Level 1 — RecognitionPhysical Chemistry (Advanced)

Physical Chemistry (Advanced)

⏱ 20 minutes30 marksprintable — key stays hidden on paper

Time limit: 20 minutes
Total marks: 30
Instructions: Answer all questions. For True/False items, a correct verdict without justification earns half marks. Use $...$ notation where relevant.

Section A — Multiple Choice (1 mark each)

Q1. For a particle in a 1-D box of length $L$ , the energy of level $n$ is:

(a) $E_n = \dfrac{n^2 h^2}{8mL^2}$
(b) $E_n = \dfrac{n h^2}{8mL^2}$
(c) $E_n = \dfrac{n^2 h^2}{2mL^2}$
(d) $E_n = \dfrac{h^2}{8mL^2 n^2}$

Q2. The ground-state energy of the hydrogen atom is:

(a) $-3.4$ eV (b) $-13.6$ eV (c) $-27.2$ eV (d) $+13.6$ eV

Q3. The variational principle guarantees that the trial energy $E_{trial}$ satisfies:

(a) $E_{trial} \le E_0$ (b) $E_{trial} = E_0$ (c) $E_{trial} \ge E_0$ (d) no relation to $E_0$

Q4. In the Hartree–Fock method, the electron–electron interaction is treated via:

(a) exact correlation (b) a mean (average) field (c) neglect entirely (d) classical point charges only

Q5. For a rigid rotor, the rotational energy levels are given by ( $B$ = rotational constant):

(a) $E_J = BJ$ (b) $E_J = hcB\,J(J+1)$ (c) $E_J = hcB\,J^2$ (d) $E_J = hcB(J+\tfrac12)$

Q6. The vibrational energy levels of a harmonic oscillator are:

(a) $E_v = hν\,v$ (b) $E_v = hν(v+\tfrac12)$ (c) $E_v = hν\,v^2$ (d) $E_v = hν(v^2+\tfrac12)$

Q7. The Langmuir isotherm assumes:

(a) multilayer adsorption (b) monolayer adsorption on equivalent sites (c) infinite adsorption energy (d) adsorbate–adsorbate attraction

Q8. The BET isotherm is used to describe:

(a) monolayer only (b) multilayer adsorption / surface area determination (c) chemisorption only (d) liquid–liquid emulsions

Q9. The critical micelle concentration (CMC) is the concentration above which:

(a) surfactant precipitates (b) micelles begin to form (c) surface tension increases sharply (d) emulsion breaks

Q10. At high overpotential, the Butler–Volmer equation reduces to the:

(a) Nernst equation (b) Tafel equation (c) Langmuir equation (d) Arrhenius equation

Q11. The Stark–Einstein law states that:

(a) one photon can excite many molecules (b) one absorbed photon activates one molecule (c) quantum yield is always 1 (d) light intensity equals rate

Q12. In a semiconductor, conductivity increases with temperature because:

(a) the band gap widens (b) more electrons are promoted across the gap (c) phonon scattering decreases (d) the Fermi level vanishes

Section B — Matching (½ mark each, 4 marks total)

Q13. Match each partition function to its dominant physical origin.

Column X	Column Y
(i) $q_{trans}$	(P) spacing of vibrational modes
(ii) $q_{rot}$	(Q) mass and box volume
(iii) $q_{vib}$	(R) degeneracy of ground electronic state
(iv) $q_{elec}$	(S) moment of inertia

Q14. Match photophysical process to description.

Column X	Column Y
(i) Fluorescence	(P) $T_1 \to S_0$ , spin-forbidden, long-lived
(ii) Phosphorescence	(Q) $S_1 \to S_0$ , spin-allowed, fast
(iii) Internal conversion	(R) non-radiative $S_1 \to S_0$
(iv) Intersystem crossing	(S) $S_1 \to T_1$ , spin change

Section C — True/False with Justification (2 marks each)

Q15. "The energy levels of a particle in a box are non-degenerate and increase in spacing as $n$ increases." — True or False? Justify.

Q16. "The Freundlich isotherm ( $x/m = k\,p^{1/n}$ ) predicts saturation of the surface at high pressure." — True or False? Justify.

Q17. "A quantum yield greater than 1 is impossible under any circumstances." — True or False? Justify.

Q18. "In the Morse potential, vibrational levels converge as $v$ increases and the molecule can dissociate." — True or False? Justify.

Q19. "A superconductor below its critical temperature expels magnetic flux (Meissner effect) and has zero DC resistance." — True or False? Justify.

