(a) The volume of a certain gas is found to be 5.0×10⁻⁴ m³ mol⁻¹ at 273 K and 3.0×10⁶ Pa. This gas obeys the van der Waals' equation with a = 0.50 m⁶ Pa mol⁻².
(i) Calculate the other van der Waals' constant, b.
(ii) What is the compression factor fo

Question

(a) The volume of a certain gas is found to be 5.0×10⁻⁴ m³ mol⁻¹ at 273 K and 3.0×10⁶ Pa. This gas obeys the van der Waals' equation with a = 0.50 m⁶ Pa mol⁻².
(i) Calculate the other van der Waals' constant, b.
(ii) What is the compression factor for this gas at the same temperature and pressure?
(iii) Comment on the nature of the molecular interactions of this gas. 20

(b) A 15 cm drinking straw is inserted in a glass of water at an angle 45°. What pressure difference (in torr) must be generated across the length of the straw to drink water? The density of water is 1.0 g cm⁻³. 10

(c) A capillary tube of radius r is inserted into a liquid to blow a bubble of the same radius r. If the excess pressure required to blow the bubble is 2·16 torr, then what is the diameter of the capillary tube in cm? The surface tension of the liquid is 0·072 N m⁻¹. 10

(d) Pyroxenes, amphiboles and phyllosilicates are well-known groups of silicates that occur in crust of the earth. Write the empirical formulae and draw the basic structural units of the above-mentioned silicates. 10

UPSC Answer Check · Accepted Answer

This is primarily a calculation-based question demanding precise numerical work across thermodynamics and surface chemistry, with a structural component for silicates. Allocate approximately 40% of time to part (a) given its 20 marks and triple sub-parts requiring van der Waals manipulation; 20% each to parts (b) and (c) for pressure calculations involving hydrostatic and Laplace equations; and 20% to part (d) for silicate structures. Begin with clear statement of relevant equations, show systematic substitution with units, and conclude with brief physical interpretation where asked.
- For (a)(i): Correct rearrangement of van der Waals equation and calculation of b = 8.314×10⁻⁵ m³ mol⁻¹ (or 8.3×10⁻⁵ m³ mol⁻¹) showing proper unit handling
- For (a)(ii): Calculation of compression factor Z = PV/RT or Z = PVₘ/RT yielding Z ≈ 0.66, with recognition that Z < 1 indicates attractive forces dominate
- For (a)(iii): Physical interpretation that negative deviation from ideality (Z < 1) means attractive intermolecular forces predominate over repulsive at these conditions
- For (b): Correct application of hydrostatic pressure ΔP = ρgh with effective vertical height h = 15×sin45° cm, yielding ΔP ≈ 7.8 torr (or ~8 torr)
- For (c): Application of Laplace equation for spherical bubble ΔP = 4γ/r (not 2γ/r, as bubble has two surfaces), solving for diameter d = 2r ≈ 0.27 cm
- For (d): Empirical formulae: pyroxene (SiO₃)₂ⁿ⁻ or MgSiO₃, amphibole (Si₄O₁₁)₆ⁿ⁻, phyllosilicate (Si₂O₅)₂ⁿ⁻; with correct 1D chain, 2D double chain, and 2D sheet structures respectively
- Recognition that silicate structures relate to Indian geological context: pyroxenes in Deccan traps, amphiboles in Himalayan metamorphic rocks, phyllosilicates (mica) in Jharkhand/Bihar pegmatites

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Demonstrates flawless grasp of van der Waals corrections (a for attraction, b for excluded volume), distinguishes single vs double surface bubbles for Laplace pressure, correctly identifies effective vertical height component for inclined straw, and accurately classifies silicate polymerization degrees	Shows basic understanding of van der Waals equation and surface tension concepts but confuses bubble geometry (uses 2γ/r instead of 4γ/r) or makes minor errors in silicate classification	Fundamental misconceptions such as treating van der Waals gas as ideal, using full straw length instead of vertical component, or completely misidentifying silicate structural types
Mechanism / equation	20%	10	States all equations explicitly before substitution: van der Waals (P + a/Vₘ²)(Vₘ - b) = RT, compression factor Z = PVₘ/RT, hydrostatic ΔP = ρgh sinθ, Laplace ΔP = 4γ/r for bubble, and silicate general formula (SiO₄)₄₋₂ₙⁿ⁻; shows clear logical flow from equation to solution	Uses correct equations but presents them embedded in working without clear statement, or makes minor errors in algebraic rearrangement that are later corrected	Missing key equations, incorrect formula substitution (e.g., using ideal gas law for part a), or no derivation shown for silicate empirical formulae from basic SiO₄ tetrahedron
Numerical accuracy	25%	12.5	All calculations correct to appropriate significant figures: b ≈ 8.3×10⁻⁵ m³ mol⁻¹, Z ≈ 0.66, pressure difference ≈ 7.8 torr (or 780 Pa converted correctly), capillary diameter ≈ 0.27 cm; demonstrates proper unit conversion throughout (Pa, m³, torr, N m⁻¹)	Correct method but arithmetic errors or unit conversion mistakes (e.g., factor of 10 error in torr conversion, cm/m confusion) yielding answers within 20% of correct value	Major calculation errors, incorrect order of magnitude, missing units, or failure to convert between SI and non-SI units where required
Diagram / structure	20%	10	Clear, labeled diagrams for all three silicate structures: pyroxene showing single chain of SiO₄ tetrahedra sharing two corners, amphibole showing double chain with alternating two and three shared oxygens, phyllosilicate showing hexagonal sheet with three shared oxygens per tetrahedron; correct empirical formulae derived from diagrams	Attempts diagrams but lacks clarity in showing corner-sharing pattern, or provides formulae without structural justification; diagrams present but poorly labeled	No diagrams for silicate structures, or diagrams that misrepresent the connectivity (e.g., showing isolated tetrahedra instead of chains/sheets)
Application context	15%	7.5	Provides insightful physical interpretation: for (a)(iii) explains that Z < 1 means gas is more compressible than ideal due to attractive forces dominating at high pressure; connects silicate structures to Indian geology (Deccan basalt mineralogy, Himalayan metamorphism, mica belts); relates surface tension to everyday drinking and bubble formation	Brief comment on Z < 1 meaning attractive forces, generic mention of silicate importance without Indian context, minimal physical interpretation	No interpretation of numerical results, purely mechanical calculation without physical meaning, or completely irrelevant geological references

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