L6 - Nernst Equation and Eh-pH Diagrams

Redox Reactions Continued

Review of Redox Reactions

• Recap of yesterday's topics:

  • Introduction to redox reactions.

  • Oxidation number: calculation and significance.

  • Balancing redox reactions.

  • Potential or electromotive force (EMF): measure of electrochemical energy in redox reactions.

    • Related to Gibbs free energy.

    • Convenient way to measure energy flow associated with electron transfer.

  • Standard hydrogen potential: potential of a half-reaction coupled with the standard hydrogen electrode.

  • Redox reactions in natural environments.

Today's Focus

• Introducing the Nernst equation: its derivation, meaning, and usage.

• Eh-pH diagrams: their utility in aqueous geochemistry.

  • Analogous to mineral stability diagrams (covered last week).

  • Two-dimensional diagrams with Eh (redox potential) as one axis and pH as the other.

The Nernst Equation

• Derivation:

  • Based on known equations.

  • Relationship between Gibbs free energy and potential (from yesterday):

    • $\Delta G = -nFE$

  • Under standard conditions (25°C, 1 bar, activities = 1):

    • $\Delta G^0 = -nFE^0$

  • Relationship between free energy, standard free energy, and reaction quotient (from earlier lectures):

    • $\Delta G = \Delta G^0 + RT \ln{Q}$

  • Combining these equations gives the Nernst equation.

• Nernst Equation:

  • $E = E^0 - \frac{RT}{nF} \ln{Q}$

• Simplified version (at 25°C, converting to base-10 logarithm):

  • $E = E^0 - \frac{0.0592}{n} \log_{10}{Q}$

Uses of the Nernst Equation

• Calculates the potential of a redox reaction under non-standard conditions.

• Example: Reduction of iron(III) to iron(II) by hydrogen gas.

  • $Fe^{3+} + \frac{1}{2}H_2 \rightleftharpoons Fe^{2+} + H^{+}$

  • Calculate the standard potential ($E^0$) from standard free energies of formation.

    • $E^0$ (for this reaction) = $0.77\text{ V}$.

  • Nernst equation:

    • $E = E^0 - \frac{RT}{nF} \ln{Q}$

    • $E=E^0-\frac{RT}{nF}\ln{\frac{a_{Fe^{2+}}\cdot a_{H^+}}{a_{Fe^{3+}}\cdot\sqrt{P_{H_2}}}}$

    • If you measure concentration of iron, pH, and hydrogen partial pressure, then you can estimate the potential.

Standard Hydrogen Potential

• Potential of a redox couple (half-reaction) measured against the standard hydrogen electrode.

• Standard hydrogen electrode: solution with protons at 1 mol/L and hydrogen gas at 1 atm.

• For the $Fe^{3+}/Fe^{2+}$ couple:

  • $E_{H}=E_{H}^0-\frac{RT}{nF}\ln{\frac{a_{Fe^{3+}}}{a_{Fe^{2+}}}}$

    • Subscript H indicates measurement against standard hydrogen electrode.

• Generalized form:

  • $E_{H} = E^0 + \frac{RT}{nF} \ln{\frac{\text{activity product of oxidized species}}{\text{activity product of reduced species}}}$

  • $a_{e^-}$ is equal to 1.

  • $E_{H} = E^0 + \frac{0.0592}{n} \log{\frac{\text{activity product of oxidized species}}{\text{activity product of reduced species}}}$

  • Note that the fraction in the log has been flipped because of the sign change.

• Example application:

  • Reducing environment (e.g., sediments): Measure ${E}_{H} = -0.2 V$

    

    • Means massively more iron(II) than iron(III).

  • Oxidizing environment (e.g., surface seawater): Measure $\text{E}_{H} = 0.8 V$

    

    • Means more iron(III) than iron(II).

Eh-pH Diagrams

• Eh and pH are important environmental variables.

  • pH: Controls proton concentration; important for acid-base reactions, carbonate systems, mineral precipitation/dissolution.

  • Eh: Indicates how oxidized or reducing the environment is; controls distribution and concentration of redox-sensitive elements.

• Construction:

  • Similar to mineral stability diagrams.

  • Lines represent equilibrium between species.

  • Diagrams are drawn for a specific pressure and temperature (typically 25°C and 1 bar).

  • Always include stability limits of water.

