Chemistry 2024 Paper I 50 marks Compulsory Calculate

Q1

(a) (i) Determine the distance from the nucleus at which the electron is most expected in the hydrogen atom in its ground state. [Given : The normalized radial function for hydrogen-like systems is R₁₀(1s) = 2(Z/a₀)^(3/2) · e^(-ρ/2) where ρ = 2Zr/na₀ and a₀ is the first Bohr orbit radius. Other notations have their usual meanings.] (5 marks) (ii) Sodium bromide and sodium iodide have higher lattice energies than expected from theoretical calculations. Justify. (5 marks) (b) (i) The ²³⁵U isotope undergoes fission when bombarded with neutrons. However, its natural abundance is only 0·72 percent. To separate it from more abundant ²³⁸U isotope, U is first converted to UF₆, which is easily vaporized above room temperature. The mixture of ²³⁵UF₆ and ²³⁸UF₆ gases is then subjected to many stages of effusion. Calculate the separation factor, that is enrichment of ²³⁵U relative to ²³⁸U after one stage of effusion. (5 marks) (ii) Define 'unit cell'. Draw all the Bravais lattices for a cubic system. (5 marks) (c) Use the following data to determine the normal boiling point of mercury. What assumptions must you make in order to do the calculations ? Hg (l) ΔH°f = 0, S° = 77·4 J/K mol; Hg (g) ΔH°f = 60·78 kJ/mol, S° = 174·7 J/K mol (10 marks) (d) (i) The compound dichlorodifluoromethane (CCl2F2) has a normal boiling point of – 30°C, a critical temperature of 112°C, and a corresponding critical pressure of 40 atm. If the gas is compressed to 18 atm at 20°C, will the gas condense ? Give your answer on the basis of graphical presentation. (5 marks) (ii) Define overvoltage. Mention the applications of overvoltage. (5 marks) (e) The activation energy for the decomposition of hydrogen peroxide 2H2O2 (aq) → 2H2O (l) + O2 (g) is 42 kJ/mol, whereas when the reaction is catalyzed by enzyme catalase, it is 7·0 kJ/mol. Calculate the temperature that would cause the uncatalyzed reaction to proceed as rapidly as the enzyme catalized decomposition at 20°C. Assume the frequency factor A to be the same in both cases. (10 marks)

