- , the mean can be expressed as +
+
+
=+
BA
BA ZZ
,
[45]
and we can write
+= aRT ln0
[46]
where a+ is the mean ionic activity. For an aqueous electrolyte species, the mean ion activity coefficient and molality are related to the solute
parameters as follows.
( )
/1++
+
++
+
===
== aaaa[47,48]
( ) ( ) /1)( +++ ===+
mmmmmm [49]
The term ( +) +( -)
- will be unity for 1:1 electrolyte (NaCl), 4 for 2:1and 1:2 electrolytes (Na2SO4, CaCl2) and 27 for 3:1 and 1:3 (Na3PO4,
AlCl3) electrolytes. These relationship are important to keep in mind when it becomes necessary to convert between solute activities, solute
molalities and mean salt values. The mean ion activity coefficients for salts in aqueous solutions ( +) are obtained from measurement of i)
freezing point depression, ii) boiling point elevation, iii) water vapor pressure, iv) osmotic pressure, v) solubility and distribution and vi) transport
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properties such as conductivity and diffusion. In methods i to iv the properties of the salts are obtained directly from its effects on the properties of
water. Methods v and vi provide more direct measure of the salt's mean ion activity coefficient. In the same way as mentioned above, the mean ion
activity coefficient of a 2:1 salt K2SO4 can be expressed in the following way, considering the dissociation K2SO42 K+ + SO42- for which
( ) ( ) +++ == 24
24
24
222)(
SOSOKKSOKspmmaaK and ( ) ( )[ ] 3/12, 2
442+= SOKSOK [50]
Before proceeding further, it would be useful to have a look at an important parameter of aqueous solution known as ionic strength denoted
conventionally as I. It is a cumulative effect of all ionic species dissolved in solution and since the interaction between ions is largely Coulombic,
it is proportional to the charge on the ion. It is expressed as ( )= 22
1iiZmI and represented in molal unit since it involves concentration of
ionic species in the same unit. When we compute ionic strength with this formula, we necessarily assume complete dissolution of the electrolyte
species, which is not always true as we discussed earlier and also follow up later. While working out the energetics of ionic solution from the
potential energy associated with each ion and the surrounding cloud of other ions, this term automatically comes in the expression (see Barrow,
1992, Chap. 9 for details). Long before this theoretical treatment however, activity coefficient was empirically found out to be dependent on the
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ionic strength. Fig.13 shows the relationship of + and ionic strength for a number of electrolyte species. Since activity coefficient is a measure of
nonideality due to increasing short-range
interactions among solutes, the starting point would be to consider very dilute solutions where only long range interactions prevail.. For solutions
with TDS up to 5000 ppm (I=0.1m) activity coefficients of monovalent and divalent cations can be computed through the theoretical Debye-
Huckel equation (given below) where the inherent assumptions are : i) ion interactions are purely Coulombic; ii)ion size does not vary with I and
iii) ions of sign do not interact.
2/1
2/12
1log
IBa
IAZ
i
ii +
= [51]
where A=1.824928 x 106 01/2( T)-3/2, B=50.3( T)-1/2, 0 is the density of water, is the dielectric constant whose temperature and pressure
dependence is known from empirical equations (available in literature) for calculation of the same and in turn, the activity coefficient at higher
temperatures and pressures. ai is effective size of the hydrated ion in , which is a function of the degree of hydration that is in turn is a function
of the Z/r ratio as has been mentioned
38
Relationship of mean ion activity coeff. Of
salts and ionic strength of solution for a
number of electrolytes
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earlier. Tables containing values of a I for ions of different valency (both +ve and -ve) are available from literature. Fig. 14 shows the activity
coefficients for a number of ions as a function of ionic strength calculated on the basis of the above equation. The D-H equation can be used to
compute accurate activity coefficients (individual ion) for monovalent ions up to about I=0.1m, for divalent ions
39
Fig. 14 Single ion activity coeff. As function of
oinic strength using eq(51). The declining trend is
to be noted.
