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UPSC Civil Services (Mains)

Chemistry optional

  • 2 papers
  • 250 marks each
  • 3 h per paper
  • 5 of 8 questions, Q1 and Q5 compulsory

Paper I covers physical and inorganic chemistry.

UPSC Civil Services (Mains), Chemistry optional · 2017 · Paper I

Q3(c). Polonium: cubic class and cell edge from three Bragg reflections

  • 15 marks
  • Answered
Question

Polonium crystallizes in the cubic system. Using X-ray of wavelength 0.154 nm, first order Bragg reflections occur at sinθ values 0.225, 0.316 and 0.388 from (100), (110) and (111) planes respectively. Determine the class (SC or BCC or FCC) of the cubic crystal and edge length (a) of unit cell.

Answer in brief

Simple cubic (SC); a=0.344 nm (344 pm).

Key points

  • θ₁₁₀:θ₁₁₁=1:√2:√3 if all three planes reflect. Observed: 0.225:0.316:0.388=1:1.404:1.724, against 1:1.414:1.732. The three planes are first-order reflections of one cell.
  • reflections with h+k+l odd are absent, so (100) and (111) cannot appear. FCC (cubic F): only all-even or all-odd h,k,l appear, so (100) and (110) cannot appear. All three appear here, so the lattice is simple (primitive) cubic.
  • d=λ/2θ: d₁₀₀=0.154/0.450=0.3422 nm; d₁₁₀=0.2437 nm; d₁₁₁=0.1985 nm.
  • a=dₕₖₗh²+k²+l²: (100) 0.3422 nm; (110) 0.2437√2=0.3446 nm; (111) 0.1985√3=0.3437 nm.

The full model answer

Written out in full, as it should go in the answer booklet. The solution page with its diagrams follows it.

BRAGGλ=2dθCUBICdₕₖₗ=a/h²+k²+l²RESULTSC, a=0.344 nm

Given: λ=0.154 nm, n=1; θ=0.225 (100), 0.316 (110), 0.388 (111).

  1. Ratio test. θ=(λ)/(2a)h²+k²+l², so θ₁₀₀:θ₁₁₀:θ₁₁₁=1:√2:√3 if all three planes reflect. Observed: 0.225:0.316:0.388=1:1.404:1.724, against 1:1.414:1.732. The three planes are first-order reflections of one cell.
  2. Systematic absences. BCC (cubic I): reflections with h+k+l odd are absent, so (100) and (111) cannot appear. FCC (cubic F): only all-even or all-odd h,k,l appear, so (100) and (110) cannot appear. All three appear here, so the lattice is simple (primitive) cubic.
  3. Spacings: d=λ/2θ: d₁₀₀=0.154/0.450=0.3422 nm; d₁₁₀=0.2437 nm; d₁₁₁=0.1985 nm.
  4. Edge length: a=dₕₖₗh²+k²+l²: (100) 0.3422 nm; (110) 0.2437√2=0.3446 nm; (111) 0.1985√3=0.3437 nm.
  5. a=0.344 nm (mean 0.3435 nm, 344 pm)
  6. Bragg angles: θ=13.0°, 18.4° and 22.8°. The small spread in a comes from the three-figure θ data.
  7. Polonium (α-Po) is the only element that crystallises in the simple cubic structure (one atom per cell, coordination number 6, packing fraction 52 %).

Application. the same absence rules are used to index powder patterns: NaCl (F) shows (111), (200), (220) first; a BCC metal such as iron shows (110), (200), (211).

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Model answer to Q3(c), 2017 Paper I: solution page 13
Solution page 13 of 38
Read the whole 2017 solution2017 question paper

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