Chemistry 2021 Paper I 50 marks Compulsory Solve

Q1

(a) Calculate ⟨x⟩, the expectation value of position of a particle, in the ground state of one-dimensional box having length from 0 to l. (10 marks) (b) The enthalpy of formation of an ionic compound can be calculated with accuracy by Born-Haber cycle. Predict, giving valid reasons, the possibility of formation of NaCl₂ salt. The following thermodynamic data are given for NaCl and NaCl₂ : NaCl U₀ = −757 kJ mol⁻¹ ΔH₍IE₎ = +495 kJ mol⁻¹ ΔH₍EA₎ = −348 kJ mol⁻¹ ½ ΔH₍diss₎ = +121 kJ mol⁻¹ ΔH₍sub₎ = +108 kJ mol⁻¹ NaCl₂ U₀ = −2155 kJ mol⁻¹ ΔH₍IE₂₎ = +4561 kJ mol⁻¹ (10 marks) (c) For a rubber band $$ \left(\frac{\partial T}{\partial l}\right)_S = -\frac{T}{C_V}\left(\frac{\partial S}{\partial l}\right)_T $$ What would be the length of the rubber band with an increase in temperature? Explain it. (10 marks) (d) The vapour pressure of water at 95 °C is found to be 634 mm. What would be the vapour pressure at a temperature of 100 °C? The heat of vapourization in this range of temperature may be taken as 40593 J mol⁻¹. [R = 8·314 J K⁻¹ mol⁻¹] (10 marks) (e) Define electrochemical series. Give its significance. (10 marks)

हिंदी में प्रश्न पढ़ें

(a) एक-विमीय बॉक्स, जिसकी लंबाई 0 से l है, में एक कण जो कि मूल अवस्था में है, की स्थिति के अपेक्षा मान, ⟨x⟩, का परिकलन कीजिए। (10 अंक) (b) आयनी यौगिक की संभव एन्थैल्पी यथार्थता से बॉर्न-हाबर चक्र से परिकलित की जा सकती है। NaCl₂ लवण के संभव की संभावना मान्य कारणों के साथ सूचित कीजिए। निम्नलिखित ऊष्मागतिक आँकड़ा NaCl और NaCl₂ के लिए दिया गया है : NaCl U₀ = −757 kJ mol⁻¹ ΔH₍IE₎ = +495 kJ mol⁻¹ ΔH₍EA₎ = −348 kJ mol⁻¹ ½ ΔH₍diss₎ = +121 kJ mol⁻¹ ΔH₍sub₎ = +108 kJ mol⁻¹ NaCl₂ U₀ = −2155 kJ mol⁻¹ ΔH₍IE₂₎ = +4561 kJ mol⁻¹ (10 अंक) (c) एक रबड़ बैंड के लिए $$ \left(\frac{\partial T}{\partial l}\right)_S = -\frac{T}{C_V}\left(\frac{\partial S}{\partial l}\right)_T $$ तापमान बढ़ाने से इस रबड़ बैंड की लंबाई क्या होगी? व्याख्या कीजिए। (10 अंक) (d) 95 °C पर जल का वाष्प-दाब 634 mm पाया गया। 100 °C तापमान पर कितना वाष्प-दाब होगा? इस तापमान परिसर में वाष्पन ऊष्मा 40593 J mol⁻¹ ली गई है। [R = 8·314 J K⁻¹ mol⁻¹] (10 अंक) (e) वैद्युत रासायनिक श्रेणी की परिभाषा दीजिए। इसकी सार्थकता बताइए। (10 अंक)

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Directive word: Solve

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How this answer will be evaluated

Approach

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.

Key points expected

  • 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)

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness25%12.5Demonstrates flawless understanding of quantum mechanical operators, Born-Haber cycle energetics, thermodynamic potentials for elastomers, phase equilibria, and electrochemical principles; correctly identifies why NaCl₂ formation is unfavorable and explains rubber's entropy-driven elasticityShows basic familiarity with concepts but makes minor errors in interpreting thermodynamic feasibility or misapplies quantum mechanical postulates; may confuse endothermic/exothermic contributions in cycleFundamental misconceptions about expectation values, treats Born-Haber cycle as simple addition without energy cycle logic, or completely misunderstands entropy-elasticity relationship in rubber
Mechanism / equation25%12.5Writes complete and correct equations: normalized wavefunction, full Born-Haber equation ΔHf = ΔHsub + IE + ½ΔHdiss + EA + U₀, Maxwell relation for rubber, integrated Clausius-Clapeyron form, and Nernst equation context for electrochemical seriesEquations mostly correct but missing key terms (e.g., forgets lattice energy sign convention) or uses incorrect form of Clausius-Clapeyron; partial Born-Haber expressionsIncorrect or missing equations, wrong operator for position expectation value, confuses ΔG with ΔH in feasibility analysis, or uses linear approximation instead of logarithmic form
Numerical accuracy25%12.5All calculations precise: ⟨x⟩ = l/2 derived rigorously, ΔHf(NaCl) ≈ -411 kJ/mol and ΔHf(NaCl₂) ≈ +2247 kJ/mol showing instability, correct temperature conversion (368K, 373K), final vapour pressure ≈ 760 mmHg or 759.5 mmHg with proper significant figuresCorrect methodology but arithmetic errors or unit conversion mistakes (e.g., uses Celsius in Clausius-Clapeyron); correct trend but wrong magnitude in Born-Haber comparisonMajor calculation errors, incorrect integration limits, wrong lattice energy comparison, or completely unrealistic vapour pressure values; no working shown
Diagram / structure15%7.5Clear Born-Haber cycle diagram with labeled arrows for each energy term (subimation, ionization, dissociation, electron affinity, lattice energy); potential energy curve sketch for NaCl vs NaCl₂ comparison; electrochemical series table with standard valuesBasic cycle diagram present but missing labels or arrows; incomplete series representation; diagrams lack clarity but convey essential structureNo diagrams despite cycle requirement; messy or misleading representations; completely omits visual component for parts where diagrams are standard
Application context10%5Connects electrochemical series to practical Indian contexts: extraction of aluminum (BALCO, NALCO), zinc/copper displacement in metallurgy, corrosion protection of Mumbai-Ahmedabad bullet train infrastructure; explains rubber band behavior in automobile tires and engineering applicationsMentions general applications without specific Indian examples; standard textbook significance without contemporary relevanceNo application context provided; purely theoretical treatment; fails to explain why understanding these concepts matters for technology or industry

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