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up to I=0.01m and for trivalent ions up to I=0.001m. For I=0.001 to 0.0001m, curves for I versus I for ions of same valence converge on a single
curve. At this situation the second term in the denominator vanishes and the D-H equation reduces to
log I = -A Zi2 I1/2 [52]
and is known as the Debye-Huckel limiting law. Since A is proportional to -log I and A is a function of temperature (increases with temperature)
activity coefficient decreases with temperature and because of the Z i2 factor, the effect of temperature is greater for multivalent ions. At higher
concentration of solutes, (I > 0.1m), the assumptions in the D-H equation become invalid because of increased solute-solute interactions. Besides,
the D-H equation predicts that decline continuously with ionic strength which is not true. One of the several reasons for the reverse trend of
at higher ionic strength is that with increasing salt concentration, less of free water molecules are available resulting in a decrease in activity of
water (near unity in dilute solution to 0.807 in a 5 m solution of NaCl). To accommodate the increase in , some positive term must be added, as
proposed by Huckel and later workers. The equation known as empirical Davies equation is mentioned below.
+= I
I
IAZ 3.0
1log
2/1
2/12
[53]
However, this lacks the ion-size parameter though it gives accurate values for monovalent salts from I=0.1 to 0.7m and gives same valuesfor all ions of same charge at a given ionic strength.
The Truesdell-Jones (TJ) equation is a simple extension of D-H equation plus an additional term :
bIIBa
IAZ
i
i
i ++=
2/1
2/12
1log [54]
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with aI and b being empirical parameters.
Bronsted-Guggenheim-Scatchard specific interaction theory (SIT) is an ion and electrolyte specific approach of activity coefficient. It is therefore
theoretically capable of greater accuracy than Davies equation. The general SIT equation for a single ion i
( ) ( )jmIjiDZj
i += ,,log 2 [55]
where D is the D-H term, m(j) is the molality of major electrolyte ion j which is of opposite charge to ion i. Interaction parameters (i,j,I) refer to
the interaction between the ion i and major electrolyte ion j. is taken as zero for neutral species or between ions of like sign. The interaction
parameters are often assumed to be independent of ionic strength that works well for I up to 3.5 m. The SIT approach to study of natural waters is
limited by the availability of interaction parameters.
As we move on to solutions of higher ionic strength solute-solute interaction between like and unlike charges become prominent. These
interactions can be quantified in terms of binary (between like) and ternary (between three or more ions some of which will be of like sign)
interaction parameters. The most accepted model for high ionic strength is the Pitzer's model (Pitzer, 1987) which has been shown to model the
behavior of solutions accurately up to I=6.0m. The Pitzer model (see Pitzer, 1987 for details of the derivation from excess free energy of solution,
osmotic coefficient and so on) requires interaction parameters involving aqueous species of interest and major species in water. These parameters
have neen worked out for major ions (tabulated in Pitzer, 1987) but are not available for trace species including strong complexes, some of which
may be approximated to be equal to similar known species.
The activity coefficient for an individual ion in Pitzer's model approach can be written as
( ) ......ln,,
,,,
2 +++= kjkji
kjij
i
jiii mmEmIDfZ
[56]
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The first term in the r.h.s. is a modified D-H term where
( )
++
+
= /12/1
2/1
2.11l n
2.1
2
2.11
3 9 2.0 I
I
If [57]
The Pitzer equation can be thought of as a virial-type expansion. The second term is for the binary interaction involving two solute species of
opposite or same sign and the terms required to evaluate it are called binary virial coefficient. The parameters describing interaction of like species
are independent of ionic strength and those describing the interaction of unlike species are functions of ionic strength. If we neglect the terms
containing higher order interactions, the ln obtained are quite comparable to those of SIT and TJ equations below I=3.5 m. Ternary virial
coefficients account for interactions among two like and a third unlike charged species. These are usually independent of I. Pitzer model equations
are linear algebraic functions of ln I and are often extremely long involving numerous individual parameters and substitution and solution of
Pitzer's model equation is best accomplished by a computer. There are many computer codes (SOLMINEQ.88, PHRQPITZ, PHREEQC available
as freeware) which include the Pitzer model (not SUPCRT). The model is useful in the study of concentrated waters like acid-mine waters and
industrial waste water. The model works best when cation-anion interactions produce complexes whose formation constants are less than 10 2.
