UPSC Combined Geo-Scientist, Mains · 2023 · Mains Paper II
Q1(a). van der Waals equation: meaning of a and b
Question
Write the van der Waals equation and explain the physical significance of van der Waals constants a and b.
Answer in brief
(P + a/Vm²)(Vm − b) = RT; a corrects for intermolecular attraction, b is the excluded volume, four times the volume of the molecules.
Key points
- a measures intermolecular attraction: molecules near the wall are pulled back, so the observed P is lower than ideal by a/Vm².
- Larger a ⇒ the gas liquefies more easily (higher Tc): He 0.0346, H₂ 0.248, N₂ 1.370, CO₂ 3.640 bar L² mol⁻².
- b is the excluded volume (co-volume) per mole: b = 4NA·(4/3)πr³, four times the actual volume of the molecules.
- a and b are characteristic of the gas, independent of T and P; a = b = 0 gives the ideal gas.
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.
Coversvan der Waals equationvan der Waals constants a and bexcluded volumeintermolecular attractionreal gases
All parts of Question 1
- 1(a) van der Waals equation: meaning of a and b5 marks
- 1(b) Exceptional behaviour of hydrogen and helium5 marks
- 1(c) Unit cell of solid C₆₀ and its carbon atoms5 marks
- 1(d) Planes that diffract: simple cubic, a = 5 Å5 marks
- 1(e) Chemical potential and its link with Gibbs energy5 marks
- 1(f) Debye–Hückel limiting law for γ±5 marks
- 1(g) Activation energy of 2HI → H₂ + I₂5 marks
- 1(h) Best coagulant for the Sb₂S₃ sol5 marks
- 1(i) E° of Na⁺/Na from ΔrH° and ΔrS°5 marks
- 1(j) pH from a calomel–quinhydrone cell5 marks
- 1(k) Degeneracy of the level 6h²/8mL² in a cubic box5 marks
- 1(l) Is Peⁱᵏˣ + Qe⁻ⁱᵏˣ an eigenfunction?5 marks
- 1(m) Zero-point energy in joule and in cm⁻¹5 marks
- 1(n) Microwave activity of SF₆, CO, H₂, H₂O and CO₂5 marks
- 1(o) Stark–Einstein law and the quantum yield5 marks
- 1(p) Decay routes of a photoexcited molecule5 marks