Douce A.P. Thermodynamics of the Earth and Planets

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558 Dilute solutions

is a curve for total dissolved iron = m

,total

. Recall that the calculations

are only appropriate for a very dilute solution of iron in pure water, and that concentrations

in ocean water are 2–3 orders of magnitude higher. The qualitative behavior shown in

the figure is, however, applicable to seawater. The concentration of total dissolved ferric

iron in equilibrium with hematite is of course independent of oxygen fugacity, but that

of total dissolved ferrous iron is not. For the present day atmospheric oxygen fugacity

the concentration of total dissolved ferrous iron is vanishingly small, and only becomes

comparable to that of total ferric iron at oxygen fugacities ∼10

−45

bar. Below 10

−50

bar

f (O

) ferrous species become the dominant forms of dissolved iron, and iron solubility

increases rapidly with decreasing oxygen fugacity.

At 298 K magnetite becomes stable at f (O

) ∼10

−69

bar (Fig. 9.3). In order to see the

effect of this phase transition on iron solubility (see also Worked Example 6.7) we have to

write the corresponding saturation equations. This is accomplished by replacing each of the

first equations in (11.107) and (11.110) with the corresponding magnetite equations:

O +

 Fe

(

)

3aq

(III, mt) (11.113)

and:

O  FeOH

+OH

−

(II, mt) (11.114)

with equilibrium constants:

III,mt

Fe(OH)





1/12

(11.115)

and:

II,mt





1/6

·m

FeOH

·m

−

. (11.116)

Numerical values (with standard state data from Wagman et al., 1982) are: K

III,mt

1.356 × 10

−6

, K

II,mt

= 6.723 × 10

−26

. The other equations in (11.107) and (11.108)

remain unchanged, so we find the following equations for total dissolved ferric and ferrous

iron in equilibrium with magnetite:



,total



mt sat

III,mt





1/12



1 +

III,1

−pH

III,1

·K

III,2

−2pH

III,1

·K

III,2

·K

III,3

−3pH



(11.117)

and:



, total



mt sat

II,mt

−pH





1/6



1 +

II,2

−pH



. (11.118)

Equations (11.117) and (11.118), and the sum of both, are also shown in Fig. 11.6. The

curves for total dissolved iron in equilibrium with hematite and magnetite intersect at

559 11.5 Speciation in ionic solutions

log f (O

) =−68.59. This is the oxygen fugacity at which the hematite–magnetite phase

transition takes place, but the careful reader may note that there is a slight discrepancy

between this value and the one shown in Fig. 9.3. This arises from the use of standard

state properties from different data sets. I used data from Holland and Powell (1998)to

calculate the hematite-magnetite buffer in Fig. 9.3, and Wagman et al.(1982) to calculate

the equilibrium constants in this example.

At oxygen fugacity higher than that of the magnetite–hematite equilibrium the calculated

concentrations of total dissolved ferrous and ferric species in equilibrium with magnetite

are higher than those in equilibrium with hematite. This means that magnetite is not stable,

and that iron solubility is controlled by hematite saturation. There are two equivalent ways

of seeing this: you can think that with increasing iron content the solution becomes saturated

in hematite first, or that the chemical potentials of iron species at equilibrium with hematite

are lower than those at equilibrium with magnetite. At f (O

) lower than that of the phase

transition the converse is true.

If we now “clean up” the diagram, leaving only the stable saturation curves, we get the

phase diagram shown in Fig. 11.7.The fields labeled “hematite” and “magnetite” correspond

to total dissolved iron concentrations above the solubility curves. These are “prohibited

regions”, in the sense that solution compositions inside these regions are thermodynamically

unstable relative to precipitation of the corresponding crystalline phase. Conditions below

the solubility curves correspond to solutions undersaturated in iron oxides. For oxygen

fugacities lower than ∼10

−50

bar, i.e. in the region where iron solubility varies strongly

with oxygen fugacity, the solubility is controlled by oxidation of Fe

to a ferric crystalline

phase. This is so because in this region the saturation concentration of total dissolved

total dissolved iron molality

pH = 8

T= 25º C

hematite

magnetite

dissolved iron

–11

–12

–13

–10

–9

–8

–7

–6

–5

–80 –70 –60 –50 –40 –3

log f(O

) – bar

Fig.11.7

SameasFig.11.6,butshowingonlyconcentrationoftotaldissolvedironalongthemagnetiteandhematite

saturationcurves(thethickcurveinFig.11.6).Conditionsinsidetheshadedregionsaremetastablerelativeto

precipitation of the corresponding oxide phase.