Though we have discussed equations for calculation of single ion activity coefficients we have not seen how the mean ion activity coefficients are
actually calculated by any of the methods that was mentioned. Calculation of mean ion activity coefficient from difference in vapor pressure of
pure water and that of salt solutions of varying molality was done by Wood et al (1984). The details is omitted here that interested readers might
follow up.
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We have discussed the principles and equations for mean and individual ion activity coefficients. Before we see some example calculations, it
would be worthwhile to have a look at the McInnes convention, which has served as an important tool for calculation of individual ion activity
coefficients from data on mean ion activity coefficients. This makes the modeling of natural water chemistry a bit flexible. The McInnes
convention states that K+ = Cl -, based on the observation that K+ and Cl- ions are of the same charge and nearly same size; having similar
electronic structure (inert gas) and similar ionic mobilities. The McInnes convention leads to
KClClK ,== + . Individual ion activity coefficient of strong electrolytes can be measured for their mean values using +,KCl values as starting
point. All such calculations must be done with + values for KCl and other salts measured at the same ionic strength, which is not the same
molality except for monovalent salts.
For the general cation case where M is the cation
( ) ( )[ ] nnClMMCln/11
, 1
= [58]
where n=n++n-. Substituting +,KCl for Cl and solving for M
( )1,
1
)(
= nKCl
n
nM
MCl
[59]
For example, for Fe3+ from +,FeCl3,
[ ] [ ] 4/134/13, 333 KClFeClFeFeCl ++
== [60]
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and since Cl = +KCl
3
,
4
)(
)(3
3
KCl
FeCl
Fe
=+
[61]
For anion A e.g. in Kn-1A solution
( ) ( )[ ]( )
1
,
,/11
,)(
1
1
==
n
KCl
n
AK
A
n
A
n
KAKn
n
[62]
For example to obtain SO4 from +,K2SO4
( ) ( )2
,
3
,3/12
,
3/12
,42
44442
KCl
SOK
SOSOKClSOKSOK
=== +
[63]
Fig. 15 is a plot of individual ionic activity coefficient of a number of cations and anions on the basis of McInnes convention where the increase of
i with I at I> 1.0 m is shown. Fig. 16 is a plot of individual ion activity coefficients taking Ca2+ as an example with ionic strength up to 10 m,
44Fig.15 Plot of single ion activity coeff. Of a no.
of anions and cations vs ionic strength usingMcInnes convention
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where various methods are compared. As can be seen, the values obtained by different methods start diverging significantly above I>0.3m. Since
Davies and SIT models ignore ionic size, they are likely to be less accurate. The extended D-H is inaccurate above I>0.3. The mean salt and TJ
equations produce
identical values since the TJ parameters are based on mean salt data. The SIT model interaction parameters result in activity coefficient that agree
well with mean salt values over a relatively wide range of ionic strength.
45
Fig. 16 Activity coeff. Of Ca2+ using various
methods. The divergence in different
methods should be noted.
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Fig. 17 is a summary plot of individual ion activity coefficient versus I1/2 (molal) where the validity of the equations described in the ionic strength
range are shown.
Effective ionic strength:
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There is another important aspect to calculation of activity coefficients in solution. This is more so where we try to compute activity coefficients of
species in solution at high P and T. As discussed earlier, because of decreased dielectric constant of water, there is increasing ion association. For
concentrated solutions and solutions of weak electrolytes (whose dissociation constant is low) there is presence/dominance of uncharged or less
charged associated or complex species. This decreases the ionic strength and the ionic strength calculated taking the degree of association of the
electrolyte species at any desired condition, is usually referred to as 'effective ionic strength' as against the stoichiometric or total ionic strength
calculated assuming complete dissociation of all electrolytes. For example, in seawater I s = 0.718 m whereas the Ie = 0.668 m (7% lower). Thedifference increases at higher ionic strength. Fig. 17 is a summary plot of individual ion activity coefficient versus I1/2 (molal) where the validity of
the equations described in the ionic strength range are shown.
