Paper I — Q1
(a) Calculate ⟨x⟩, the expectation value of position of a particle, in the ground state of one-dimensional box having length from…
Calculate ⟨x⟩, the expectation value of position of a particle, in the ground state of one-dimensional box having length from 0 to l. 10 marks
The enthalpy of formation of an ionic compound can be calculated with accuracy by Born-Haber cycle. Predict, giving valid reasons, the possibility of formation of NaCl₂ salt. The following thermodynamic data are given for NaCl and NaCl₂ :
NaCl U₀ = −757 kJ mol⁻¹ ΔH₍IE₎ = +495 kJ mol⁻¹ ΔH₍EA₎ = −348 kJ mol⁻¹ ½ ΔH₍diss₎ = +121 kJ mol⁻¹ ΔH₍sub₎ = +108 kJ mol⁻¹
NaCl₂ U₀ = −2155 kJ mol⁻¹ ΔH₍IE₂₎ = +4561 kJ mol⁻¹ 10 marks
For a rubber band
((∂ T)/(∂ l))_S = -T/(C_V)((∂ S)/(∂ l))_T
What would be the length of the rubber band with an increase in temperature? Explain it. 10 marks
The vapour pressure of water at 95 °C is found to be 634 mm. What would be the vapour pressure at a temperature of 100 °C? The heat of vapourization in this range of temperature may be taken as 40593 J mol⁻¹.
[R = 8·314 J K⁻¹ mol⁻¹] 10 marks
Define electrochemical series. Give its significance. 10 marks
हिंदी में प्रश्न पढ़ें
एक-विमीय बॉक्स, जिसकी लंबाई 0 से l है, में एक कण जो कि मूल अवस्था में है, की स्थिति के अपेक्षा मान, ⟨x⟩, का परिकलन कीजिए। (10 अंक)
आयनी यौगिक की संभव एन्थैल्पी यथार्थता से बॉर्न-हाबर चक्र से परिकलित की जा सकती है। NaCl₂ लवण के संभव की संभावना मान्य कारणों के साथ सूचित कीजिए। निम्नलिखित ऊष्मागतिक आँकड़ा NaCl और NaCl₂ के लिए दिया गया है :
NaCl U₀ = −757 kJ mol⁻¹ ΔH₍IE₎ = +495 kJ mol⁻¹ ΔH₍EA₎ = −348 kJ mol⁻¹ ½ ΔH₍diss₎ = +121 kJ mol⁻¹ ΔH₍sub₎ = +108 kJ mol⁻¹
NaCl₂ U₀ = −2155 kJ mol⁻¹ ΔH₍IE₂₎ = +4561 kJ mol⁻¹ (10 अंक)
एक रबड़ बैंड के लिए
((∂ T)/(∂ l))_S = -T/(C_V)((∂ S)/(∂ l))_T
तापमान बढ़ाने से इस रबड़ बैंड की लंबाई क्या होगी? व्याख्या कीजिए। (10 अंक)
95 °C पर जल का वाष्प-दाब 634 mm पाया गया। 100 °C तापमान पर कितना वाष्प-दाब होगा? इस तापमान परिसर में वाष्पन ऊष्मा 40593 J mol⁻¹ ली गई है।
[R = 8·314 J K⁻¹ mol⁻¹] (10 अंक)
वैद्युत रासायनिक श्रेणी की परिभाषा दीजिए। इसकी सार्थकता बताइए। (10 अंक)
Model answer
Drafted with AI by UPSC Answer Check against this question's marking rubric, to the expected length. UPSC does not publish answers for Mains — this is one way to score well, not an official key. How we build rubrics and answers.
(a) Expectation value of position Ground state of a particle in a box (0 to l): ψ₁ = √(2/l) sin(πx/l), normalised. ⟨x⟩ = ∫₀ˡ ψ₁* x ψ₁ dx = (2/l) ∫₀ˡ x sin²(πx/l) dx. Use sin²θ = (1 − cos 2θ)/2: ⟨x⟩ = (1/l)[∫₀ˡ x dx − ∫₀ˡ x cos(2πx/l) dx]. First integral = l²/2. For the second, by parts, ∫x cos(2πx/l)dx = (l x/2π) sin(2πx/l) + (l²/4π²) cos(2πx/l); between 0 and l this equals 0. Hence ⟨x⟩ = (1/l)(l²/2) = l/2. This agrees with the symmetry of |ψ₁|² about the box centre, and with the classical average position.
(b) Born–Haber cycle: NaCl and NaCl₂ ΔHf(NaCl) = ΔHsub + ΔHIE + ½ΔHdiss + ΔHEA + U₀ = 108 + 495 + 121 − 348 − 757 = −381 kJ mol⁻¹. It is strongly exothermic, so NaCl is stable. ΔHf(NaCl₂) = ΔHsub + ΔHIE₁ + ΔHIE₂ + ΔHdiss + 2ΔHEA + U₀ = 108 + 495 + 4561 + 242 − 696 − 2155 = +2555 kJ mol⁻¹. The large positive value arises because the second ionisation removes an electron from the stable neon core (Na⁺, 2p⁶), costing 4561 kJ mol⁻¹, which even the higher lattice energy (−2155) cannot repay. Hence NaCl₂ cannot form; Na gives only Na⁺.