हिंदी में प्रश्न पढ़ें

(a) (i) मूल (निम्नतम) अवस्था में हाइड्रोजन परमाणु में इलेक्ट्रॉन की नाभिक से वह दूरी निर्धारित कीजिए जो सबसे अधिक अपेक्षित होती है। [दिया गया है : हाइड्रोजन जैसे निकायों का प्रसामान्यीकृत त्रिज्य फलन R₁₀(1s) = 2(Z/a₀)^(3/2) · e^(-ρ/2) जहाँ ρ = 2Zr/na₀ है और a₀ = प्रथम बोर (Bohr) कक्षा त्रिज्या है। अन्य सभी अंकों का सामान्य अर्थ है।] (5 अंक) (ii) सोडियम ब्रोमाइड और सोडियम आयोडाइड की जालक ऊर्जाओं के मान सैद्धांतिक परिकलनों के द्वारा निकाले गए मानों की अपेक्षा उच्चतर हैं। उचित सिद्ध कीजिए। (5 अंक) (b) (i) न्यूट्रॉनों के साथ बमबारी करने पर ²³⁵U समस्थानिक विखंडित हो जाता है। परंतु इसका प्राकृतिक बाहुल्य केवल 0·72 प्रतिशत है। इसको अपने से ज्यादा बाहुल्य वाले समस्थानिक ²³⁸U से अलग करने के लिए U को पहले UF₆ में बदलना पड़ता है जो कि कक्ष ताप के ऊपर आसानी से वाष्पित हो जाता है। फिर ²³⁵UF₆ और ²³⁸UF₆ गैसों के मिश्रण को निस्सरण करने के लिए कई चरणों के अधीन डाला जाता है। एक निस्सरण चरण के बाद, ²³⁵U की ²³⁸U से अपेक्षिक समृद्धि के पृथक्करण गुणांक का परिकलन कीजिए। (5 अंक) (ii) एकक सेल (कोषिका)' को परिभाषित कीजिए । एक घनीय निकाय के लिए समस्त ब्रेवे जालक खींचिए । (5 अंक) (c) निम्नलिखित आँकड़ों का प्रयोग करके, पारे (mercury) का सामान्य क्वथनांक निर्धारित कीजिए । परिकलन करने के लिए आपको कौन-कौन-सी कल्पनाएँ करनी पड़ेंगी ? Hg (l) ΔH°f = 0, S° = 77·4 J/K mol; Hg (g) ΔH°f = 60·78 kJ/mol, S° = 174·7 J/K mol (10 अंक) (d) (i) डाइक्लोरोडाइफ्लुओरोमेथेन (CCl2F2) यौगिक का सामान्य क्वथनांक – 30°C है, क्रांतिक ताप 112°C है, और उसके अनुरूप क्रांतिक दाब 40 atm है । यदि गैस को 20°C ताप पर 18 atm दाब तक संपीड़ित किया जाए, तो क्या गैस संघनित होगी ? आलेखीय निरूपण के आधार पर अपना उत्तर दीजिए । (5 अंक) (ii) अधिवोल्टता को परिभाषित कीजिए । अधिवोल्टता के अनुप्रयोगों का उल्लेख कीजिए । (5 अंक) (e) हाइड्रोजन परॉक्साइड के अपघटन 2H2O2 (aq) → 2H2O (l) + O2 (g), की सक्रियण ऊर्जा 42 kJ/mol है लेकिन जब अभिक्रिया को एंजाइम कैटलेज से उत्प्रेरित किया जाता है तो इसकी सक्रियण ऊर्जा 7·0 kJ/mol है । उस ताप का परिकलन कीजिए जिस पर अनुत्प्रेरित अभिक्रिया उसी शीघ्रता पर चले, जिस पर एंजाइम उत्प्रेरित अपघटन 20°C ताप पर चलता है । मान लीजिए आवृत्ति गुणक A, दोनों मामलों में अभिन्न/एक ही है । (10 अंक)

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Directive word: Calculate

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Approach

This multi-part question requires systematic calculation and derivation across six sub-parts spanning quantum chemistry, solid state, thermodynamics and kinetics. Allocate approximately 15% time to each 5-mark part (a)(i), (a)(ii), (b)(i), (b)(ii), (d)(i), (d)(ii) and 25% to each 10-mark part (c) and (e). Begin with concise definitions and stated assumptions, present derivations stepwise with proper units, and conclude with physical interpretation of results.

Key points expected

  • For (a)(i): Derive r_max = a₀/Z for 1s orbital by maximizing radial probability density P(r) = 4πr²|R₁₀|², showing dP/dr = 0 yields r = a₀ for hydrogen ground state
  • For (a)(ii): Explain polarizability of Br⁻ and I⁻ ions causing partial covalent character via Fajan's rules, increasing experimental lattice energy over theoretical Born-Landé values
  • For (b)(i): Apply Graham's law to calculate separation factor α = √(M₂₃₈/M₂₃₅) = √(352/349) ≈ 1.0043, showing minimal enrichment per stage necessitating multi-stage cascade
  • For (b)(ii): Define unit cell as smallest repeating unit showing full crystal symmetry; draw and label simple cubic, body-centered cubic, and face-centered cubic Bravais lattices
  • For (c): Calculate ΔH°vap = 60.78 kJ/mol and ΔS°vap = 97.3 J/K·mol; set ΔG° = 0 at equilibrium to find Tb = ΔH°/ΔS° ≈ 625 K; state assumptions of standard state, ideal gas behavior, and temperature-independent enthalpy/entropy
  • For (d)(i): Construct or describe P-T phase diagram showing critical point (112°C, 40 atm), normal boiling point (-30°C, 1 atm), and locate point (20°C, 18 atm) in liquid region confirming condensation
  • For (d)(ii): Define overvoltage as excess potential beyond theoretical for electrode reaction; cite applications in electrolytic refining, corrosion protection, and battery design
  • For (e): Apply Arrhenius equation with equal A factors; set k_uncat(T) = k_cat(293K) to derive ln(k_cat/k_uncat) = (Ea,uncat - Ea,cat)/RT_cat = Ea,uncat/RT; solve for T ≈ 335 K or 62°C

