Douce A.P. Thermodynamics of the Earth and Planets

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548 Dilute solutions

Since we required that all thermodynamic potentials of (B

)

be given by an equation

of the form of (11.83), the standard state chemical potential of the bulk aqueous electrolyte

is given by:

0,B

≡βµ

0,B+

+αµ

0,A−

. (11.86)

This definition of standard state properties is the crucial difference with the treatment of

weak electrolytes. A consequence of equation (11.86) is that we can think of the dissoci-

ation constant of a strong electrolyte as being identically equal to 1 at all pressures and

temperatures for which the electrolyte undergoes complete dissociation. Equation (11.86)

is, however, never valid for a weak electrolyte in which, as we saw, the standard state is

defined in the usual way for a dilute solute.

From (11.85) and (11.86) we find that the activity of the bulk electrolyte is given by:



B+





A−



. (11.87)

Note that this is different from the activity of an undissociated weak electrolyte, given

by (11.81). Substituting mean ionic molality and mean ionic activity coefficient, (11.87)

becomes:





β+α

, (11.88)

which should be compared with equation (11.81) for a weak electrolyte. Exercise 11.3 asks

you to apply these relations to the equations that relate activity and osmotic coefficients

(Section 11.2.3), and may be helpful in clarifying the meaning of (11.87) and (11.88).

11.4.3 The reference for standard state properties of ionic species

A problem that arises in the treatment of ionic species is that their concentrations cannot

be varied independently of those of ions of opposite charge. It is possible to determine

the infinite dilution standard state properties of dissolved molecular species for a weak

electrolyte, or the standard state properties of (fictive) bulk strong electrolytes, for example

by measurements of enthalpies of dissolution. Once these data are available one knows

the sum of the standard state properties of the corresponding anion and cation, e.g. via

equations (11.78)or(11.86). In all cases we have one equation with two free parameters,

so that there is no unique solution unless one specifies the value of one of the parameters.

The convention is to make the standard state enthalpy, entropy, Gibbs free energy and heat

capacity of the H

cation in aqueous solution at infinite dilution equal to zero, i.e.:



0,H

=

0,H

=0. (11.89)

Because the heat capacity is set equal to zero equation (11.89) is valid at all temperatures.

Of course, the entropy of protons at any temperature other than zero is not zero, but since all

we care about when performing thermodynamic calculations are entropy (or free energy)

differences at constant temperature and pressure the convention does not introduce any

difficulties. Also, H

ions in aqueous solutions are always hydrated, i.e. attached electro-

statically to one or more H

O molecules, and do not exist as free protons. The symbol H

in equation (11.89) and all subsequent discussions must be understood as thermodynamic

shorthand for the actual molecular entity that exists in solution.

549 11.4 Thermodynamic formulation of electrolyte solutions

11.4.4 Dissociation of water and pH

Water is a polar liquid, so it should not be surprising that H

O molecules dissociate in liquid

water to some extent (Section 11.3). We can write this dissociation reaction as follows:

(

)

liq

 H

+OH

−

. (11.90)

Choosing the standard state of water as the pure liquid at the temperature and pressure of

interest, we get the following equation for the dissociation constant of water, K

(compare

equation (11.80) for the dissociation of a weak electrolyte):

·a

−



·m

−





·γ

−



. (11.91)

At 298.15 K the constant K

, also known as the ionization product of water, has a value

of K

= 1.011 × 10

−14

, which is commonly rounded off to K

≈ 10

−14

. The ionic

concentration in pure water is very low. Therefore, as a first approximation that is consistent

with the infinite dilution standard state, we can assume that the activity coefficients of the

ionic species are unity. The charge balance constraint, equation (11.69), then requires:

−

≈10

−7

. (11.92)

This was the basis of the original definition of pH as the negative of the decimal logarithm

of the hydrogen ion concentration:

pH =−log

(11.93)

and the definition of a neutral solution, with pH = 7, as one in which the hydrogen ion

molality is 10

−7

. Rigorously, however, pH is defined on the basis of hydrogen ion activity:

pH ≡−log

=−log





(11.94)

and a neutral solution is one in which the hydrogen ion activity is 10

−7

. In dilute electrolyte

solutions the difference between (11.93) and (11.94) is negligible, but in concentrated

solutions it may be necessary to calculate the activity coefficient of H

and use the rigorous

definitions (11.91) and (11.94).

