UPSC Chemistry 2021 — Paper I

Solve each of the five sub-parts systematically with clear step-by-step derivations and calculations. Begin with the quantum mechanical expectation value problem, followed by thermodynamic feasibility analysis using Born-Haber cycle, then thermodynamic relation for rubber elasticity, Clausius-Clapeyron calculation for vapour pressure, and finally definition with practical significance of electrochemical series. Conclude each part with the final answer clearly stated.

• Part (a): Correct wavefunction ψ₁ = √(2/l) sin(πx/l), integration of ⟨x⟩ = ∫ψ*xψ dx from 0 to l yielding l/2 with proper mathematical steps
• Part (b): Complete Born-Haber cycle construction for both NaCl and NaCl₂, calculation of ΔHf values, comparison showing NaCl₂ is thermodynamically unfavorable due to high second ionization energy
• Part (c): Application of Maxwell relation and thermodynamic identity, recognition that (∂S/∂l)T > 0 for rubber, leading to negative (∂T/∂l)S and conclusion that length decreases with temperature increase
• Part (d): Correct application of Clausius-Clapeyron equation ln(P₂/P₁) = (ΔHvap/R)(1/T₁ - 1/T₂) with temperature conversion to Kelvin
• Part (e): Precise definition of electrochemical series as arrangement of elements by standard reduction potentials, with significance including prediction of displacement reactions, corrosion tendency, and industrial applications like extraction of metals (e.g., aluminum production in India)

Calculate the Rydberg transition parameters for part (a), draw and label hybridization geometries for part (b), and perform density-based unit cell calculations for part (c). Allocate approximately 20% time to (a), 40% to (b) for four detailed diagrams, and 40% to (c) for the multi-step crystallographic calculation with comparison. Begin each part with the relevant formula, show stepwise working, and conclude with physical interpretation.

• For (a)(i): Apply Rydberg formula 1/λ = R_H(1/n₁² - 1/n₂²) with n₁=100, n₂=101 to find wavelength in cm/m range and frequency via c/λ
• For (a)(ii): Explain transition difficulty due to extremely small energy gap (~10⁻⁴ eV), thermal broadening, spontaneous emission probability ∝ ν³, and competition from collisional de-excitation
• For (b): Draw trigonal bipyramidal (sp³d, d_z²), octahedral (sp³d², d_z² and d_x²-y²), square planar (dsp², d_x²-y²), and tetrahedral (sd³, no d-orbital from valence shell—note this is hypothetical/invalid)
• For (c): Calculate atomic radius r = (√3/4)a from b.c.c. density, derive a = (2M/N_Aρ)^(1/3), obtain r ≈ 1.24 Å, then compare f.c.c. packing efficiency (74%) vs b.c.c. (68%) to conclude contraction occurs
• For (c) continuation: Explicitly calculate f.c.c. edge length from same atomic radius, show density increases to ~8.6 g/cm³, confirming structural contraction despite same atomic radius assumption

Begin with a brief introduction linking surface phenomena, electrochemistry, and thermodynamics as core physical chemistry topics. For part (a), apply the surface energy formula with proper unit conversion from CGS to SI or consistent use of dynes/cm; for (b), define ion-selective electrodes with emphasis on glass electrode mechanism and Nernst equation application; for (c)(i), sketch the enthalpy of vaporization curve showing its decrease from Tp to Tc with proper labeling; for (c)(ii), prove T_final = T_initial using Joule-Thomson expansion concepts and demonstrate entropy increase for irreversibility. Allocate approximately 15-20% time to (a), 35-40% to (b), 20% to (c)(i), and 25% to (c)(ii), ensuring all numerical derivations show intermediate steps.

• Part (a): Conservation of volume to find radius of small drops (r = R/5 = 0.08 cm), calculation of initial and final surface areas, application of ΔE = γ × ΔA = 4πγ(n·r² - R²) yielding 4πγ × 4R² = 16πγR² = 579.6 ergs or ~58 μJ
• Part (b): Definition of ion-selective electrodes (ISEs) as membrane electrodes responding selectively to specific ions; glass electrode construction (Ag/AgCl internal reference, thin glass membrane, internal buffer); Nernst equation E = E° - 0.0591 pH at 25°C; calibration using standard buffers (pH 4, 7, 10) and temperature compensation
• Part (c)(i): Enthalpy of vaporization (ΔHvap) decreases from maximum at triple point to zero at critical point; curve shape concave downward due to weakening intermolecular forces; mention of Watson correlation or Trouton's rule context; proper axes labeling (T on x-axis, ΔHvap on y-axis)
• Part (c)(ii): Free expansion of ideal gas into vacuum; ΔU = 0 implies T_final = T_initial for ideal gas; calculation of entropy change ΔS = nR ln[(V₁+V₂)/V₁] > 0 proving irreversibility; mention that real gases show Joule-Thomson cooling
• Integration: Recognition that parts (a) and (c)(ii) both involve energy considerations but with different constraints (surface vs. bulk thermodynamics), while (b) connects electrochemical potential to chemical potential concepts

Begin with a brief introduction acknowledging the interconnected nature of physical chemistry principles across electrochemistry, transport phenomena, and phase equilibria. Allocate approximately 15 minutes (20%) to part (a) for precise Nernst equation calculation; 20 minutes (25%) to part (b)(i)-(ii) covering thermal conductivity analysis and real gas comparisons; and 35 minutes (45%) to part (c) requiring detailed phenol-water phase diagram with tie line construction. Conclude by synthesizing how non-ideality manifests across electrochemical, gaseous, and liquid-liquid systems.

