Level 1 — RecognitionThermodynamics & Statistical Mechanics (Advanced)

Thermodynamics & Statistical Mechanics (Advanced)

⏱ 20 minutes30 marksprintable — key stays hidden on paper

Level 1 — Recognition Test

Time Limit: 20 minutes Total Marks: 30

Section A — Multiple Choice (1 mark each)

Choose the single best answer.

Q1. The Helmholtz free energy is defined as: (a) $F = U + PV$ (b) $F = U - TS$ (c) $F = H - TS$ (d) $F = U + TS$

Q2. The natural variables of the Gibbs free energy $G$ are: (a) $(S, V)$ (b) $(S, P)$ (c) $(T, V)$ (d) $(T, P)$

Q3. Boltzmann's entropy formula is: (a) $S = k_B \Omega$ (b) $S = k_B \ln \Omega$ (c) $S = \ln(k_B \Omega)$ (d) $S = k_B / \ln \Omega$

Q4. In the canonical ensemble, the probability of a microstate with energy $E_i$ is proportional to: (a) $E_i$ (b) $e^{+\beta E_i}$ (c) $e^{-\beta E_i}$ (d) $1/E_i$

Q5. The Clausius–Clapeyron equation describes: (a) the slope of a coexistence curve $dP/dT$ (b) the heat capacity ratio (c) partition function scaling (d) Fermi energy

Q6. The equipartition theorem assigns to each quadratic degree of freedom an average energy of: (a) $k_B T$ (b) $\tfrac{3}{2}k_B T$ (c) $\tfrac{1}{2}k_B T$ (d) $\tfrac{1}{2}k_B T^2$

Q7. Fermi–Dirac statistics apply to particles that: (a) are bosons (b) obey the Pauli exclusion principle (c) are distinguishable (d) have integer spin

Q8. The Gibbs phase rule is: (a) $F = C - P + 2$ (b) $F = C + P - 2$ (c) $F = P - C + 2$ (d) $F = C - P - 2$

Q9. The average energy in the canonical ensemble is obtained from $Z$ via: (a) $\langle E\rangle = \partial \ln Z/\partial \beta$ (b) $\langle E\rangle = -\partial \ln Z/\partial \beta$ (c) $\langle E\rangle = k_B T \ln Z$ (d) $\langle E\rangle = Z^2$

Q10. The chemical potential is defined as: (a) $\mu = (\partial G/\partial N)_{T,P}$ (b) $\mu = (\partial G/\partial T)_{N,P}$ (c) $\mu = (\partial U/\partial S)_{V,N}$ (d) $\mu = (\partial F/\partial V)_{T,N}$

Q11. The Helmholtz free energy is related to the partition function by: (a) $F = k_B T \ln Z$ (b) $F = -k_B T \ln Z$ (c) $F = -\ln Z$ (d) $F = k_B T / Z$

Q12. Bose–Einstein condensation is the macroscopic occupation of: (a) the highest energy state (b) the single lowest-energy quantum state (c) all states equally (d) the Fermi level

Section B — Matching (2 marks each)

Q13. Match each thermodynamic potential to its natural variables:

Potential		Natural Variables
(i) $U$		(A) $(T,P)$
(ii) $H$		(B) $(S,V)$
(iii) $F$		(C) $(T,V)$
(iv) $G$		(D) $(S,P)$

Q14. Match each distribution/statistics to its feature:

Statistics		Feature
(i) Maxwell–Boltzmann		(A) $\langle n\rangle = 1/(e^{\beta(\epsilon-\mu)}+1)$
(ii) Bose–Einstein		(B) distinguishable classical particles
(iii) Fermi–Dirac		(C) Planck blackbody spectrum
(iv) Photon gas		(D) $\langle n\rangle = 1/(e^{\beta(\epsilon-\mu)}-1)$

Section C — True/False WITH Justification (2 marks each: 1 verdict + 1 justification)

Q15. "A Maxwell relation $\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V$ can be derived from the internal energy $U(S,V)$ ." True or False? Justify.

Q16. "In an isolated system, the equilibrium macrostate corresponds to the smallest number of microstates $\Omega$ ." True or False? Justify.

Q17. "The Legendre transform $H = U + PV$ replaces the natural variable $V$ by $P$ ." True or False? Justify.

Q18. "The Gibbs–Helmholtz equation is $\left(\frac{\partial (G/T)}{\partial T}\right)_P = -\frac{H}{T^2}$ ." True or False? Justify.

Q19. "For a system in contact with a heat and particle reservoir, fermions can occupy the same quantum state without restriction." True or False? Justify.

