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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 · 2020 · Paper I

Q6(b). Coordination number of the cations in NaBr and KBr from radius ratios

  • 10 marks
  • Answered
Question

The ionic radii of Na⁺, K⁺ and Br⁻ are 137, 148 and 195 pm, respectively. Determine the coordination number and geometry of cations in NaBr and KBr.

Answer in brief

NaBr: r₊/r₋=137/195=0.703, between 0.414 and 0.732: CN 6, octahedral (rock-salt). KBr: 148/195=0.759, between 0.732 and 1: CN 8, cubic (CsCl type) predicted from the given radii, although real KBr has the rock-salt structure (CN 6).

Key points

  • a cation takes the largest number of anions that can touch it without the anions touching (overlapping) each other. The limit for each arrangement is fixed by geometry: 0.225 (tetrahedral, CN 4), 0.414 (octahedral, CN 6),…
  • four anions in a square around the cation touch along the diagonal: 2(r₊+r₋)=2√2 r₋, so r₊/r₋=√2-1=0.414. Cubic limit: along the body diagonal 2(r₊+r₋)=√3 (2r₋), so r₊/r₋=√3-1=0.732.
  • r₊r₋=(137)/(195)=0.703 (0.414<0.703<0.732): Na⁺ in NaBr: CN 6, octahedral (NaCl type)
  • r₊r₋=(148)/(195)=0.759 (0.732<0.759<1): K⁺ in KBr: CN 8, cubic (CsCl type), by the rule

The full model answer

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

  1. Radius-ratio rule: a cation takes the largest number of anions that can touch it without the anions touching (overlapping) each other. The limit for each arrangement is fixed by geometry: 0.225 (tetrahedral, CN 4), 0.414 (octahedral, CN 6), 0.732 (cubic, CN 8).
  2. Octahedral limit: four anions in a square around the cation touch along the diagonal: 2(r₊+r₋)=2√2 r₋, so r₊/r₋=√2-1=0.414. Cubic limit: along the body diagonal 2(r₊+r₋)=√3 (2r₋), so r₊/r₋=√3-1=0.732.
  3. NaBr: r₊r₋=(137)/(195)=0.703 (0.414<0.703<0.732): Na⁺ in NaBr: CN 6, octahedral (NaCl type)
  4. KBr: r₊r₋=(148)/(195)=0.759 (0.732<0.759<1): K⁺ in KBr: CN 8, cubic (CsCl type), by the rule
  5. Comparison with the real crystals: both NaBr and KBr actually have the rock-salt structure (CN 6) at ordinary pressure. The rule treats ions as hard spheres and predicts correctly in only about two-thirds of cases; KBr lies close to the 0.732 limit, where the CN 6 and CN 8 structures differ little in lattice energy.
  6. The printed radius of Na⁺ (137 pm) is much larger than the usual 6-coordinate value (116 pm, Shannon crystal radius, with K⁺ 152 and Br⁻ 182 pm); with those values the ratios are 0.637 (CN 6) and 0.835.

Application. radius ratios explain why CsCl, CsBr and CsI (large cation) take the CN 8 structure while most alkali halides are rock salt, and they guide the choice of host lattices for ionic conductors and pigments.

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Model answer to Q6(b), 2020 Paper I: solution page 30
Solution page 30 of 45
Read the whole 2020 solution2020 question paper

All parts of Question 6