Worked Example 11.3 Atmospheric CO

and the pH of rainwater

Dissolution of atmospheric gases in water is described by equation (11.18). In Worked

Example 11.1 we calculated the concentrations of the molecular species N

, Ar and

in water at equilibrium with the terrestrial atmosphere. When CO

dissolves in water,

however, it reacts with H

O and produces bicarbonate and carbonate ions, according to:

2aq

O  HCO

−

3aq

(I )

HCO

−

3aq

 CO

2−

3aq

(II). (11.95)

If the solution is dilute and we take the activity coefficients to be unity then we can write

the dissociation constants as:

ds,I

HCO

−

(11.96)

550 Dilute solutions

and:

ds,II

2−

HCO

−

. (11.97)

At equilibrium the chemical potential of gaseous CO

in the atmosphere is the same as

the chemical potential of molecular CO

dissolved in water. Therefore, by choosing the

standard state of CO

as the pure gas at the temperature of interest and 1 bar, rather than

infinitely dilute CO

in aqueous solution, we can re-write equation (11.96) as follows:

ds,I

HCO

−

. (11.98)

Equations (11.96) and (11.98) are equivalent, differing only in the choice of standard state

for CO

and, therefore, in the numerical value of the dissociation constant. Although it is

customary to use an equation of the form of (11.96), I believe that (11.98) is preferable, as this

gets around the problem of calculating activity coefficients for dissolved molecular gases.

If one wishes to calculate the concentration of dissolved gases this is simply accomplished

by means of Henry’s law (equation (11.18)), i.e. in this case:

·K

H ,s

. (11.99)

There are two additional equations that must be satisfied by the molalities of dissolved

species at equilibrium. One is the water ionization equation (11.91) (taking the activity

coefficients to be unity), and the other is the charge balance equation (11.69) which in this

case is:

−m

HCO

−

−2m

2−

−m

−

=0. (11.100)

If we fix the fugacity of CO

in equilibrium with water then equations (11.91) (setting

the activity coefficients equal to 1), (11.97), (11.98) and (11.100) constitute a system of

four equations in which the four aqueous molalities are the unknowns. The equations are

non-linear, but a numerical solution with Maple is straightforward (Software Box 11.1).

The two ionic dissociation constants can be calculated from standard state properties for

aqueous species given, for example, by Wagman et al.(1982) or Robie and Hemingway

(1995). If one wishes to calculate equilibrium at 298.15 K then the dissociation constants

are simply obtained from the listed standard state Gibbs free energy of formation of the

species, as in equation (11.79). Equilibrium at other temperatures can be calculated from

listed standard state enthalpy and entropy values: 

=

−T

, including heat

capacity integrals if the C

values are known. The constants at 298.15 K and 1 bar are:

ds,I

=1.465 ×10

−8

(with a standard state of pure CO

gas) and K

ds,II

=4.685 ×10

−11

Software Box 11.1 Calculation of carbonate speciation in aqueous solution, assuming ideal

behavior

Procedure zeco2 in Maple worksheet aq_spec_ideal.mw calculates molalities

of dissolved carbonate, bicarbonate and carbon dioxide, in equilibrium witha1bar

atmosphere at 298.15 K. The aqueous solution is assumed to be ideal (see Worked

Example 11.3). A range of CO

atmospheric concentrations (in ppm) is specified in the

551 11.4 Thermodynamic formulation of electrolyte solutions

procedure call, as well as the number of intermediate values to calculate between these

boundaries, and a file name to send the output to. Output fields are: CO

atmospheric

concentration (ppm), pH , m

, m

HCO

−

, m

2−

, total dissolved CO

molality.