560 Dilute solutions

ferric iron remains very low and constant, at close to 10

−12

molal, until magnetite starts

precipitating, where it decreases further with decreasing oxygen fugacity. Above ∼10

−50

bar oxygen fugacity the concentration of total dissolved ferrous iron becomes negligibly

small and iron solubility is determined by the solubility of ferric species.

The total concentration of dissolved iron in Archaean seawater in equilibrium with an

anoxic atmosphere in which f (O

) was less than about 10

−50

bar may have been several

orders of magnitude higher than the solubility of iron in today’s shallow oceans. Dissolved

iron in Archaean oceans would have been overwhelmingly present as ferrous species, that

could precipitate to solid ferric phases upon oxidation. Iron solubility remains essentially

unchanged over an f (O

) range of ∼40 orders of magnitude, from 10

−40

bar to its present-

day value. Deposition of banded iron formations by oxidation of dissolved ferrous iron

must have taken place under oxidation conditions (f (O

) <∼10

−50

bar) that are not even

remotely comparable to those that are needed for aerobic metabolism (Chapter 14).

The equations show that iron solubility in water is a function not only of oxygen fugacity

but also of pH, and the question arises, could the deposition of banded iron formations

primarily reflect a change in oceanic pH, rather than in oxygen fugacity? In Exercise 11.6

you can explore this and see why, although possible, this is a very unlikely explanation –

BIF deposition must be the response to oxidation of the Earth’s surficial environments.

Figure 11.7 shows a subtle detail that is not evident from the schematic phase diagrams

that we constructed in Worked Example 6.7. This is the fact that formation of magnetite

iron formations requires more concentrated iron solutions than formation of hematite iron

formations. For a total iron concentration lower than that at the magnetite–hematite transi-

tion (∼6×10

−8

molar in this simplified model) magnetite iron formations cannot form at

any oxygen fugacity, but hematite iron formations would form by oxidation.

In Worked Example 6.7 we saw that magnetite iron formations are sometimes associated

with siderite, and that some Precambrian BIFS are in fact composed chiefly of siderite. In

order to study the conditions that lead to the formation of siderite iron formations we begin

by writing the following magnetite–siderite equilibrium:

2Fe

+6CO

 6FeCO

. (11.119)

The oxygen fugacity at which siderite replaces magnetite as the iron solubility-limiting

phase depends on the fugacity of carbon dioxide. Using the pure gases at the temperature of

interest and 1 bar as the standard states for O

and CO

we write the equilibrium condition

for (11.119) as follows:





mt−sd

(11.120)

with K

mt−sd

= exp(−

/RT ) = 2.887 ×10

−70

at 298 K (data from Wagman et al.,

1982). As an example, let us assume that CO

fugacity in the Archaean atmosphere was

0.02 bar. A value considerably higher than today’s is suggested by the fact that Archaean

glaciations appear to be non-existent, despite the fact that the early Sun was fainter than

today’s (Sagan & Mullen, 1972; Kasting, 1987; Gilliland, 1989). From this CO

fugacity

we calculate an oxygen fugacity at the magnetite–siderite transition of ∼1.9 × 10

−80

bar,

i.e. well below the hematite phase transition. An important question is whether CO

is stable

at this low oxygen fugacity, relative to, for example, CO. The answer, which you are asked

to prove in Exercise 11.7, is yes.

561 11.5 Speciation in ionic solutions

It would appear that we could now simply add another phase boundary to Fig. 11.7,

showing the stability of siderite below this oxygen fugacity. The problem is that this diagram

is calculated for pH = 8, which is approximately correct for the present day atmospheric

concentration but not for the higher concentration that we are now assuming. Under

0.02 bar of CO

the ocean must have been more acidic than today. The effect of pH on the

saturation boundaries is small (Exercise 11.6) but not entirely negligible. We therefore must

first calculate the pH of seawater in equilibrium with 0.02 bar of CO

, and then adjust Fig.

