1.(a) Compare average and most probable values of position of an electron in the ground state of hydrogen atom. Explain with the help of drawing, why two values differ. (10 marks)

1.(b) Consider the following atomic orbitals of A and B atoms in a he

Question

1.(a) Compare average and most probable values of position of an electron in the ground state of hydrogen atom. Explain with the help of drawing, why two values differ. (10 marks)

1.(b) Consider the following atomic orbitals of A and B atoms in a heterodiatomic AB molecule.

A
2s
2px

B
2py
2pz

Depict the nonbonding interactions among these atomic orbitals with reason. (10 marks)

1.(c) (i) F-centre defect can be introduced in several ways. Regardless of the method used, the colour produced in any particular crystal is always same. Explain the reasons.

(ii) How would the formation of F-centre affect the density of the crystal ? (10 marks)

1.(d) Calculate ΔG at 298 K for the formation of O3 from O2 in urban smog, where [O2] = 0·21 atm and [O3] = 5×10⁻⁷ atm.
Given; ΔG°f(O3) = 163 kJ mol⁻¹.
R = 8·314 J mol⁻¹K⁻¹,
F = 96485 C mol⁻¹. (10 marks)

1.(e) For a cell reaction
Cu²⁺(aq) + Zn(s) → Cu(s) + Zn²⁺(aq)
the standard electrode potential for Zn/Zn²⁺ = 0·339 V and for Cu²⁺/Cu = 0·762 V at 25°C. Calculate standard change in free energy ΔG° and equilibrium constant K for the reaction.
[R = 8·314 J mol⁻¹ k⁻¹] (10 marks)

UPSC Answer Check · Accepted Answer

Begin with a brief introduction acknowledging the diverse nature of the five sub-parts spanning quantum mechanics, chemical bonding, solid state, and electrochemistry. Allocate approximately 20% time each to parts (a), (b), (c), (d), and (e) since all carry equal marks. For (a), compare and r_mp with radial probability distribution diagrams; for (b), apply symmetry rules for orbital overlap; for (c), explain electron trapping and density changes; for (d) and (e), show step-by-step thermodynamic calculations with proper units. Conclude by briefly connecting how quantum mechanical principles underpin macroscopic observations in electrochemistry and materials science. - Part (a): Average position = 3a₀/2 and most probable position r_mp = a₀ for 1s hydrogen; radial probability distribution 4πr²|ψ|² peaks at a₀ while integrates over all space weighted by r - Part (b): Nonbonding interactions arise from zero net overlap: A(2s) with B(2py, 2pz) due to symmetry mismatch; A(2px) with B(2py, 2pz) due to orthogonality; only σ/π bonding possible if symmetry allows - Part (c)(i): F-centre colour depends only on crystal lattice parameter and electron trap depth (Mollwo-Ivey relation), not method of introduction; trapped electron absorbs specific wavelength - Part (c)(ii): Density decreases because anion vacancies create mass deficit without proportional volume change; Schottky-type defect mechanism - Part (d) and (e): Correct application of ΔG = ΔG° + RTlnQ and ΔG° = -nFE° = -RTlnK with proper unit conversions (kJ to J, atm handling, n values)

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Precisely distinguishes <r> vs r_mp in (a); correctly identifies all symmetry-forbidden overlaps in (b); explains why F-centre colour is method-independent using quantum confinement in (c); applies correct thermodynamic sign conventions in (d) and (e)	States correct values but confuses physical meaning of average vs most probable; identifies some nonbonding interactions but misses symmetry arguments; gives partial explanation for colour; makes minor sign or unit errors in calculations	Confuses <r> and r_mp completely; fails to apply symmetry rules for orbital overlap; cannot explain why colour remains constant; applies wrong thermodynamic relations or ignores standard state corrections
Mechanism / equation	20%	10	Writes complete radial probability equation P(r) = 4πr²(1/πa₀³)e^(-2r/a₀) for (a); uses group theory or symmetry-based overlap integrals for (b); states Mollwo-Ivey equation for (c); shows ΔG = ΔG° + RTlnQ and ΔG° = -nFE° with correct n values for (d)-(e)	Writes basic equations but misses normalization constants or limits; describes overlap qualitatively without integrals; states qualitative trends without Mollwo-Ivey; shows main equations but skips intermediate steps	Missing key equations entirely; incorrect orbital overlap rules; no thermodynamic equations shown; confuses ΔG with ΔG° or uses wrong n values
Numerical accuracy	20%	10	Part (d): ΔG = 163000 + 8.314×298×ln(5×10⁻⁷/0.21^1.5) ≈ 163 - 33.5 ≈ 129.5 kJ/mol (or correct stoichiometry-adjusted calculation); Part (e): E°cell = 0.762 - 0.339 = 0.423 V, ΔG° = -81.6 kJ/mol, K ≈ 10^14; all unit conversions correct	Correct final answers for (e) but minor arithmetic errors in (d); correct E°cell but wrong n; correct approach but slips in logarithm calculation or pressure exponent	Order of magnitude errors; wrong sign for ΔG°; confuses atm with bar or ignores pressure term entirely; calculates K from wrong equation; no numerical working shown
Diagram / structure	20%	10	Clear radial probability distribution plot for 1s showing peak at a₀ and tail extending to infinity with <r> marked; orbital orientation diagrams for (b) showing orthogonal axes and zero overlap; F-centre schematic with trapped electron in anion vacancy; cell diagram for (e)	Sketchy diagrams with correct features but poor labeling; 2D instead of 3D orbital representations; missing key annotations like a₀ or nodal planes	No diagrams despite explicit 'explain with drawing' instruction; incorrect radial distribution shape (e.g., node in 1s); confusing orbital overlap diagrams suggesting bonding where none exists
Application context	20%	10	Connects (a) to spectroscopic observables and atomic size definitions; links (b) to molecular orbital theory and heteronuclear diatomics like CO/NO; relates (c) to alkali halide color centers and solid-state lighting; contextualizes (d) to Delhi/Mumbai smog chemistry and (e) to Daniell cell commercial applications	Mentions applications superficially without specific Indian examples; generic statements about industrial importance without linking to question specifics	No real-world connections; fails to mention environmental relevance of ozone calculation or technological importance of F-centres in lasers/photography

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