Thus, we might find it convenient to define an activity for the electron. For this reason, chemists have defined an analogous parameter to pH, called pε, which is the negative log of the activity of electrons in solution:
(3.112)
The log of the equilibrium constant for eqn. 3.101 may then be written as:
Upon rearranging we have:
(3.113)
When the activities of reactants and products are in their standard states (i.e., a = 1), then:
(3.114)
(where z again is the number of electrons exchanged: 1 in reaction 3.102). pε° values are empirically determined and may be found in various tables. Table 3.3 lists values for some of the more important reactions. For any state other than the standard state, pε is related to the standard state pε° by:
(3.115)
pε and EH are related by the following equation:
(3.116)
(the factor 2.303 arises from the switch from natural log units to base 10 log units).
In defining electron activity and representing it in log units, there is a clear analogy between pε and pH. However, the analogy is purely mathematical, and not physical. Natural waters do not contain significant concentrations of free electrons. Also, although a system at equilibrium can have only one value for pε, just as it will have only one value of pH, redox equilibrium is often not achieved in natural waters. The pε of a natural system is therefore often difficult to determine. Thus, pε is a hypothetical unit, defined for convenience of incorporating a representation of redox state that fits readily into established thermodynamic constructs such as the equilibrium constant. In this sense, eqn. 3.116 provides a more accurate definition of pε than does eqn. 3.112.
The greater the pε, the greater the tendency of species to lose their transferable, or valence, electrons. In a qualitative way, we can think of the negative of pε as a measure of the availability of electrons. pε can be related in a general way to the relative abundance of electron acceptors. When an electron acceptor, such as oxygen, is abundant relative to the abundance of electron donors, the pε is high and electron donors will be in electron-poor valence states (e.g., Mn4+ instead of Mn2+). pε, and EH, are particularly useful concepts when combined with pH to produce diagrams representing the stability fields of various species. We will briefly consider how these are constructed.
3.11.1.3 pε–pH diagrams
pε–pH and EH–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
