Redox chemistry: reduction potentials, Latimer and Frost diagrams

Manganese can be found in compounds in every oxidation state from +2 to +7. Purple permanganate, MnO₄⁻, is a powerful oxidizing agent used to treat drinking water; brown manganese dioxide, MnO₂, is the cathode of ordinary alkaline batteries; pale pink Mn²⁺ is the form found in most minerals and in living cells. Which of these states is stable in water, and which will oxidize or reduce something else, or simply fall apart into two other states? Inorganic chemists answer such questions with a few compact diagrams built from standard reduction potentials. This lesson reviews the thermodynamics of redox reactions from CHEM-102, shows how potentials combine, and introduces the Latimer, Frost and Pourbaix diagrams that summarize the redox chemistry of an element on a single page.

Reduction potentials and free energy

A standard reduction potential E° measures the tendency of a half-reaction to proceed as a reduction relative to the standard hydrogen electrode, 2H⁺(aq) + 2e⁻ → H₂(g), which is assigned E° = 0. The link to thermodynamics is

where n is the number of electrons transferred and F = 96 485 C/mol is the Faraday constant. A positive E° means a negative ΔG°, a spontaneous reduction. A strong oxidizing agent has a large positive E° (F₂/F⁻ +2.87 V, MnO₄⁻/Mn²⁺ +1.51 V); a strong reducing agent has a large negative one (Li⁺/Li −3.04 V, Na⁺/Na −2.71 V). The Nernst equation gives the potential under non-standard conditions:

For half-reactions involving H⁺ or OH⁻, Q contains [H⁺], so the potential depends on pH. For MnO₄⁻ + 8H⁺ + 5e⁻ → Mn²⁺ + 4H₂O, the potential falls by 8 × 0.05916/5 = 0.095 V per pH unit when the other concentrations are at their standard values. Permanganate is a much stronger oxidizing agent in acid than in neutral or basic solution, where it is reduced only to MnO₂.

Combining potentials

Potentials are intensive: they do not add directly. To find the potential of a half-reaction from two others, combine the free energies, which are extensive, and divide by the total number of electrons:

Latimer diagrams

Wendell Latimer, in his 1938 book on oxidation potentials, introduced a compact way to list the potentials of an element. The species are written in order of decreasing oxidation state from left to right, with the standard potential for each step written above the arrow joining them. For manganese in acid solution:

Each arrow stands for a balanced half-reaction, with H⁺ and H₂O added as needed; for example, MnO₂ + 4H⁺ + e⁻ → Mn³⁺ + 2H₂O for the +0.95 V step. Potentials for non-adjacent species are found by the combination rule: from MnO₄⁻ to Mn²⁺ (5 electrons), E° = (0.56 + 2 × 2.26 + 0.95 + 1.51)/5 = 7.54/5 = 1.51 V; from MnO₄⁻ to MnO₂ (3 electrons), E° = (0.56 + 4.52)/3 = 1.69 V.

A Latimer diagram reveals disproportionation at a glance. A species disproportionates (reacts with itself to give a higher and a lower oxidation state) when the potential on its right is greater than the potential on its left. For Mn³⁺, the potential to its right (+1.51 V, reduction to Mn²⁺) is greater than that on its left (+0.95 V, the reduction of MnO₂ to Mn³⁺), so 2Mn³⁺ + 2H₂O → MnO₂ + Mn²⁺ + 4H⁺ is spontaneous: E° = 1.51 − 0.95 = +0.56 V. Manganate, MnO₄²⁻, likewise disproportionates in acid (2.26 > 0.56) to MnO₄⁻ and MnO₂, which is why it is stable only in strongly basic solution. The reverse process, in which a higher and a lower state react to form an intermediate one, is called comproportionation.

Frost diagrams

Arthur Frost proposed in 1951 a graphical form that makes these relationships even clearer. A Frost diagram plots nE° (the free energy of a species relative to the element, in volt-equivalents, equal to −ΔG°/F) against oxidation state N. Each point is calculated by adding the nE° contributions from the element up to that species. For manganese, starting from Mn (0, 0): Mn²⁺ is at 2 × (−1.18) = −2.36 V; Mn³⁺ at −2.36 + 1.51 = −0.85; MnO₂ at −0.85 + 0.95 = +0.10; MnO₄²⁻ at 0.10 + 2 × 2.26 = 4.62; and MnO₄⁻ at 4.62 + 0.56 = 5.18 V.

