This Electrochemistry formula sheet provides a concise and comprehensive overview of essential concepts, formulas, and results crucial for JEE Main preparation. Mastering these topics will significantly enhance your problem-solving abilities and performance in chemical kinetics related questions.
Key Concepts
Electrochemistry is the branch of chemistry that deals with the interconversion of electrical and chemical energy. This involves redox reactions, which are characterized by the transfer of electrons.
Electrochemical Cells: Devices that convert chemical energy into electrical energy (voltaic or galvanic cells) or electrical energy into chemical energy (electrolytic cells).
Galvanic Cells: Spontaneous redox reactions produce electrical energy. Anode is negative, cathode is positive. Electrons flow from anode to cathode in the external circuit. Current flows from cathode to anode.
Electrolytic Cells: Non-spontaneous redox reactions are driven by external electrical energy. Anode is positive, cathode is negative.
Electrode Potential: The potential difference developed between an electrode and the electrolyte when they are in contact. Standard electrode potential ($E^\circ$) is measured at 298 K, 1 atm pressure, and 1 M concentration for ions.
Standard Hydrogen Electrode (SHE): Reference electrode with a defined $E^\circ = 0$ V for all temperatures.
Nernst Equation: Relates electrode potential to ion concentrations, allowing calculation of cell potential under non-standard conditions.
Electrolytic Conduction: Movement of ions in a solution due to an applied electric field. Governed by factors like ion concentration, size, charge, solvent viscosity, and temperature.
Conductance (G): Reciprocal of resistance (R). Measured in Siemens (S). Conductivity ($\kappa$ or C): Reciprocal of resistivity ($\rho$). Defined as conductance of a solution of unit length and unit cross-sectional area. Measured in S cm$^{-1}$ or S m$^{-1}$. Molar Conductivity ($\Lambda_m$): The conductivity of a solution containing 1 mole of electrolyte when placed between two electrodes 1 cm apart with sufficient area to contain the entire solution. Measured in S cm$^2$ mol$^{-1}$.
Kohlrausch's Law: At infinite dilution, the molar conductivity of an electrolyte is the sum of the individual contributions of the cations and anions.
Faraday's Laws of Electrolysis: Quantitative relationships between the amount of substance deposited or liberated at an electrode and the quantity of electricity passed.
Batteries/Accumulators: Electrochemical cells used as sources of electrical energy. Primary batteries are non-rechargeable (e.g., dry cell), while secondary batteries are rechargeable (e.g., lead-acid battery, Li-ion battery).
Corrosion: Electrochemical process involving the deterioration of a material (usually metal) due to its reaction with the environment. Example: Rusting of iron.
Formulas & Results
1. Redox Reactions & Cell Potentials
Standard Cell Potential: $$E^\circ_{\text{cell}} = E^\circ_{\text{cathode}} - E^\circ_{\text{anode}}$$ (Both $E^\circ$ are standard reduction potentials)
Nernst Equation (for a single electrode): For reaction $M^{n+}(aq) + ne^- \to M(s)$ $$E = E^\circ - \frac{RT}{nF} \ln \frac{[\text{M(s)}]}{[M^{n+}(aq)]} = E^\circ - \frac{0.0591}{n} \log \frac{1}{[M^{n+}(aq)]} \quad (\text{at 298 K})$$
Nernst Equation (for a full cell): For a general reaction $aA + bB \to cC + dD$ $$E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{RT}{nF} \ln Q = E^\circ_{\text{cell}} - \frac{0.0591}{n} \log Q \quad (\text{at 298 K})$$ Where $Q$ is the reaction quotient: $Q = \frac{[C]^c[D]^d}{[A]^a[B]^b}$
2. Thermodynamics of Electrochemical Cells
- Gibbs Free Energy Change: $$\Delta G = -nFE_{\text{cell}}$$
- Standard Gibbs Free Energy Change: $$\Delta G^\circ = -nFE^\circ_{\text{cell}}$$
- Relationship between $\Delta G^\circ$ and Equilibrium Constant ($K_c$): $$\Delta G^\circ = -RT \ln K_c = -2.303 RT \log K_c$$ From Nernst equation at equilibrium ($E_{\text{cell}} = 0$): $$E^\circ_{\text{cell}} = \frac{RT}{nF} \ln K_c = \frac{0.0591}{n} \log K_c \quad (\text{at 298 K})$$
3. Conductance & Conductivity
- Resistance (R): $$R = \rho \frac{l}{A}$$ Where $\rho$ is resistivity, $l$ is length, $A$ is cross-sectional area.
