UPSC Combined Geo-Scientist, Mains · 2023 · Mains Paper II
Q1(c). Unit cell of solid C₆₀ and its carbon atoms
Question
C₆₀ molecules are cubic close-packed to form the solid C₆₀ unit cell. Draw the unit cell and find the number of carbon atoms present in the unit cell.
Answer in brief
fcc (ccp) cell with C₆₀ at 8 corners and 6 face centres: 4 C₆₀ per cell, so 4 × 60 = 240 carbon atoms.
Key points
- Cubic close packing = face-centred cubic lattice; one C₆₀ molecule at each lattice point.
- Corners 8 × 1/8 = 1, face centres 6 × 1/2 = 3 ⇒ Z = 4 C₆₀ per cell.
- C atoms per cell = 4 × 60 = 240.
- Check with a = 14.17 Å: nearest C₆₀–C₆₀ distance a/√2 = 10.02 Å; density 1.68 g cm⁻³ (measured 1.65 to 1.7).
- Each cell has 4 octahedral and 8 tetrahedral holes; K⁺ in all 12 gives K₃C₆₀, a superconductor (Tc ≈ 18 K).
The full model answer
As written for the answer booklet: the diagram carries the working, the lines stay short. Read here only; it is not offered as a download.
CoversC60 fullerenecubic close packingfcc unit cellatoms per unit cellK3C60
All parts of Question 1
- 1(a) van der Waals equation: meaning of a and b5 marks
- 1(b) Exceptional behaviour of hydrogen and helium5 marks
- 1(c) Unit cell of solid C₆₀ and its carbon atoms5 marks
- 1(d) Planes that diffract: simple cubic, a = 5 Å5 marks
- 1(e) Chemical potential and its link with Gibbs energy5 marks
- 1(f) Debye–Hückel limiting law for γ±5 marks
- 1(g) Activation energy of 2HI → H₂ + I₂5 marks
- 1(h) Best coagulant for the Sb₂S₃ sol5 marks
- 1(i) E° of Na⁺/Na from ΔrH° and ΔrS°5 marks
- 1(j) pH from a calomel–quinhydrone cell5 marks
- 1(k) Degeneracy of the level 6h²/8mL² in a cubic box5 marks
- 1(l) Is Peⁱᵏˣ + Qe⁻ⁱᵏˣ an eigenfunction?5 marks
- 1(m) Zero-point energy in joule and in cm⁻¹5 marks
- 1(n) Microwave activity of SF₆, CO, H₂, H₂O and CO₂5 marks
- 1(o) Stark–Einstein law and the quantum yield5 marks
- 1(p) Decay routes of a photoexcited molecule5 marks