13.6 Rotational (Microwave) and Raman Spectra
★★★ 2021–2025: 26 parts, 170 marks (Civil Services 3 parts / 25 m · IFoS 4 / 25 m · Geo-Scientist P-II 19 / 120 m). Geo-Scientist P-II has asked it 4–6 times every year, often as numericals.
Syllabus lines. Civil Services: “Rotational: diatomic molecules; isotopic substitution and rotational constants”. IFoS: “Rotational spectra: diatomic molecules; isotopic substitution and rotational constants”. Geo-Scientist P-II: “Condition of resonance … Rotational spectroscopy of diatomic molecules: rigid rotor model, selection rules, spectrum, characteristic features of spectral lines. Determination of bond length, effect of isotopic substitution … Raman effect: characteristic features and conditions of Raman activity … Rotational and vibrational Raman spectra”.
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CSE 2024 Q7(a) = asked there; tap to open the solved answerSN = line kept in the short notes
13.6.1 Resonance, discrete levels and units ★
- A molecule has only discrete energies; it absorbs or emits a photon only when hν = Ej − Ei (resonance, Bohr condition). Together these give a line spectrum; continuous levels would give a continuum Geo 2025 Q1(m) 5m.SN
- How the photon couples (HCl): the permanent dipole Hδ+–Clδ− rotates, so along any fixed axis it oscillates as μ cos ωt; when ω matches the radiation, the field exerts a torque and energy is exchanged (J → J + 1). The photon carries one unit of angular momentum: ΔJ = ±1 Geo 2022 Q1(n) 5m.SN
13.6.2 The rigid rotor ★★★
- Lines are equally spaced, levels are not: levels at 0, 2B, 6B, 12B … (gaps 2B, 4B, 6B grow), but each allowed line is the difference of neighbours, 2B(J + 1), so successive lines differ by 2B Geo 2025 Q15(b) 5m.SN
- J = 3 → 4 at 0.50 cm⁻¹: 8B = 0.50 ⇒ B = 0.0625 cm⁻¹; J = 6 → 7 = 14B = 0.875 cm⁻¹ Geo 2021 Q15(a) 10m.SN
- Diatomic B values: H₂ 60.9, HCl 10.6, CO 1.93 cm⁻¹: lines in the far-infrared or microwave region.
13.6.3 Who shows a microwave spectrum ★★★
- Gross selection rule: the molecule needs a permanent dipole moment. Specific rule: ΔJ = ±1 (linear molecules).SN
| Active (μ ≠ 0) | Inactive (μ = 0) |
|---|---|
| HCl, CO, BrF, NO, H₂O (C₂ᵥ), NH₃ (C₃ᵥ), CH₃Cl (C₃ᵥ) | H₂, N₂, O₂ (homonuclear); CO₂, C₂H₂ (D∞h); CH₄ (Td); SF₆ (Oh); BCl₃ (D₃h); XeF₄ (D₄h); C₆H₆ (D₆h) |
Asked: CSE 2025 Q5(d) 10m Geo 2025 Q1(n) 5m Geo 2023 Q1(n) 5m Geo 2021 Q1(m) 5m CSE 2023 Q8(b)(i) 10m Geo 2024 Q1(n) 5m
13.6.4 Bond lengths from the rotational constant ★★★ numericals
- Read B: line spacing = 2B; first line (J = 0 → 1) = 2B. If the spacing is in Hz, divide by c (in cm s⁻¹) to get cm⁻¹.
- I = h/(8π²cB) with c in cm s⁻¹ and B in cm⁻¹ (SI: I in kg m²).
- μ in kg = (m₁m₂/(m₁ + m₂)) × 1.661 × 10⁻²⁷; then r = √(I/μ).SN
Solved PYQs Geo 2025 Q15(c) 10m Geo 2024 Q15(a) 10m CSE 2024 Q8(c)(iii) 5m
- H³⁵Cl, lines 6.26 × 10¹¹ Hz apart: B = 3.13 × 10¹¹ Hz = 10.44 cm⁻¹; I = h/(8π²BHz) = 2.68 × 10⁻⁴⁷ kg m²; μ = 1.008 × 34.97/35.98 u = 1.627 × 10⁻²⁷ kg; r = 1.28 × 10⁻¹⁰ m.
- B = 10.26 cm⁻¹: I = 2.73 × 10⁻⁴⁷ kg m², r = 129.5 pm. Given I = 2.70 × 10⁻⁴⁷ kg m² (masses 1 and 35.5): μ = 1.615 × 10⁻²⁷ kg, r = 1.29 Å.SN
Solved PYQs IFoS 2025 Q8(a)(i) 10m IFoS 2022 Q5(d) 5m
- CO, first line 3.8424 cm⁻¹: B = 1.9212 cm⁻¹; I = h/(8π²cB) = 6.625 × 10⁻²⁷/(8π² × 3 × 10¹⁰ × 1.9212) = 1.456 × 10⁻³⁹ g cm².
- μ = (12 × 16/28)/NA = 1.139 × 10⁻²³ g; r = √(I/μ) = 1.131 Å (spacing 3.84 cm⁻¹ gives 1.13 Å).SN
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