5.3 Electrochemical Cells, Reduction Potentials & the Nernst Equation

5.3 Electrochemical Cells, Reduction Potentials & the Nernst Equation

Electrochemistry analyzes chemical-electrical energy interconversion in redox reactions.

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Oxidation-Reduction Fundamentals and Half-Reaction Balancing

Redox reactions involve electron transfer:

• Oxidation: Loss of electrons ($e^-$); oxidation state increases.
• Reduction: Gain of electrons ($e^-$); oxidation state decreases.
• Mnemonics: OIL RIG (Oxidation Is Loss, Reduction Is Gain) and LEO says GER (Lose Electrons Oxidation, Gain Electrons Reduction).
• Oxidizing Agent (Oxidant): Accepts electrons; becomes reduced.
• Reducing Agent (Reductant): Donates electrons; becomes oxidized.

Oxidation State Assignment Rules

1. Free elements = 0 ($\text{Na}, \text{O}_2, \text{Fe}$).
2. Monatomic ions = ionic charge ($\text{Fe}^{3+} = +3, \text{Cl}^- = -1$).
3. Group 1 = +1; Group 2 = +2; Fluorine = -1.
4. Hydrogen = +1 with nonmetals, -1 with metal hydrides ($\text{NaH}$).
5. Oxygen = -2 in oxides, -1 in peroxides ($\text{H}_2\text{O}_2$), +2 in $\text{OF}_2$.
6. Sum of oxidation numbers equals net species charge.

Half-Reaction Balancing Method

1. Separate oxidation and reduction half-reactions.
2. Balance elements other than H and O.
3. Balance O using $\text{H}_2\text{O}$; balance H using $\text{H}^+$.
4. Balance charge adding $e^-$ to the more positive side.
5. Multiply half-reactions by integers to equalize electrons ($n$).
6. Sum half-reactions and cancel identical species.
7. In base, neutralize $\text{H}^+$ with $\text{OH}^-$ to form $\text{H}_2\text{O}$, then simplify waters.

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Electrochemical Cell Architectures: Galvanic vs. Electrolytic

All electrochemical cells follow two universal rules:

• "AN OX": Oxidation occurs at the ANODE.
• "RED CAT": Reduction occurs at the CATHODE.
• Electron Flow: Electrons travel from ANODE to CATHODE through the external circuit.
• Current Flow: Conventional electrical current flows oppositely, from CATHODE to ANODE.

Galvanic (Voltaic) Cells

Galvanic cells exploit spontaneous redox reactions ($\Delta G^\circ < 0, E^\circ_{\text{cell}} > 0$) to generate electrical work:

• Anode Polarity: Negative ($-$); oxidation releases electrons, creating high negative charge density.
• Cathode Polarity: Positive ($+$); consumes electrons during reduction.
• Salt Bridge: Contains inert electrolyte (e.g., $\text{KNO}_3$). Anions migrate into the anode compartment to neutralize generated cations; cations migrate into the cathode compartment to replace consumed cations. Without a salt bridge, rapid charge buildup halts current.

Electrolytic Cells

Electrolytic cells use an external power supply to force a non-spontaneous reaction ($\Delta G^\circ > 0, E^\circ_{\text{cell}} < 0$):

• Anode Polarity: Positive ($+$); connected to the battery's positive terminal, which forcibly removes electrons.
• Cathode Polarity: Negative ($-$); connected to the battery's negative terminal, which pumps electrons into the electrode.

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Standard Reduction Potentials ($E^\circ$) and Cell Voltage

Potentials are measured relative to the Standard Hydrogen Electrode (SHE), defined as $0.00\text{ V}$ ($2\text{H}^+(aq, 1.0\text{ M}) + 2e^- \rightleftharpoons \text{H}_2(g, 1.0\text{ atm})$):

• More positive $E^\circ$: Greater electron affinity; stronger oxidizing agent, readily reduced ($\text{F}_2 + 2e^- \to 2\text{F}^-$, $E^\circ = +2.87\text{ V}$).
• More negative $E^\circ$: Lower electron affinity; oxidized form difficult to reduce, reduced form is a strong reducing agent ($\text{Li}^+ + e^- \to \text{Li}$, $E^\circ = -3.04\text{ V}$).

The standard cell potential is:

$$E^\circ_{\text{cell}} = E^\circ_{\text{cathode}} - E^\circ_{\text{anode}}$$

where both values are standard reduction potentials.

INTENSIVE PROPERTY RULE: Reduction potential is an intensive property ($\text{V} = \text{J/C}$). NEVER multiply $E^\circ$ by stoichiometric coefficients when balancing electrons!