A second procedure, zecalcite, adds calcite saturation (equation (11.106), see

Worked Example 11.4) but is otherwise identical to zeco2. An additional field at the

end of each line of output contains m

The results of the calculations as a function of f

for values ranging from 0 to 5 ×

−4

bar (≈ 500 ppm CO

by volume in a 1 bar atmosphere) are shown in Fig. 11.5. The

top panel shows molalities of dissolved molecular CO

(calculated from f

and Henry’s

law constant given by Sander, 1999, K

=0.034) and of carbonate and bicarbonate ions,

as well as the sum of all three species, which is the total amount of dissolved CO

, called

the carbonate analytical concentration. At very low CO

fugacity the dominant species is

the bicarbonate anion, but as CO

concentration increases and the solution becomes more

acidic (bottom panel) the increased H

concentration displaces equilibrium (I) to the left and

limits dissociation, so that molecular CO

is the most abundant species. The bottom diagram

–4

2*10

–4

3*10

–4

4*10

–4

5*10

–4

–11

–10

–9

–8

–7

–6

–5

f (CO

) – bar

–4

2*10

–4

3*10

–4

4*10

–4

5*10

–

f (CO

) – bar

molality

5.5

6.0

6.5

7.0

2–

HCO

–

total

Carbonate speciation in rain water at 25º C

pH of rain water at 25º C

Fig. 11.5 Carbonate species distribution (top) and pH (bottom) in “pure” water (∼rainwater) in equilibrium with the

terrestrial atmosphere at 25

◦

C. The analytical CO

concentration equals the total concentration of CO

-bearing

species =m

HCO

(−)

(2−)

552 Dilute solutions

shows that the pH of water in equilibrium with the present-day terrestrial atmosphere

(380 ppm CO

) should be about 5.6. This is approximately the pH of rainwater in regions

far removed from sources of pollution (particularly burning of high-sulfur coal and diesel

fuel). It is not the pH of ocean water, nor of surface waters in general, as these waters contain

other solutes in addition to CO

and may also be in equilibrium with solid phases.

11.4.5 Equilibrium between electrolyte solutions and solid phases

Aquestion that repeatedly comes up when studying aqueous solutions is whether the solution

becomes saturated in one or more crystalline phases. This is the same problem that is at

the core of igneous petrology, but aqueous solutions are generally better understood from a

theoretical point of view than silicate melts. We can represent equilibrium between an ionic

crystalline solid, (B

)

, and its dissociation products by:





 β

(

B+

)

+α

(

A−

)

. (11.101)

This equilibrium is equally valid whether the substance is a strong or weak electrolyte.

In the latter case we could also, if we wished, write an equilibrium equation with the

undissociated aqueous species. As usual, the standard state for the crystalline phase is the

pure solid at the temperature and pressure of interest. Note that this is not equal to the

standard state chemical potential of the molecular species (B

)

for a weak electrolyte.

The equilibrium condition for (11.101) is:

0,B

=βµ

B+

+αµ

A−

=βµ

0,B+

+αµ

0,A−

+RT ln





B+





A−





. (11.102)

Let us call the equilibrium constant for this reaction K

, so that:

=exp



−





=exp



−

βµ

0,B+

+αµ

0,A−

−µ

0,B



(11.103)

and:



B+





A−



(

B+

)

(

A−

)





β+α

. (11.104)

These equations are valid in general, but it is convenient to distinguish between relatively

soluble and relatively insoluble substances. For the former the molalities at saturation are

large enough that the activity coefficients cannot be ignored. For relatively insoluble elec-

trolytes the molalities are small enough that it may be acceptable to ignore the activity

coefficients. In such cases K

is known as the solubility product and it provides a conve-

nient way of checking for saturation of specific phases. At a given temperature and pressure

the solubility product is a constant for each phase in equilibrium with an aqueous solution.

It is independent of the composition of the solution, as it is a combination of standard state

properties only (see equation (11.103), but recall that if the solvent is a liquid other than

water then the standard states for the solutes will be different, and so will the solubility prod-

uct). At saturation the product of the activities (or molalities if the solution is sufficiently

553 11.4 Thermodynamic formulation of electrolyte solutions

dilute) of the ionic species, with the stoichiometric coefficients as exponents, is equal to

the solubility product. If the activity (or molality) product is less than the solubility product

then the solution is not saturated in that particular phase. If it is greater then the solution is

supersaturated with respect to the phase, which is a non-equilibrium condition.