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies that radial probability maximum differs from wavefunction maximum for (a)(i); recognizes polarizability-covalency link for (a)(ii); understands effusion as kinetic phenomenon for (b)(i); distinguishes 7 Bravais lattices reducing to 3 cubic types for (b)(ii); applies phase equilibrium condition ΔG° = 0 for (c); interprets critical phenomena for (d)(i); relates overvoltage to activation energy of electrode processes for (d)(ii); recognizes equal A factor cancels in Arrhenius ratio for (e)States correct formulas but with minor conceptual gaps (e.g., confuses radial node with probability maximum, or states Graham's law without explaining isotope separation relevance)Fundamental misconceptions such as using Bohr radius directly without derivation, claiming ionic radii alone explain lattice energy discrepancies, or equating effusion with diffusion
Mechanism / equation20%10Writes complete derivations: radial probability P(r) = 4πr²|R|² with proper differentiation for (a)(i); Born-Landé equation reference for (a)(ii); Graham's law rate ratio v₁/v₂ = √(M₂/M₁) for (b)(i); precise thermodynamic cycle ΔG° = ΔH° - TΔS° = 0 for (c); Clausius-Clapeyron or van der Waals context for (d)(i); Tafel equation mention for overvoltage in (d)(ii); Arrhenius ratio with algebraic isolation of unknown temperature for (e)Presents final equations without showing intermediate derivation steps, or uses correct equations with inconsistent notationMissing critical equations (e.g., no probability density expression, no Graham's law, no Arrhenius equation) or uses entirely wrong relationships
Numerical accuracy20%10Precise calculations: r_max = a₀ (0.529 Å) for (a)(i); α = 1.0043 or enrichment factor ~1.004 for (b)(i); Tb = 624-626 K (351-353°C) for (c); T ≈ 335 K (62°C) for (e); proper unit conversions (kJ to J, °C to K) and significant figures maintained throughoutCorrect method but arithmetic errors (e.g., factor of 10 in entropy conversion, incorrect molar mass for UF₆ isotopologues) yielding slightly wrong final answersOrder-of-magnitude errors, missing unit conversions, or no numerical working despite theoretical setup
Diagram / structure20%10Clear labeled diagrams: three cubic Bravais lattices with lattice points and basis vectors for (b)(ii); P-T phase diagram with critical point, vapor pressure curve, triple point reference, and marked process path showing (20°C, 18 atm) lies in liquid region for (d)(i); schematic of gaseous diffusion cascade for (b)(i) contextDiagrams present but incomplete labeling (e.g., missing axes labels, unclear lattice point representation) or verbally described without visualNo diagrams where required (Bravais lattices, phase diagram) or entirely incorrect representations (e.g., drawing unit cells instead of lattice points)
Application context20%10Connects to Indian/UPSC-relevant applications: Kudankulam/Tarapur nuclear fuel enrichment relevance for (b)(i); mercury use in thermometers, barometers, and chlor-alkali industry hazards for (c); CFC refrigerant phase-out under Montreal Protocol for (d)(i); catalase in biological systems and industrial H₂O₂ decomposition for (e); overvoltage in India's aluminum smelting and hydrogen economy initiatives for (d)(ii)Mentions generic applications without specific Indian context or contemporary relevanceNo application context provided; purely theoretical treatment without real-world significance

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