11.7 accordingly.

Suppose that the Archaean ocean was saturated in both calcite and siderite. We can then

add two more equations to the system of equations that we solved in Worked Example 11.4.

One of them is the solubility product of siderite:

FeCO

 Fe

+CO

2−

3aq

(11.121)

for which K

sp,sd

=3.13 ×10

−11

. The other is the Fe

hydrolysis reaction:

O  FeOH

, (11.122)

which can be converted to the second dissociation reaction in (11.110) using the ionization

product of water (equation (11.91)). We must also modify the charge balance equation

(11.105) to include the molalities of Fe

and FeOH

. Modifying the Maple routine to do

the calculations is easy, and is left as an exercise for the reader (Exercise 11.8).

ForaCO

fugacity of 0.02 bar we calculate the pH of our simplified ocean to be 7.13.

The solution set of the system of equations for siderite saturation (Exercise 11.8) also yields

the total concentration of dissolved ferrous iron, which in this case is 4.11 × 10

−5

molal.

Figure 11.8 shows the hematite and magnetite saturation boundaries redrawn for pH =7.13

(compare Fig. 11.7). The concentration of ferrous aqueous species in equilibrium with

siderite is independent of oxygen fugacity (equation (11.121)), so the siderite saturation

boundary is parallel to the f (O

) axis, at a total ferrous iron molality of 4.11 × 10

−5

. This

line intersects the magnetite saturation boundary at f (O

) = 1.9 × 10

−80

bar, which is

the same oxygen fugacity that we calculated for the magnetite–siderite equilibrium from

equation (11.120). Consistency between two sets of calculations is always reassuring.

The resulting diagram shows the stability fields of the three most common phases in

banded iron formations. Reaction (11.119) determines the maximum oxygen fugacity that

allows siderite precipitation, which varies as the sixth power of CO

fugacity. An increase in

fugacity expands the siderite field both towards the right (at the expense of magnetite)

and downwards, by lowering the concentration of Fe

required for siderite crystallization

(solubility product for reaction (11.121)). We can calculate with equation (11.120) the

fugacity needed to eliminate the magnetite field altogether, i.e. so that the siderite

field adjoins the hematite field. Making log f (O

) =−68.59 (the oxygen fugacity at

the magnetite–hematite transition) we get f (CO

) = 1.44 bar. The rarity of hematite–

siderite iron formations suggests that this is a reasonable upper bound for CO

fugacity in

the Archaean atmosphere at the time at which banded iron formations precipitated.

All of these calculations ignore the activity coefficients of aqueous species, which are

not negligible in a solution as concentrated as seawater. The rest of this chapter is focused

on this problem.

562 Dilute solutions

–90 –80 –70 –60 –5

–10

–8

–6

–4

–2

log f(O

) – bar

total dissolved iron molality

f(CO

) = 0.02 bar, pH = 7.13, T = 25º C

siderite magnetite hematite

dissolved iron

Fig. 11.8

Same as Fig. 11.7, but at pH =7.13, corresponding to a calcite-saturated solution at f(CO

) =0.02 bar. Increasing

fugacity expands the siderite ﬁeld downwards and rightwards, along the magnetite saturation curve. The

magnetite ﬁeld disappears at f (CO

) ≈1.44 bar. This represents the minimum chemical potential of CO

at which

siderite and hematite can precipitate at equilibrium. The rarity of this assemblage in terrestrial banded iron

formations places an upper bound on Archaean atmospheric CO

fugacity.

11.6 Activity coefﬁcients in electrolyte solutions

The key difficulty when calculating chemical equilibrium in electrolyte solutions is in

estimating reliable values for the activity coefficients of the many charged and neutral

species that exist in solution. The preceding section shows that even in a highly idealized

solution of Fe in water one must deal with almost ten species, and the number of species

grows rapidly with the number of components in the solution. The problem is not unlike that

of calculating speciation in a gas phase (Chapter 9), but it is more complicated because of (i)

the greater number of species, (ii) the existence of a solvent whose properties change with

temperature and (iii) the nature of the interactions among charged particles, and between

ions and solvent molecules.