Worked Example 11.4 Limestone saturation and the pH of ocean water

Shallow ocean water is saturated in calcium carbonate, or at least nearly so. We can be

reasonably certain of this because, for instance, the shells of marine invertebrates do not

dissolve when the animals die. We will study how calcium carbonate saturation affects our

calculation of carbonate speciation and pH (Worked Example 11.3). Three of the equations

that we used in those calculations, (11.97), (11.98) and the water ionization equation (11.91),

remain unchanged. Because now we must also consider aqueous Ca

ions, the electrical

neutrality equation (11.100) is modified as follows:

+2m

−m

HCO

−

−2m

2−

−m

−

=0 (11.105)

and we have an additional equation, which is the solubility product of calcite:

sp,cc

·m

2−



±,CaCO



. (11.106)

There are now five equations in five unknowns: the five ionic molalities listed in equation

(11.105).Thesolubilityproductofcalciteat298.15KcalculatedwithdatafromWagman

et al.(1982)isK

sp,cc

= 4.965 ×10

−9

(using aragonite instead of calcite makes a small

difference). For now we will assume that we can ignore the activity coefficient in equation

(11.106), and will return to this in Section 11.6. Modifying the Maple procedure discussed in

Software Box 11.1 to include the additional equation (11.106) is straightforward (Exercise

11.4).

For an atmospheric CO

concentration of 380 ppm we calculate an oceanic pH of

∼8.3, which is similar to typical values measured in the Earth’s oceans. The agreement

is remarkably good, considering that we have ignored all excess mixing properties. Table

11.1 compares the calculated solution compositions with and without calcite saturation

(Worked Example 11.3), for a fixed atmospheric CO

concentration of 380 ppm. The con-

centration of dissolved molecular CO

is of course the same in both cases, as this is fixed by

equilibrium with the gas phase. Total dissolved carbonate, however, is almost two orders of

magnitude higher in water saturated with calcite. The calculated Ca

molality is ∼5.4 ×

−4

, which is more than one order of magnitude lower than measured molality in seawater

(∼10

−2

). The discrepancy arises to a large extent from ignoring the activity coefficient in

equation (11.106).

Why is the pH of seawater controlled by CaCO

saturation? The answer is that among

the most abundant species in seawater carbonate is the only weak electrolyte, for which it is

possible to write a partial dissociation reaction such as (11.95)(II). Table 11.2 (taken from

Millero, 2004) shows the concentrations of the most abundant ions in seawater. The four

most abundant cations are strong bases, and the halide and sulfate anions are strong acids.

None of these ions react with H

O to consume or produce H

ions. The pH of seawater

is therefore fixed by the solid carbonate for which the solubility product is first exceeded,

and for present day seawater this is calcite.

554 Dilute solutions

Table 11.1 Calculated ionic speciation in simpliﬁed natural waters

Rainwater Calcite-saturated water

ideal Debye–Hückel ideal Debye–Hückel

1.29 ×10

−5

1.29 ×10

−5

1.29 × 10

−5

1.29 × 10

−5

HCO

−

2.36 ×10

−6

2.36 ×10

−6

1.05 × 10

−3

1.15 × 10

−3

2−

4.68 ×10

−11

4.71 ×10

−11

9.28 × 10

−6

1.22 × 10

−5

Total CO

1.53 ×10

−5

1.53 ×10

−5

1.07 × 10

−3

1.18 × 10

−3

−−5.35 × 10

−4

5.89 × 10

−4

Ionic strength 2.36 ×10

−6

2.37 ×10

−6

1.61 × 10

−3

1.78 × 10

−3

pH 5.63 5.63 8.28 8.30

Calculations are based on 380 ppm atmospheric CO

at 1 bar and 25

◦

C, assuming either ideal

aqueous solutions or Debye–Hückel activity coefficients. Concentrations given in molality.

Table 11.2 Most abundant ionic species in seawater (after Millero, 2004)

Cation Molality Anion Molality

0.4691 Cl

−

0.5459

0.0528 SO

2−

0.0282

0.0103 HCO

−

0.0018

0.0102 Br

−

0.0008

−

0.0007

2−

0.0003

11.5 Speciation in ionic solutions. Iron solubility in ocean water

as an example

Iron cations occur in two oxidation states, Fe

and Fe

, that have vastly different solu-

bilities in water. Both ferrous and ferric ions react with water to form several ionic species

whose relative abundances are a function of pH . These properties make iron aqueous solu-

tions an excellent test subject on which to apply the concepts that we have discussed so

far. We will estimate the solubility of iron in ocean water, and study how this may have

varied with changes in atmospheric composition. The problem was introduced in Worked

Example 6.7, where we discussed banded iron formations and the information that they

preserve about conditions in the early Earth.

