Chemistry 2024 Paper I 50 marks Solve

Q2

(a) State Heisenberg's uncertainty principle. Show that for a particle in a one-dimensional box having length from 0 to L, the two normalised eigen functions corresponding to the eigenvalues E₁ and E₂ (characterised by quantum numbers 1 and 2 respectively) are orthogonal to each other. (10 marks) (b) (i) Calculate the bond order for the following : (I) Oxygen, (II) Superoxide, (III) Peroxide, (IV) Dioxygenyl ion. Which will have the highest stability ? (10 marks) (ii) Draw the molecular orbital diagram for CO. (10 marks) (c) (i) Calculate the limiting radius ratio for the crystals with coordination number 3 and 6. (10 marks) (ii) Explain stoichiometric defects with an example. (10 marks)

हिंदी में प्रश्न पढ़ें

(a) हाइजेनबर्ग अनिश्चितता सिद्धांत का उल्लेख कीजिए । प्रदर्शित कीजिए कि एक-विमीय बॉक्स, जिसकी लंबाई 0 से L तक है, उसमें एक कण के लिए दो प्रसामान्यीकृत आइगेन फलन जो कि आइगेनमान E₁ और E₂ (क्रमशः क्वांटम संख्या 1 और 2 द्वारा विशेषीकृत) के तदनुसार हैं, एक दूसरे के लंबकोणीय हैं । (10 अंक) (b) (i) निम्नलिखित के लिए आबंध क्रम का परिकलन कीजिए : (I) ऑक्सीजन, (II) सुपरऑक्साइड, (III) परोक्साइड, (IV) डाइऑक्सिजेनिल आयन। इनमें से किसका स्थायित्व उच्चतम होगा ? (10 अंक) (ii) CO के आणविक कक्षक आरेख को खींचिए । (10 अंक) (c) (i) समन्वय संख्या 3 और 6 वाले क्रिस्टलों के सीमांत त्रिज्या अनुपात का परिकलन कीजिए । (10 अंक) (ii) एक उदाहरण सहित स्टॉइकियोमीट्री दोषों की व्याख्या कीजिए । (10 अंक)

Evaluate your answer to this question → See all 2024 Chemistry questions

Directive word: Solve

This question asks you to solve. The directive word signals the depth of analysis expected, the structure of your answer, and the weight of evidence you must bring.

See our UPSC directive words guide for a full breakdown of how to respond to each command word.

How this answer will be evaluated

Approach

Begin with a concise statement of Heisenberg's uncertainty principle in part (a), then rigorously prove orthogonality of ψ₁ and ψ₂ wavefunctions through integration. For part (b), systematically calculate bond orders using molecular orbital configurations, explicitly showing electron counts for O₂, O₂⁻, O₂²⁻ and O₂⁺, then construct the MO diagram for CO with proper energy level ordering and mixing. In part (c), derive limiting radius ratios geometrically for CN=3 (trigonal planar) and CN=6 (octahedral), then explain Schottky and Frenkel defects with NaCl/AgBr examples. Allocate approximately 20% time to (a), 40% to (b), and 40% to (c) based on mark distribution.

Key points expected

  • Part (a): Correct statement of Heisenberg's uncertainty principle (Δx·Δp ≥ h/4π) and complete mathematical proof of orthogonality showing ∫₀ᴸ ψ₁ψ₂ dx = 0 using trigonometric identities
  • Part (b)(i): Accurate MO electron configurations for all four oxygen species, correct bond order calculations (O₂=2, O₂⁺=2.5, O₂⁻=1.5, O₂²⁻=1), and correct identification of O₂⁺ (dioxygenyl ion) as most stable
  • Part (b)(ii): Proper MO diagram for CO showing: (a) correct energy ordering with σ2p below π2p due to s-p mixing, (b) 10 valence electrons properly filled, (c) bond order = 3, (d) polarity indication with electron density shift toward oxygen
  • Part (c)(i): Geometric derivation of limiting radius ratios: r⁺/r⁻ = 0.155 for CN=3 (planar triangular void) and 0.414 for CN=6 (octahedral void), with clear geometric constructions
  • Part (c)(ii): Clear distinction between Schottky defect (equal cation-anion vacancies, e.g., NaCl) and Frenkel defect (cation displacement to interstitial sites, e.g., AgBr, ZnS), with effects on density and conductivity
  • Cross-connection: Mention of how MO theory explains paramagnetism of O₂ (unpaired electrons in π*2p orbitals) and its industrial relevance in steel manufacturing
  • Cross-connection: Application of radius ratio rules to predict structures of common Indian minerals like calcite (CaCO₃) or corundum (Al₂O₃)
  • Cross-connection: Brief mention of how crystal defects enable ionic conductivity in solid-state batteries relevant to India's energy storage research

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Precise statement of uncertainty principle with correct mathematical form; accurate MO energy level ordering for both homonuclear (O₂) and heteronuclear (CO) diatomics; correct classification of defects with proper thermodynamic reasoningCorrect principle statements but minor errors in MO energy ordering (e.g., missing s-p mixing in CO) or incomplete defect classification without distinguishing Schottky/Frenkel mechanismsConfused statements of uncertainty principle; reversed MO energy levels; conflation of defect types or incorrect examples
Mechanism / equation20%10Complete step-by-step derivation: integration of ψ₁ψ₂ = √(2/L)sin(πx/L) × √(2/L)sin(2πx/L) with proper use of 2sinA sinB identity; explicit MO electron filling with σ1s² σ*1s²... notation; geometric derivation of radius ratios using trigonometric relationsCorrect final equations but skipped derivation steps; incomplete MO notation; stated radius ratio values without geometric proofMissing or incorrect derivations; wrong integration approach; no MO electron configurations; no geometric basis for radius ratios
Numerical accuracy20%10All calculations verified: orthogonality integral evaluates to exactly zero; bond orders correct to one decimal place (2.5, 2, 1.5, 1); radius ratios precise to three decimal places (0.155, 0.414); correct stability orderingMinor arithmetic errors in one calculation; correct bond order values but wrong stability conclusion; approximate radius ratiosMultiple calculation errors; incorrect bond orders; wrong radius ratio values; confused stability ranking
Diagram / structure20%10Clear MO diagram for CO with: (a) atomic orbitals on left (C) and right (O) with correct relative energies, (b) molecular orbitals in center with proper σ/π labeling and symmetry, (c) 10 electrons shown with arrows, (d) bond order calculation shown; neat geometric sketches for radius ratio derivationsMO diagram present but missing s-p mixing indication or incorrect energy spacing; electron count correct but spin directions unclear; geometric sketches without labelsAbsent or severely flawed MO diagram; no geometric constructions; illegible or chemically incorrect orbital drawings
Application context20%10Explicit connections: paramagnetism of O₂ to liquid oxygen's magnetic properties; CO MO diagram explaining its toxicity (lone pair donation) and use in steel production; radius ratio rules predicting structures of Indian minerals (corundum from Odisha, calcite from Rajasthan); defect chemistry linked to solid electrolytes for batteriesBrief mention of one application (e.g., O₂ paramagnetism) without elaboration; generic statements about defect importance without specific examplesNo real-world applications mentioned; purely theoretical treatment without context of materials properties or industrial relevance

Practice this exact question

Write your answer, then get a detailed evaluation from our AI trained on UPSC's answer-writing standards. Free first evaluation — no signup needed to start.

Evaluate my answer →

More from Chemistry 2024 Paper I