Reading a Frost diagram takes a few rules:

1. The lowest point is the most thermodynamically stable oxidation state. For manganese in acid it is Mn²⁺, which is why manganese is found as Mn²⁺ in many minerals and natural waters.

2. The slope of the line joining two points equals the standard potential of the couple. A steep positive slope (as from MnO₂ to MnO₄²⁻) means the higher state is a strong oxidizing agent.

3. A species that lies above the line joining its two neighbors is unstable with respect to disproportionation into them. Mn³⁺ and MnO₄²⁻ are such species.

4. A species that lies below the line joining its neighbors is stable toward disproportionation, and those neighbors will comproportionate to form it. MnO₂ lies below the line from Mn³⁺ to MnO₄²⁻.

5. Comparing diagrams for acid and base shows how pH changes the picture. In basic solution, the high oxidation states of manganese are less strongly oxidizing and MnO₂ and Mn(OH)₂ become important.

Disproportionation in practice: chlorine bleach

The Latimer diagram for chlorine in basic solution explains the chemistry of household bleach:

Chlorine has a larger potential on its right (+1.36) than on its left (+0.42), so in base it disproportionates: Cl₂ + 2OH⁻ → Cl⁻ + ClO⁻ + H₂O. This is how sodium hypochlorite bleach is made, by passing chlorine into cold sodium hydroxide. Hypochlorite is itself unstable toward further disproportionation, slowly at room temperature and faster when heated: 3ClO⁻ → 2Cl⁻ + ClO₃⁻. That is why bleach loses strength in a warm cupboard and why hot alkali converts chlorine to chlorate. In acid the situation reverses: the potential for HClO → Cl₂ (+1.63) exceeds that for Cl₂ → Cl⁻ (+1.36), so HClO and Cl⁻ comproportionate to chlorine, which is why mixing bleach with an acidic cleaner releases toxic chlorine gas.

A worked computation

The steps from a Latimer diagram to a Frost diagram, and the tests for disproportionation, are easy to automate. The code below builds the Frost diagram for manganese and for copper, flags any species that lies above the line joining its neighbors, and computes the pH dependence of the permanganate potential.

# Latimer -> Frost conversion and disproportionation tests (acid solution, 25 C).
def frost(latimer):
    """latimer: list of (species, oxidation state) from high to low, and potentials between them.
    Returns points (N, nE) with the element (N = 0) at nE = 0."""
    species, potentials = latimer
    pts = {species[-1][0]: (species[-1][1], 0.0)}
    nE = 0.0
    for i in range(len(species) - 2, -1, -1):
        (name, N), (_, N_low) = species[i], species[i + 1]
        nE += (N - N_low) * potentials[i]
        pts[name] = (N, nE)
    return pts

def report(title, latimer):
    pts = frost(latimer)
    print(title)
    for name, (N, nE) in sorted(pts.items(), key=lambda kv: kv[1][0]):
        print(f'   {name:7} N = {N:+d}   nE = {nE:+6.2f} V')
    order = sorted(pts.items(), key=lambda kv: kv[1][0])
    for (a, (Na, Ea)), (b, (Nb, Eb)), (c, (Nc, Ec)) in zip(order, order[1:], order[2:]):
        line = Ea + (Ec - Ea) * (Nb - Na) / (Nc - Na)
        verdict = 'disproportionates' if Eb > line else 'stable to disproportionation'
        print(f'   {b:7}: {Eb:+.2f} V vs line {line:+.2f} V -> {verdict}')
    lowest = min(pts.items(), key=lambda kv: kv[1][1])[0]
    print('   most stable species:', lowest)

mn = ([('MnO4-', 7), ('MnO4 2-', 6), ('MnO2', 4), ('Mn3+', 3), ('Mn2+', 2), ('Mn', 0)],
      [0.56, 2.26, 0.95, 1.51, -1.18])
cu = ([('Cu2+', 2), ('Cu+', 1), ('Cu', 0)], [0.153, 0.521])
report('Manganese, acid', mn)
report('Copper, acid', cu)

pts = frost(mn)
E_mno4_mn2 = (pts['MnO4-'][1] - pts['Mn2+'][1]) / 5
print(f'\nE(MnO4-/Mn2+) from the diagram: {E_mno4_mn2:.3f} V')
for pH in (0, 3, 7):
    E = E_mno4_mn2 - 0.05916 * 8 / 5 * pH
    print(f'   at pH {pH}: E = {E:.2f} V')