The solubility of iron in the present day shallow oceans, i.e. in ocean water that is in

equilibrium with the atmosphere, is limited by precipitation of Fe

species. This fol-

lows from the fact that the present day atmospheric oxygen fugacity is about 70 orders of

555 11.5 Speciation in ionic solutions

magnitude higher than the oxygen fugacity along the hematite–magnetite buffer at room

temperature (Fig. 9.3). We can expect the solubility-limiting phase to be either ferric oxide

or ferric hydroxide. The Gibbs free energy change for the reaction 2 Fe(OH)

→ Fe

Oat298Kis∼−30 kJ mol

−1

, which means that hematite is the stable phase.

The phase that usually precipitates from seawater is not hematite, however, but rather

an amorphous solid or a hydrated crystalline ferric oxide with a composition intermedi-

ate between those of hematite and ferric hydroxide. Over time the precipitate dehydrates

and becomes hematite, and this may have been the mechanism by which Precambrian

hematite iron formations formed. This is a kinetic problem, that we will not address here

(see Chapter 12). We will use hematite in our calculations, in order to be consistent with

Worked Example 6.7. Because the metastable phase that actually forms has higher Gibbs

free energy than hematite, the results of our calculations are an absolute minimum of

solubility.

In these calculations we will only consider dissolved iron species that exist in an aqueous

solution with no other solutes, except carbonate anions. The effect of this simplification is

to underestimate the solubility of iron relative to the actual solubility in seawater, in which

formation of other Fe-bearing ionic species is possible. We will also assume that the activity

of dissolved iron species is ideal (i.e. a = m). Our estimated total iron contents in solution

will be about two orders of magnitude lower than the analytical concentrations of iron in

present day seawater (∼10

−10

molal in oxygen-rich near-surface water). The qualitative

behavior with changes in oxygen fugacity and pH that we will uncover are, however, robust,

and will illuminate the significance of banded iron formations (see also Worked Example

6.7). A rigorous discussion of iron solubility in seawater is given, for example, by Millero

et al.(1995) and Liu and Millero (2002).

We can think of dissolution of Fe ionic species in water as occurring in one of two ways.

The one that is perhaps more intuitively appealing is to think of iron hydroxides as weak

bases. Just as dissolved CO

, which is a weak acid, undergoes successive ionization reactions

(Worked Example 11.3), so a molecular aqueous species such Fe(OH)

3aq

can be thought

to dissociate in steps to Fe(OH)

, FeOH

and finally Fe

. All of these stoichiometric

species are known to exist in ferric iron solutions. Alternatively, we may imagine that Fe

exists in solution as free ions that associate electrostatically with OH

−

groups to form

complex ions with the same stoichiometries as the molecular species that would form by

dissociation. The difference is that whereas in the former case the intermediate ionic species,

as well as neutral Fe(OH)

, are taken to be covalently bonded molecules, in the latter case the

ionic species are complexes in which Fe

and OH

−

ions are attached electrostatically. The

more accurate physical picture is the latter one, but from a macroscopic thermodynamic

point of view the two approaches are equivalent. I will therefore use the “weak base”

model, because of its intuitive appeal. According to this model dissolution of hematite in

water proceeds as follows:

O  Fe

(

)

3aq

(III,hm)

(

)

3 aq

 Fe

(

)

2 aq

+OH

−

(III,1)

(

)

2 aq

 FeOH

+OH

−

(III,2)

FeOH

 Fe

+OH

−

(III,3)

(11.107)

556 Dilute solutions

for which the equilibrium constants are (assuming an ideal electrolyte solution):

III,hm

Fe(OH)

III,1

Fe(OH)

·m

−

Fe(OH)

III,2

FeOH

·m

−

Fe(OH)

