(a) Calculate the number of collisions that oxygen makes per second on 1·00 cm² of the surface of the vessel containing them if the pressure is 1·00 × 10⁻⁶ atm and the temperature is 25°C. 10

(b) Suppose that 10·0 J of work is required to create dro

Question

(a) Calculate the number of collisions that oxygen makes per second on 1·00 cm² of the surface of the vessel containing them if the pressure is 1·00 × 10⁻⁶ atm and the temperature is 25°C. 10

(b) Suppose that 10·0 J of work is required to create droplets of uniform size from a mole of water in bulk at 25°C and 1 atm pressure.

(i) Assuming that surface tension is independent of area, calculate the radius of the droplets.

(ii) Calculate the number of water molecules in a droplet.

Given : Surface tension of water = 0·072 J/m² 15

(c) You are given the following data for butane :
Normal melting point = – 138°C
Normal boiling point = 0°C
Critical temperature = 152°C
Critical pressure = 38 atm
Assume that the triple point is slightly lower in temperature than the melting point and that the vapour pressure at the triple point is 3 × 10⁻⁵ torr.

(i) Sketch a phase diagram for butane.

(ii) Butane at 1 atm and 140°C is compressed to 40 atm. Are two phases present at any time during this process ?

(iii) Butane at 1 atm and 200°C is compressed to 40 atm. Are two phases present at any time during this process ? 10

(d) A container with 100 g of ice at 0°C is placed in a humid room whose temperature is 40°C. The ice melts as water vapour condenses into the container. Assuming that all the heat transferred to the container comes from the condensation, how much water will have condensed in the container once all the ice is melted and has reached 40°C ?

Given : Heat of fusion of ice = 334 Jg⁻¹
Heat of vaporization of water = 2260 Jg⁻¹
Heat capacity of water = 4184 J kg⁻¹ K⁻¹ 10

(e) Explain why crystalline solids are generally more defective as a result of increasing temperature. 5

UPSC Answer Check · Accepted Answer

Begin with the directive to calculate across all sub-parts, showing systematic problem-solving. Allocate approximately 20% time to part (a) on collision theory, 30% to part (b) on surface tension and droplet formation, 20% to part (c) on phase diagrams with careful sketching, 20% to part (d) on thermal equilibrium calculations, and 10% to part (e) on crystal defects. Structure as: brief statement of principles → step-by-step calculations with units → labeled diagram for (c) → concluding physical interpretation of results.
- Part (a): Apply kinetic theory of gases using Z = (P/√(2πmkT)) × N_A to find collision frequency per unit area, converting pressure to SI units and using O₂ molecular mass
- Part (b)(i): Use work of surface creation W = γ × ΔA = γ × n × 4πr² with n droplets from total surface area to solve for droplet radius
- Part (b)(ii): Calculate molecules per droplet using droplet volume, water density, and Avogadro's number
- Part (c)(i): Sketch phase diagram with correctly positioned triple point (-138°C, 3×10⁻⁵ torr), critical point (152°C, 38 atm), and phase boundaries showing solid-liquid line with negative slope
- Part (c)(ii)-(iii): Analyze compression paths relative to critical point to determine phase coexistence, noting 140°C < T_c and 200°C > T_c
- Part (d): Set up energy balance: heat from condensation = heat for fusion + heating water, solving for condensed mass using latent heats and specific heat
- Part (e): Explain Schottky and Frenkel defect formation with Arrhenius-type temperature dependence, citing Boltzmann factor for defect concentration
- Physical interpretation: Connect numerical results to real phenomena (e.g., droplet stability in clouds, LPG storage conditions, humidity effects)

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Demonstrates precise understanding of kinetic theory assumptions for (a), surface thermodynamics for (b), phase rule and critical phenomena for (c), calorimetry principles for (d), and defect thermodynamics for (e); correctly identifies when ideal gas approximations apply and when they fail	Shows basic familiarity with concepts but confuses collision frequency with collision rate per molecule, or misidentifies phase regions in (c); partial understanding of defect types	Fundamental misconceptions such as using rms speed instead of mean speed for flux, or treating supercritical fluid as two-phase in (c)(iii)
Mechanism / equation	20%	10	States and correctly applies: kinetic theory flux equation for (a); surface work and area-volume relations for (b); Clausius-Clapeyron reasoning and lever rule for (c); first law energy balance for (d); Arrhenius defect formation for (e); all equations dimensionally consistent	Uses correct final equations but omits derivations or states them incompletely; minor errors in geometric relations for droplet surface area	Missing key equations, uses wrong formulas (e.g., ideal gas law for collision frequency), or applies thermodynamic relations to inappropriate conditions
Numerical accuracy	25%	12.5	All calculations with correct unit conversions (atm→Pa, cm²→m², °C→K, torr→atm), proper significant figures, and physically reasonable magnitudes; final answers: (a) ~2.7×10²³ collisions/s, (b)(i) r~1.9 nm, (b)(ii) ~3×10⁴ molecules, (d) ~16.8 g condensed	Correct method but arithmetic errors or unit conversion mistakes (e.g., forgetting 10⁻⁶ in pressure, or cm³ vs m³ confusion); order of magnitude correct	Gross numerical errors, wrong powers of ten, or physically impossible results accepted without comment; missing essential data substitutions
Diagram / structure	15%	7.5	Clear phase diagram for (c) with labeled axes (P vs T), correctly positioned critical point, triple point, three phase boundaries, and marked process paths for (ii) and (iii); compression paths clearly distinguished as crossing vs not crossing vapor-liquid coexistence curve	Rough sketch with correct general shape but misplaced critical point or incorrect slope of solid-liquid line; missing process path labels	No diagram, or diagram with serious errors (e.g., triple point above critical point, missing phase labels, or confused axes)
Application context	20%	10	Interprets results physically: collision frequency relevance to vacuum technology and surface catalysis; droplet size significance for cloud seeding and nanotechnology; phase diagram application to LPG storage safety in Indian households; humidity calculation to condensation harvesting; defect concentration to semiconductor processing	Brief mention of applications without elaboration; generic statements about 'industrial importance' without specific examples	Purely mathematical treatment with no physical interpretation; or irrelevant applications that misrepresent the phenomena

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