An example:
Here is an example (from Langmuir, 1997) to demonstrate the effective ionic strength calculation. Let's say a fluid is saturated with gypsum
without any other solute. There are only Ca2+ and SO42-. The solubility product of gypsum Ksp = 10
-4.59. At gypsum saturation, Ca= SO4 =
0.0154. Here the stoichiometric or total ionic strength would be
Is = 1/2(2 x 0.0154 x 4) = 0.0616m
The effective or true ionic strength approach assumes ion pair such as CaSO40 in the present case. Now the mass balance equations become
04
24
4
4 CaSOSO
CaSOCa
mmSO
mmCa
+=
+=
[64,65]
47
Equations for calculation of single ion activity coeff.
eir ionic strength range of applicability
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The solubility reaction for gypsum is CaSO4 2H2O = Ca2+ + SO4
2- + 2H2O. Calcium and sulfate ions form the ion pair CaSO40 with the association
constant (= 10 2.31, you can calculate given the thermodynamic data) .
+
=24
2
04
SOCa
CaSO
assocaa
aK [66]
Since at saturation, product of the two ionic species is 10-4.59, the value of activity of the neutral ion pair can be obtained which is 10-2.28. Assuming
activity coefficient of the neutral ion pair is unity (that is a reasonable assumption since neutral species will not experience much of interaction),
from eq( ) we can find mCa- = mSO4
2- = 1.02 x 10-2 m. Now we can calculate the effective ionic strength
Ie = 1/2(2 x 1.02 x 10-2 x 4) = 0.0408 m. (Is is about 50% higher than Ie) From this Ie we can further calculate the activity coefficients of the two
ions and in turn their individual activities. It can be checked that the activities calculated are the same in both cases but activity coefficients are
different.
Metal Solubility and Complexing
As has been discussed in the introductory section, one of the most important implications of studying the thermodynamics of aqueous electrolytes
is to compute solubility of metals. This is important from ore genetic point of view i.e. to assess the metal carrying capability of hydrothermal
fluid whose gross chemistry has been assessed by other means such as fluid inclusion studies and mineral-fluid equlibria. Study of metal solubility
has also tremendous implications for practical problems such as radioactive waste disposal and assessing environmental hazards. It is established
long since that metals, particularly transition metals, are carried in natural solutions as complexes rather than simple ions, involving a metal and a
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ligand. Spectroscopic studies and phase equilibria experiments have demonstrated the type of complexing that occurs, though much is yet to be
known. Table-1 is a a list of ligands and their
Ligand Concentartion
range (moles/kgH2O)
Cl-SO42-S2-NH32-CO32-acetate
propionateoxalatemalonate
7
10-10
to 0.3
10-4
to 0.1
10-3.9
to 10-1.5
10-3.4
to 1
0 to 0.17
0 to 0.059
0 to 0.005
0 to 0.025
49
Table-1. Ligands and their concentration in natural hydrothermal fluid indicating the dominant mode
of transport of metals.
