Level 2 — RecallThermodynamics & Statistical Mechanics (Advanced)

Thermodynamics & Statistical Mechanics (Advanced)

⏱ 30 minutes40 marksprintable — key stays hidden on paper

Level 2 Paper — Recall & Standard Problems

Time Limit: 30 minutes
Total Marks: 40

Instructions: Answer all questions. Use $k_B$ for Boltzmann's constant, $R$ for the gas constant. Show working where required.

Q1. State the four thermodynamic potentials $U$ , $H$ , $F$ , $G$ and write each in terms of $U$ using the appropriate variables. (4 marks)

Q2. Starting from $dF = -S\,dT - P\,dV$ , derive the Maxwell relation associated with the Helmholtz free energy. (4 marks)

Q3. State the Gibbs phase rule and use it to determine the number of degrees of freedom for a single-component system at its triple point. (4 marks)

Q4. Write down Boltzmann's entropy formula. A system has $\Omega = 4$ equally likely microstates in state A and $\Omega = 8$ in state B. Compute the entropy difference $S_B - S_A$ in units of $k_B$ . (4 marks)

Q5. For a two-level system with energies $0$ and $\varepsilon$ , write down the canonical partition function $Z$ for a single particle and derive an expression for the average energy $\langle E\rangle$ . (5 marks)

Q6. State the equipartition theorem. Hence give the total internal energy and the molar heat capacity $C_V$ of one mole of an ideal diatomic gas (translational + rotational degrees of freedom only). (5 marks)

Q7. Write the Clausius–Clapeyron equation and state the physical meaning of each symbol. (4 marks)

Q8. Write down the Bose–Einstein and Fermi–Dirac occupation-number distributions. State one key physical difference between the two. (5 marks)

Q9. Write the Planck distribution for the spectral energy density $u(\nu)$ of blackbody radiation and state the limit it reduces to at low frequency. (5 marks)

End of Paper

Answer keyMark scheme & solutions

Q1. (4 marks)

$U$ = internal energy, natural variables $(S,V)$ . (1)
$H = U + PV$ , natural variables $(S,P)$ . (1)
$F = U - TS$ , natural variables $(T,V)$ . (1)
$G = U + PV - TS = H - TS$ , natural variables $(T,P)$ . (1)

Why: Each is a Legendre transform swapping a natural variable pair to make different variables independent.

Q2. (4 marks)

$dF = -S\,dT - P\,dV$ so $S = -\left(\partial F/\partial T\right)_V$ , $P = -\left(\partial F/\partial V\right)_T$ . (1)
Equality of mixed second partials: $\frac{\partial^2 F}{\partial V\,\partial T} = \frac{\partial^2 F}{\partial T\,\partial V}$ . (1)
$-\left(\frac{\partial S}{\partial V}\right)_T = -\left(\frac{\partial P}{\partial T}\right)_V$ . (1)
Result: $\boxed{\left(\dfrac{\partial S}{\partial V}\right)_T = \left(\dfrac{\partial P}{\partial T}\right)_V}$ . (1)

Q3. (4 marks)

Gibbs phase rule: $F = C - P + 2$ , where $C$ = components, $P$ = phases, $F$ = degrees of freedom. (2)
Triple point: $C=1$ , $P=3$ → $F = 1 - 3 + 2 = 0$ . (1)
Interpretation: invariant point, no freedom to vary $T$ or pressure. (1)

Q4. (4 marks)

$S = k_B \ln \Omega$ . (1)
$S_A = k_B\ln 4$ , $S_B = k_B\ln 8$ . (1)
$S_B - S_A = k_B\ln(8/4) = k_B\ln 2$ . (1)
$\approx 0.693\,k_B$ . (1)

Q5. (5 marks)

