Chemistry 2022 Paper I 50 marks Calculate

Q2

2.(a) An automobile tyre contains air at 320×10³ Pa at 20°C. The stem valve is removed and the air is allowed to expand adiabatically against a constant external pressure of 100×10³ Pa until P = P_external. For air, C_v, m = 5/2 R. Calculate the final temperature of the gas in the tyre. Assume ideal gas behaviour. (10 marks) 2.(b) A particle is in the nth energy state, φ_n(x), of an infinite square well potential with width L (box size from O to L). Calculate the probability that the particle is confined to the first 1/a of the width of the well. (20 marks) 2.(c) In a certain material of simple cubic structure, (100) diffraction is obtained at θ = 14·88° with radiation of λ = 1·541 Å. Can this material accommodate an atom of 1·08 Å radius interstitially in void space without lattice distortion ? [Sin 14·88 = 0·257] (20 marks)

हिंदी में प्रश्न पढ़ें

2.(a) एक ऑटोमोबाइल टायर में 20°C पर वायु का दबाव 320×10³ Pa है । स्टेम वाल्व को हटाने पर वायु को एक अपरिवर्ती बाह्य दबाव, 100×10³ Pa के विरुद्ध रुद्धोष्म के आधार पर प्रसारित होने की छूट मिल गई, जब तक वायु का दबाव, P, अपरिवर्ती बाह्य दबाव, P_external, के समान नहीं हो जाता, (P = P_external) । वायु के लिए C_v, m = 5/2 R टायर में गैस के अंतिम तापमान का परिकलन कीजिए । आदर्श गैस आचरण मान लीजिए । (10 अंक) 2.(b) एक अंत वर्ग विभव कूप, जिसकी चौड़ाई, L, है (बॉक्स का आकार O से L है), उसमें एक कण अपनी nth ऊर्जा अवस्था, φ_n(x), में है । अगर कण कूप की चौड़ाई के प्रथम 1/a भाग में सीमित है, तो कण की इस भाग में होने की प्रायिकता का परिकलन कीजिए । (20 अंक) 2.(c) सरल घन संरचना के किसी पदार्थ में विकिरण, जिसकी तरंगदैर्घ्य, λ = 1·541 Å के साथ, (100) विवर्तन, θ = 14·88° पर प्राप्त किया गया । बिना जालक विक्षेपण के क्या यह पदार्थ, एक परमाणु, जिसका त्रिज्या 1·08 Å है को अपने अंतराकाशी रिक्त स्थान में समायोजित कर सकता है ? [Sin 14·88 = 0·257] (20 अंक)

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Directive word: Calculate

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How this answer will be evaluated

Approach

Calculate requires systematic numerical problem-solving across three distinct physical chemistry domains. Structure the answer by tackling each sub-part sequentially: (a) adiabatic irreversible expansion using first law, (b) quantum probability integration for infinite square well, and (c) Bragg's law analysis with void geometry. Present clear step-wise derivations with proper unit handling and physical reasoning before final numerical answers.

Key points expected

  • Part (a): Apply first law for adiabatic irreversible expansion against constant external pressure: q=0, so ΔU = -W_ext, leading to nCv(T2-T1) = -P_ext(V2-V1) with ideal gas substitution
  • Part (b): Set up probability integral P = ∫₀^(L/a) |φₙ(x)|²dx = (2/L)∫₀^(L/a) sin²(nπx/L)dx and evaluate to [1/a - sin(2nπ/a)/(2nπ)]
  • Part (c): Apply Bragg's law nλ = 2d sinθ to find a = λ/(2sinθ) = 3.00 Å, then compare octahedral void radius (0.414a = 1.24 Å) vs tetrahedral (0.225a = 0.68 Å) with atomic radius 1.08 Å
  • Clear identification of irreversible adiabatic vs reversible process in part (a) - must NOT use TV^(γ-1) = constant
  • Proper handling of quantum number n as general variable in part (b), not assuming ground state

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness25%12.5Correctly identifies irreversible adiabatic expansion in (a) distinguishing from reversible adiabatic; recognizes probability interpretation of |ψ|² in (b); correctly identifies simple cubic structure and void types in (c) with proper radius ratio geometryMixes up reversible and irreversible adiabatic conditions in (a); treats probability classically or misses normalization in (b); confuses SC with BCC/FCC void geometry in (c)Applies isothermal or reversible adiabatic equations to (a); fundamental misunderstanding of quantum probability or uses classical probability in (b); incorrect structure identification or void geometry in (c)
Mechanism / equation25%12.5Uses correct first law equation nCv,m(T₂-T₁) = -P_ext(nRT₂/P₂ - nRT₁/P₁) for (a); sets up proper integral with correct wavefunction φₙ(x) = √(2/L)sin(nπx/L) for (b); applies Bragg's law with correct d-spacing for (100) planes in SCCorrect final equation but unclear derivation; minor errors in wavefunction form or integration limits; correct Bragg's law but confusion about which planesUses wrong thermodynamic relation (e.g., PV^γ = constant for irreversible); incorrect wavefunction or missing normalization; wrong diffraction equation or plane identification
Numerical accuracy30%15Part (a): T₂ ≈ 241 K (-32°C) via correct algebraic solution; Part (b): clean evaluation yielding probability = 1/a - sin(2nπ/a)/(2nπ); Part (c): a = 3.00 Å, void radii calculated correctly, clear conclusion that 1.08 Å fits octahedral voidCorrect method but arithmetic errors (e.g., sign errors, calculator mistakes); correct integral form but evaluation errors; correct lattice parameter but wrong void radius calculationMajor calculation errors or missing steps; completely wrong numerical answers; unit conversion errors (Å to m, kPa to Pa)
Diagram / structure10%5Clear P-V diagram showing irreversible adiabatic vs reversible path for (a); sketch of infinite square well with shaded probability region for (b); simple cubic unit cell with labeled void positions for (c)Minimal or unclear diagrams; missing labels; generic sketches not tied to specific problemNo diagrams where clearly needed; completely wrong or misleading diagrams
Application context10%5Brief physical interpretation: tyre cooling in (a) connects to real tyre pressure monitoring; quantum confinement relevance to nanomaterials/quantum dots in (b); interstitial alloys like steel (C in Fe) or solid electrolytes in (c)Generic statements without specific connection; mentions applications but not clearly linked to calculationsNo physical interpretation or context; purely mathematical treatment with no physical insight

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