III,3

·m

−

FeOH

(11.108)

The values of the equilibrium constants at 298.15 K and 1 bar, calculated with data from

Wagman et al.(1982) are: K

III,hm

=1.499 ×10

−12

, K

III,1

=5.997 ×10

−12

, K

III,2

=1.011×

−9

, K

III,3

=1.515 ×10

−12

. From equations (11.108) it follows that the solubility product

for hematite is K

sp,hm

III,hm,

·K

III,1

·K

III,2

·K

III,3

= 1.38 ×10

−44

. If we were to calculate

the solubility of hematite from this figure we would get a molality of Fe

of ∼10

−23

pH =7. This is the correct concentration of Fe

cations in equilibrium with hematite, but

the analytical concentration of dissolved ferric iron is many orders of magnitude higher than

this, as the solubility product fails to account for the abundances of the other ionic species.

This is a point that one must always be careful with when calculating solubilities: failure to

account for speciation leads to grossly erroneous results. In fact, at pH =7 the most abundant

species is neutral Fe(OH)

, and the second most abundant one, with a concentration over

four orders of magnitude lower, is Fe(OH)

.Fe

becomes the dominant aqueous species

only in very acidic solutions (Exercise 11.5).

The total concentration (= analytical concentration) of dissolved ferric iron is obtained

by solving each equation in (11.108) for one of the species molalities, and adding up the

resulting equations. Using the ionization product of water (equation (11.91)) to convert

−

to m

we get the following equation for total dissolved ferric iron, m

,total

,in

equilibrium with hematite, as a function of pH:



,total



hm sat

III,hm



1 +

III,1

−pH

III,1

·K

III,2

−2pH

III,1

·K

III,2

·K

III,3

−3pH



(11.109)

Although in the present day terrestrial oceans the concentration of Fe

is vanishingly

small (at least in those parts of the ocean that are close to equilibrium with the atmosphere),

the same may not have been true during theArchaean. To account for this we write equations

for equilibrium between hematite and dissolved ferrous iron, which will necessarily involve

oxygen. In contrast to Fe(OH)

, there is no experimental evidence for the existence of the

neutral species Fe(OH)

in aqueous solution. We therefore write the reactions for hematite

saturation from Fe

as follows:

O  FeOH

+OH

−

(II, hm)

FeOH

 Fe

+OH

−

(II, 2). (11.110)

557 11.5 Speciation in ionic solutions

Taking the standard state of oxygen as the pure gas at the pressure and temperature of

interest the equilibrium constants are:

II,hm





1/4

·m

FeOH

·m

−

II,2

·m

−

FeOH

(11.111)

and the values of the equilibrium constants (calculated with data from Wagman et al., 1982)

are K

II,hm

= 7.431 × 10

−32

and K

II,2

= 5.920 × 10

−8

. We can now derive an equation for

total dissolved ferrous iron in equilibrium with hematite, analogous to (11.109), which is:



, total



hm sat

II,hm

−pH





1/4



1 +

II,2

−pH



. (11.112)

Equations (11.109) and (11.112) are plotted in Fig. 11.6 as a function of oxygen fugacity,

at a constant pH =8, which is approximately that of present day ocean water. Also shown

–80 –60 –40 –20

log f(O

) – bar

molality

, hematite

, magnetite

, hematite

, magnetite

Fe total, hematite

Fe total, magnetite

magnetite

hematite

Dissolved iron at pH = 8 and 25º C

Fe total, hematite

–11

–12

–13

–10

–9

–8

–7

–6

–5

Fig. 11.6

Concentration of dissolved iron species in water at 25

◦

C as a function of oxygen fugacity at pH =8(∼present day

seawater). The diagram shows total Fe

species (Fe

analytical concentration), total Fe

species (Fe

analytical

concentration) and total dissolved iron (=Fe

+Fe

species analytical concentrations) in equilibrium with either

hematite or magnetite. The stable phase is the one which attains saturation at lower concentration, shown with the

solid curves. The dashed curves show calculated concentrations in equilibrium with the metastable phase. The thick

solid curve shows total dissolved iron in equilibrium with the stable phase, which is hematite at log f (O

) > −68.59

and magnetite at log f (O

) < −68.59.