Redox equilibria

Every battery in your phone or car is a pair of redox half-reactions kept apart so that the electrons must travel through a wire. This topic shows how standard electrode potentials measure the pull of each half-reaction for electrons, how they predict which reactions can happen, and how redox titrations put oxidation and reduction to work in quantitative analysis.

Electrochemical cells

A half-cell is an element in two oxidation states in contact with each other, such as a strip of zinc dipping into a solution of zinc ions. Each half-cell sets up its own equilibrium, with the reduced form on the right by convention:

Zn²⁺(aq) + 2e⁻ ⇌ Zn(s)      Cu²⁺(aq) + 2e⁻ ⇌ Cu(s)

Zinc has a greater tendency than copper to lose electrons and form ions, so more electrons build up on the zinc strip. Join the two half-cells and there is a potential difference between them: the zinc is the negative electrode and the copper the positive one. For standard Zn and Cu half-cells it is 1.10 V.

• Salt bridge: completes the circuit by letting ions move between the two solutions, without mixing them. It is usually filter paper soaked in saturated potassium nitrate, because K⁺ and NO₃⁻ do not react with the electrodes or solutions or form precipitates. (Potassium chloride would be unsuitable with silver or copper ions.) A metal wire cannot be used, because it would set up half-cells of its own.
• High-resistance voltmeter: draws almost no current, so the reactions do not proceed and the measured voltage is the maximum potential difference, the cell e.m.f.
• When current flows (a bulb instead of a voltmeter): the more positive electrode undergoes reduction (Cu²⁺ + 2e⁻ → Cu), the more negative one undergoes oxidation (Zn → Zn²⁺ + 2e⁻), electrons flow through the wire from negative to positive, and the voltage gradually falls as the reactants are used up.

Cell diagrams

A cell is written in shorthand, with a single vertical line for a boundary between phases and a double line for the salt bridge. The oxidised forms are written next to the salt bridge, and by convention the positive electrode is on the right:

Zn(s) | Zn²⁺(aq) || Cu²⁺(aq) | Cu(s)    E_cell = +1.10 V

• No metal in the half-reaction? Use an inert platinum electrode, which conducts electrons but does not react. For Fe³⁺ + e⁻ ⇌ Fe²⁺ the half-cell is || Fe³⁺(aq), Fe²⁺(aq) | Pt, with a comma between species in the same phase and both ions at 1.00 mol dm⁻³.
• Gases also need platinum: || Cl₂(g) | Cl⁻(aq) | Pt. H⁺ and H₂O that only balance a half-equation are usually left out: || MnO₄⁻(aq), Mn²⁺(aq) | Pt.

Standard electrode potentials

You cannot measure the potential of one half-cell on its own, only the difference between two. So every half-cell is measured against the same reference, the standard hydrogen electrode (SHE), whose potential is defined as exactly 0 V:

Pt | H₂(g) | H⁺(aq)     2H⁺(aq) + 2e⁻ ⇌ H₂(g)

• The SHE: hydrogen gas at 100 kPa bubbling over a platinum electrode coated in finely divided platinum (platinum black, which catalyses the equilibrium and adsorbs the gas), dipping into a solution with [H⁺] = 1.00 mol dm⁻³ (usually 1.00 mol dm⁻³ HCl), at 298 K.
• Standard electrode potential, E^⦵: the e.m.f. of a cell made from the half-cell and a standard hydrogen electrode, under standard conditions: 298 K, all solutions at 1.00 mol dm⁻³, gases at 100 kPa, and no current flowing.
• Why standard conditions? Each half-cell is an equilibrium, so its potential changes with concentration and temperature. Adding more Zn²⁺ pushes Zn²⁺ + 2e⁻ ⇌ Zn to the right, making its potential more positive.
• Secondary standards: the SHE is awkward to use, so ordinary measurements use electrodes calibrated against it, such as the silver/silver chloride electrode (+0.22 V) or the calomel electrode (+0.27 V).

The more negative E^⦵, the more readily the right-hand species loses electrons: the strongest reducing agents (Li, Mg) are at the top right. The more positive E^⦵, the more readily the left-hand species gains electrons: the strongest oxidising agents (F₂, MnO₄⁻) are at the bottom left.

Cell e.m.f. and feasibility

\[ E^{\ominus}_{\text{cell}} = E^{\ominus}_{\text{right}} - E^{\ominus}_{\text{left}} = E^{\ominus}_{\text{reduced}} - E^{\ominus}_{\text{oxidised}} \]

In a working cell, the half-cell with the more positive E^⦵ goes forwards (reduction) and the one with the more negative E^⦵ goes backwards (oxidation). A reaction is feasible (thermodynamically possible) when E^⦵_cell for it is positive.

Disproportionation

In disproportionation one species is both oxidised and reduced. Electrode potentials predict it: the species must appear on the left of one half-equation and the right of another with a less positive E^⦵.

• Copper(I): Cu⁺ + e⁻ ⇌ Cu (+0.52 V) and Cu²⁺ + e⁻ ⇌ Cu⁺ (+0.15 V). E^⦵_cell = 0.52 − 0.15 = +0.37 V, so in aqueous solution 2Cu⁺ → Cu + Cu²⁺: copper goes from +1 to 0 and to +2. This is why copper(I) compounds such as Cu₂O give copper metal and a blue solution with dilute sulfuric acid.
• Hydrogen peroxide: H₂O₂ + 2H⁺ + 2e⁻ ⇌ 2H₂O (+1.77 V) and O₂ + 2H⁺ + 2e⁻ ⇌ H₂O₂ (+0.68 V). E^⦵_cell = +1.09 V, so 2H₂O₂ → 2H₂O + O₂, oxygen going from −1 to −2 and 0.

