UPSC Combined Geo-Scientist, Mains · 2021 · Mains Paper II
Q1(n). Highest vibrational frequency among H₂, D₂ and HD
Question
Which of the following molecules has the highest fundamental frequency of vibration? Explain. H₂, D₂, HD
Answer in brief
H₂ (4161 cm⁻¹). All three have the same force constant; H₂ has the smallest reduced mass. Order: H₂ > HD > D₂.
Key points
- Born–Oppenheimer: isotopes have the same electrons ⇒ the same potential curve and the same force constant (k ≈ 575 N m⁻¹); only μ changes.
- Reduced mass (u): H₂ 0.504, HD 0.672, D₂ 1.007.
- ν̃ = (1/2πc)√(k/μ) ∝ 1/√μ ⇒ smallest μ, highest frequency: H₂ > HD > D₂.
- Observed fundamentals: 4161, 3632, 2994 cm⁻¹.
- H₂ and D₂ have no dipole (IR inactive); their frequencies come from Raman spectra.
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.
Coversvibrational frequencyisotope effectreduced massforce constantH2 D2 HD
All parts of Question 1
- 1(a) Conditions for ideal behaviour of real gases5 marks
- 1(b) Compressibility factor and Z against P plots5 marks
- 1(c) Formula from corner and face-centre positions5 marks
- 1(d) Crystal systems from the unit-cell constants5 marks
- 1(e) Electrophoretic effect in the Debye–Hückel–Onsager theory5 marks
- 1(f) Activity and activity coefficient: units and magnitudes5 marks
- 1(g) Why a finely powdered adsorbent works better5 marks
- 1(h) Half-life of a first order reaction5 marks
- 1(i) Role of H₃PO₄ in the dichromate titration of Fe²⁺5 marks
- 1(j) Why a voltmeter does not give the EMF5 marks
- 1(k) Well-behaved functions: two tests5 marks
- 1(l) Uncertainty in the velocity of an electron5 marks
- 1(m) Microwave activity of C₂H₂, CH₃Cl, C₆H₆ and CO₂5 marks
- 1(n) Highest vibrational frequency among H₂, D₂ and HD5 marks
- 1(o) Photosensitised reactions5 marks
- 1(p) Energy of one mole of 2500 Å photons5 marks