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range of concentrations in natural hydrothermal fluid where we can see the dominance of chloride. Table -2 lists the metals and ligands on the
basis of hard-soft classification proposed by Pearson to explain the dominant type of complexing of metals in natural solutions. Mnay hypotheses
have been proposed in this regard. Fig .18 is a plot of the Pauling's electronegativity versus Z/r ratio of some metals and ligands where fields of
dominance of ligands are depicted. This more or less explains the ligand-metal specificity in solutions within the constraint of availability of a
particular ligand. For example, though gold forms stable bisulfide complexes,
Hard Borderline Soft AcidsH+
Li+ > Na+ > K+ >Rb+>Cs+Fe2+,Mn2+,Co2+,Ni2+ Au+,>Ag+>Cu+
50
Fig.18. Pauling's electronegativity vs Z/r explaining
metal-ligand specificity
Table-2 Pearson's hard-soft classification of metals and ligands
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Be2+>Mg2+>Ca2+>Sr2+>Ba2+
Al3+>Ga3+
Sc3+>Y3+; REE(Lu3+>La3+);
Ce4+; Sn4+;
Ti4+>Ti3+; Zr4+=Hf4+
Cr6+>Cr3+; Mo5+>Mo4+;
W6+>W4+;Nb5+; Ta5+Re7+>
Re6+
>Re4+
; V6+
>V5+
>V4+
;Mn4+; Fe3+;Co3+;As5+Sb5+
Th4+;U6+>U4+
PGE6+>PGE4+
BasesF-;H2O,OH
-, O2-, NH3; NO3
-
CO32-> HCO3
-;SO42->HSO4
-
Cu2+,Zn2+,Pb2+,Sn2+
As3+, Sb3+,Bi3+
Cl-
Hg2+>Cd2+
I->Br,CN
S2->HS->H2S
chloride and other complexes do also form in the absence of bisulfide. Fig. 19 is a plot of the ligand-field stabilization energy (LFSE) versus Z/r
of a number of metals. Though details of the LFSE of transition metal complexes can not be dealt here, the figure clearly explains the reason
behind occurrence of some metals as sulfides. For example, the case of Cr, Mn, W etc. are noteworthy.
We usually express the reaction for formation of a ligand in the following way:
51
Fig. 19. LFSE vs Z/r of metals explaining the type of deposit formed by them.
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M+q + n L-p = MLnq-np [67]
And refer to the equilibrium constant as formation constant ( ). Metals may form complexes in steps and we represent their overall formation
constant as n and write
....211
KKKn
i
in ==
= [68]
We shall see the case of Pb in details. It implies that in order to undertake such exercise, we must have the thermodynamic data for the species in
consideration and appropriate formulation for extrapolation to higher temperature and pressure, and, also the activity-composition relationship in
aqueous solutions. Thus our discussions in the previous sections look justified.
In order to calculate concentrations of species (speciation) of species from such equilibrium constants, we require to have the activity coefficients
of all species and in turn, we must have the true or effective ionic strength of the solution. An example in this regard will give a glimpse of what it
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really is. Suppose we want to calculate the solubility of galena (PbS) and sphalerite in an NaCl solution. If they form their chloride complexes we
can write the following complex forming reaction and their formation constants in the following way.
nMClnClM + =+ 22 [69]
for which we ca write
( )( )( ) n
n
n
n
ClM
MCl
+
=2
2
[70]
and also for the hydrolysis of the minerals we can write the following reaction and equilibrium constant as
)(222 aqSHMHMS +=+++
for which 22
2
)(
))((+
+
=H
SHMK . [71]
+=+= +++n
ClMClMM
n
n nnaaaMClMM // 22
22 [72]
Substituting for activity of M2+ and MCln2-n we can obtain
+= +
+
+
n MCl
M
n
n
M n
Cl
SH
HKM
2
2
)(1
)(
)(
2
2
[73]
This is the expression for total concentration of Pb and Zn as a function of dissolved H 2S, Cl- activity and pH. Now we shall be requiring to
calculate the activity coefficients of the dissolved species and in turn we have to compute the ionic strength. If we have to take the association of
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NaCl to account, we must know the formation constant of NaCl at the temperature of interest so that we can estimate for the actual concentration
of Na+ and Cl- in the solution. We shall see a more elaborate method of solubility calculation below. This discussion is mostly taken from Wood
and Samson(1998). For solubility of galena in a solution of sufficiently low pH so that Pb2+ and H2S are the only species present we can write
PbS(s) + 2 H+ = Pb2+ + H2S [R3]
