MChem Master of Chemistry Master of Chemistry: Physical Chemistry 3 — Questions and Answers
Question 1: The canonical partition function Q for a system of N distinguishable, independent particles is related to the single-particle partition function q by:
- Q = Nq
- Q = q^N (Correct answer)
- Q = q^N / N!
- Q = N! · q
Correct answer: Q = q^N
For distinguishable independent particles, the system partition function is simply the product of individual partition functions: Q = q^N.
Question 2: At high temperatures, the heat capacity at constant volume of a diatomic ideal gas approaches (per molecule) according to the equipartition theorem:
- ³∕₂ k_B
- ⁵∕₂ k_B
- ⁷∕₂ k_B (Correct answer)
- 4k_B
Correct answer: ⁷∕₂ k_B
A diatomic molecule has 3 translational + 2 rotational + 2 vibrational (kinetic+potential) degrees of freedom, giving C_V = 7/2 k_B at high T.
Question 3: The Boltzmann entropy formula S = k_B ln(Ω) implies that the entropy of a perfect crystal at 0 K is zero because:
- All atomic vibrations cease
- There is only one accessible microstate (Ω=1) (Correct answer)
- The heat capacity becomes zero
- The partition function equals N
Correct answer: There is only one accessible microstate (Ω=1)
A perfect crystal at 0 K has a unique ground state arrangement, so Ω = 1 and S = k_B ln(1) = 0, consistent with the Third Law.
Question 4: The Maxwell-Boltzmann speed distribution for gas molecules predicts that the most probable speed v_p scales with temperature as:
- v_p ∝ T
- v_p ∝ T^(1/2) (Correct answer)
- v_p ∝ T^(3/2)
- v_p ∝ T^2
Correct answer: v_p ∝ T^(1/2)
The most probable speed v_p = √(2RT/M) is proportional to the square root of absolute temperature.
Question 5: For an ideal gas undergoing a reversible adiabatic expansion, which relationship between pressure P and volume V holds (γ = C_p/C_v)?
- PV = constant
- PV^γ = constant (Correct answer)
- P/V^γ = constant
- P^γV = constant
Correct answer: PV^γ = constant
A reversible adiabatic (isentropic) process for an ideal gas obeys PV^γ = constant, derived from the first law with dq = 0.
Question 6: The residual entropy of ice at 0 K arises from:
- Lattice defects in the crystal structure
- Random orientation of hydrogen bonds following the ice rules (Correct answer)
- Isotopic disorder of hydrogen and oxygen
- Nuclear spin degeneracy of oxygen-17
Correct answer: Random orientation of hydrogen bonds following the ice rules
Pauling showed that hydrogen-bond proton positions in ice are not unique at 0 K, giving S_residual = Nk_B ln(3/2) per mole.
Question 7: The Sackur-Tetrode equation gives the absolute entropy of a monatomic ideal gas. Doubling the volume at constant temperature and N changes the entropy by:
- Nk_B ln 2 (Correct answer)
- Nk_B / 2
- 2Nk_B ln 2
- −Nk_B ln 2
Correct answer: Nk_B ln 2
From S = Nk_B[ln(V/N) + ...], doubling V increases S by Nk_B ln(2), consistent with a reversible isothermal expansion.
MChem Master of Chemistry Master of Chemistry: Physical Chemistry 3
The canonical partition function Q for a system of N distinguishable, independent particles is related to the single-particle partition function q by: