Electrochemistry – Chemistry Study Notes

Electrochemical Cells and Electrode Potentials

An electrochemical system fundamentally consists of two half-cells where oxidation and reduction occur simultaneously. The overall cell reaction is a composite of these two half-cell reactions.

The tendency of an electrode to lose or gain electrons when it is in contact with its own ions in solution is quantified as its electrode potential.

By international convention established by IUPAC, standard electrode potentials are expressed as reduction potentials ($E^\circ$). The standard hydrogen electrode (SHE) is assigned an arbitrary potential of exactly 0.00 V at all temperatures, serving as the universal reference electrode against which all other half-cell potentials are measured.

Standard Cell Potential ($E^\circ_{\text{cell}}$): Calculated as the difference between the reduction potential of the cathode and the anode: $E^\circ_{\text{cell}} = E^\circ_{\text{cathode}} – E^\circ_{\text{anode}}$. A positive $E^\circ_{\text{cell}}$ indicates a spontaneous chemical process under standard conditions.

The electrochemical series arranges elements in order of their standard reduction potentials. Elements with highly negative reduction potentials are strong reducing agents because they undergo oxidation easily.

Conversely, elements with highly positive reduction potentials act as strong oxidizing agents.

The Nernst Equation and Gibbs Free Energy

Real-world reactions rarely occur under standard state conditions of 1 molar concentration and 1 bar pressure. The Nernst equation bridges this gap by relating actual cell potential to concentrations of reactants and products, temperature, and the reaction quotient ($Q$).

For a general reduction half-reaction, $M^{n+} + ne^- \rightarrow M(s)$, the Nernst equation is expressed as:

• $E = E^\circ – \frac{RT}{nF} \ln Q$
• At 298 K, converting natural logarithm to base 10: $E = E^\circ – \frac{0.0591}{n} \log \frac{1}{[M^{n+}]}$
• For a complete cell reaction, $aA + bB \rightleftharpoons cC + dD$, the potential is given by: $E_{\text{cell}} = E^\circ_{\text{cell}} – \frac{0.0591}{n} \log \frac{[C]^c[D]^d}{[A]^a[B]^b}$

The electrical work done by a galvanic cell is directly tied to thermodynamics via Gibbs free energy change ($\Delta G$). The maximum work obtainable from a reversible electrochemical cell equals the decrease in free energy: $-\Delta G = W_{\text{max}} = nFE_{\text{cell}}$. Under standard conditions, this relates to the standard cell potential through $\Delta G^\circ = -nFE^\circ_{\text{cell}}$.

Furthermore, at chemical equilibrium, the cell potential becomes zero ($E_{\text{cell}} = 0$). This allows us to connect standard cell potential directly to the equilibrium constant ($K_{\text{eq}}$) via the fundamental relation: $\Delta G^\circ = -RT \ln K_{\text{eq}} = -2.303 RT \log K_{\text{eq}}$.

Electrolytic Conductance and Molar Conductivity

Unlike metallic conduction, electrolytic conduction involves the actual physical migration of ions through a liquid medium. Ohm’s law is applicable to electrolytic solutions, where resistance ($R$) is proportional to length ($l$) and inversely proportional to cross-sectional area ($A$): $R = \rho \frac{l}{A}$, where $\rho$ is the resistivity (specific resistance).

The reciprocal of resistance is conductance ($G$), measured in Siemens (S) or $\Omega^{-1}$. The reciprocal of resistivity is conductivity ($\kappa$), defined as $\kappa = \frac{1}{\rho} \frac{l}{A}$. Conductivity measures the conducting power of all ions present in a unit volume of solution.

To compare the conducting power of different solutions, competitive exams heavily emphasize molar conductivity ($\Lambda_m$) and equivalent conductivity ($\Lambda_{eq}$):

• Molar Conductivity ($\Lambda_m$): Defined as $\Lambda_m = \frac{\kappa \times 1000}{M}$, where $M$ is the molarity of the solution. Its standard unit is $\text{S cm}^2 \text{ mol}^{-1}$.
• Equivalent Conductivity ($\Lambda_{eq}$): Defined as $\Lambda_{eq} = \frac{\kappa \times 1000}{N}$, where $N$ is the normality. Its standard unit is $\text{S cm}^2 \text{ equiv}^{-1}$.

When an electrolyte solution is diluted, the distance between ions increases, reducing interionic attractive forces. For strong electrolytes, molar conductivity increases gradually with dilution and approaches a limiting value known as molar conductivity at infinite dilution ($\Lambda_m^\circ$).

For weak electrolytes, molar conductivity increases sharply at high dilutions due to an increase in the degree of dissociation ($\alpha$).

Kohlrausch’s Law of Independent Migration of Ions

Friedrich Kohlrausch formulated a foundational law stating that the limiting molar conductivity of an electrolyte can be represented as the sum of the individual contributions of its constituent anions and cations.

