STATISTICAL PHYSICS COMPREHENSIVE
EXAM QUESTION AND CORRECT
ANSWERS (VERIFIED ANSWERS) PLUS
RATIONALES 2026 Q&A INSTANT
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1. Which ensemble describes a system at fixed energy, volume, and particle
number?
A. Canonical
B. Grand canonical
C. Microcanonical
D. Isothermal–isobaric
Answer: C
Rationale: The microcanonical ensemble is defined by fixed E, V, and N
with equal a priori probabilities.
2. The partition function for the canonical ensemble is defined as
A. Z = Σe^{+βE}
B. Z = Σe^{−βE}
C. Z = ΣE e^{−β}
D. Z = ln Σe^{−βE}
Answer: B
Rationale: The canonical partition function sums Boltzmann factors
e^{−βE_i}.
3. β in statistical mechanics is equal to
A. kT
B. 1/kT
C. T/k
, D. −1/kT
Answer: B
Rationale: β ≡ 1/(k_B T), relating temperature to energy scale.
4. Entropy in the microcanonical ensemble is given by
A. S = k ln Z
B. S = −k Σ p ln p
C. S = k ln Ω
D. S = −∂F/∂T
Answer: C
Rationale: Boltzmann’s formula defines entropy in terms of the number of
accessible microstates Ω.
5. The Helmholtz free energy is defined as
A. F = U − TS
B. F = U + TS
C. F = H − TS
D. F = G − PV
Answer: A
Rationale: Helmholtz free energy measures useful work at constant T and
V.
6. Which quantity is minimized at equilibrium in the canonical ensemble?
A. Entropy
B. Internal energy
C. Helmholtz free energy
D. Enthalpy
Answer: C
Rationale: At fixed T and V, equilibrium corresponds to minimum
Helmholtz free energy.
7. The grand canonical ensemble allows fluctuations in
A. Energy only
B. Particle number only
C. Energy and volume
, D. Energy and particle number
Answer: D
Rationale: In the grand canonical ensemble, both E and N fluctuate while
T, V, and μ are fixed.
8. The grand partition function depends on
A. T, V, N
B. T, V, μ
C. E, V, N
D. P, T, N
Answer: B
Rationale: The grand partition function is a function of temperature,
volume, and chemical potential.
9. The chemical potential represents
A. Energy per unit volume
B. Heat capacity
C. Energy change when adding a particle
D. Pressure per particle
Answer: C
Rationale: Chemical potential is defined as the change in free energy with
respect to particle number.
10.Which distribution applies to indistinguishable particles with no restriction
on occupancy?
A. Maxwell–Boltzmann
B. Fermi–Dirac
C. Bose–Einstein
D. Gibbs
Answer: C
Rationale: Bose–Einstein statistics allow multiple particles in the same
quantum state.
11.Fermi–Dirac statistics apply to particles with
A. Integer spin
EXAM QUESTION AND CORRECT
ANSWERS (VERIFIED ANSWERS) PLUS
RATIONALES 2026 Q&A INSTANT
DOWNLOAD PDF
1. Which ensemble describes a system at fixed energy, volume, and particle
number?
A. Canonical
B. Grand canonical
C. Microcanonical
D. Isothermal–isobaric
Answer: C
Rationale: The microcanonical ensemble is defined by fixed E, V, and N
with equal a priori probabilities.
2. The partition function for the canonical ensemble is defined as
A. Z = Σe^{+βE}
B. Z = Σe^{−βE}
C. Z = ΣE e^{−β}
D. Z = ln Σe^{−βE}
Answer: B
Rationale: The canonical partition function sums Boltzmann factors
e^{−βE_i}.
3. β in statistical mechanics is equal to
A. kT
B. 1/kT
C. T/k
, D. −1/kT
Answer: B
Rationale: β ≡ 1/(k_B T), relating temperature to energy scale.
4. Entropy in the microcanonical ensemble is given by
A. S = k ln Z
B. S = −k Σ p ln p
C. S = k ln Ω
D. S = −∂F/∂T
Answer: C
Rationale: Boltzmann’s formula defines entropy in terms of the number of
accessible microstates Ω.
5. The Helmholtz free energy is defined as
A. F = U − TS
B. F = U + TS
C. F = H − TS
D. F = G − PV
Answer: A
Rationale: Helmholtz free energy measures useful work at constant T and
V.
6. Which quantity is minimized at equilibrium in the canonical ensemble?
A. Entropy
B. Internal energy
C. Helmholtz free energy
D. Enthalpy
Answer: C
Rationale: At fixed T and V, equilibrium corresponds to minimum
Helmholtz free energy.
7. The grand canonical ensemble allows fluctuations in
A. Energy only
B. Particle number only
C. Energy and volume
, D. Energy and particle number
Answer: D
Rationale: In the grand canonical ensemble, both E and N fluctuate while
T, V, and μ are fixed.
8. The grand partition function depends on
A. T, V, N
B. T, V, μ
C. E, V, N
D. P, T, N
Answer: B
Rationale: The grand partition function is a function of temperature,
volume, and chemical potential.
9. The chemical potential represents
A. Energy per unit volume
B. Heat capacity
C. Energy change when adding a particle
D. Pressure per particle
Answer: C
Rationale: Chemical potential is defined as the change in free energy with
respect to particle number.
10.Which distribution applies to indistinguishable particles with no restriction
on occupancy?
A. Maxwell–Boltzmann
B. Fermi–Dirac
C. Bose–Einstein
D. Gibbs
Answer: C
Rationale: Bose–Einstein statistics allow multiple particles in the same
quantum state.
11.Fermi–Dirac statistics apply to particles with
A. Integer spin
