Redox Reactions and Electrochemistry

1. Electronic Concept of Redox & Oxidation Number Rules

• Redox Mechanics: Oxidation corresponds to de-electronation (loss of electrons / increase in oxidation number); reduction corresponds to electronation (gain of electrons / decrease in oxidation number). An oxidising agent (oxidant) accepts electrons; a reducing agent (reductant) donates electrons.
• Rules for Assigning Oxidation Number (O.N.):
  1. Elementary state atoms bear an O.N. of $0$ ($\text{H}_2, \text{O}_2, \text{Cl}_2, \text{P}_4, \text{S}_8, \text{Na}$).
  2. Monatomic ions equal their net charge ($\text{Na}^+ = +1, \text{Al}^{3+} = +3, \text{Cl}^- = -1$). Alkali metals are strictly $+1$; alkaline earths are $+2$; Fluorine is $-1$ across all compounds.
  3. Oxygen is generally $-2$, except in peroxides where it is $-1$ ($\text{H}_2\text{O}_2, \text{Na}_2\text{O}_2$), superoxides where it is $-\frac{1}{2}$ ($\text{KO}_2$), and oxygen fluorides where it is $+2$ ($\text{OF}_2$) or $+1$ ($\text{O}_2\text{F}_2$).
  4. Hydrogen is $+1$, except in binary metal hydrides where it is $-1$ ($\text{NaH}, \text{CaH}_2$).
• The Paradox of Fractional Oxidation Numbers:
  • Carbon suboxide ($\text{C}_3\text{O}_2$): $\text{O}=\overset{+2}{\text{C}}=\overset{0}{\text{C}}=\overset{+2}{\text{C}}=\text{O} \implies \text{Average O.N.} = +\frac{4}{3}$.
  • Tribromooctaoxide ($\text{Br}_3\text{O}_8$): Two terminal $\text{Br}$ atoms are $+6$; central $\text{Br}$ is $+4 \implies \text{Average} = +\frac{16}{3}$.
  • Tetrathionate ion ($\text{S}_4\text{O}_6^{2-}$): Two terminal $\text{S}$ atoms are $+5$; two central bridging $\text{S}$ atoms are $0 \implies \text{Average} = +2.5$.
• Types of Redox Reactions:
  • Combination: $\text{C} + \text{O}_2 \rightarrow \text{CO}_2$
  • Decomposition: $2\text{KClO}_3 \rightarrow 2\text{KCl} + 3\text{O}_2$
  • Displacement: $\text{Zn}(s) + \text{CuSO}_4(aq) \rightarrow \text{ZnSO}_4(aq) + \text{Cu}(s)$
  • Disproportionation: A single element in an intermediate oxidation state is simultaneously oxidised and reduced:

    $$2\text{H}_2\text{O}_2(aq) \longrightarrow 2\text{H}_2\text{O}(l) + \text{O}_2(g) \quad (\text{O: } -1 \rightarrow -2 \text{ and } 0)$$

    $$\text{P}_4(s) + 3\text{OH}^- + 3\text{H}_2\text{O} \longrightarrow \text{PH}_3(g) + 3\text{H}_2\text{PO}_2^-(aq) \quad (\text{P: } 0 \rightarrow -3 \text{ and } +1)$$

    $$\text{Cl}_2(g) + 2\text{OH}^-(aq) \longrightarrow \text{ClO}^-(aq) + \text{Cl}^-(aq) + \text{H}_2\text{O}(l)$$

2. Balancing Redox Equations & Quantitative Titrimetry

Ion-Electron Balancing Protocol (Acidic Medium)

1. Separate the skeleton equation into oxidation and reduction half-reactions.
2. Balance all atoms other than $\text{O}$ and $\text{H}$.
3. Balance $\text{O}$ by adding $\text{H}_2\text{O}$ molecules; balance $\text{H}$ by adding $\text{H}^+$ ions.
4. Balance charges by adding electrons ($e^-$) to the more positive side.
5. Multiply half-reactions by suitable integers to equalize electrons, then add:

   $$\text{Cr}_2\text{O}_7^{2-} + 14\text{H}^+ + 6\text{Fe}^{2+} \longrightarrow 2\text{Cr}^{3+} + 6\text{Fe}^{3+} + 7\text{H}_2\text{O}$$

   $$2\text{MnO}_4^- + 5\text{C}_2\text{O}_4^{2-} + 16\text{H}^+ \longrightarrow 2\text{Mn}^{2+} + 10\text{CO}_2 + 8\text{H}_2\text{O}$$

• For Basic Medium: Carry out acidic balancing, then add equivalent $\text{OH}^-$ to both sides to combine with $\text{H}^+$ into water molecules.