Kohlrausch’s Law: At infinite dilution, where dissociation is complete, each ion migrates independently of its co-ion and contributes a definite share to the total molar conductivity of the electrolyte, regardless of the nature of the other ion with which it is associated.

Mathematically, for a general electrolyte $A_x B_y$:

$\Lambda_m^\circ(A_x B_y) = x\lambda^\circ_+ + y\lambda^\circ_-$

where $\lambda^\circ_+$ and $\lambda^\circ_-$ are the limiting molar ionic conductivities of the cation and anion respectively.

Primary Applications for Exams:

1. Calculation of $\Lambda_m^\circ$ for weak electrolytes (like acetic acid, $CH_3COOH$) which cannot be obtained directly by extrapolation of experimental measurement curves. By combining strong electrolyte values (e.g., $HCl$, $CH_3COONa$, $NaCl$), the weak electrolyte value is evaluated.
2. Determination of the degree of dissociation ($\alpha$) of a weak electrolyte at a given concentration: $\alpha = \frac{\Lambda_m^c}{\Lambda_m^\circ}$.
3. Calculation of the dissociation constant ($K_a$) of a weak electrolyte: $K_a = \frac{c\alpha^2}{1-\alpha} = \frac{c(\Lambda_m^c / \Lambda_m^\circ)^2}{1 – (\Lambda_m^c / \Lambda_m^\circ)}$.

Electrolysis and Faraday’s Laws

Electrolysis is a non-spontaneous process where electrical energy is converted into chemical energy using an external direct current source. Oxidation occurs at the anode (positive terminal in electrolytic cells) and reduction occurs at the cathode (negative terminal).

Michael Faraday quantified the relationship between the amount of substance deposited or liberated at an electrode and the quantity of electrical charge passed through the electrolyte:

• First Law of Electrolysis: The mass ($w$) of a substance deposited or liberated at any electrode is directly proportional to the total charge ($Q$) passed through the solution. $w = Z \cdot Q = Z \cdot I \cdot t$, where $Z$ is the electrochemical equivalent.
• Second Law of Electrolysis: When the same quantity of electricity is passed through different electrolytic solutions connected in series, the masses of substances liberated at the respective electrodes are directly proportional to their chemical equivalent masses ($E$): $\frac{w_1}{w_2} = \frac{E_1}{E_2}$.

The charge carried by one mole of electrons is known as the Faraday constant ($F$), approximately equal to $96,487 \text{ C mol}^{-1}$ (often rounded to $96500 \text{ C mol}^{-1}$ for problem-solving in exams). The number of equivalents of a substance produced during electrolysis equals the number of Faradays of charge passed.

Important Facts / Formulas

Previous Year Question Hints

• Hint 1 (Nernst & Equilibrium): When asked for the equilibrium constant of a redox cell at 298 K, use the shortcut formula $E^\circ_{\text{cell}} = \frac{0.0591}{n} \log K_{eq}$. Pay close attention to the number of electrons ($n$) transferred in the balanced ionic equation.
• Hint 2 (Kohlrausch Applications): If finding $\Lambda_m^\circ$ for $NH_4OH$ (a weak base), construct an algebraic sum using strong electrolytes containing $NH_4^+$ ($NH_4Cl$), $OH^-$ ($NaOH$), and $Cl^-$ ($NaCl$) such that unwanted ions cancel out: $\Lambda_m^\circ(NH_4OH) = \Lambda_m^\circ(NH_4Cl) + \Lambda_m^\circ(NaOH) – \Lambda_m^\circ(NaCl)$.
• Hint 3 (Faraday Calculations): For gas evolution at electrodes during electrolysis (e.g., $O_2$ at anode or $H_2$ at cathode), remember that 1 mole of electrons (1 Faraday) discharges 1 equivalent of substance. For instance, liberating 1 mole of $O_2$ gas from water requires 4 moles of electrons ($4F$).

Quick Revision Summary

• Standard reduction potentials determine spontaneous redox directions; higher positive values favor reduction and make better oxidizing agents.
• The Nernst equation correlates non-standard cell potentials with reactant and product concentrations via the reaction quotient $Q$.
• Gibbs free energy change links electrochemistry directly to thermodynamics: $\Delta G = -nFE$.
• Conductivity ($\kappa$) decreases with dilution due to decreasing ion concentration per unit volume, whereas molar conductivity ($\Lambda_m$) increases with dilution.
• Strong electrolytes show a linear variation of $\Lambda_m$ with $\sqrt{C}$, whereas weak electrolytes exhibit a sharp upward curve at high dilutions.
• Kohlrausch’s Law allows the computation of limiting molar conductivities for weak acids and bases using linear combinations of strong electrolytes.
• The degree of dissociation ($\alpha$) and acid/base dissociation constants ($K_a/K_b$) can be extracted directly from conductivity data.
• Faraday’s laws quantify mass deposition: $1 \text{ Faraday} = 96500 \text{ C}$ deposits 1 gram-equivalent of any chemical species.