For which, taking the activity of PbS as unity, the equilibrium constant reduces to the solubility product (we shall use square bracket for
concentration and normal bracket for activity)
( )( )2)(
22
+
+=H
SHPb
spa
faK [74]
and taking the activity coefficient of the Pb ion, the concentration in molaity can be written as
( )+++ = 22
2 )/()(2
PbSHHspPbfaKm [75]
If pH, fugacity of H2S and activity coefficient of Pb remain constant, the molality of Pb is fixed at constnt P and T. If additionally, the solution
contains enough chloride ion and the Pb-chloride complexes form as follows
Pb2+ + Cl- = PbCl+ (R4)
Pb2+ + 2 Cl- = PbCl0 (R5)
Pb2+ + 3 Cl- = PbCl-
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)///
//1()()(
2432
22
44
4
33
3
22
2
1
12
+++
+++
+= PbClClClPbClClClPbClClCl
PbClClClPbSHHsp
mmm
mfaKPb
[79]
We can calculate the single ion activity coefficient with any suitable formula described earlier. The individual ion activity coefficients calculated
by using Davies equation (taking 0.2 instead of 0.3 and taking the values of A from Helgeson et al., 1981) the activity coefficients of the species
are calculated to be : Pb2+ = Pb4+ = 0.107 ; Cl-= PbCl+ = PbCl3- = 0.572. Substitution of these values to equation ( ), the molality of Pb is
calculated to be 2.23 x 10-3 m (=462 mg/kg of water) that means less than half of what we got assuming ideality. We can still move a bit further to
get a feel of solubility calculation and speciation in natural solutions - say a geothermal brine. In addition to the eight species we have already
considered, let us include Na+, HCl0, NaCl0, OH-, NaOH0, HS-, S2-, NaHS0, Pb(OH)+, Pb(OH)20, Pb(OH)3
0, Pb(HS)3-, Pb(HS)2
0 and Pb(H2S)20. These
species in fact occur in natural brines though some of them may be insignificant. Now, our compositional variables have increased (to 22) and in
order to solve for the concentration, we must have 22 equations. We already have five mass action equations and can have thirteen more keeping
the constraint that no reaction should be a linear combination of any two other reactions.
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H+ + C1- = HCI0
Na++ Cl- = NaCI0
H2O (l) = H+ + OH-
Na+ + OH- = NaOH0
H2S0 = H+ + HS-
HS- = H+ + S2-
Na+ + HS- = NaHS0
PB2+ + H2O = PbOH+ + H+
Pb2+ + 2H2O = Pb (OH) 20 + 2H+
Pb2+ + 3H2O = Pb(OH).3- +.3H+
PbS + H2S0 = Pb(HS)2
0
PbS + H2S0 + HS- = Pb (HS) 3
-
PbS + 2H2S0 = PbS(H2S)2
0
Now we shall be needing four more equations. Since our solution has to be electrically neutral,
we shall frame the charge balance equation, equating all the positive charges with the negative
charges, and it will be again in the concentration unit abiding by the stoichiometric coefficients of
charge that was discussed earlier.
The charge balance equation can be written as
+++++
++
+++++=++++
33324
322
)()(2
22
HSPbOHPbPbCl
PbClSHSOHClPbOHNaHPbClPb
mmm
mmmmmmmmmm
[80]
We can have more equations if we know the pH (in this case pH is taken as 3), S (total sulfur,suppose for this calculation it is known to be 10 -3 m) and Cl (total chloride, in this case 1 m).
The S will be the sum of all S-bearing species and Cl will be the sum of all chloride-bearing
species. Since pH= - log aH+ , we can write mH
+ as equal to 10-pH/ H+. The three mass balances for
this exaplme can be written as:
+
++
++++++=
=
++++++=
2432
22232
2
432
/10
323 )()()(
PbClPbClPbClPbClNaClHClCl
H
pH
H
SHPbSHSPbHSPbNaHSSHSSH
mmmmmmmCl
m
mmmmmmmS
[81,82,83]
At this stage the calculation scheme is quite complicated even without considering activity
coefficients. Solution of 22 unknowns from 22 equations would be too lengthy and tedious to do
manually and hence, you will be needing a computer. There are several numerical methods and
computer routines that implement those methods available or can be developed if one is good at
it. In this case, even without considering activity coefficients of all species, the activity
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coefficient of the H+ ion has been incorporated. It is actually a case of more unknowns (23) than
equations (22), hence some initial approximation and iterative procedure has to be adopted. The
result obtained in the present case is 1.02 x 10-3 m of Pb i.e., 211 mg/kg (ppm). Thus, competition
for Cl- by other cations (Na in the present case) has greatly reduced the solubility.