• Part (a): Correct identification of half-reactions (H₂ → 2H⁺ + 2e⁻ and Cu²⁺ + 2e⁻ → Cu), application of Nernst equation E = E° - (RT/nF)lnQ with proper substitution of concentrations and partial pressure, yielding Ecell ≈ 0.34 - 0.0591/2 log(0.01²/0.1) = 0.34 + 0.0887 = 0.4287 V
• Part (b)(i): Explanation of thermal conductivity dependence on molecular weight (κ ∝ 1/√M, lower M increases conductivity, worsening insulation), degrees of freedom (polyatomic vs diatomic affecting specific heat and energy transfer), and mean free path; overall conclusion that air replacement degrades insulating performance
• Part (b)(ii): Derivation that smaller Tc and larger Pc indicate smaller 'a' (intermolecular forces) and 'b' (molecular size), hence NO has smaller a and b than CCl₄; NO behaves more ideally at 300K/10atm due to lower Tc and reduced significance of attractive forces at T >> Tc
• Part (c): Construction of temperature-composition phase diagram for phenol-water showing upper consolute temperature (~66°C, 34% phenol), two-phase region below CST, and conjugate solutions; explicit demonstration of tie lines connecting equilibrium liquid compositions and their use in lever rule calculations for phase amounts
• Integration point: Recognition that all parts involve deviation from ideality—electrochemical (activity coefficients), gaseous (van der Waals corrections), and liquid-liquid (partial miscibility)—with practical relevance to materials science and chemical engineering applications in Indian industrial contexts

Begin with the directive 'derive' for part (a), applying systematic derivation methodology across all five parts. Allocate approximately 20% time to each part given equal 10-mark weighting: (a) derive zero-order kinetics with integrated rate law and half-life proof; (b) state and derive Beer-Lambert law with extinction coefficient significance; (c) explain heme-oxygen binding mechanism with proximal histidine role and globin protection; (d) analyze geometrical and optical isomerism for M(AB)b₂c₂ with clear counting; (e) complete five xenon reactions showing hydrolysis patterns and fluorinating behavior. Structure as five distinct sections without introduction or conclusion.

• Part (a): Derivation of rate constant k = [A]₀ - [A]/t for zero-order reaction; proof that t₁/₂ = [A]₀/2k showing direct proportionality to initial concentration
• Part (b): Statement of Beer-Lambert law (A = εcl or I = I₀10^(-εcl)); derivation from differential form -dI/dl = k'Ic; definition of molar extinction coefficient ε and its units
• Part (c): Mechanism of heme Fe(II) oxidation to Fe(III) hematin via μ-oxo dimer formation; role of distal and proximal histidine in globin preventing autoxidation; mention of picket-fence porphyrin model by Collman
• Part (d): Analysis of M(AB)b₂c₂ showing 5 geometrical isomers; identification of which geometrical isomers possess optical isomerism leading to total stereoisomer count
• Part (e): (i) XeF₄ + 12H₂O → Xe + ½O₂ + 4HF + 11H₂O or Xe + 2O₂ + 4HF; (ii) XeF₆ + OPF₃ or H₂O; (iii) XeOF₄ + SiO₂ → 2XeO₂F₂ + SiF₄; (iv) 3XeF₂ + 2S₂O₆F₂ → 3Xe + 6SF₂ + 6O₂; (v) (C₆H₅)₂S + XeF₂ → (C₆H₅)₂SF₂ + Xe

Begin with the directive 'derive' for part (c), the highest-weighted section, while addressing 'how' in part (a) and 'what/why' in part (b). Allocate approximately 20% time to part (a) on bonding comparisons, 40% to part (b) on kinetic theory limitations and transition state explanations, and 40% to part (c) for rigorous derivation of Langmuir isotherm with limiting condition proofs. Structure as: comparative bonding analysis → theory critique with mechanism → mathematical derivation with graphical verification.

