I. Recap of reactions and activities discussed so far
Activities: for ions [Y] = g (Y); for water in aqueous solution [H2O] = 1; for uncharged solutes [X] = (X) (activity coefficient = 1 for uncharged solute); for a gas phase [Zgas] = PZ (activity equals partial pressure).
Equilibrium constants: water or acid or base dissociation K; gas exsolution KH Henry's Law constant in atm/mol/L; ion association (complexation) either K for individual step or b (stability constant) for net reaction of multiple steps.
Note that gas exsolution reactions are not necessarily fast enough to assume instantaneous equilibrium (demonstrated with bottle of fizzy water).
II. Precipitation-dissolution reactions
This group of reactions has been studied extensively by geochemists. Rates of some dissolution reactions are very slow, so we cannot necessarily expect these to reach equilibrium during groundwater transport. Nevertheless, an equilibrium approach provides a starting point to examine the net direction and extent of the "water-rock" interactions.
A simple example is the dissolution of the mineral halite (NaCl) for which the dissolution reaction can be written as
with the equilibrium mass-law relation
K = [Na+][Cl-]/[NaClsolid]
Note that we have not yet defined the activity of a mineral phase. It turns out that the activity of a mineral phase (when it is present) is unity. This means that the extent to which solutes react with the mineral phase is independent of how much of the mineral is in the system. Thus, the equilibrium constant is equal to a product of the activities of the reactants only when the reaction has reached equilibrium. Equilibrium constants for precipitation dissolution reactions are often called "solubility product" constants and are represented by Ksp. Thus, for halite
Ksp = [Na+][Cl-]
The equilibrium relation above holds ONLY if the reaction has reached equilibrium. Much more commonly, one might find that
[Na+][Cl-] < Ksp
In this case, the measured activities are used to compute the "ion-activity product" (really an activity quotient, but for most dissolution reactions this becomes an activity product since the denominator is generally equal to one).
IAP = [Na+]measured x [Cl-]measured
If IAP < Ksp, the solution is "undersaturated" with respect to the mineral and there will be net dissolution if the reaction continues. On the other hand, if IAP > Ksp, the solution is "supersaturated" with respect to the mineral and there will be net precipitation if the reaction continues. This comparison between IAP and K can also be quantified as a saturation index, SI equal to the log of the ratio IAP/K.
III. Redox reactions
Overall reactions and half-reactions
These reactions involve transfer of electrons from on species to another, changing the "oxidation states". Consider the transfer of electrons from iron to oxygen in the following reaction:
In this reaction, iron gave electrons and was oxidized from the +II oxidation state to the +III oxidation state, while oxygen in the gas accepted the electrons given up by iron and was reduced from the 0 oxidation state to -II. The equilibrium relation for this reaction can be written as
The net transfer involves at least two steps: an oxidation in which an electron is given off and a reduction in which the electron is taken up by another species. These two steps are often referred to as "half-reactions" and can be written separately as
with equlibrium relation and
with equilibrium relation
S&Z list a number of redox half-reactions (all written as reductions) in Table 18.7.
Free electrons do not exist in solution, so the activity of electrons, [e-], that appears in the equilibria for the two have reactions is not a real effective concentration, but rather a "hypothetical activity". This hypothetical activity disappears in the equilibrium equation for the net reaction. Despite the fact that free electrons do not exist, it is useful to compute the hypothetical electron activity for each of the half-reactions in order to determine if the net reaction is at equilibrium, and, if it is not at equilibrium, the direction in which the reaction must proceed to reach equilibrium.
The parameter pe, equal to the negative log of the hypothetical electron activity [e-], is used in a similar manner as the parameter pH is used to quantify [H+].
For a given half reaction, the pe can be computed by rearranging the equilibrium expression to solve for [e-] and by taking the log of each side of the equation. This will generate an equation of the form of 18.29 in S&Z. But NOTE, the terms [Red] and [Ox] in that equation represent do not necessarily represent the activities of individual reduced and oxidized species, respectively, but rather the ion activity products of the reduced and oxidized species.
Another parameter used to quantify [e-] is known as EH, which has units of volts. The conversion between EH and pe is temperature dependent. At 25 oC
EH = 0.059 pe
There are electrodes that can be used to measure a voltage associated with certain redox couples in water, and these are often used as measures of solution EH (and hence pe or [e-]). However, unless all redox reactions are at equilibrium, there is no unique [e-] for a given water sample, and the EH measured with an electrode may have little meaning for a particular redox couple of interest, such as the iron species discussed above. There has been considerable discussion in the literature about the potential problems with using such measurements to characterize the redox conditions of natural waters.
Application to determining speciation of solutes of interest
The hypothetical [e-] calculated from one half reaction can be used, assuming equilibrium, to solve for concentrations of unmeasured ions. For example, using the [e-] calculated from measured dissolved oxygen and pH, you could use this value of [e-] to calculate the ratio of dissolved iron species activities. If you had measured total Fe in solution, you could use ratio of iron species activities to determine concentrations of the individual species. This approach is only valid IF the net reaction is at equilibrium.
Diagrams that show the dominant ionic or solid phase forms of redox reaction participants as a function of pH and EH (or of pH and pe) are useful ways of summarizing equilibrium speciation for a variety of conditions. See for example Figure 18.13 of S&Z.
Note that the equilibrium constants for reduction half-reactions are listed in Table 18.7 of S&Z as peo= log K. For the same reactions written in reverse (as oxidations) the log of the equilibrium constant would be - peo.