(c) Rubber band Stretching uncoils the polymer chains, so entropy falls: (∂S/∂l)_T < 0. In the given relation (∂T/∂l)_S = −(T/C)(∂S/∂l)_T > 0, so reversible adiabatic stretching warms the band (Gough–Joule effect). At constant tension, heating shortens it: (∂l/∂T)_f = −(∂f/∂T)_l/(∂f/∂l)_T. Since f = −T(∂S/∂l)_T > 0 rises with T, (∂f/∂T)_l > 0, and (∂f/∂l)_T > 0, so (∂l/∂T)_f < 0. The band contracts on heating, unlike metals. Elasticity is entropic: higher temperature increases the tendency of chains to coil.
(d) Vapour pressure at 100 °C Clausius–Clapeyron: ln(P₂/P₁) = (ΔHvap/R)(1/T₁ − 1/T₂). T₁ = 368.15 K, T₂ = 373.15 K. ΔHvap/R = 40593/8.314 = 4882.6 K. 1/T₁ − 1/T₂ = 3.640 × 10⁻⁵ K⁻¹. ln(P₂/P₁) = 0.1777, so P₂/P₁ = 1.194. P₂ = 634 × 1.194 ≈ 757 mm Hg, close to 760 mm, the vapour pressure at the normal boiling point.
(e) Electrochemical series It is the arrangement of elements (or their ions) in order of their standard electrode (reduction) potentials, measured at 25 °C, 1 M and 1 atm against the standard hydrogen electrode (E° = 0). Significance: (i) the more negative the E°, the stronger the reducing agent (Li, K, Zn); the more positive, the stronger the oxidising agent (F₂, Au³⁺); (ii) a metal displaces any lower one from solution (Zn displaces Cu²⁺); (iii) metals above hydrogen liberate H₂ from dilute acids; (iv) cell emf = E°cathode − E°anode, and ΔG° = −nFE°, predicting spontaneity; (v) it helps choose sacrificial metals against corrosion and explains order of discharge at electrodes.
How this answer will be evaluated
What earns full marks: Full derivations, correct signs, clear physical reasoning, and accurate calculations.
Evaluation rubric
Each sub-part is marked on its own, against the marks and word limit printed on the paper.
- (a) Derive the expectation value of position for a particle in a 1D box. 10 marks
calculate— given → formula → substitution → result with units → interpretation
Must cover
- State ground state wavefunction ψ₁ = √(2/l)sin(πx/l)
- Set up integral ⟨x⟩ = ∫₀ˡ x|ψ₁|² dx
- Show integration by parts or substitution steps
- Final result ⟨x⟩ = l/2
Loses marks
- Using wrong wavefunction (e.g., cos)
- Missing integration steps
Earns more
- Mention symmetry argument for l/2
- Correct limits of integration
Extra mark
- Comparison with classical probability distribution
- (b) Predict formation of NaCl₂ using Born-Haber cycle data. 10 marks
justify— claim → 3-4 reasons → evidence → conclusion
Must cover
- Calculate ΔHf for NaCl = -381 kJ/mol (negative)
- Calculate ΔHf for NaCl2 = +2555 kJ/mol (uses 2 x EA and full ΔHdiss; positive)
- Compare stability based on enthalpy signs
- Conclude NaCl₂ is not stable
Loses marks
- Arithmetic errors in summation
- Ignoring the sign of enthalpy
Earns more
- Explicit Born-Haber cycle equation
- Mention high 2nd IE of Na
Extra mark
- Reference to actual lattice energy trends
- (c) Explain length change of rubber band with temperature increase. 10 marks
explain— definition/context → points in order → small example → short close
Must cover
- Identify (∂S/∂l)_T as negative (entropy decreases on stretch)
- Deduce (∂T/∂l)_S is positive
- State that at constant tension rubber contracts on heating (negative thermal expansion); adiabatic stretching warms it
- Link to entropic elasticity of polymer chains
Loses marks
- Claiming rubber expands like metals
- Confusing isothermal and adiabatic conditions
Earns more
- Mention adiabatic heating on stretching
- Reference to molecular coiling
Extra mark
- Gough-Joule effect reference
- (d) Calculate vapour pressure of water at 100 °C. 10 marks
calculate— given → formula → substitution → result with units → interpretation
Must cover
- Use Clausius-Clapeyron equation
- Convert temperatures to Kelvin (368K, 373K)
- Substitute ΔH_vap = 40593 J/mol
- Calculate P2 ≈ 757 mm (close to 760 mm at normal boiling point)
Loses marks
- Using Celsius in gas constant term
- Sign error in log equation
Earns more
- Correct use of log(P₂/P₁) form
- Unit consistency check
Extra mark
- Mentioning standard boiling point
- (e) Define electrochemical series and its significance. 10 marks
define— precise definition → the distinguishing feature → one example
Must cover
- Define as list of elements by standard reduction potential
- Mention reference to Standard Hydrogen Electrode (SHE)
- Explain use in predicting cell EMF
- Explain use in predicting displacement reactions
Loses marks
- Confusing oxidation and reduction potential order
- Vague definition without standard conditions
Earns more
- Mentioning galvanic vs electrolytic cells
- Example of a specific reaction
Extra mark
- Reference to corrosion protection
What "Solve" is asking you to do
Choose the method, then carry it through to a final answer. Identifying what kind of problem this is and why that method applies is the first thing marked; a correct figure arrived at invisibly earns almost nothing.
Structure that answers it
Given data and what is required → method chosen, with the reason it applies → set-up (equation, circuit, free body, trial balance) → working, step by step → answer with units and any condition of validity
Where marks are lost
Doing the middle steps mentally and writing only the result. In mathematics papers, a further loss comes from giving a decimal where the exact value in surds or fractions was wanted, or from skipping the justification a part explicitly asks for.
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