1. Nernst Equation Consider the half reaction:
NO H+ + 8e-  NH4+ + 3H2O(l)
We can calculate the Eh if the activities of H+, NO3-, and NH4+ are known. The general Nernst equation is
The Nernst equation for this reaction at 25°C is
The Eh or pe of a natural water can be calculated if we know the activities of species involved in a half reaction. In the case illustrated, we will use the nitrate (NO3-)-ammonium(NH4+) redox couple to calculate Eh. First, we must write a correctly balanced half reaction involving these species. Next, we use an equation called the Nernst equation. This equation relates the ion activity product (IAP) for the half reaction and the standard electrode potential for the half-reaction (E0) to the Eh of the solution. The general form of the Nernst equation is given as the first equation above; this form can be applied to any half reaction. The second equation gives the Nernst equation specific to this reaction at 25°C. Note that, at 25°C, the collection of constants 2.303RT/ = In the Nernst equation, n refers to the number of electrons transferred.
WARNING: When using the Nernst equation, you need to be careful with regard to conventions. The forms of the Nernst equation and the equation relating E0 and Gr° used by Kehew (2001) and in my lecture notes are correct for reactions written as reductions (i.e., electrons on the left side of the half reaction). Different sign conventions apply if we write the reactions as oxidation reactions. Note that, the convention employed by Kehew (2001) is just the opposite of the convention employed by the textbooks used in my GEOL 423 and GEOL 478/578 classes. To avoid confusion and errors, when you are using the Nernst equation in this class, ALWAYS WRITE YOUR HALF-CELL REACTIONS AS REDUCTIONS WITH THE ELECTRONS ON THE LEFT SIDE OF THE HALF REACTION!!!!

2. First, we must make use of the relationship
Let’s assume that the concentrations of NO3- and NH4+ have been measured to be 10-5 M and 310-7 M, respectively, and pH = 5. What are the Eh and pe of this water?
First, we must make use of the relationship
For the reaction of interest
rG° = 3(-237.1) + (-79.4) - (-110.8)
= kJ mol-1
Let us now assume that we have measured the pH and the concentrations of NO3- and NH4+ in our natural water sample, and we obtained the values shown above. We can now plug these into the Nernst equation to get an estimate of the Eh of our water sample (assuming that the half reaction has attained equilibrium and is controlling the solution Eh). However, before we can do this, we must obtain the value of E0. For some simple half reaction, the standard electrode potential is available in reference tables in many chemical handbooks. However, in most cases we have to calculate E0 from Gr° using the first equation above, keeping in mind that Gf° = 0 for H+ and e-.

3. The Nernst equation now becomes
substituting the known concentrations (neglecting activity coefficients)
and
Now, with our calculated value of E0, and the given values of pH and concentrations of NH4+ and NO3-, we plug everything into the Nernst equation and obtain an estimate of Eh. Note that we have assumed activity coefficients are unity. The Eh estimate can readily be converted to a pe value using the equation previously introduced. As we will see, in natural waters, Eh values calculated from redox couples and the Nernst equation may be more accurate and more relevant than Eh values measured directly with a platinum electrode.

4. Reaction directions for 2 different redox couples brought together??
Biology’s view  upside down?
Reaction directions for 2 different redox couples brought together??
More negative potential  reductant // More positive potential  oxidant
Example – O2/H2O vs. Fe3+/Fe2+  O2 oxidizes Fe2+ is spontaneous!

5. Stability Limits of Water
H2O  2 H+ + ½ O2(g) + 2e-
Using the Nernst Equation:
Must assign 1 value to plot in x-y space (PO2)
Then define a line in pH – Eh space

6. UPPER STABILITY LIMIT OF WATER (Eh-pH)
To determine the upper limit on an Eh-pH diagram, we start with the same reaction
1/2O2(g) + 2e- + 2H+  H2O
but now we employ the Nernst eq.
In the next three slides, we repeat the calculation of the stability limits for water, but this time in Eh-pH coordinates. There are two reasons for repeating the calculation in this way. The first is to show how pe-pH diagrams and Eh-pH diagrams relate to each other, and to show the slight differences in how they are calculated. The second reason is so that we can use a published Eh-pH diagram to show where different geological environments sit with respect to pH.
The reaction governing the upper stability limit for water on an Eh-pH diagram is exactly the same reaction that we used for the pe-pH diagram. The difference is that, instead of using the mass-action (equilibrium constant) expression, we use the Nernst equation. To employ the Nernst equation given the convention Kehew (2001) has adopted, we must right the reaction as a reduction reaction.
Once the appropriate Nernst equation is written, we can proceed to the next step. Note that, in employing the Nernst equation, the convention employed is that ae- = 1. This is a major difference compared to the pe-pH diagram, where ae- is not necessarily equal to 1. The pe = -log ae-, so if we set ae- = 1 we could not plot a pe-pH diagram, because all pe values would be zero. However, such a situation would make many students happy!

7. As for the pe-pH diagram, we assume that pO2 = 1 atm. This results in
This yields a line with slope of
We need to calculate E0 from the value of Gr° as shown above, and then rearrange the Nernst equation so that we have an equation of a straight line with Eh on the Y-axis and pH on the X-axis. We find that we still have to fix the value of the partial pressure of oxygen to plot the boundary, and so we choose 1 atm as for the pe-pH diagram.

8. LOWER STABILITY LIMIT OF WATER (Eh-pH)
Starting with
H+ + e-  1/2H2(g)
we write the Nernst equation
We set pH2 = 1 atm. Also, Gr° = 0, so E0 = 0. Thus, we have
Again, the lower limit of water solubility in Eh-pH space is governed by the same reaction as in pe-pH space. We repeat the same steps as before, except that for this reaction, E0 = 0, because all the reactants and products have free energies of formation that are zero by convention.

10. O2/H2O
C2HO

11. Making stability diagrams
For any reaction we wish to consider, we can write a mass action equation for that reaction
We make 2-axis diagrams to represent how several reactions change with respect to 2 variables (the axes)
Common examples: Eh-pH, PO2-pH, T-[x], [x]-[y], [x]/[y]-[z], etc

12. Construction of these diagrams
For selected reactions:
Fe H2O  FeOOH + e- + 3 H+
How would we describe this reaction on a 2-D diagram? What would we need to define or assume?

13. How about:
Fe H2O  FeOOH(ferrihydrite) + 3 H+
Ksp=[H+]3/[Fe3+]
log K=3 pH – log[Fe3+]
How would one put this on an Eh-pH diagram, could it go into any other type of diagram (what other factors affect this equilibrium description???)

14. Redox titrations
Imagine an oxic water being reduced to become an anoxic water
We can change the Eh of a solution by adding reductant or oxidant just like we can change pH by adding an acid or base
Just as pK determined which conjugate acid-base pair would buffer pH, pe determines what redox pair will buffer Eh (and thus be reduced/oxidized themselves)

15. Redox titration II
Let’s modify a bjerrum plot to reflect pe changes