(a) (i) The melting point of tin is 232°C, its latent heat of fusion is 14 cal/g and the specific heat of solid and molten tin are 0·055 and 0·064 cal/g °C respectively. Calculate the change in entropy when 1·0 gm of tin is heated from 100°C to 300°C

Question

(a) (i) The melting point of tin is 232°C, its latent heat of fusion is 14 cal/g and the specific heat of solid and molten tin are 0·055 and 0·064 cal/g °C respectively. Calculate the change in entropy when 1·0 gm of tin is heated from 100°C to 300°C.

(ii) Calculate the efficiency of an engine having compression ratio 13·8 and expansion ratio 6 and working on diesel cycle. Given γ = 1·4. (10+5 marks)

(b) (i) Write the expression for the Fermi-Dirac distribution. Plot the Fermi-Dirac distribution at T = 0 and for T₁ > T₂ > 0. Now from the plot propose two alternative definitions of the Fermi level.

(ii) Calculate the probability of an electron occupying an energy level 0·02 eV above the Fermi level at T = 300 K. (15+5 marks)

(c) Given an infinite line charge of charge density 2 nCm⁻¹ parallel to the y-axis and passing through the point (3, 0, 4) m and an infinite sheet of charge of charge density 4 nCm⁻² parallel to the x-y plane and passing through the point (0, 0, 6) m. Calculate the electric field intensity at the point (10, 10, 10) m. Assume free space. (15 marks)

UPSC Answer Check · Accepted Answer

This is a multi-part numerical problem demanding precise calculation across thermodynamics, statistical mechanics and electrostatics. Allocate approximately 30% time to part (a) covering entropy change and diesel cycle efficiency, 40% to part (b) on Fermi-Dirac distribution with its conceptual plots and probability calculation, and 30% to part (c) on vector superposition of electric fields from line and sheet charges. Begin each part with stated assumptions and relevant formulas, show step-by-step calculations with proper units, and conclude with physical interpretation of results.
- For (a)(i): Calculate entropy change in three stages—heating solid tin from 100°C to 232°C, phase change at 232°C, and heating liquid tin from 232°C to 300°C using ΔS = ∫dQ/T and ΔS = mL/T
- For (a)(ii): Apply diesel cycle efficiency formula η = 1 - (1/γ)[(ρ^γ - 1)/(r^(γ-1)(ρ - 1))] where r = compression ratio, ρ = cutoff ratio = r/r_expansion = 13.8/6
- For (b)(i): State Fermi-Dirac distribution f(E) = 1/[exp((E-E_F)/k_BT) + 1], sketch three curves showing step function at T=0 and thermal broadening at T₁ > T₂ > 0, define Fermi level as E where f(E)=0.5 or as chemical potential
- For (b)(ii): Calculate f(E_F + 0.02 eV) using given values, showing substitution of k_BT at 300K ≈ 0.0259 eV
- For (c): Calculate electric field from infinite line charge E_line = λ/(2πε₀r) perpendicular to line, and from sheet E_sheet = σ/(2ε₀) perpendicular to sheet, then vectorially sum at point (10,10,10)m with proper distance calculations

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies all thermodynamic processes in (a), states Fermi-Dirac distribution with quantum mechanical basis in (b), and applies superposition principle with correct field directions in (c); no conceptual errors in any sub-part	Minor errors in identifying process stages or field directions; correct formulas but muddled physical reasoning in one sub-part	Fundamental misconceptions such as treating entropy as state function incorrectly, confusing Fermi-Dirac with Maxwell-Boltzmann, or adding fields as scalars in (c)
Derivation rigour	20%	10	Shows complete step-by-step derivations for entropy integrals, diesel cycle efficiency formula manipulation, and vector field components with clear algebraic steps; justifies all approximations	Skips intermediate steps or assumes results without derivation in one sub-part; mostly correct but with gaps in logical flow	Missing derivations, jumps to final formulas without setup, or contains algebraic errors in key steps like cutoff ratio calculation
Diagram / FBD	15%	7.5	Accurate Fermi-Dirac plots in (b)(i) showing T=0 step function, thermal smearing at finite T with T₁ > T₂ > 0 clearly distinguished, labeled axes (E/E_F vs f(E)), and E_F marked; diesel cycle P-V diagram optional but helpful	Rough sketches with correct qualitative features but poor scaling or missing labels; omits one temperature curve	Missing plots, incorrect curves (e.g., Maxwell-Boltzmann shape), or unlabeled axes making interpretation impossible
Numerical accuracy	30%	15	Precise calculations: entropy change ≈ 0.042 cal/K (or 0.176 J/K), diesel efficiency ≈ 56-58%, probability ≈ 0.24, field magnitude ≈ 45-47 V/m with correct vector components; proper unit conversions throughout	Correct method but arithmetic errors or wrong unit conversions (e.g., °C vs K in entropy, nm vs m in distances); final answers within 10-20% of correct value	Order-of-magnitude errors, wrong formulas applied numerically, or missing units; demonstrates poor calculation discipline
Physical interpretation	15%	7.5	Interprets entropy increase as irreversibility and disorder growth in phase change; explains why diesel efficiency exceeds Otto cycle; discusses how Fermi level shifts with temperature in real materials; validates field direction geometrically	Brief concluding statements without deep insight; mentions physical meaning but doesn't connect to broader principles	Purely mathematical treatment with no physical insight; fails to interpret what calculated values signify

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