Answer keyMark scheme & solutions

Section A (1 mark each)

Q1 — (b) $F = U - TS$ . Helmholtz free energy Legendre-transforms $U$ in $S$ . (1)

Q2 — (d) $(T,P)$ . $dG = -S\,dT + V\,dP$ , so $G=G(T,P)$ . (1)

Q3 — (b) $S = k_B \ln \Omega$ . Entropy grows as the logarithm of the multiplicity. (1)

Q4 — (c) $e^{-\beta E_i}$ . Boltzmann factor; higher energy → lower probability. (1)

Q5 — (a) slope $dP/dT$ of a phase-coexistence curve. (1)

Q6 — (c) $\tfrac{1}{2}k_B T$ per quadratic degree of freedom. (1)

Q7 — (b) Pauli exclusion (half-integer spin fermions). (1)

Q8 — (a) $F = C - P + 2$ . (1)

Q9 — (b) $\langle E\rangle = -\partial \ln Z/\partial \beta$ . Sign from $e^{-\beta E}$ . (1)

Q10 — (a) $\mu = (\partial G/\partial N)_{T,P}$ . (1)

Q11 — (b) $F = -k_B T \ln Z$ . (1)

Q12 — (b) macroscopic occupation of the single lowest-energy quantum state. (1)

Section B (2 marks each)

Q13. (i)→(B), (ii)→(D), (iii)→(C), (iv)→(A). Reason: $dU=TdS-PdV\Rightarrow(S,V)$ ; $dH=TdS+VdP\Rightarrow(S,P)$ ; $dF=-SdT-PdV\Rightarrow(T,V)$ ; $dG=-SdT+VdP\Rightarrow(T,P)$ . All four correct = 2; two–three correct = 1; else 0. (2)

Q14. (i)→(B), (ii)→(D), (iii)→(A), (iv)→(C). MB = distinguishable classical; BE has $-1$ in denominator; FD has $+1$ ; photons give Planck spectrum. All four correct = 2; two–three correct = 1; else 0. (2)

Section C (2 marks each: verdict 1 + justification 1)

Q15 — TRUE. (verdict 1) From $dU=TdS-PdV$ : $T=(\partial U/\partial S)_V$ , $-P=(\partial U/\partial V)_S$ . Equality of mixed second partials $\partial^2 U/\partial V\partial S=\partial^2 U/\partial S\partial V$ gives $(\partial T/\partial V)_S=-(\partial P/\partial S)_V$ . (justification 1)

Q16 — FALSE. (verdict 1) Since $S=k_B\ln\Omega$ and the second law maximizes entropy, equilibrium corresponds to the largest $\Omega$ , not the smallest. (justification 1)

Q17 — TRUE. (verdict 1) $H=U+PV$ is the Legendre transform of $U(S,V)$ with respect to $V$ ; since $P=-(\partial U/\partial V)_S$ , adding $PV$ swaps the conjugate pair so $H=H(S,P)$ , replacing $V$ by $P$ . (justification 1)

Q18 — TRUE. (verdict 1) Using $G=H-TS$ and $S=-(\partial G/\partial T)_P$ : $\big(\partial(G/T)/\partial T\big)_P=\frac{1}{T}(\partial G/\partial T)_P-\frac{G}{T^2}=\frac{-S}{T}-\frac{G}{T^2}=\frac{-TS-G}{T^2}=\frac{-H}{T^2}$ (since $G+TS=H$ ). (justification 1)

Q19 — FALSE. (verdict 1) Fermions obey the Pauli exclusion principle; each quantum state holds at most one fermion, reflected in the FD occupation $\langle n\rangle=1/(e^{\beta(\epsilon-\mu)}+1)\le 1$ . (justification 1)

[
  {"claim":"F = U - TS is the Helmholtz definition; check dF = -S dT - P dV structure gives natural vars (T,V)",
   "code":"S,T,U,V,P,Fsym=symbols('S T U V P F'); F=U-T*S; dF=diff(F,T); result=(dF== -S)"},
  {"claim":"Gibbs-Helmholtz: d/dT (G/T) = -H/T**2 with G=H-T*S and S=-dG/dT",
   "code":"T=symbols('T',positive=True); G=Function('G'); H,S=symbols('H S'); Gt=G(T); expr=diff(Gt/T,T); sub=expr.subs(diff(Gt,T),-S); target=(-(Gt+T*S)/T**2); result=simplify(sub-target)==0"},
  {"claim":"Average energy = -d(lnZ)/dbeta for Z=exp(-beta*E) two-level model gives Boltzmann weighted energy",
   "code":"beta,E1,E2=symbols('beta E1 E2',positive=True); Z=exp(-beta*E1)+exp(-beta*E2); Eavg=-diff(ln(Z),beta); check=(E1*exp(-beta*E1)+E2*exp(-beta*E2))/Z; result=simplify(Eavg-check)==0"},
  {"claim":"Gibbs phase rule F=C-P+2 gives F=2 for single component single phase",
   "code":"C,P=1,1; Fphase=C-P+2; result=(Fphase==2)"}
]