Finally we can consider an example from a natural ore deposit (Panigrahi, 1992), where in the
absence of any direct chemistry of the fluid, how the mechanism of transport and deposition of
metals can be speculated and worked out semiquantitatively. Recall the reactions (R1 to R3) and
the isothermal-isobaric abridged activity diagram depicting phase relationship in the Cu-Fe-S-O-
H system. In addition to the sulfides and oxides, we also have silicates (K-feldspar, biotite,
chlorite, epidote, quartz and so on) present in the assemblage from which we can chose the
subsystem from mineral paragenesis. For example, for biotite-chalcopyrite (taking the annite
component) assemblage, we can write a reaction like this involving Cu ion.
2 KFe3Si3AlO10(OH)2 + 6 Cu+ + 12 H2S + 1.5 O2 = 2 Kal Si3O8 + 6 CuFeS2 + 6 H
+ + 11 H2O
Now for this reaction we can have the ratio of actvities of Cu+ and H+ worked out at a particular
pressure, temperature, activity of the H2S(aq) and fugacity of oxygen. In fact we can get a range of
the activity ratio from different such assemblage because of varying X annite in biotite. Now if you
also have magnetite in the assemblage, you can write an oxidation reaction of the type
KFe3Si3AlO10(OH)2 + 0.5 O2 = Kal Si3O8 + Fe3O4 + H2O
In fact the fugacity of oxygen in the previous reaction can be fixed from this reaction in the
availability of the appropriate assemblage. Subsequently, you could constrain the chloride ion
activity in the solution by considering equilibrium of the fluid with biotite considering exchange
of OH - Cl between biotite and fluid
Cl-Bt + H2O(fl) = OH
-Bt + HCl(fl)
This gives the ratio of the fugacities of HCl and H2O in the fluid. From this reaction
log aCl- = log K - log (fH2O/fHCl) + log aOH
-
The above reaction has been modeled in terms of fugacity ratio of H 2O and HCl as a function of
temperature and composition of biotite. If we have some constraint on the pH of the fluid, we can
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find out the activity of the Cu ion in the fluid and hence the activity product of Cu + and Cl- and
compare with the formation of constant of CuCl to know whether chloride complexing was the
dominant mode of transport. We can use any sulfide-silicate-oxide assemblage for the purpose of
reconstructing the physicochemical environment in the fluid. We must have the required
thermodynamic data and activity - composition relationship in the minerals and fluid.
To conclude, what we have discussed would still be very preliminary in terms of concept and
coverage. If you aim to climb the mountain, this document will take you to the base, to muster
enough courage to move up, the success will largely depend upon how much of obstruction you
are able to clear up the mountain - each may experience unique situations.
Selected References: (for further readings please contact the author)
Tanger JC and Helgeson HC(1988) Am. J. Sci., 288, 19-98
Shock E. and Helgeson HC(1988) Goechim. Cosmochim Acta, 52, 2009-2036
Shock E., Helgeson HC and Sverjensky D(1989) Geochim. Cosmochim Acta, 53, 2157-2183
Hegeson HC Kirkham DH and Flowers GC(1981) Am. J. Sci., 281, 1249-1516
Brimhall GH and Crerar DA(1987) Rev. in Min. v17, Min. Soc. Am., 235-321
Wood SA and Samson IM(1998) Rev. in Econ. Geol., v10, 33-80
Wood SA, Crerar DA, Brantley SL and Borcsik(1984) Am. J. Sci., 284, 668-705
Nordstorm DK and Munoz JL (1985) Geochemical Thermodynamics, The Benjamin/Cummings
Pub.
Langmuir D(1997) Aqueous environmental Geochemistry. Prentice Hall
Barrow G.M.(1992) Physical Chemistry, Tata Mc Graw Hill
Pitzer KS(1987) Rev. in Min. v17, Min Soc. Am., Chap-6