The fact that electrolyte solutions behave differently from solutions of neutral species was

recognized early in the twentieth century. The different behaviors are compared schemat-

ically in Fig. 11.9, which shows the logarithm of the activity coefficient as a function of

molality for very dilute solutions of a neutral species (lnγ ) and an electrolyte (lnγ

). In

both cases the activity coefficient becomes unity at m =0. It is important to recall that this is

conventional, and that it arises from choosing to define the standard state at infinite dilution.

In other words, and as we discussed in Section 11.1.2, any excess Gibbs free energy that

may exist at infinite dilution remains unknown and is lumped into the standard state Gibbs

free energy. What matters now is that, whereas at very great dilution the activity coefficient

563 11.6 Activity coefﬁcients in electrolyte solutions

solute concentration

ln γ

neutral species (ln γ)

electrolyte (ln γ ±)

m << 1

Fig. 11.9

Schematic diagram showing the contrasting behaviors of dilute solutions of a neutral species and an electrolyte. The

key difference is that, whereas in a dilute solution of a neutral species lnγ varies linearly with concentration, the

variation in a dilute electrolyte solution is of the form ln γ =−αI

, with α>0 and 0 <β<1.

of neutral species varies linearly with molality, the same is not true for electrolytes. The

activity coefficient of electrolytes in dilute solutions varies in a strongly non-linear fashion

with concentration, and approaches the limiting value at infinite dilution with an infinite

slope. For neutral species the activity coefficient is generally greater than 1, whereas for

electrolytes activity coefficients in dilute solutions are less than one, but may become greater

than one at higher concentrations.

The distinct behavior of electrolytes can be thought of as arising from the formation

of “ionic atmospheres”. These are regions around each ion that, owing to electrostatic

attraction, carry an excess of charge of a sign opposite to that of the central ion. As long as

the individual ionic atmospheres remain distant, i.e. at low concentration, they shield the

central ions and make it less likely that they will interact with other ions. This is expressed

macroscopically as a decrease in their chemical potential. As the solution becomes less

dilute the ionic atmospheres interfere with one another and shielding becomes less effective.

Particles of a neutral solute, in contrast, are not surrounded by ionic atmospheres so one

would expect that any energetic effect arising from interactions among them would vary

more or less linearly, or at least monotonically, with concentration.

11.6.1 Debye–Hückel theory

The first theory capable of predicting activity coefficients for electrolytes on the basis of

a rigorous physical description of the microscopic structure of electrolyte solutions was

proposed by Debye and Hückel in 1923. The place of Debye–Hückel theory in the ther-

modynamics of electrolytes is not unlike that of the van der Waals equation of state in the

thermodynamics of fluids. The theory is quantitatively accurate only for dilute solutions,

564 Dilute solutions

however. For example, it is already grossly inaccurate for seawater, in which the total con-

centration of dissolved electrolytes is less than 1 mol kg

−1

. It is, however, the basis for many

of the more elaborate formulations that are used to calculate activity coefficients in concen-

trated electrolyte solutions. In contrast to most of the latter formulations, which are empirical

to varying extents, the Debye–Hückel approach has a strong theoretical foundation.

A rigorous derivation of the remarkably simple final equation of Debye and Hückel for

the mean ionic activity coefficient of an electrolyte is beyond the scope of this book, but

it is instructive to construct a semi-formal a posteriori justification of their equation. We

begin by defining the ionic strength of a solution, symbolized by I , as:

I =



, (11.123)

where z

is the charge of ion i, m

its molality, and the sum is over all of the ionic species

present in the solution. The ionic strength is a measure of the total concentration of dissolved

electrolytes. If we apply (11.123) to a solution of a single electrolyte we see that the behavior

of an electrolyte solution as it approaches the infinite dilution limit (Fig. 11.9) can be

represented by a function of the form:

ln γ =−αI

, α>0, 0 <β<1. (11.124)

The derivative of this function relative to I diverges as I goes to zero, as required by the

observed behavior of very dilute electrolyte solutions schematized in Fig. 11.9. Debye and

