UPSC Combined Geo-Scientist, Mains · 2026 · Mains Paper II
Q1(m). Normal modes of vibration
Question
How many normal modes of vibrations are possible for the following molecules? (i) C₆H₆ (ii) C₄H₂ (iii) Os(CO)₅ (iv) SiCl₄ (v) CS₂
Answer in brief
(i) 30 (ii) 13 (iii) 27 (iv) 9 (v) 4
The full model answer
Worked out step by step, as it should go in the answer booklet. The solution sheet follows it.
Non-linear: 3N-6; linear: 3N-5.
| Molecule | N | Shape | Formula | Modes |
| C₆H₆ | 12 | planar, non-linear | 3(12)-6 | 30 |
| C₄H₂ (H–C≡C–C≡C–H) | 6 | linear | 3(6)-5 | 13 |
| Os(CO)₅ | 11 | trigonal bipyramid | 3(11)-6 | 27 |
| SiCl₄ | 5 | tetrahedral | 3(5)-6 | 9 |
| CS₂ | 3 | linear | 3(3)-5 | 4 |
Coversvibrational modesNormal modes of vibration
All parts of Question 1
- 1(a) Compressibility factor and its significance5 marks
- 1(b) Andrew's isotherms and the critical-point conditions5 marks
- 1(c) First-order Bragg angle for the (110) planes5 marks
- 1(d) Intrinsic semiconductor and its conversion to extrinsic5 marks
- 1(e) Fugacity against pressure of a non-ideal gas5 marks
- 1(f) Kₚ and K_c for A+B⇌ C5 marks
- 1(g) Half-life of a nuclide with two parallel first-order decays5 marks
- 1(h) Soap and detergent5 marks
- 1(i) Galvanic cell for 2 Cu⁺leftharpoonsCu²⁺+Cu5 marks
- 1(j) pH when a weak base is 65% neutralised5 marks
- 1(k) Uncertainty in an energy eigenstate5 marks
- 1(l) Expectation value and eigenvalue5 marks
- 1(m) Normal modes of vibration5 marks
- 1(n) Raman activity and polarisability ellipsoids of CO₂5 marks
- 1(o) The incorrect statement on electronic spectra5 marks
- 1(p) Non-radiative decay competing with fluorescence5 marks