(a) Using the Gibbs equation for a closed system in the absence of non-expansion work at constant composition, answer the following :

(i) Deduce the thermodynamic relations for the variation of G with T, and with P.

(ii) What are the implications o

Question

(a) Using the Gibbs equation for a closed system in the absence of non-expansion work at constant composition, answer the following :

(i) Deduce the thermodynamic relations for the variation of G with T, and with P.

(ii) What are the implications of the above relations?

(iii) Draw G versus T graph and identify the phase transition temperatures, if any.

(iv) Explain how the presence of (1) attractive and (2) repulsive molecular interactions affects the molar Gibbs free energy of a gas relative to its normal value.

(b) Explain why the energy of a free particle can vary continuously but the energy of a particle in a box is quantized.

(c) Consider a primitive cubic lattice structure of an element.

(i) How many lattice points are present in this unit cell?

(ii) What is the coordination number of the atom present in this structure?

(iii) What is the percentage void volume of this structure?

(iv) If the radius of the atom present in this lattice is 178·1 pm, then find the radius of the sphere that can fit in the centre of this cubic unit cell.

(v) What is the coordination number of this sphere? 10

(d) Consider the equilibrium reaction A₂(g) → 2A(g), in which A₂ gas is 18·5% dissociated at 25 °C and 1 bar.

(i) Calculate K_eq at 25 °C.

(ii) Calculate K_eq at 100 °C.

Given that ΔH° = 57·2 kJ mol⁻¹ (at the above temperature range).

(iii) What is the effect of compression on this reaction? 10

UPSC Answer Check · Accepted Answer

Derive the thermodynamic relations in part (a) starting from dG = VdP − SdT, then apply these to explain phase behavior and molecular interactions. For (b), contrast the boundary conditions of free particle versus particle in a box to explain quantization. Part (c) requires systematic calculation of primitive cubic parameters and void geometry. Part (d) involves equilibrium calculations using van't Hoff equation. Allocate ~35% effort to (a), ~20% to (b), ~25% to (c), and ~20% to (d), ensuring all nine sub-parts are addressed.
- Part (a)(i): Derivation of (∂G/∂T)_P = −S and (∂G/∂P)_T = V from dG = VdP − SdT with proper Maxwell relation justification
- Part (a)(ii)-(iv): Implications including spontaneity criterion, G vs T plot showing melting/boiling points as discontinuities, and explanation of how attractive interactions lower G while repulsive interactions raise G relative to ideal gas
- Part (b): Explanation that free particle has no boundary conditions allowing continuous k values, while particle in a box has quantized k = nπ/L leading to discrete energy levels E_n = n²h²/8mL²
- Part (c): Primitive cubic has 1 lattice point, coordination number 6, 47.6% void volume, body cavity radius = 0.732r = 130.4 pm, and cavity coordination number 8
- Part (d): Calculation of K_p = 4α²/(1−α) = 0.168 at 25°C, then K_p at 100°C using van't Hoff equation giving ~1.12, with compression favoring reverse reaction (Le Chatelier)

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	2	Correctly identifies that G decreases with T at constant P and increases with P at constant T; recognizes that phase transitions appear as slope changes in G-T plot; correctly distinguishes boundary condition effects on quantization; identifies primitive cubic as 1 atom/cell with 6 coordination; understands that dissociation increases with temperature for endothermic reaction	States correct relations but confuses signs or misidentifies phase transition points; gives generic statement about quantization without specific boundary condition analysis; calculates coordination number as 8 or void percentage incorrectly; makes conceptual error in Le Chatelier application	Fundamental confusion between G, H, and S; claims energy is always quantized or never quantized regardless of system; identifies primitive cubic as BCC or FCC; applies van't Hoff equation with wrong sign for ΔH or confuses K_p with K_c
Mechanism / equation	20%	2	Properly derives both partial derivatives from dG = VdP − SdT with clear steps; sets up Schrödinger equation for particle in a box with correct boundary conditions ψ(0)=ψ(L)=0; uses correct formula for void volume % = (V_cell − V_atoms)/V_cell × 100; correctly applies van't Hoff equation ln(K₂/K₁) = (ΔH°/R)(1/T₁ − 1/T₂)	Writes final relations without showing derivation steps; mentions wavefunction but doesn't apply boundary conditions; uses correct void formula but with calculation errors; uses van't Hoff equation but with temperature in Celsius or incorrect ΔH sign	Writes incorrect differential form or confuses with dU, dH relations; no mention of boundary conditions or incorrect quantization condition; uses wrong formula for atomic packing factor; completely wrong approach to equilibrium constant calculation
Numerical accuracy	20%	2	Correct values: α = 0.185 gives K_p(298K) = 4×(0.185)²/(1−0.185²) ≈ 0.168; K_p(373K) ≈ 1.12 using van't Hoff; void volume = 47.6%; cavity radius = (√3 − 1)r/2 or equivalent = 130.4 pm; coordination number of cavity = 8	Correct method but arithmetic errors leading to K_p within 20% of correct value; void percentage calculation with minor error; cavity radius using approximate formula; correct K_p trend but wrong absolute values	Order of magnitude errors in K_p; negative K_p or K_p > 1 without justification; void volume > 100% or < 0%; cavity radius larger than atomic radius; completely wrong temperature dependence
Diagram / structure	20%	2	Clear G vs T plot with negative slopes for all phases, steeper slopes for gas, correct identification of T_m and T_b as intersection points where ΔG = 0 for phase transitions; labeled primitive cubic unit cell showing 8 corner atoms with proper shading; clear indication of body cavity position	G vs T plot with correct general shape but missing phase labels or incorrect slope ordering; primitive cubic diagram without clear atom positions or missing unit cell boundaries; cavity position indicated but not dimensioned	No diagram for (a)(iii); confusing G vs P with G vs T; incorrect phase ordering (solid above liquid above gas); no unit cell diagram; diagram of BCC or FCC instead of primitive cubic
Application context	20%	2	Connects (∂G/∂P)_T = V to geological phase transitions under pressure; relates attractive/repulsive interactions to real gas behavior and Joule-Thomson coefficient; links primitive cubic voids to interstitial alloy formation (e.g., steel); connects equilibrium shift to industrial synthesis conditions like Haber process optimization	Mentions that pressure affects phase stability without specific example; states that real gases deviate from ideality without quantitative link; mentions interstitial sites without naming specific material; states Le Chatelier principle without industrial context	No application context provided; irrelevant examples; confuses thermodynamic relations with kinetic control; no mention of practical significance of any calculation

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