Hückel proved rigorously that the actual equation valid as the solution approaches infinite

dilution is:

log

=−A|z

i−

1/2

, (11.125)

where z

and z

i−

are the charges of the cation and anion in electrolyte i, and A is a constant

that depends on the solvent only, but varies with temperature and pressure. I is the ionic

strength, given by (11.123), and includes the concentration of i as well as of any other ions

present in the solution. Equation (11.125) is known as Debye–Hückel’s limiting law. It is

only accurate for ionic strengths lower than ∼10

−3

, but at such low concentrations it is

accurate enough that it is routinely used to extrapolate measurements of thermodynamic

functions done at very low dilutions to the infinite dilution limit. Its main use for our

purposes will be to understand the meaning of the parameter A and, with it, some of the

physical aspects of Debye–Hückel theory. Their theory shows that A is given by:

A =



2πNρs



1/2

, (11.126)

where N is Avogadro’s number, ρ is the density of the solvent, and s is a length given by:

s =

4π/

(11.127)

with e the elementary unit of charge, /

the permittivity of free space, ε

the dielectric

constant of the solvent and k

Boltzmann’s constant. The meaning of the length s becomes

clear if we re-write (11.127) as follows:

T =

4π/

(11.128)

565 11.6 Activity coefﬁcients in electrolyte solutions

and note that the equation now has units of energy. The length s is the distance between

two units of charge at which their electrostatic energy (see also equations (11.65)) equals

their thermal energy. The higher the temperature, or the dielectric constant, the closer the

charges have to be in order for the electrostatic force to overcome thermal agitation. We

next note that we can re-write the product N ρ as:

Nρ =

V/N

≈

, (11.129)

where M is the molecular weight of the solvent, V its molar volume and λ is a length of

the order of the intermolecular distance in the solvent. The parameter A is a function of the

ratio (s/λ), i.e.:

A ∼



2πM







1/2

. (11.130)

If (s/λ) is small then the distance over which electrostatic forces are effective is small com-

pared to the distance between solvent molecules, and formation of an ionic atmosphere

would tend to be restricted: the value of the parameter A decreases and the activity coef-

ficient approaches unity (equation (11.125)). On the other hand, a large value of the ratio

(s/λ) implies that electrostatic forces are effective over distances greater than intermolec-

ular separation in the solvent, allowing the formation of ionic atmospheres that lower the

chemical potential of the ion.

Before we proceed note two formal aspects of equation (11.125). First, it is customary

to write it in terms of decimal logarithm, rather than natural logarithm. Second, there is the

usual problem with units that arises when expressing concentration as molality. As written,

equation (11.126) implies thatA has units of (kg mol

−1

)

. If we express molality in mol kg

−1

then the units of I

1/2

are (mol kg

−1

)

and the logarithm is dimensionless, as it must be.

Alternatively, if we choose to use dimensionless molality (Section 11.1.1) then the term

in parentheses in equation (11.126) must be multiplied by 1 mol kg

−1

(see also equation

(11.4)) and A becomes a dimensionless parameter. This is the usual convention. Numerically

it makes no difference, but mathematical rigor requires that we worry about this.

Debye–Hückel’s parameter A is a property of the solvent only (and temperature, both

directly, see equation (11.127), and via the effect of T on the dielectric constant and density

of the solvent). The limiting law, equation (11.125), does not contain any solute properties

besides their charges. In particular, it ignores ionic radius. We can expect this to be a

reasonable model at very high dilution, where ions are separated by distances that are large

enough compared to their radii that they can be thought of as point charges.As concentration

increases the radius of the ions becomes significant relative to their separation and we must

expect the activity coefficient to become a function of ionic radius. This leads to the full

Debye–Hückel equation:

log

−A|z

i−

1/2

1 +å

1/2

, (11.131)

whereå

,knownasthedistance of maximumapproach,isafunction oftheeffectiveionicradii

of the ions that constitute electrolyte i, and B is a parameter that depends only on the identity

of the solvent and temperature. The product BI

1/2

has dimension of length

−1

(necessarily,

as å

has dimension of length) and can be thought of as the inverse of the effective radius of

the ionic atmosphere. If the distance of maximum approach is much smaller than this radius

then the denominator of (11.131) approaches 1 and we recover the limiting law, equation

