(a) (i) Find the wavelength of the emitted light when 1.0×10⁻²⁷ g particle in a one-dimensional box of length 3 Å goes from nₓ = 2 to nₓ = 1 level.

(ii) Explain the Heisenberg uncertainty principle. 10

(b) Draw Lewis dot structure of [Br₃]⁻ and H₂N

Question

(a) (i) Find the wavelength of the emitted light when 1.0×10⁻²⁷ g particle in a one-dimensional box of length 3 Å goes from nₓ = 2 to nₓ = 1 level.

(ii) Explain the Heisenberg uncertainty principle. 10

(b) Draw Lewis dot structure of [Br₃]⁻ and H₂NCSNH₂ (central C atom is bonded to both the N atoms and to the S atom). Does thiourea contain polar bonds? If yes, which is the most polar bond? 10

(c) (i) Derive an expression that relates the wavelength of the X-rays with the distance between the layers of atoms in a crystal.

(ii) The X-rays of wavelength 220 pm are diffracted from an ionic crystal at an angle of 23°. What is the distance between the layers that are responsible for this diffraction? 10

(d) (i) Find the temperature at which the water molecules can have the root-mean-square speed of 719 m s⁻¹. 5

(ii) What is the root-mean-square velocity of water molecules at 473 K? 5

(e) (i) Calculate the E° value for the half-reaction Cr³⁺(aq)+3e⁻ → Cr

Given that at 25 °C
Cr³⁺(aq)+e⁻ → Cr²⁺(aq)    E° = –0.424 V
Cr²⁺(aq)+2e⁻ → Cr    E° = –0.90 V

(ii) The surface tension of liquid A is seven times higher than that of liquid B.

(1) Which liquid is expected to have higher contact angle with glass?

(2) 10 mL of each of these liquids are placed in separate 100 mL glass beakers. How do these liquids respond if the gravitational field is switched off?

UPSC Answer Check · Accepted Answer

Solve this multi-part numerical and theoretical problem by allocating time proportionally to marks: spend ~20% on (a)(i) particle-in-a-box calculation, ~10% on (a)(ii) Heisenberg principle explanation, ~20% on (b) Lewis structures and polarity analysis, ~20% on (c) Bragg's law derivation and calculation, ~20% on (d) RMS velocity calculations, and ~10% on (e) electrochemistry and surface chemistry. Begin each part with the relevant formula, show step-by-step working with proper units, and conclude with brief conceptual explanations where asked.
- (a)(i) Apply particle-in-a-box energy formula ΔE = (n₂²-n₁²)h²/8mL² and use E=hc/λ to find emitted wavelength (~657 nm or similar)
- (a)(ii) State Heisenberg uncertainty principle Δx·Δp ≥ h/4π with physical interpretation and mention its role in establishing quantum mechanical nature of particles
- (b) Draw correct Lewis structures: [Br₃]⁻ as linear with 3 lone pairs on central Br and single bonds; thiourea with C=S double bond, C-N single bonds, and lone pairs; identify C=S as most polar bond due to electronegativity difference
- (c)(i) Derive Bragg's law nλ = 2d sinθ from constructive interference of X-rays scattered by crystal planes with path difference analysis
- (c)(ii) Calculate interplanar spacing d = λ/2sinθ = 220 pm/(2×sin23°) ≈ 282 pm
- (d)(i) Use v_rms = √(3RT/M) to find T = Mv²_rms/3R ≈ 373 K (100°C) for water
- (d)(ii) Calculate v_rms = √(3×8.314×473/0.018) ≈ 808 m s⁻¹ at 473 K
- (e)(i) Apply ΔG° = -nFE° and Hess's law: E°(Cr³⁺/Cr) = [1×(-0.424)+2×(-0.90)]/3 = -0.741 V
- (e)(ii)(1) Liquid B has higher contact angle (lower surface tension → poorer wetting); (2) Liquid A forms more spherical cap/lower meniscus, liquid B spreads more in zero gravity

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies quantum confinement in 1-D box for (a), resonance and electronegativity effects in thiourea for (b), constructive interference condition for Bragg's law in (c), Maxwell-Boltzmann distribution for RMS velocity in (d), and Gibbs free energy-electrode potential relationship in (e); no conceptual errors across any sub-part	Most concepts correctly identified but minor errors such as confusing RMS with average velocity, or incomplete explanation of why C=S is most polar; some conceptual gaps in surface tension-contact angle relationship	Fundamental misconceptions like treating particle-in-a-box as hydrogen atom, confusing Bragg's law with Beer-Lambert law, or incorrect relationship between surface tension and wetting behavior
Mechanism / equation	20%	10	All equations stated correctly before use: E_n = n²h²/8mL² with correct substitution for (a), proper path difference derivation for Bragg's law in (c), v_rms = √(3RT/M) with M in kg mol⁻¹ for (d), and nFE° = ΣnᵢE°ᵢ for (e); proper unit conversions throughout	Correct final equations but missing intermediate steps or inconsistent units (e.g., using g instead of kg for mass, or Å not converted to m); derivation of Bragg's law incomplete or hand-waved	Wrong equations used (e.g., using kinetic energy formula ½mv² directly for quantum system), missing critical equations, or algebraic manipulation errors that propagate through calculations
Numerical accuracy	25%	12.5	All numerical answers correct with proper significant figures: (a)(i) λ ≈ 657 nm or 6.57×10⁻⁷ m, (c)(ii) d ≈ 282 pm, (d)(i) T ≈ 373 K, (d)(ii) v_rms ≈ 808 m s⁻¹, (e)(i) E° ≈ -0.741 V; clear unit handling and dimensional analysis	Correct methodology but calculation errors leading to answers within 10-20% of correct value; common errors like factor of 2 in Bragg's law, or using Celsius instead of Kelvin in gas calculations	Numerical answers orders of magnitude wrong due to unit conversion failures (e.g., not converting Å to m, or g to kg), or completely wrong formulas yielding nonsensical results
Diagram / structure	15%	7.5	Clear Lewis structures for [Br₃]⁻ showing 22 valence electrons with correct formal charges (central Br +1, terminal Brs -1 each) and linear geometry; thiourea with correct connectivity (C central, bonded to two N and S), proper double bond placement, and all lone pairs shown; clear labeling of most polar bond	Structures drawn but with minor errors like missing lone pairs, incorrect formal charges, or wrong connectivity in thiourea (e.g., S-C-N-N instead of N-C-S-C-N type structure)	Structures completely wrong, missing, or showing impossible bonding (e.g., expanded octet on nitrogen, wrong number of electrons); no identification of polar bonds or incorrect polarity assignment
Application context	20%	10	Connects particle-in-a-box to quantum dots/nanomaterials, Bragg's law to XRD in mineral identification (e.g., Indian zeolites) or pharmaceutical polymorphism, RMS velocity to atmospheric escape and planetary atmospheres, surface tension to microgravity fluid behavior and space applications; mentions Indian contributions like C.V. Raman's work on light scattering	Brief mention of applications without elaboration, or generic statements about 'use in industry'; some real-world connections but lacking specificity or relevance to Indian context	No application context provided; purely mechanical problem-solving without any physical interpretation or real-world relevance; fails to explain why results matter

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