- Conductance (G): $$G = \frac{1}{R} = \frac{1}{\rho} \frac{A}{l} = \kappa \frac{A}{l}$$
- Cell Constant (G):* $$G^* = \frac{l}{A}$$ Units: cm$^{-1}$ or m$^{-1}$ $$G = \kappa / G^* \implies \kappa = G \times G^*$$
- Molar Conductivity ($\Lambda_m$): $$\Lambda_m = \frac{\kappa \times 1000}{C}$$ Where $\kappa$ is in S cm$^{-1}$ and C is in mol L$^{-1}$ (molarity). Units: S cm$^2$ mol$^{-1}$ If $\kappa$ is in S m$^{-1}$ and C is in mol m$^{-3}$: $$\Lambda_m = \frac{\kappa}{C}$$ Units: S m$^2$ mol$^{-1}$ (Note: 1 S m$^2$ mol$^{-1}$ = 10$^4$ S cm$^2$ mol$^{-1}$)
4. Kohlrausch's Law
Molar Conductivity at Infinite Dilution: $$\Lambda_m^\circ = \nu_+ \lambda_+^\circ + \nu_- \lambda_-^\circ$$ Where $\nu_+$ and $\nu_-$ are the number of cations and anions per formula unit of electrolyte, and $\lambda_+^\circ$ and $\lambda_-^\circ$ are their respective limiting molar ionic conductivities.
Degree of Dissociation ($\alpha$): $$\alpha = \frac{\Lambda_m}{\Lambda_m^\circ}$$
Dissociation Constant ($K_a$ or $K_c$ for weak electrolytes): $$K_c = \frac{C\alpha^2}{(1-\alpha)}$$
5. Faraday's Laws of Electrolysis
- First Law: The mass of a substance deposited or liberated at an electrode is directly proportional to the quantity of electricity passed through the electrolyte. $$m \propto Q \implies m = ZQ$$ Where $Q = I \times t$ (Current $\times$ Time) $Z$ is the electrochemical equivalent.
- Second Law: When the same quantity of electricity is passed through different electrolytes connected in series, the masses of the substances produced at the electrodes are directly proportional to their chemical equivalents (equivalent weights). $$\frac{m_1}{m_2} = \frac{E_1}{E_2}$$
- Charge of one mole of electrons (1 Faraday): $$1 F = 96485 \text{ C mol}^{-1} \approx 96500 \text{ C mol}^{-1}$$
- Mass deposited/liberated (from First Law & Definition of Faraday): $$m = \frac{E}{F} \text{ (grams per Faraday)} = \frac{\text{Molar Mass}}{nF} Q = \frac{\text{Molar Mass} \times I \times t}{n \times 96500}$$ Where $n$ is the number of electrons involved in the electrode reaction (valency factor).
Worked Examples
Example 1: Nernst Equation Calculation
Question: Calculate the electrode potential of a copper electrode dipped in 0.1 M CuSO$_4$ solution at 298 K. The standard electrode potential of Cu$^{2+}$/Cu is +0.34 V.
Solution: The electrode reaction is $Cu^{2+}(aq) + 2e^- \to Cu(s)$. Given: $E^\circ = +0.34 \text{ V}$, $[Cu^{2+}] = 0.1 \text{ M}$, $n = 2$. Using the Nernst equation for a single electrode at 298 K: $E = E^\circ - \frac{0.0591}{n} \log \frac{1}{[Cu^{2+}]}$ $E = 0.34 - \frac{0.0591}{2} \log \frac{1}{0.1}$ $E = 0.34 - 0.02955 \log(10)$ $E = 0.34 - 0.02955 \times 1$ $E = 0.34 - 0.02955$ $E = 0.31045 \text{ V}$
Example 2: Relationship between $\Delta G^\circ$ and $K_c$
Question: For the reaction $2Ag^+(aq) + Cu(s) \to 2Ag(s) + Cu^{2+}(aq)$, the standard cell potential $E^\circ_{\text{cell}} = 0.46 \text{ V}$ at 298 K. Calculate the equilibrium constant $K_c$.