    • Water is stable within a certain range of Eh and pH conditions.

    • Limits are determined by the oxidation of water to oxygen.

  • Upper limit:

    • $H_2O \rightleftharpoons \frac{1}{2}O_2 + 2H^+ + 2e^-$

    • The upper limit is constrained by the oxidation of water to dioxygen

• Stability limits of water can be represented by equation. The upper and lower limits of water in the environment are:

  • Upper Limit: $E^{Upper}_{H,H_2O}=1.23-0.0592pH$

  • Lower Limit: $E^{Lower}_{H,H_2O} = -0.0592pH$

Example: Eh-pH Diagram for Iron-Oxygen-Water System

• System: Iron, oxygen, and water at 25°C and 1 bar.

• Minerals considered: Magnetite ($Fe3O4$) and Hematite ($Fe2O3$).

• Dissolved species: $Fe^{2+}$ and $Fe^{3+}$.

• Total activity of the dissolved Fe species: $a_{Fe}=10^{-6}$

• Gibbs energy of formation for all species:

  

• Assume a total activity of dissolved iron species (e.g., $10^{-6}$ mol/L).

Steps to Build the Diagram

• Boundary between $Fe^{2+}$ and $Fe^{3+}$

• Oxidation numbers are different, so it's a redox reaction.

  1. Half-reaction: $Fe^{3+} + e^- \rightleftharpoons Fe^{2+}$

  2. Calculate standard free energy of the reaction:

     • $\Delta G^0 = \Delta G_f^0(Fe^{2+}) - \Delta G_f^0(Fe^{3+}) = -73.3 \text{ kJ/mol}$

  3. Calculate standard potential:

     • $E_H^0 = -\frac{\Delta G^0}{nF} = \frac{73300 \text{ J/mol}}{(1)(96485 \text{ C/mol})} = 0.76 \text{ V}$

  4. Apply the Nernst equation:

     • $E_H = E_H^0 + \frac{0.0592}{n} \log{\frac{a_{Fe^{3+}}}{a_{Fe^{2+}}}}$

     • Looking at this equation, we can see that

  5. Boundary condition: activities of $Fe^{3+}$ and $Fe^{2+}$ are equal, so the log term equals 0

     • $E=0.76\text{ V}$

     • Horizontal line on the Eh-pH diagram.

  6. $Fe^{3+}$ stable above the line (more oxidizing conditions), $Fe^{2+}$ stable below.

• Boundary between Magnetite and Hematite

  • Oxidation numbers of iron are different, so it's a redox reaction.

  • Balanced redox reaction:

    • $3Fe_2O_3 + 2H^+ + 2e^- \rightleftharpoons 2Fe_3O_4 + H_2O$

  • Calculate standard free energy of reaction using standard free energy of formations.

  • Calculate the standard potential: $E_H^0 = 0.152 \text{ V}$

  • Apply the Nernst equation:

    • $E_H = E_H^0 + \frac{0.0592}{n} \log{a_{H^+}^2}$

    • $E = 0.152 - 0.0592pH$

  • Plot line: y-intercept = 0.152, negative slope with pH.

  • Hematite is more oxidized, so it's above the line; magnetite is below.

• Boundary between $Fe^{3+}$ and Hematite

  • Oxidation number of iron is the same in both species (Fe(III)), so it's not a redox reaction. This means that the reaction is independent of $E_H$

  • Mineral dissolution reaction: $Fe_2O_3 + 6H^+ \rightleftharpoons 2Fe^{3+} + 3H_2O$

  • Calculate the free energy using $\Delta G_r^0=-RT \ln{K}$ and rearranging for K.

  • Write law of mass action:

    • $K = \frac{a_{Fe^{3+}}^2}{a_{H^+}^6}$

    • $log(K) = 2log(a_{Fe^{3+}}) + 6pH$

  • Boundary condition: activity of $Fe^{3+}$ equals total dissolved iron activity.

  • Solve for pH to plot the vertical line.

  • Hematite stable at high Ph.

• Boundary between $Fe^{2+}$ and Hematite

  • Redox or not?

  • Apply Nernst equation.

  • Remember that at the boundary I have this boundary condition where the activity of the solute is 10 to the -6.

Uses of Eh-pH Diagrams

• Predict what is likely to happen in a system.

• Determine conditions in the geological record.