Redox Indicators

• $\text{KMnO}_4$ (Self-Indicator): The deeply purple permanganate ion acts as its own indicator; the end-point is signaled by the first permanent light pink tinge at concentrations as low as $10^{-6}\text{ M}$.
• $\text{K}_2\text{Cr}_2\text{O}_7$ (External Indicator): Not a self-indicator; titrated using diphenylamine indicator, which is oxidised to an intense blue compound at the equivalence point.
• Iodometry: Titration of liberated $\text{I}_2$ with standard sodium thiosulphate ($\text{Na}_2\text{S}_2\text{O}_3$):

  $$\text{I}_2 + 2\text{S}_2\text{O}_3^{2-} \longrightarrow 2\text{I}^- + \text{S}_4\text{O}_6^{2-}$$

  Starch solution serves as an indicator, producing a deep blue complex that turns colourless when all iodine is reduced.

3. Galvanic Cells & The Nernst Equation

• Daniell Cell Architecture: Zinc rod in $\text{ZnSO}_4$ (anode, negative polarity) and copper rod in $\text{CuSO}_4$ (cathode, positive polarity) connected via an agar-agar $\text{KCl}$ salt bridge:

  $$\text{Zn}(s) \mid \text{Zn}^{2+}(aq) \parallel \text{Cu}^{2+}(aq) \mid \text{Cu}(s)$$

  $$E^\circ_{\text{cell}} = E^\circ_{\text{cathode}} - E^\circ_{\text{anode}} = +0.34\text{ V} - (-0.76\text{ V}) = \mathbf{1.10\text{ V}}$$

• Standard Hydrogen Electrode (SHE): Assigned a potential of $0.00\text{ V}$ at all temperatures:

  $$\text{Pt}(s) \mid \text{H}_2(g, 1\text{ bar}) \mid \text{H}^+(aq, 1\text{ M})$$

• The Nernst Equation at 298 K:
  For any general cell reaction $a\text{A} + b\text{B} \xrightarrow{ne^-} c\text{C} + d\text{D}$:

  $$E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{2.303 RT}{nF} \log Q = E^\circ_{\text{cell}} - \frac{0.0591}{n} \log\left(\frac{[\text{C}]^c [\text{D}]^d}{[\text{A}]^a [\text{B}]^b}\right)$$

4. Thermodynamics of Electrochemical Cells

• Reversible Electrical Work & Free Energy:

  $$\Delta_r G = -nFE_{\text{cell}} \quad \text{and} \quad \Delta_r G^\circ = -nFE^\circ_{\text{cell}}$$

  Where $n$ is the moles of electrons transferred and $F = 96487\text{ C mol}^{-1} \approx 96500\text{ C mol}^{-1}$.
  • For a cell to operate spontaneously: $E_{\text{cell}} > 0 \implies \Delta_r G < 0$.
• Equilibrium Constant ($K_c$) from $E^\circ_{\text{cell}}$:
  At dynamic equilibrium: $E_{\text{cell}} = 0$ and $Q = K_c$:

  $$E^\circ_{\text{cell}} = \frac{0.0591}{n} \log K_c \quad \implies \quad \log K_c = \frac{n E^\circ_{\text{cell}}}{0.0591}$$

5. Conductance, Molar Conductivity & Kohlrausch's Law

• Resistance & Conductance:

  $$R = \rho \frac{l}{A} \quad \implies \quad G = \frac{1}{R} = \kappa \frac{A}{l}$$

  Where $G^* = \frac{l}{A} = R \times \kappa$ is the cell constant (units: $\text{cm}^{-1}$ or $\text{m}^{-1}$).
• Specific Conductivity ($\kappa$) and Molar Conductivity ($\Lambda_m$):

  $$\Lambda_m = \frac{\kappa}{C} = \frac{\kappa(\text{S cm}^{-1}) \times 1000}{\text{Molarity}(\text{mol L}^{-1})} \quad (\text{Units: }\text{S cm}^2\text{ mol}^{-1})$$

• Variation with Dilution:
  • Strong Electrolytes (Debye-Hückel-Onsager Equation):

    $$\Lambda_m = \Lambda_m^\circ - A \sqrt{c}$$

    Plot of $\Lambda_m$ vs $\sqrt{c}$ yields a straight line with intercept $\Lambda_m^\circ$.
  • Weak Electrolytes: Dissociation increases steeply on dilution, making direct extrapolation to $\Lambda_m^\circ$ impossible.
• Kohlrausch's Law of Independent Migration:

  $$\Lambda_m^\circ = \nu_+ \lambda_+^\circ + \nu_- \lambda_-^\circ$$

  • Degree of ionisation: $\alpha = \frac{\Lambda_m}{\Lambda_m^\circ}$.
  • Dissociation constant: $K_a = \frac{c \alpha^2}{1 - \alpha} = \frac{c \Lambda_m^2}{\Lambda_m^\circ (\Lambda_m^\circ - \Lambda_m)}$.

