Chemistry 2021 Paper I 50 marks Calculate

Q2

(a) Hydrogen atoms are observed to have radiative transitions from n = 101 to n = 100 to occur. (i) What are the frequency and wavelength of the radiation emitted in this transition? (ii) Why is it difficult to observe this transition? (10 marks) (b) Draw the geometrical arrangements for the following hybridized systems and identify the type of d-orbitals involved in each system : sp³d, sp³d², dsp², sd³ (20 marks) (c) Iron crystallizes in a b.c.c. unit cell at room temperature (ρ = 7·86 g/cm³). Calculate the radius of an iron atom in this crystal. At temperatures more than 910 °C, iron prefers to be in f.c.c. If we neglect the temperature dependence of radius of iron on the grounds that it is negligible, use this information to determine whether iron expands or contracts when it undergoes transformation from b.c.c. to f.c.c. structure. The atomic mass of iron is 55·845 u. (20 marks)

हिंदी में प्रश्न पढ़ें

(a) हाइड्रोजन परमाणु में विकिरणी संक्रमण n = 101 से n = 100 पाए जाने का अवलोकन किया गया। (i) इस संक्रमण में उत्सर्जित विकिरण की आवृत्ति और तरंगदैर्घ्य क्या है? (ii) इस संक्रमण का अवलोकन करना क्यों मुश्किल है? (10 अंक) (b) निम्नलिखित संकरित समुदायों के लिए ज्यामितीय व्यवस्थाओं को खींचिए और अभिनिर्धारित कीजिए कि प्रत्येक समुदाय में किस प्रकार का d-क्षक सम्मिलित है : sp³d, sp³d², dsp², sd³ (20 अंक) (c) लोहा कक्ष ताप पर b.c.c. एकक सेल में क्रिस्टलित होता है (ρ = 7·86 g/cm³)| इस क्रिस्टल में लोहा परमाणु की त्रिज्या का परिकलन कीजिए। 910 °C तापमान से ऊपर लोहा f.c.c. को प्राथमिकता/तर्जीह देता है। अगर हम लोहे की त्रिज्या की तापमान पर निर्भरता को इस आधार पर छोड़ दें कि वह उपेक्षणीय है, इस जानकारी का प्रयोग करके निर्धारित कीजिए कि लोहा जब b.c.c. से f.c.c. संरचना में रूपांतरण करेगा, तो वह प्रसारित होगा या आकुंचित। लोहे का परमाणविक द्रव्यमान 55·845 u है। (20 अंक)

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Approach

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.

Key points expected

  • 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

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies Rydberg series for high-n hydrogen (radio frequency transitions), distinguishes between valence d-orbital participation vs core d-orbital usage in hybridization, and understands that b.c.c. to f.c.c. is a reconstructive phase transition with coordination number change from 8 to 12States basic Rydberg formula and hybridization shapes but confuses which d-orbitals participate (e.g., claims d_xy for sp³d) or misidentifies sd³ as valid; recognizes b.c.c. and f.c.c. differ but cannot explain why contraction occursApplies Rydberg formula to atoms other than hydrogen, invents non-existent hybridizations without orbital basis, or claims expansion occurs due to higher temperature without analyzing packing efficiency
Mechanism / equation20%10Writes complete Rydberg equation with reduced mass correction mentioned, shows ΔE = hν derivation, explicitly writes density formula ρ = zM/(N_Aa³) for both structures, and relates atomic radius to unit cell edge through correct geometric factors (√3/4 for b.c.c., √2/4 for f.c.c.)Uses correct formulas but with minor errors (e.g., forgets factor of 2 in b.c.c. atoms per cell, uses wrong geometric factor), or states formulas without derivationUses Bohr radius formula incorrectly for transition energy, applies molecular orbital theory to hybridization, or invents incorrect relationship r = a/2 for both structures
Numerical accuracy20%10Obtains λ ≈ 0.045 m (4.5 cm, radio wave), ν ≈ 6.7 GHz for (a); calculates a_bcc ≈ 2.87 Å, r_Fe ≈ 1.24 Å; computes a_fcc ≈ 3.51 Å and shows density ratio or explicit percentage contraction (~2.5% in linear dimension, ~7% volume decrease)Correct order of magnitude for wavelength (cm range) and radius (~1.2 Å) with minor calculation errors; attempts f.c.c. calculation but arithmetic errors in final comparisonWavelength in visible/UV range (fundamental error), radius > 2 Å or < 0.5 Å, or concludes expansion without calculation; unit conversion errors (g/cm³ to kg/m³) causing 10³ errors
Diagram / structure20%10Four clear 3D diagrams with labeled axes, bond angles (90°, 120°, 180° for sp³d; 90° for sp³d²; 90° for dsp²; 109.5° for sd³), axial/equatorial distinction, and explicit d-orbital labeling (d_z², d_x²-y²); uses wedge-dash or ball-and-stick representationTwo-dimensional representations of geometries with correct shapes but missing angle labels or orbital specifications; diagrams understandable but lack clarity in 3D perspectiveSingle generic 'd-orbital' label without specification, confuses tetrahedral with square planar, or draws linear/sp geometries instead of requested hybridizations; no diagrams for part (b)
Application context20%10Connects (a) to radio astronomy (HI 21-cm line, high-n recombination lines in HII regions like Orion Nebula); links (b) to VSEPR examples (PCl₅, SF₆, [Ni(CN)₄]²⁻); relates (c) to steel phase transitions (α-Fe to γ-Fe at 910°C), industrial heat treatment, and why f.c.c. γ-Fe dissolves more carbon (interstitial sites)Mentions one relevant example per part (e.g., only PCl₅ for sp³d, only steel for iron) without elaborating significance; states applications without connecting to calculated resultsNo real-world connections, or invents irrelevant applications (e.g., claims hydrogen transition used in medical imaging, or hybridization explains nuclear fission)

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