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pε–pH and E_H–pH diagrams are commonly used tools of aqueous geochemistry, and it is important to become familiar with them. An example, the pε–pH diagram for iron, is shown in Figure 3.19. pε–pH diagrams look much like phase diagrams, and indeed there are many similarities. There are, however, some important differences. First, labeled regions do not represent conditions of stability for phases; rather they show which species will predominate under the pε–pH conditions within the regions. Indeed, in Figure 3.19 we consider only a single phase: an aqueous solution. The bounded regions are called predominance areas. Second, species are stable beyond their region: boundaries represent the conditions under which the activities of species predominating in two adjoining fields are equal. However, since the plot is logarithmic, activities of species decrease rapidly beyond their predominance areas.

More generally, a pε–pH diagram is a type of activity or predominance diagram, in which the region of predominance of a species is represented as a function of activities of two or more species or ratios of species. We will meet variants of such diagrams in later chapters.

Let's now see how Figure 3.19 can be constructed from basic chemical and thermodynamic data. We will consider only a very simple Fe-bearing aqueous solution. Thus, our solution contains only species of iron, the dissociation products of water and species formed by reactions between them. Thermodynamics allow us to calculate the predominance region for each species. To draw boundaries on this plot, we will want to obtain equations in the form of pε = a + b × pH. With an equation in this form, b is a slope and a is an intercept on a pε–pH diagram. Hence we will want to write all redox reactions so that they contain e^– and all acid–base reactions so that they contain H⁺.

Figure 3.19 pε–pH diagram showing predominance regions for ferric and ferrous iron and their hydrolysis products in aqueous solution at 25°C and 0.1 MPa.

In Figure 3.18, we are only interested in the region where water is stable. So to begin construction of our diagram, we want to draw boundaries outlining the region of stability of water. The upper limit is the reduction of oxygen to water:

The equilibrium constant for this reaction is:

(3.117)

Expressed in log form:

The value of log K is 41.56 (at 25°C and 0.1 MPa). In the standard state, the activity of water and partial pressure of oxygen are 1 so that 3.117 becomes:

(3.118)

Equation 3.118 plots on a pε–pH diagram as a straight line with a slope of −1 intersecting the vertical axis at 20.78. This is labeled as line ➀ on Figure 3.19.

Similarly, the lower limit of the stability of water is the reduction of hydrogen:

Because ΔG°_r = 0 and log K = 0 (by convention), we have pε = −pH for this reaction: a slope of 1 and intercept of 0. This is labeled as line ➁ on Figure 3.19. Water is stable between these two lines (region shown in gray on Figure 3.19).

Now let's consider the stabilities of a few simple aqueous iron species. One of the more important reactions is the hydrolysis of Fe³⁺:

The equilibrium constant for this reaction is 0.00631. The equilibrium constant expression is then:

Region boundaries on pε–pH diagrams represent the conditions under which the activities of two species are equal. When the activities of FeOH²⁺ and Fe³⁺ are equal, the equation reduces to:

Thus, this equation defines the boundary between regions of predominance of Fe³⁺ and Fe(OH)²⁺. The reaction is independent of pε (no oxidation or reduction is involved), and it plots as a straight vertical line pH = 2.2 (line ➂ on Figure 3.19). Boundaries between the successive hydrolysis products, such as and , can be similarly drawn as vertical lines at the pH equal to their equilibrium constants, and occur at pH values of 3.5, 7.3, and 8.8. The boundary between Fe²⁺ and Fe(OH)^– can be similarly calculated and occurs at a pH of 9.5.

Now consider equilibrium between Fe²⁺ and Fe³⁺ (eqn. 3.102). The pε° for this reaction is 13.0 (Table 3.3), hence from eqn. 3.112 we have:

(3.119)

When the activities are equal, this equation reduces to:

and therefore plots as a horizontal line at pε = 13 that intersects the FeOH²⁺–Fe³⁺ line at an invariant point at pH = 2.2 (line ➃ on Figure 3.19).

The equilibrium between Fe²⁺ and Fe(OH)²⁺ is defined by the reaction:

Two things are occurring in this reaction: reduction of ferric to ferrous iron, and reaction of H+ ions with the OH– radical to form water. Thus, we can treat it as the algebraic sum of the two reactions we just considered:

or:

This boundary has a slope of −1 and an intercept of 15.2 (line ➄ on Figure 3.19). Slopes and intercepts of other reactions may be derived in a similar manner.