Q20. "In density functional theory (DFT), the ground-state energy is a functional of the electron density $\rho(\mathbf{r})$ ." — True or False? Justify.

Answer keyMark scheme & solutions

Section A (1 mark each)

Q1 — (a). From solving the Schrödinger equation with infinite walls, $E_n = n^2 h^2/(8mL^2)$ ; energy scales as $n^2$ . ✔

Q2 — (b) $-13.6$ eV. $E_n = -13.6/n^2$ eV; ground state $n=1$ gives $-13.6$ eV. ✔

Q3 — (c) $E_{trial} \ge E_0$ . Any trial wavefunction gives an expectation energy that is an upper bound to the true ground-state energy. ✔

Q4 — (b) mean field. Each electron feels the averaged field of all others; correlation beyond this is neglected. ✔

Q5 — (b) $E_J = hcB\,J(J+1)$ . Standard rigid-rotor result. ✔

Q6 — (b) $E_v = hν(v+\tfrac12)$ . Equally spaced levels with zero-point energy $\tfrac12 hν$ . ✔

Q7 — (b) monolayer on equivalent sites. Langmuir assumes uniform sites, no lateral interactions, monolayer coverage. ✔

Q8 — (b) multilayer / surface area. BET extends Langmuir to multilayers; used for surface-area measurement. ✔

Q9 — (b) micelles begin to form. Above CMC surfactant aggregates into micelles; surface tension plateaus. ✔

Q10 — (b) Tafel. At large $|\eta|$ one exponential term dominates → linear $\eta$ vs $\log j$ . ✔

Q11 — (b) one photon activates one molecule. Primary photochemical act is 1:1. ✔

Q12 — (b) more electrons promoted across gap. Thermal excitation increases carrier concentration → intrinsic semiconductors more conductive when hot. ✔

Section B

Q13 (½ each): (i)–Q, (ii)–S, (iii)–P, (iv)–R.
Reasoning: $q_{trans}$ depends on mass/volume; $q_{rot}$ on moment of inertia $I$ ; $q_{vib}$ on mode spacing $hν$ ; $q_{elec}$ on ground-state degeneracy.

Q14 (½ each): (i)–Q, (ii)–P, (iii)–R, (iv)–S.

Section C (verdict 1 + justification 1)

Q15 — TRUE. Levels $E_n \propto n^2$ are non-degenerate in 1-D; spacing $E_{n+1}-E_n = (2n+1)h^2/(8mL^2)$ grows with $n$ . (verdict 1, justification 1)

Q16 — FALSE. Freundlich is a power law with no plateau; $x/m$ keeps rising with $p$ , so it does not predict saturation (Langmuir does). (1+1)

Q17 — FALSE. In chain reactions (e.g. H₂+Cl₂), one photon can initiate a chain giving quantum yields ≫ 1. (1+1)

Q18 — TRUE. Morse anharmonicity makes levels converge ( $\Delta E$ decreases with $v$ ) toward the dissociation limit $D_e$ . (1+1)

Q19 — TRUE. Both zero DC resistance and Meissner flux expulsion are defining properties below $T_c$ . (1+1)

Q20 — TRUE. By the Hohenberg–Kohn theorem, ground-state energy is a unique functional of $\rho(\mathbf{r})$ . (1+1)

[
  {"claim":"PIB level spacing between n=2 and n=1 equals 3h^2/8mL^2 (coefficient 3)","code":"h,m,L,n=symbols('h m L n',positive=True); E=lambda k: k**2*h**2/(8*m*L**2); diff=simplify(E(2)-E(1)); coeff=simplify(diff/(h**2/(8*m*L**2))); result=(coeff==3)"},
  {"claim":"Hydrogen ground state -13.6/n^2 gives -13.6 eV at n=1 and -3.4 at n=2","code":"n=symbols('n'); E=lambda k:-Rational(136,10)/k**2; result=(E(1)==Rational(-136,10)) and (E(2)==Rational(-34,10))"},
  {"claim":"Harmonic oscillator spacing is constant (independent of v)","code":"h,nu,v=symbols('h nu v',positive=True); E=lambda k:h*nu*(k+Rational(1,2)); s1=simplify(E(v+1)-E(v)); s2=simplify(E(v+2)-E(v+1)); result=simplify(s1-s2)==0"},
  {"claim":"Rigid rotor J=1 to J=0 gives energy 2hcB","code":"h,c,B,J=symbols('h c B J',positive=True); E=lambda k:h*c*B*k*(k+1); result=simplify(E(1)-E(0))==2*h*c*B"}
]