6. Electrolysis & Quantitative Faraday Laws

• Faraday's First Law: Mass of substance deposited is proportional to total charge passed:

  $$w = Z \cdot Q = Z \cdot I \cdot t = \frac{E \cdot I \cdot t}{96500}$$

  Where electrochemical equivalent $Z = \frac{\text{Equivalent weight}}{96500} = \frac{\text{Molar mass}}{n \times 96500}$.
• Faraday's Second Law: When identical current passes through cells in series:

  $$\frac{w_1}{w_2} = \frac{E_1}{E_2}$$

• Selective Discharge Criteria during Aqueous Electrolysis:
  • Cathode: Species with the more positive reduction potential discharges first.
  • Anode: Species with the lower reduction potential oxidises first, subject to gas overpotentials.

7. Commercial Batteries, Fuel Cells & Corrosion

Primary Batteries (Irreversible)

• Dry Cell (Leclanché): Anode: Zinc container; Cathode: Graphite rod coated with $\text{MnO}_2$; Electrolyte: Moist paste of $\text{NH}_4\text{Cl} + \text{ZnCl}_2$. Potential $\approx 1.5\text{ V}$.

  $$\text{Anode: } \text{Zn}(s) \rightarrow \text{Zn}^{2+} + 2e^-$$

  $$\text{Cathode: } \text{MnO}_2 + \text{NH}_4^+ + e^- \rightarrow \text{MnO(OH)} + \text{NH}_3$$

• Mercury Button Cell: Anode: $\text{Zn-Hg}$ amalgam; Cathode: Paste of $\text{HgO} + \text{C}$; Electrolyte: $\text{KOH} + \text{ZnO}$ paste. Potential $\approx 1.35\text{ V}$.

  $$\text{Overall: } \text{Zn(Hg)} + \text{HgO}(s) \longrightarrow \text{ZnO}(s) + \text{Hg}(l)$$

Secondary Batteries (Rechargeable)

• Lead Storage Battery: Anode: Spongy lead; Cathode: Grid of lead packed with $\text{PbO}_2$; Electrolyte: $38\% \text{ H}_2\text{SO}_4$ ($d = 1.30\text{ g mL}^{-1}$).

  $$\text{Discharging: } \text{Pb}(s) + \text{PbO}_2(s) + 2\text{H}_2\text{SO}_4(aq) \longrightarrow 2\text{PbSO}_4(s) + 2\text{H}_2\text{O}(l)$$

  $$\text{Recharging: } 2\text{PbSO}_4(s) + 2\text{H}_2\text{O}(l) \longrightarrow \text{Pb}(s) + \text{PbO}_2(s) + 2\text{H}_2\text{SO}_4(aq)$$

Fuel Cells & Corrosion Prevention

• $\text{H}_2-\text{O}_2$ Apollo Fuel Cell: Converts heat of combustion directly into electricity with $\sim 70\%$ thermodynamic efficiency:

  $$\text{Anode: } 2\text{H}_2(g) + 4\text{OH}^-(aq) \longrightarrow 4\text{H}_2\text{O}(l) + 4e^-$$

  $$\text{Cathode: } \text{O}_2(g) + 2\text{H}_2\text{O}(l) + 4e^- \longrightarrow 4\text{OH}^-(aq)$$

  $$\text{Net: } 2\text{H}_2(g) + \text{O}_2(g) \longrightarrow 2\text{H}_2\text{O}(l)$$

• Corrosion of Iron (Rusting): Electrochemical process in the presence of water and oxygen:

  $$\text{Anode Spot: } 2\text{Fe}(s) \longrightarrow 2\text{Fe}^{2+} + 4e^- \quad (E^\circ = -0.44\text{ V})$$

  $$\text{Cathode Spot: } \text{O}_2(g) + 4\text{H}^+ + 4e^- \longrightarrow 2\text{H}_2\text{O}(l) \quad (E^\circ = +1.23\text{ V})$$

  Rust is hydrated ferric oxide: $\text{Fe}_2\text{O}_3 \cdot x\text{H}_2\text{O}$. Prevented by galvanisation (coating with sacrificial Zinc) or cathodic protection with Magnesium sacrificial blocks.