566 Dilute solutions

(11.125). Note that the radius of the ionic atmosphere is a function both of a solvent property

(B) and of the ionic strength of the solution. For historical reasons, the unit that is universally

used for å

and B is the Ångstrom = 10

−10

m. Values of the å parameter for many common

ions were tabulated by Kielland (1937) and this is still the standard reference more than

seven decades later. The values of the A and B parameters for water have been calculated

over a large temperature range by Helgeson and Kirkham (1974).

Although equation (11.131) has theoretical justification it does not reproduce the observed

behavior of electrolytes at ionic strengths greater than ∼0.1. Figure 11.10 shows measured

activity and osmotic coefficients for four strong electrolytes at 298 K (data from Robinson

& Stokes, 1959). Also shown are activity coefficients calculated with the Debye–Hückel

equation (11.131), and osmotic coefficients calculated by integrating (11.131) according to

(11.43) (Exercise 11.9). The discrepancy above I ∼0.1 is very large. Note that, as equation

(11.131) does not have a minimum for any value of I, it is incapable of reproducing the

behavior of actual electrolytes, which generally show a minimum in γ at moderate ionic

strengths.

012345

0.2

0.4

0.6

γφ

0.8

solute molality

012 3 4 5

solute molality

Measured values NaCl NaClO

Mg(ClO

)

0.6

0.7

0.8

0.9

1.1

1.2

NaClO

NaCl

Mg(ClO

)

Mg(ClO

)

NaCl

NaClO

Fig. 11.10

Measured activity and osmotic coefﬁcients (symbols) in aqueous solutions of sodium chloride, sulfate and perchlorate,

andmagnesiumperchlorate(datafromRobinson&Stokes,1959),comparedtothevaluesoftheactivitycoefﬁcients

calculatedwithDebye–Hückelequation(11.131),andosmoticcoefﬁcientscalculatedbyintegrating(11.131)

accordingto(11.43).

567 11.6 Activity coefﬁcients in electrolyte solutions

Worked Example 11.5 The pH of natural waters revisited and a foray into soda lakes

We can refine the calculations in Worked Examples 11.3 and 11.4 by incorporating activity

coefficients calculated with the full Debye–Hückel equation, (11.131). In order to calculate

the pH of rainwater we re-write the first dissociation reaction, (11.98) as follows:

ds,I

HCO

−

·γ

HCO

−

·γ

(11.132)

or, using the definition of mean ionic activity coefficient, equation (11.75)

ds,I

HCO

−



HCO

−H



(11.133)

where:

log



HCO

−H



−A|z

HCO

−

1/2

1 +



+å

HCO

−



1/2

. (11.134)

The distance of maximum approach between two ions equals the sum of their radii, but

Kielland (1937) and all subsequent writers use ionic diameters – hence the division by 2 in

the denominator of (11.134). The second dissociation reaction, (11.97), becomes

ds, II

2−

HCO

−

2−

·γ

HCO

−

. (11.135)

In contrast to (11.132), the individual ion activity coefficients in (11.135) do not work out

to a mean ionic activity coefficient. We have two options to deal with this situation. We can

choose to calculate single ion activity coefficients by applying (11.131) to each ion, so that

the product of charges in the numerator becomes the square of the charge of the ion, and

the distance of maximum approach is the ionic diameter. Or we can multiply the numerator

and denominator in (11.135)byγ

H+

and re-write this equation as:

ds,II

2−

HCO

−



2−





HCO

−H



. (11.136)

Mathematically the two approaches are identical. Physically, however, equation (11.136)is

preferable because single ion activity coefficients cannot be measured, whereas mean ionic

activity coefficients can. Finally, we include activity coefficients in the water ionization

reaction, (11.91), which becomes:

·m

−



−



(11.137)

and we calculate pH with equation (11.94).

We are now ready to re-do the calculation in Worked Example 11.3, by using equations

(11.133), (11.136) and (11.137), plus the charge balance equation (11.100), which of course