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Thermodynamics of Electrochemical Cells: $\Delta G^\circ$, $E^\circ_{\text{cell}}$ & $K_{\text{eq}}$

Electrical work connects free energy to cell potential:

$$\Delta G^\circ = -n F E^\circ_{\text{cell}}$$

where $n$ is moles of electrons transferred and $F$ is Faraday's constant ($96,485\text{ C/mol }e^- \approx \mathbf{96,500\text{ C/mol }e^-}$).

• $E^\circ_{\text{cell}} > 0 \implies \Delta G^\circ < 0$ (Spontaneous, Galvanic).
• $E^\circ_{\text{cell}} < 0 \implies \Delta G^\circ > 0$ (Non-spontaneous, Electrolytic).
• $E^\circ_{\text{cell}} = 0 \implies \Delta G^\circ = 0$ (Dynamic Equilibrium).

Linking $\Delta G^\circ = -RT\ln K_{\text{eq}}$ and $\Delta G^\circ = -nFE^\circ_{\text{cell}}$ at $25^\circ\text{C}$ gives:

$$E^\circ_{\text{cell}} = \frac{0.0592}{n}\log_{10} K_{\text{eq}}$$

A cell with $K_{\text{eq}} > 1$ yields a positive standard voltage.

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The Nernst Equation and Non-Standard Conditions

Dynamic concentrations alter cell voltage via the Nernst equation:

$$E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{RT}{nF}\ln Q = E^\circ_{\text{cell}} - \frac{0.0592}{n}\log_{10} Q \approx E^\circ_{\text{cell}} - \frac{0.06}{n}\log_{10} Q$$

• $Q < 1$ (reactants high, products low): $\log_{10} Q < 0 \implies E_{\text{cell}} > E^\circ_{\text{cell}}$ (voltage rises).
• $Q > 1$ (products accumulated): $\log_{10} Q > 0 \implies E_{\text{cell}} < E^\circ_{\text{cell}}$ (voltage falls).
• At Equilibrium ($Q = K_{\text{eq}}$): $E_{\text{cell}} = 0.00\text{ V}$ and $\Delta G = 0$ (battery discharges completely).

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Concentration Cells and Membrane Potentials

A concentration cell contains identical half-cells differing solely in solute concentrations. Because electrodes are chemically identical, $\mathbf{E^\circ_{\text{cell}} = 0.00\text{ V}}$, driven purely by the entropy of dilution:

• Anode: Dilute compartment ($[M^{n+}]_{\text{dilute}}$); oxidation increases ion concentration.
• Cathode: Concentrated compartment ($[M^{n+}]_{\text{conc}}$); reduction consumes ions.
• Voltage: $E_{\text{cell}} = \frac{0.0592}{n}\log_{10}\left(\frac{[M^{n+}]_{\text{conc}}}{[M^{n+}]_{\text{dilute}}}\right)$.

Biological membranes function as concentration cells. The biological Nernst potential at $37^\circ\text{C}$ calculates single-ion equilibrium voltages:

$$V_{\text{eq}} = \frac{61.5\text{ mV}}{z}\log_{10}\left(\frac{[\text{ion}]_{\text{out}}}{[\text{ion}]_{\text{in}}}\right)$$

For potassium ($[\text{K}^+]_{\text{in}} = 140\text{ mM}, [\text{K}^+]_{\text{out}} = 4\text{ mM}$), $V_K \approx -95\text{ mV}$, driving the resting membrane potential ($V_m \approx -70\text{ mV}$).

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Faraday's Law of Electrolysis

Faraday's Law relates electric charge to electrodeposition:

$$q = I \times t \implies n_{e^-} = \frac{I \times t}{F}$$ $$\text{Moles of Metal Plated } (n_{\text{metal}}) = \frac{I \times t}{n \times F}$$ $$\text{Mass Plated } (m) = \frac{I \times t \times \text{Molar Mass}}{n \times F}$$

Worked Example: Plating copper metal ($\text{Cu}$, $63.55\text{ g/mol}$) from $\text{CuSO}_4$ ($\text{Cu}^{2+} + 2e^- \to \text{Cu}$, $n = 2$) using $5.0\text{ A}$ for $1930\text{ seconds}$:

1. Charge: $q = 5.0\text{ A} \times 1930\text{ s} = 9650\text{ C}$.
2. Moles of electrons: $n_{e^-} = 9650\text{ C} / 96,500\text{ C/mol }e^- = 0.100\text{ mol }e^-$.
3. Moles of $\text{Cu}$: $n_{\text{Cu}} = 0.100 / 2 = 0.0500\text{ mol }\text{Cu}$.
4. Mass plated: $m = 0.0500\text{ mol} \times 63.55\text{ g/mol} = 3.18\text{ g}$.