• Part (a): Contrast phosphazene's dπ-pπ bonding with benzene's delocalized π-system and borazine's partial aromaticity; note bond length equality in phosphazene vs. bond alternation in borazine; mention skeletal electron counting (6π vs. 6π vs. 6π but different orbital contributions)
• Part (b): List collision theory limitations—ignores molecular orientation, energy distribution assumptions, no account for activation energy details; explain how transition state theory introduces the activated complex, potential energy surface, and thermodynamic formulation of rate constants via partition functions or thermodynamic parameters (ΔH‡, ΔS‡, ΔG‡)
• Part (c): Derive Langmuir isotherm from kinetic equilibrium (rate of adsorption = rate of desorption) or statistical mechanics; define θ = KP/(1+KP); prove low P limit: θ ≈ KP (first-order, Henry's law region) and high P limit: θ ≈ 1 (zero-order, saturation)
• Part (c) continued: Show mathematical steps clearly—start with assumptions (monolayer, uniform surface, no interaction), set up equilibrium expression, solve for surface coverage θ
• Cross-part integration: Connect surface chemistry in (c) to catalytic applications relevant to Indian industry (e.g., Haber-Bosch ammonia synthesis, heterogeneous catalysis in refineries); relate transition state theory in (b) to enzyme kinetics and pharmaceutical development

Begin with a brief introduction acknowledging the diverse nature of the three parts covering bioinorganic, organometallic, and photochemistry domains. Allocate approximately 20% time/space to part (a) on iron-sulphur proteins, 40% to part (b) on fluxional molecules with detailed NMR analysis for both compounds, and 40% to part (c) with systematic calculation showing all steps. For (b), explicitly state the directive 'How would you account for bonding' requires explaining the dynamic processes and temperature-dependent NMR coalescence phenomena. Conclude with a brief synthesis if time permits, though not mandatory.

• Part (a): Structures of [2Fe-2S], [4Fe-4S], and [3Fe-4S] clusters with correct oxidation states (Fe²⁺/Fe³⁺) and redox couples (e.g., [2Fe-2S]²⁺/⁺, [4Fe-4S]²⁺/⁺)
• Part (a): Recognition that ferredoxins and high-potential iron proteins (HiPIPs) represent different redox families with distinct cluster types
• Part (b)(i): Explanation of ring-whizzing/ring rotation in (C₅H₅)₄Ti with η¹↔η⁵ hapticity interchange, showing single ¹H NMR signal at room temperature due to rapid exchange
• Part (b)(ii): Analysis of C₃(CH₃)₄Fe(CO)₄ as a trimethylenemethane (TMM) complex with η⁴-bonding, explaining fluxionality via Berry pseudorotation or TMM rotation, and temperature-dependent NMR showing methyl equivalence
• Part (c): Correct calculation of photon energy E = hc/λ = (6.626×10⁻³⁴ × 3×10⁸)/(2500×10⁻¹⁰) = 7.95×10⁻¹⁹ J per photon
• Part (c): Moles of oxalic acid decomposed = (0.05-0.04) × 0.01 = 10⁻⁴ mol; moles of photons absorbed = 80/(7.95×10⁻¹⁹ × 6.022×10²³) = 1.67×10⁻⁴ mol; quantum yield Φ = 0.6
• Part (c): Recognition that uranyl sulphate acts as photosensitizer in the uranyl-oxalate actinometer system, a classic photochemical application

Begin with a brief introduction acknowledging the diverse nature of the three sub-parts covering interhalogen chemistry, lanthanide magnetism, and coordination chemistry. Allocate approximately 20% effort to part (a) on BrF₃ solvent chemistry, 40% to part (b) on lanthanide magnetic moments including calculations, and 40% to part (c) on NO bonding modes with structures. For each part, define key terms first, then provide explanations with equations or diagrams, and conclude with specific examples. Use chemical equations for acid-base reactions in (a), show μeff calculations for (b), and draw structures for (c).

• Part (a): BrF₃ undergoes autoionization as 2BrF₃ ⇌ BrF₂⁺ + BrF₄⁻, establishing it as an aprotic ionizing solvent; identification of BrF₂⁺ as acid and BrF₄⁻ as base
• Part (a): Specific acid-base reaction examples such as SbF₅ + BrF₃ → BrF₂⁺ + SbF₆⁻ (acid) and KF + BrF₃ → K⁺ + BrF₄⁻ (base), plus neutralization: BrF₂⁺ + BrF₄⁻ → 2BrF₃
• Part (b): Explanation that lanthanide magnetic moments differ due to strong spin-orbit coupling (J states) versus quenched orbital contribution in first-row TM; μeff = gJ√[J(J+1)] for lanthanides vs spin-only μso = √[n(n+2)] BM for first-row TM
• Part (b): Lanthanide ions with spin-only values: La³⁺ (4f⁰), Gd³⁺ (4f⁷, L=0, J=S=7/2), Lu³⁺ (4f¹⁴); diamagnetic ions: La³⁺ and Lu³⁺ (both μ = 0 BM)
• Part (c): Linear NO bonding as NO⁺ (nitrosyl, 2-electron donor, triple bond character, M-N≡O ~180°) with M-N bond order ~2-3; bent NO as NO⁻ (nitrosyl anion, 1-electron donor, M-N=O ~120°) with M-N bond order ~1-2
• Part (c): Application of Enemark-Feltham notation to [Ru(PPh₃)₂Cl(NO)₂]⁺: {Ru(NO⁺)(NO⁻)}⁺ or {Ru(NO)₂}⁷ configuration; electron counting shows one NO⁺ (linear) and one NO⁻ (bent) to satisfy 18-electron rule with Ru(II) or Ru(III) oxidation state analysis