$Z = e^{0} + e^{-\beta\varepsilon} = 1 + e^{-\beta\varepsilon}$ , with $\beta = 1/(k_BT)$ . (2)
$\langle E\rangle = -\dfrac{\partial \ln Z}{\partial\beta}$ . (1)
$\dfrac{\partial \ln Z}{\partial\beta} = \dfrac{-\varepsilon e^{-\beta\varepsilon}}{1+e^{-\beta\varepsilon}}$ . (1)
$\langle E\rangle = \dfrac{\varepsilon e^{-\beta\varepsilon}}{1+e^{-\beta\varepsilon}} = \dfrac{\varepsilon}{e^{\beta\varepsilon}+1}$ . (1)

Q6. (5 marks)

Equipartition: each quadratic degree of freedom contributes $\tfrac12 k_BT$ to average energy. (1)
Diatomic (trans + rot): $f = 3 + 2 = 5$ degrees of freedom. (1)
Per mole: $U = \tfrac{5}{2}RT$ . (2)
$C_V = \left(\partial U/\partial T\right)_V = \tfrac{5}{2}R$ . (1)

Q7. (4 marks)

$\dfrac{dP}{dT} = \dfrac{L}{T\,\Delta V}$ (or $\dfrac{\Delta S}{\Delta V}$ ). (2)
$\dfrac{dP}{dT}$ = slope of coexistence curve; $L$ = latent heat of transition; $T$ = transition temperature; $\Delta V$ = volume change between phases. (2)

Q8. (5 marks)

Bose–Einstein: $\bar{n} = \dfrac{1}{e^{(\varepsilon-\mu)/k_BT} - 1}$ . (2)
Fermi–Dirac: $\bar{n} = \dfrac{1}{e^{(\varepsilon-\mu)/k_BT} + 1}$ . (2)
Difference: fermions obey Pauli exclusion ( $\bar n \le 1$ , sign $+1$ ); bosons can multiply-occupy a state (sign $-1$ ). (1)

Q9. (5 marks)

Planck: $u(\nu)\,d\nu = \dfrac{8\pi h\nu^3}{c^3}\dfrac{1}{e^{h\nu/k_BT}-1}\,d\nu$ . (3)
Low-frequency (Rayleigh–Jeans) limit: $e^{h\nu/k_BT}-1 \approx h\nu/k_BT$ , giving $u(\nu) \to \dfrac{8\pi\nu^2}{c^3}k_BT$ . (2)

[
  {"claim":"S_B - S_A = k_B ln 2 (coefficient ln2)","code":"import sympy as sp\nkB=sp.Symbol('kB',positive=True)\nSA=kB*sp.log(4); SB=kB*sp.log(8)\nresult = sp.simplify(SB-SA - kB*sp.log(2))==0"},
  {"claim":"Two-level average energy equals eps/(e^{beta eps}+1)","code":"beta,eps=sp.symbols('beta eps',positive=True)\nZ=1+sp.exp(-beta*eps)\nE=-sp.diff(sp.log(Z),beta)\nresult = sp.simplify(E - eps/(sp.exp(beta*eps)+1))==0"},
  {"claim":"Diatomic (5 dof) molar CV = 5R/2","code":"R,T=sp.symbols('R T',positive=True)\nU=sp.Rational(5,2)*R*T\nCV=sp.diff(U,T)\nresult = sp.simplify(CV - sp.Rational(5,2)*R)==0"},
  {"claim":"Triple point degrees of freedom F=0 for C=1,P=3","code":"C,P=1,3\nF=C-P+2\nresult = (F==0)"},
  {"claim":"Planck low-freq limit gives Rayleigh-Jeans 8 pi nu^2 kB T / c^3","code":"h,nu,kB,T,c=sp.symbols('h nu kB T c',positive=True)\nu_planck=(8*sp.pi*h*nu**3/c**3)/(sp.exp(h*nu/(kB*T))-1)\nlim=sp.limit(u_planck,h,0)\nresult = sp.simplify(lim - 8*sp.pi*nu**2*kB*T/c**3)==0"}
]