PX2231 THERMAL AND STATISSTICAL
MECHANICS EXAM WITH QUESTIONS AND
ANSWERS|| GUARANTEED PASS|| LATEST
VERSION 2026
What is the method for calculating thermodynamic quantities from a partition
function? - ANSWER-(1) Find ZSP for one particle. (2) Combine for N
particles (care about distinguishability). (3) Derive U and F. (4) Derive S, p,
CV, H, G from U and F.
For N distinguishable vs. indistinguishable particles - ANSWER-how does ZN
differ?,Distinguishable: ZN = ZSP^N. Indistinguishable (classical limit): ZN =
ZSP^N / N! (divide by N! to avoid overcounting).
Derive U for a two-level system with energies ±Δ/2. - ANSWER-ZSP =
2cosh(βΔ/2). U = -d(lnZSP)/dβ = -(Δ/2)tanh(βΔ/2).
What is the CV for a two-level system? - ANSWER-CV =
kB(βΔ/2)²·sech²(βΔ/2).
What is paramagnetism in terms of statistical mechanics? - ANSWER-An
electron spin in a magnetic field is a two-level system with energies ±μBB (Δ =
2μBB). The system can be cooled by adiabatic demagnetisation.
What is adiabatic demagnetisation? - ANSWER-A cooling technique:
isothermally increase B (reducing entropy), then adiabatically decrease B
(entropy constant, B/T constant, so T drops).
,Write the modified first law including chemical potential. - ANSWER-dU =
TdS - pdV + μdN. For multiple species: dU = TdS - pdV + Σᵢ μᵢdNᵢ.
How is chemical potential related to F and G? - ANSWER-μ = (∂F/∂N){V,T} =
(∂G/∂N){p,T}. Most useful: μ = G/N (Gibbs free energy per particle).
What is the condition for particle exchange equilibrium between two systems? -
ANSWER-μ1 = μ2. Chemical potential plays the same role in particle exchange
as temperature does in heat exchange.
What is the entropy condition for particle flow? - ANSWER-dS = (μ1/T1 -
μ2/T2)dN ≥ 0. If T1 = T2: particles flow from higher to lower chemical
potential.
Write the chemical potential for an ideal gas as a function of
pressure/concentration. - ANSWER-μ = kBT·ln(p/p0) = kBT·ln(ρ/ρ0) =
kBT·ln(x/x0). Used in osmosis.
What is the Gibbs factor? - ANSWER-The modified Boltzmann factor for the
grand canonical ensemble: exp(-β(εi - μN)). Accounts for energy μ per particle
added.
What is the Grand Partition Function Z? - ANSWER-Z = Σᵢ gᵢ·exp(-β(εᵢ - μNᵢ)).
Used when particle number is not conserved.
What is the Fermi-Dirac distribution? - ANSWER-n_FD = 1/(exp(β(ε - μ)) + 1).
Gives the mean occupancy of a state for fermions (quantum spin ½, obey Pauli
exclusion).
What is the Bose-Einstein distribution? - ANSWER-n_BE = 1/(exp(β(ε - μ)) -
1). Gives the mean occupancy of a state for bosons (integer spin, no limit on
occupancy).
, What is the Maxwell-Boltzmann distribution in the context of occupation
numbers? - ANSWER-n_MB = 1/exp(β(ε - μ)) = exp(-β(ε - μ)). The classical
limit (n ≪ nQ), valid for both fermions and bosons when occupancy is very
low.
What are fermions and examples? - ANSWER-Particles with half-integer
quantum spin (e.g., ½). Obey Pauli exclusion (max 1 per state). Examples:
electrons, protons, neutrons, quarks.
What are bosons and examples? - ANSWER-Particles with integer quantum
spin (0 or 1). No limit on occupancy. Examples: photons, gluons, Higgs bosons.
What is the Pauli exclusion principle? - ANSWER-No two identical fermions
can occupy the same quantum state simultaneously.
What is electron degeneracy pressure? - ANSWER-The pressure arising from
the Pauli exclusion principle preventing electrons occupying the same state.
Supports white dwarf stars against gravitational collapse (mass < 1.4 M_Sun,
Chandrasekhar limit).
What is neutron degeneracy pressure? - ANSWER-Similar to electron
degeneracy but for neutrons. Supports neutron stars (mass ~1.4-2.4 M_Sun)
formed in supernovae.
Write the energy levels of a quantum harmonic oscillator (SHO). - ANSWER-
εn = (n + ½)ℏω.
Write the partition function for a single quantum SHO. - ANSWER-ZSP =
exp(-βℏω/2) / (1 - exp(-βℏω)).
