UPSC Combined Geo-Scientist, Mains · 2019 · Mains Paper II
Q1(n). Bohr quantisation from matter waves
Question
Use electron wave properties to derive Bohr' s angular-momentum postulate L=mₑvr=nħ for a circular hydrogen-atom orbit.
Answer in brief
A standing wave requires 2π r=nλ; with λ=h/(mₑv) this gives mₑvr=nh/(2π)=nħ.
Key points
- An electron wave around a closed orbit must join continuously with itself after one circumference. Constructive self-interference requires an integer number of wavelengths, n=1,2,….
- Combining 2π r=nλ with the de Broglie relation p=mₑv=h/λ gives 2π rp=nh.
- Hence the orbital angular momentum is L=rp=nħ.
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.
CoversBohr quantisation from matter wavesAtomic structure
All parts of Question 1
- 1(a) Atomicity from heat capacities5 marks
- 1(b) Units of attraction parameter5 marks
- 1(c) Surface energy of cube and sphere5 marks
- 1(d) Atoms in a BCC sample5 marks
- 1(e) Volume independence of ideal-gas heat capacity5 marks
- 1(f) Entropy on doubling volume5 marks
- 1(g) Single-ion activity coefficients5 marks
- 1(h) Ammonia buffer5 marks
- 1(i) Pure-component vapour pressures5 marks
- 1(j) Zero-order half-life5 marks
- 1(k) Physical and chemical adsorption5 marks
- 1(l) Iron redox-electrode potential5 marks
- 1(m) Energy of one einstein5 marks
- 1(n) Bohr quantisation from matter waves5 marks
- 1(o) Spectral wavelength shifts5 marks
- 1(p) Bromobutane with one NMR signal5 marks