Physics 2022 Paper II 50 marks Calculate

Q4

(a) (i) In a diatomic molecule when one constituent atom is replaced by one of its heavier isotopes, what change takes place in the rotational spectrum ? (ii) Calculate the change in rotational constant B when hydrogen is replaced by deuterium in the hydrogen molecule. (iii) Draw the spectra of rigid and non-rigid rotors by using the schematic representation of the rotational energy levels and comment on it. 20 marks (b) (i) Briefly explain the effect of anharmonicity on the vibrational spectra of diatomic molecules. (ii) Calculate the average period of rotation of HCl molecule if it is in the J = 3 state. The internuclear distance and the moment of inertia of HCl are 0·1274 nm and 0·0264×10⁻⁴⁵ kg.m² respectively. 15 marks (c) Obtain the normalized eigenvectors of σₓ and σᵧ matrices. 15 marks

हिंदी में प्रश्न पढ़ें

(a) (i) एक द्विपरमाणुक अणु में जब एक घटक परमाणु को उसके भारी समस्थानिकों में से एक द्वारा प्रतिस्थापित किया जाता है तो घूर्णी स्पेक्ट्रम में क्या परिवर्तन होते हैं ? (ii) जब हाइड्रोजन अणु में हाइड्रोजन को ड्यूटेरियम द्वारा प्रतिस्थापित किया जाता है तो घूर्णी स्थिरांक B में परिवर्तन की गणना कीजिए । (iii) घूर्णी ऊर्जा स्तरों के योजनाबद्ध निरूपण का उपयोग करके कठोर और गैर-कठोर रोटरों का स्पेक्ट्रा बनाइए और उस पर टिप्पणी कीजिए । 20 अंक (b) (i) द्विपरमाणुक अणुओं के कंपनिक स्पेक्ट्रा पर अप्रसंवादिता (एनहार्मोनिसिटी) के प्रभाव को संक्षेप में समझाइए । (ii) HCl अणु के घूमने (घूर्णन) की औसत अवधि की गणना कीजिए यदि यह J = 3 अवस्था में है । HCl की अंतरा-अणुक दूरी और जड़त्व-आघूर्ण क्रमशः: 0·1274 nm और 0·0264×10⁻⁴⁵ kg.m² है । 15 अंक (c) σₓ और σᵧ आव्यूहों के सामान्यीकृत अभिलक्षणिक सदिशों को प्राप्त कीजिए । 15 अंक

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Directive word: Calculate

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How this answer will be evaluated

Approach

Begin with the directive 'calculate' for the numerical sub-parts (a)(ii), (b)(ii), and (c), while using 'explain', 'draw', and 'obtain' for the descriptive sub-parts. Allocate approximately 40% of time/effort to part (a) given its 20 marks, 30% each to parts (b) and (c). Structure as: brief conceptual introduction → systematic treatment of each sub-part with clear labeling → concluding synthesis on how isotope effects and quantum mechanical treatments reveal molecular structure.

Key points expected

  • For (a)(i): Explain that isotopic substitution increases reduced mass μ, decreases rotational constant B (B ∝ 1/μ), and compresses rotational energy levels leading to closer-spaced spectral lines
  • For (a)(ii): Calculate B_H₂ = ℏ/(4πcI_H₂) and B_D₂ = ℏ/(4πcI_D₂), showing I_D₂ = 2I_H₂ due to doubled reduced mass, hence B_D₂ = B_H₂/2, or explicitly compute numerical values
  • For (a)(iii): Draw energy level diagrams showing equally spaced levels for rigid rotor vs. diverging levels for non-rigid rotor (centrifugal distortion), with corresponding spectral line spacing
  • For (b)(i): Explain anharmonicity introduces cubic term in potential, causes energy level convergence, overtone frequencies as ν₀(1-2x_e), (ν₀-4ν₀x_e), etc., and decreased spacing at higher v
  • For (b)(ii): Calculate rotational period T = 2π/ω = 2πI/√[J(J+1)]ℏ using given I = 0.0264×10⁻⁴⁵ kg.m² and J=3, obtaining T ≈ 1.3×10⁻¹³ s or equivalent
  • For (c): Derive normalized eigenvectors of σₓ = (0 1; 1 0) as (1/√2)(1; ±1) and σᵧ = (0 -i; i 0) as (1/√2)(1; ±i), showing explicit normalization and orthogonality

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies that isotopic substitution affects rotational constant through reduced mass (not total mass), distinguishes rigid vs non-rigid rotor models physically, explains anharmonicity as deviation from parabolic potential, and correctly interprets Pauli matrices as spin observablesIdentifies general trends (heavier isotope → smaller B) but confuses total mass with reduced mass; mentions centrifugal distortion without clear physical cause; describes anharmonicity vaguely; treats σ matrices as abstract without spin connectionFundamental errors like claiming isotopic substitution doesn't affect rotational spectrum, confusing vibrational and rotational effects, or treating σ matrices as regular 2×2 matrices without quantum significance
Derivation rigour20%10Shows complete derivation: E_J = ℏ²J(J+1)/2I for rigid rotor, includes centrifugal distortion term -D̃J²(J+1)² for non-rigid; derives eigenvalue equations det(σₓ-λI)=0 and det(σᵧ-λI)=0 explicitly; all steps logically connectedStates key formulas without full derivation; skips intermediate steps in eigenvalue problem; presents results without showing characteristic equation solution; some logical gaps but correct final expressionsNo derivations shown, only final formulas stated; or contains major errors in derivation (wrong characteristic equation, incorrect eigenvalue substitution, algebraic mistakes in normalization)
Diagram / FBD15%7.5Clear schematic for (a)(iii) showing: rigid rotor with equal spacing (E ∝ J(J+1)) and spectrum with constant Δν; non-rigid rotor with level spacing decreasing at high J and correspondingly converging spectral lines; properly labeled axes and transitionsRough sketches present but missing key distinctions; or only one diagram shown; labels incomplete; fails to show spectral consequences (absorption lines) alongside energy levelsNo diagrams despite explicit 'draw' instruction; or completely incorrect diagrams (e.g., showing vibrational levels instead of rotational, or linear energy spacing for rigid rotor)
Numerical accuracy25%12.5Precise calculations: (a)(ii) B ratio exactly 1/2 or B values ~60.8 cm⁻¹ and ~30.4 cm⁻¹; (b)(ii) period calculation with correct J(J+1)=12 factor, proper unit conversion of I and r, result ~1.3×10⁻¹³ s; eigenvector normalization factors exactly 1/√2Correct method but arithmetic errors; or correct final answers with missing units; order-of-magnitude correct but significant figures mishandled; eigenvectors correct but normalization shown incompletelyOrder-of-magnitude errors (wrong powers of 10); incorrect formulas used (e.g., using B = ℏ²/2I without c for wavenumber); completely wrong numerical results; eigenvectors unnormalized or with wrong components
Physical interpretation20%10Interprets isotope shift as probe of bond strength and molecular structure; connects non-rigid rotor to real molecular bond stretching; explains anharmonicity leads to dissociation limit; relates σ matrices to spin-½ measurements and Stern-Gerlach experiment; mentions applications like IR spectroscopy of atmospheric molecules or H/D isotope effects in astrochemistrySome physical interpretation present but superficial; mentions applications without elaboration; or interpretation confined to one sub-part while others remain purely mathematicalPurely mathematical/computational answer with no physical insight; fails to connect any results to observable phenomena or experimental techniques

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