(a) Find the probability of existence of a particle in a one-dimensional box of length 'a' in the region 0 ≤ x ≤ a/4 for the states n = 1, 2 and 3. (15 marks)

(b) The standard reduction potential of oxygen under acidic conditions at 298 K is +1.23 V

Question

(a) Find the probability of existence of a particle in a one-dimensional box of length 'a' in the region 0 ≤ x ≤ a/4 for the states n = 1, 2 and 3. (15 marks)

(b) The standard reduction potential of oxygen under acidic conditions at 298 K is +1.23 V. What is the standard reduction potential for the four-electron reduction of O₂(g) under basic conditions? (15 marks)

(c) The radii of Zn²⁺ and S²⁻ ions are 0·74 Å and 1·84 Å respectively. Determine the most stable form of arrangement of ions in ZnS crystal lattice. Draw the CCP (Cubic Close Packing) structure of ZnS. (15 marks)

(d) In a sample of NaCl, one of every 10,000 sites, normally occupied by Na⁺, is occupied instead by Ca²⁺. Assuming that all of the Cl⁻ sites are fully occupied, what is the stoichiometry of the sample? (5 marks)

UPSC Answer Check · Accepted Answer

Solve each sub-part systematically with clear mathematical derivations and logical reasoning. For (a), derive the probability using wavefunction integration (~30% time); for (b), apply Nernst equation with pH adjustment (~25% time); for (c), calculate radius ratio and identify structure type with diagram (~30% time); for (d), apply charge neutrality for defect chemistry (~15% time). Present calculations step-wise with proper units and significant figures.
- For (a): Set up probability integral P = ∫₀^(a/4) |ψₙ|²dx using ψₙ = √(2/a) sin(nπx/a), evaluate for n=1,2,3 obtaining values ~0.091, 0.25, 0.303 respectively
- For (b): Write half-reactions for acidic vs basic conditions, apply E°(basic) = E°(acidic) - (0.0591×4/4)log[H⁺]⁴ or use E° = 1.23 - 0.0591×pH at pH=14 to get ~0.40 V
- For (b): Alternative correct approach using ΔG° = -nFE° and Kw relationship to find E° = +0.40 V for O₂ + 2H₂O + 4e⁻ → 4OH⁻
- For (c): Calculate radius ratio r⁺/r⁻ = 0.74/1.84 = 0.402, identify range 0.414-0.732 for octahedral but note ZnS has tetrahedral coordination (zinc blende/wurtzite)
- For (c): Draw CCP (fcc) structure of ZnS showing S²⁻ at lattice points and Zn²⁺ in alternate tetrahedral voids, or vice versa, with correct coordination numbers
- For (d): Apply charge balance: for 10,000 Na⁺ sites, 9,999 Na⁺ + 1 Ca²⁺ requires 10,001 Cl⁻ for neutrality, giving stoichiometry Na₀.₉₉₉₉Ca₀.₀₀₀₁Cl₁.₀₀₀₁ or approximately Na₂CaCl₃ when scaled

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	22%	11	Correctly identifies quantum mechanical probability interpretation, Nernst equation applicability to pH-dependent potentials, radius ratio rules with tetrahedral exception for ZnS, and charge compensation in non-stoichiometric compounds	Identifies most concepts correctly but confuses radius ratio ranges or misapplies charge neutrality condition in (d)	Fundamental misconceptions: treats probability classically, ignores pH effect on reduction potential, or fails to recognize defect chemistry principles
Mechanism / equation	20%	10	Writes complete wavefunctions with correct normalization, sets up proper integrals with limits; writes balanced half-reactions; applies correct radius ratio formula; establishes charge balance equation for defects	Writes most equations correctly but misses normalization constant or makes sign errors in Nernst application	Missing key equations, incorrect half-reactions, or no mathematical setup for probability calculation
Numerical accuracy	24%	12	Accurate values: P₁≈0.091, P₂=0.25, P₃≈0.303; E°(basic)≈+0.40 V; r⁺/r⁻=0.402; correct stoichiometry with proper significant figures throughout	Correct methods but arithmetic errors in integration or final potential value; approximate stoichiometry without precise calculation	Major calculation errors, wrong orders of magnitude, or no numerical working shown
Diagram / structure	18%	9	Clear CCP/fcc unit cell of ZnS showing both ion positions, tetrahedral voids occupied, correct coordination numbers (4:4), labeled axes, and distinction from NaCl structure	Basic diagram drawn but missing key features like void positions or incorrect labeling of ion sites	No diagram, or confused with simple cubic/NaCl structure, missing tetrahedral coordination entirely
Application context	16%	8	Connects (a) to quantum confinement in nanomaterials, (b) to fuel cells/corrosion in Indian coastal environments, (c) to ZnS as phosphor in displays, (d) to doped NaCl for solid-state batteries; shows awareness of CSIR-NPL standards	Brief mention of one application or generic context without specific Indian relevance	No contextual linkage; treats as purely academic exercise without real-world or national laboratory relevance

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