UPSC Civil Services (Mains), Chemistry optional · 2023 · Paper I
Q1(c). Bragg equation and the layer spacing
Question
(i) Derive an expression that relates the wavelength of the X-rays with the distance between the layers of atoms in a crystal. (ii) X-rays of wavelength 220 pm are diffracted from an ionic crystal at an angle of 23°. What is the distance between the layers that are responsible for this diffraction?
Answer in brief
(i) nλ = 2d sin θ (Bragg equation); (ii) d = 220/(2 sin 23°) = 282 pm (first order).
Key points
- A crystal is a stack of parallel layers a distance d apart; X-rays strike them at a glancing angle θ and each layer reflects part of the beam.
- The ray reflected from the second layer travels an extra path AB + BC = 2d sin θ.
- Reflected rays reinforce only when the path difference is a whole number of wavelengths: nλ = 2d sin θ.
- Since sin θ ≤ 1, reflection needs λ ≤ 2d, which is why X-rays (about 100 pm) are used.
- With λ = 220 pm, sin 23° = 0.3907 and n = 1: d = 281.5 pm ≈ 282 pm (2.82 Å).
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.
CoversBragg equationX-ray diffractioninterplanar spacingconstructive interferencecrystal lattice
All parts of Question 1
- 1(a) Light emitted by a particle in a box; the uncertainty principle10 marks
- 1(b) Lewis structures of [Br₃]⁻ and thiourea; the most polar bond10 marks
- 1(c) Bragg equation and the layer spacing10 marks
- 1(d) Root-mean-square speed of water molecules10 marks
- 1(e)(i) E° of Cr³⁺/Cr from two couples5 marks
- 1(e)(ii) Surface tension, contact angle and loss of gravity5 marks