Manganese, acid
   Mn      N = +0   nE =  +0.00 V
   Mn2+    N = +2   nE =  -2.36 V
   Mn3+    N = +3   nE =  -0.85 V
   MnO2    N = +4   nE =  +0.10 V
   MnO4 2- N = +6   nE =  +4.62 V
   MnO4-   N = +7   nE =  +5.18 V
   Mn2+   : -2.36 V vs line -0.57 V -> stable to disproportionation
   Mn3+   : -0.85 V vs line -1.13 V -> disproportionates
   MnO2   : +0.10 V vs line +0.97 V -> stable to disproportionation
   MnO4 2-: +4.62 V vs line +3.49 V -> disproportionates
   most stable species: Mn2+
Copper, acid
   Cu      N = +0   nE =  +0.00 V
   Cu+     N = +1   nE =  +0.52 V
   Cu2+    N = +2   nE =  +0.67 V
   Cu+    : +0.52 V vs line +0.34 V -> disproportionates
   most stable species: Cu

E(MnO4-/Mn2+) from the diagram: 1.508 V
   at pH 0: E = 1.51 V
   at pH 3: E = 1.22 V
   at pH 7: E = 0.85 V

The copper result explains a familiar observation: Cu⁺ disproportionates in water to Cu²⁺ and copper metal (E° = 0.521 − 0.153 = +0.37 V). Copper(I) survives only when it is stabilized by precipitation (CuCl, CuI), by complexation with soft ligands such as cyanide or by a non-aqueous solvent such as acetonitrile, a nice application of the HSAB principle of lesson 8.

The stability field of water

Water itself can be oxidized and reduced, which sets limits on what can exist in it. At 25 °C, O₂ + 4H⁺ + 4e⁻ → 2H₂O has E = 1.229 − 0.0592 pH V, and 2H⁺ + 2e⁻ → H₂ has E = −0.0592 pH V. Any species with a reduction potential above the first line can, in principle, oxidize water to oxygen; any species with a potential below the second can reduce water to hydrogen. Fluorine, cobalt(III) aqua ions and permanganate in acid are thermodynamically able to oxidize water, and the alkali metals reduce it vigorously. Many reactions that are thermodynamically allowed are slow, however, because electron transfer to and from water needs a large overpotential, typically 0.4 to 0.6 V at many electrodes. This is why acidic permanganate solutions can be kept for weeks, and why aqueous batteries can operate at voltages somewhat above the 1.23 V window.

Pourbaix diagrams

Marcel Pourbaix extended these ideas in the 1940s into potential–pH diagrams, now called Pourbaix diagrams, which show the most stable form of an element over a range of potential and pH. Lines mark the boundaries between regions where one species dominates: horizontal lines for electron transfers that do not involve protons, vertical lines for acid–base reactions with no electron transfer, and sloping lines for reactions involving both. Pourbaix diagrams for iron show the conditions under which steel corrodes (Fe²⁺ or Fe³⁺ regions), stays passive under an oxide film (Fe₂O₃ and Fe₃O₄ regions) or is immune (Fe metal). They are the standard tool of corrosion engineers and of geochemists studying how metals move in natural waters.

Redox in metal extraction

The same thermodynamics, applied at high temperature, governs how metals are won from their ores. Harold Ellingham plotted in 1944 the standard free energy of formation of oxides against temperature. A metal can reduce the oxide of any metal whose line lies above its own on the diagram, and carbon, whose CO line slopes downward (because 2C + O₂ → 2CO increases the amount of gas), becomes a better reducing agent as the temperature rises. This is why iron oxides are reduced by coke and carbon monoxide in a blast furnace at about 1500 °C, while aluminum oxide, whose line lies far below carbon's at any practical temperature, must be reduced electrolytically in molten cryolite by the Hall–Héroult process, invented independently by Charles Martin Hall and Paul Héroult in 1886. Magnesium, titanium and the alkali metals likewise need electrolysis or reduction by an even more reactive metal.