Now let's consider some solid phases of iron as well, specifically hematite (Fe₂O₃) and magnetite (Fe₃O₄). First, let's consider the oxidation of magnetite to hematite. We could write this reaction as we did in eqn. 3.101, however, that reaction does not explicitly involve electrons, so that we would not be able to derive an expression containing pε or pH from it. Instead, we'll use water as the source of oxygen and write the reaction as:

(3.120)

Assuming unit activity of all phases, the equilibrium constant expression for this reaction is:

(3.121)

From the free energy of formation of the phases (ΔG_f = −742.2 kJ/mol for hematite, −1015.4 kJ/mol for magnetite, and −237.2 kJ/mol for water), we can calculate ΔG_r using Hess's law and the equilibrium constant using eqn. 3.86. Doing so, we find log K = −5.77. Rearranging eqn. 3.121 we have:

The boundary between hematite and magnetite will plot as a line with a slope of −1 and an intercept of 2.88. Above this line (i.e., at higher pε) hematite will be stable; below that magnetite will be stable (Figure 3.20). Thus, this line is equivalent to a phase boundary.

Next let's consider the dissolution of magnetite to form Fe²⁺ ions. The relevant reaction is:

The equilibrium constant for this reaction is 7 × 10²⁹. Written in log form:

or:

We have assumed that the activity of water is 1 and that magnetite is pure and therefore that its activity is 1. If we again assume unit activity of Fe²⁺, the predominance area of magnetite would plot as the line:

that is, a slope of −4 and intercept of 0.58. However, such a high activity of Fe²⁺ would be highly unusual in a natural solution. A more relevant activity for Fe²⁺ would be perhaps 10^–6. Adopting this value for the activity of Fe²⁺, we can draw a line corresponding to the equation:

This line represents the conditions under which magnetite is in equilibrium with an activity of aqueous Fe²⁺ of 10⁻⁶. For any other activity, the line will be shifted, as illustrated in Figure 3.20. For higher concentrations, the magnetite region will expand, while for lower concentrations it will contract.

Figure 3.20 Stability regions for magnetite and hematite in equilibrium with an iron-bearing aqueous solution. Thick lines are for a Fe_aq activity of 10⁻⁶, finer lines for activities of 10⁻⁴ and 10^–8. The latter is dashed.

Now consider the equilibrium between hematite and Fe²⁺. We can describe this with the reaction:

The equilibrium constant (which may again be calculated from ΔG_r) for this reaction is 23.79.

Expressed in log form:

Using an activity of 1 for Fe²⁺, we can solve for pε as:

For an activity of Fe²⁺ of 10⁻⁶, this is a line with a slope of 3 and an intercept of 17.9. This line represents the conditions under which hematite is in equilibrium with = 10⁻⁶. Again, for any other activity, the line will be shifted as shown in Figure 3.20.

Finally, equilibrium between hematite and Fe³⁺ may be expressed as:

The equilibrium constant expression is:

For a Fe³⁺ activity of 10^–6, this reduces to:

Since the reaction does not involve transfer of electrons, this boundary depends only on pH.

The boundary between predominance of Fe³⁺ and Fe²⁺ is independent of the Fe concentration in solution and is the same as eqn. 3.119 and Figure 3.18, namely pε = 13.

Examining this diagram, we see that for realistic dissolved Fe concentrations, magnetite can be in equilibrium only with a fairly reduced, neutral to alkaline solution. At pH of about 7 or less, it dissolves and would not be stable in equilibrium with acidic waters unless the Fe concentration were very high. Hematite is stable over a larger range of conditions and becomes stable over a wider range of pH as pε increases. Significant concentrations of the Fe³⁺ ion (>10⁻⁶ m) will be found only in very acidic, oxidizing environments.

Figure 3.21 pε and pH of various waters on and near the surface of the Earth. After Garrels and Christ (1965).

Figure 3.21 illustrates the pH and pε values that characterize a variety of environments on and near the surface of the Earth. Comparing this figure with pH–pε diagrams allows us to predict the species we might expect to find in various environments. For example, Fe³⁺ would be a significant dissolved species only in the acidic, oxidized waters that sometimes occur in mine drainages (the acidity of these waters results from high concentrations of sulfuric acid that is produced by oxidation of sulfides). We would expect to find magnetite precipitating only from reduced seawater or in organic-rich, highly saline waters.