E_cell, entropy and the equilibrium constant

E^⦵_cell is directly proportional to the total entropy change, and to ln K, for the cell reaction:

\[ E^{\ominus}_{\text{cell}} \propto \Delta S_{\text{total}} \propto \ln K \]

So a positive E^⦵_cell means a positive ΔS_total and K greater than 1, and the larger E^⦵_cell is, the further the reaction goes towards completion.

Limitations of predictions from E^⦵

• Kinetics: a positive E^⦵_cell says a reaction can happen, not that it will be fast. If the activation energy is high it may be too slow to observe; copper does not react with water and oxygen at a noticeable rate, for example.
• Non-standard conditions: real concentrations, temperatures and pressures shift the electrode potentials (Le Chatelier applied to each half-equation). Increasing the concentration of a reactant in the cell reaction, or decreasing a product, increases E_cell; most cell reactions are exothermic, so raising the temperature usually lowers it. When E^⦵_cell is small (say under about 0.3 V), the change can even reverse the prediction.
• Current flowing: once a cell is used, the reactants are consumed and E_cell falls to zero when the cell reaction reaches equilibrium.

Fuel cells

A fuel cell converts the energy of a fuel reacting with oxygen directly into electricity. Unlike an ordinary battery, the reactants are fed in continuously, so the voltage stays constant instead of falling as the reactants are used up.

• Advantages over petrol or diesel engines: the only product is water, so no CO₂ or other pollutants are released where the fuel is used, and they convert energy more efficiently.
• Limitations: hydrogen is a flammable gas that is hard to store and transport (as a liquid under pressure, or adsorbed onto or absorbed into solids); fuel cells are expensive and have limited lifetimes; and most hydrogen is made from fossil fuels or by electrolysis, so it is only "green" if that electricity comes from renewable sources.
• Ethanol fuel cells avoid the storage problem: ethanol is a liquid that can be made from crops. At the negative electrode C₂H₅OH + 3H₂O → 2CO₂ + 12H⁺ + 12e⁻; overall C₂H₅OH + 3O₂ → 2CO₂ + 3H₂O. Methanol can be used in the same way.

Redox titrations with potassium manganate(VII)

Acidified potassium manganate(VII) is a powerful oxidising agent that acts as its own indicator: purple MnO₄⁻ is reduced to almost colourless Mn²⁺. Put it in the burette; the end point is the first permanent pale pink colour, showing a tiny excess of MnO₄⁻. Read the burette from the top of the meniscus, because the dark purple solution hides the bottom.

MnO₄⁻(aq) + 8H⁺(aq) + 5Fe²⁺(aq) → Mn²⁺(aq) + 4H₂O(l) + 5Fe³⁺(aq)

• Hydrogen peroxide: 2MnO₄⁻ + 6H⁺ + 5H₂O₂ → 2Mn²⁺ + 5O₂ + 8H₂O (ratio 2 : 5).
• Ethanedioate: 2MnO₄⁻ + 16H⁺ + 5C₂O₄²⁻ → 2Mn²⁺ + 10CO₂ + 8H₂O (ratio 2 : 5). This reaction between two negative ions starts slowly, so the flask is warmed to about 60 °C; the Mn²⁺ formed then catalyses it.
• Iron(II) ethanedioate, FeC₂O₄: both ions are oxidised. One FeC₂O₄ gives up 1 + 2 = 3 electrons and each MnO₄⁻ takes 5, so the ratio is 3MnO₄⁻ : 5FeC₂O₄.
• For any other redox titration, write the two half-equations, balance the electrons, and use the overall ratio.

Iodine–thiosulfate titrations

Iodine reacts quickly and completely with sodium thiosulfate, which is used to measure how much iodine a reaction has made:

I₂(aq) + 2S₂O₃²⁻(aq) → 2I⁻(aq) + S₄O₆²⁻(aq)

This lets you find the amount of almost any oxidising agent: add it to excess potassium iodide, so it oxidises I⁻ to I₂, then titrate that iodine with standard thiosulfate.

• Starch indicator: the iodine solution starts yellow-brown and fades to pale yellow as thiosulfate is added. Only then add a little starch, which forms an intense blue-black complex with the remaining iodine. The end point is when the blue-black colour disappears, leaving a colourless solution.
• Why add starch late? With a lot of iodine present the starch–iodine complex is insoluble and does not release its iodine readily, so the end point would be inaccurate.

Uncertainty in titrations

• A reading is one judgement on a scale; a measurement is the difference between two readings. A titre is a measurement, so a burette with ±0.05 cm³ per reading gives ±0.10 cm³ on each titre.
• \( \text{percentage uncertainty} = \dfrac{\text{uncertainty}}{\text{quantity measured}} \times 100 \). For a 24.60 cm³ titre: 0.10 ÷ 24.60 × 100 = 0.41%. Add the percentage uncertainties of every piece of apparatus to estimate the total.
• Reducing it: measure a larger quantity (a bigger titre, by using a more concentrated solution in the flask or a more dilute one in the burette), or use apparatus with a smaller uncertainty, such as a pipette instead of a measuring cylinder.
• Comparing with an accepted value: percentage difference = (difference ÷ accepted value) × 100. If it is larger than the total apparatus uncertainty, there must be other errors in the method; if it is smaller, the apparatus uncertainty explains the gap.

Quick checks

Interactive diagrams