MECHANICS EXAM WITH QUESTIONS AND
ANSWERS|| GUARANTEED PASS|| LATEST
VERSION 2026
What is the method for calculating thermodynamic quantities from a partition
function? - ANSWER-(1) Find ZSP for one particle. (2) Combine for N
particles (care about distinguishability). (3) Derive U and F. (4) Derive S, p,
CV, H, G from U and F.
For N distinguishable vs. indistinguishable particles - ANSWER-how does ZN
differ?,Distinguishable: ZN = ZSP^N. Indistinguishable (classical limit): ZN =
ZSP^N / N! (divide by N! to avoid overcounting).
Derive U for a two-level system with energies ±Δ/2. - ANSWER-ZSP =
2cosh(βΔ/2). U = -d(lnZSP)/dβ = -(Δ/2)tanh(βΔ/2).
What is the CV for a two-level system? - ANSWER-CV =
kB(βΔ/2)²·sech²(βΔ/2).
What is paramagnetism in terms of statistical mechanics? - ANSWER-An
electron spin in a magnetic field is a two-level system with energies ±μBB (Δ =
2μBB). The system can be cooled by adiabatic demagnetisation.
What is adiabatic demagnetisation? - ANSWER-A cooling technique:
isothermally increase B (reducing entropy), then adiabatically decrease B
(entropy constant, B/T constant, so T drops).
,Write the modified first law including chemical potential. - ANSWER-dU =
TdS - pdV + μdN. For multiple species: dU = TdS - pdV + Σᵢ μᵢdNᵢ.
How is chemical potential related to F and G? - ANSWER-μ = (∂F/∂N){V,T} =
(∂G/∂N){p,T}. Most useful: μ = G/N (Gibbs free energy per particle).
What is the condition for particle exchange equilibrium between two systems? -
ANSWER-μ1 = μ2. Chemical potential plays the same role in particle exchange
as temperature does in heat exchange.
What is the entropy condition for particle flow? - ANSWER-dS = (μ1/T1 -
μ2/T2)dN ≥ 0. If T1 = T2: particles flow from higher to lower chemical
potential.
Write the chemical potential for an ideal gas as a function of
pressure/concentration. - ANSWER-μ = kBT·ln(p/p0) = kBT·ln(ρ/ρ0) =
kBT·ln(x/x0). Used in osmosis.
What is the Gibbs factor? - ANSWER-The modified Boltzmann factor for the
grand canonical ensemble: exp(-β(εi - μN)). Accounts for energy μ per particle
added.
What is the Grand Partition Function Z? - ANSWER-Z = Σᵢ gᵢ·exp(-β(εᵢ - μNᵢ)).
Used when particle number is not conserved.
What is the Fermi-Dirac distribution? - ANSWER-n_FD = 1/(exp(β(ε - μ)) + 1).
Gives the mean occupancy of a state for fermions (quantum spin ½, obey Pauli
exclusion).
What is the Bose-Einstein distribution? - ANSWER-n_BE = 1/(exp(β(ε - μ)) -
1). Gives the mean occupancy of a state for bosons (integer spin, no limit on
occupancy).
, What is the Maxwell-Boltzmann distribution in the context of occupation
numbers? - ANSWER-n_MB = 1/exp(β(ε - μ)) = exp(-β(ε - μ)). The classical
limit (n ≪ nQ), valid for both fermions and bosons when occupancy is very
low.
What are fermions and examples? - ANSWER-Particles with half-integer
quantum spin (e.g., ½). Obey Pauli exclusion (max 1 per state). Examples:
electrons, protons, neutrons, quarks.
What are bosons and examples? - ANSWER-Particles with integer quantum
spin (0 or 1). No limit on occupancy. Examples: photons, gluons, Higgs bosons.
What is the Pauli exclusion principle? - ANSWER-No two identical fermions
can occupy the same quantum state simultaneously.
What is electron degeneracy pressure? - ANSWER-The pressure arising from
the Pauli exclusion principle preventing electrons occupying the same state.
Supports white dwarf stars against gravitational collapse (mass < 1.4 M_Sun,
Chandrasekhar limit).
What is neutron degeneracy pressure? - ANSWER-Similar to electron
degeneracy but for neutrons. Supports neutron stars (mass ~1.4-2.4 M_Sun)
formed in supernovae.
Write the energy levels of a quantum harmonic oscillator (SHO). - ANSWER-
εn = (n + ½)ℏω.
Write the partition function for a single quantum SHO. - ANSWER-ZSP =
exp(-βℏω/2) / (1 - exp(-βℏω)).