Solution: Given: $E^\circ_{\text{cell}} = 0.46 \text{ V}$, $T = 298 \text{ K}$. From the reaction, $Cu \to Cu^{2+} + 2e^-$ and $2Ag^+ + 2e^- \to 2Ag$. The number of electrons transferred, $n = 2$. The relationship between $E^\circ_{\text{cell}}$ and $K_c$ at 298 K is: $E^\circ_{\text{cell}} = \frac{0.0591}{n} \log K_c$ $0.46 = \frac{0.0591}{2} \log K_c$ $\log K_c = \frac{0.46 \times 2}{0.0591}$ $\log K_c = \frac{0.92}{0.0591}$ $\log K_c \approx 15.566$ $K_c = 10^{15.566}$ $K_c \approx 3.68 \times 10^{15}$
Example 3: Faraday's Laws
Question: How many grams of aluminum (Al) can be deposited by passing a current of 10 A for 965 seconds through molten Al$_2$O$_3$? (Atomic mass of Al = 27 g/mol)
Solution: The reduction reaction for aluminum is: $Al^{3+} + 3e^- \to Al(s)$. From the reaction, $n = 3$ moles of electrons are required to deposit 1 mole of Al. Quantity of electricity ($Q$) passed: $Q = I \times t = 10 \text{ A} \times 965 \text{ s} = 9650 \text{ C}$ Mass deposited ($m$): $m = \frac{\text{Molar Mass} \times Q}{n \times F}$ $m = \frac{27 \text{ g/mol} \times 9650 \text{ C}}{3 \times 96500 \text{ C/mol}}$ $m = \frac{27 \times 9650}{289500}$ $m = \frac{27}{30} = 0.9 \text{ g}$ Thus, 0.9 grams of aluminum can be deposited.
Common Mistakes
- Sign Convention for Electrode Potentials: Always use reduction potentials when calculating $E^\circ_{\text{cell}} = E^\circ_{\text{cathode}} - E^\circ_{\text{anode}}$. Do not mix reduction and oxidation potentials unless you explicitly change the sign.
- Units in Nernst Equation: Ensure consistency in units. The value $0.0591$ is derived assuming $T=298 \text{ K}$ and $\ln(x)$ converted to $\log(x)$. If temperature is different, use the full $RT/nF$ term.
- Units for Molar Conductivity: Be careful with units of $\kappa$ (conductivity) and concentration (molarity) when calculating $\Lambda_m$. $\kappa$ in S cm$^{-1}$ needs molarity in mol L$^{-1}$ for the factor of 1000. $\kappa$ in S m$^{-1}$ needs molarity in mol m$^{-3}$ (no factor of 1000 needed).
- Stoichiometry in Faraday's Laws: Correctly determine the number of electrons ($n$) involved in the electrode reaction. This is crucial for calculating the equivalent weight or using the mass deposition formula.
- Distinguishing $\Lambda_m$ and $\Lambda_m^\circ$: Remember that $\Lambda_m$ is molar conductivity at a given concentration, while $\Lambda_m^\circ$ is limiting molar conductivity at infinite dilution. They are different and used for different purposes (e.g., calculating degree of dissociation).
- Standard vs. Non-Standard Conditions: Apply $E^\circ_{\text{cell}}$ for standard conditions and $E_{\text{cell}}$ (using Nernst equation) for non-standard conditions (different concentrations or pressures).
Quick Revision Checklist
- Understand the definitions of galvanic and electrolytic cells.
- Can you apply the Nernst equation for both electrode potential and cell potential?
- Do you know the relationship between $\Delta G^\circ$, $E^\circ_{\text{cell}}$, and $K_c$?
- Are you familiar with the formulas for resistance, conductance, resistivity, and conductivity?
- Can you calculate molar conductivity and limiting molar conductivity?
- Do you know Kohlrausch's law and its applications (e.g., degree of dissociation of weak electrolytes)?
- Can you apply Faraday's first and second laws of electrolysis to problems involving mass deposition or gas evolution?
- Do you understand the concept of 1 Faraday and its numerical value?
- Can you identify the anode and cathode in both galvanic and electrolytic cells and the direction of electron flow?
