Questions: Canonical Ensemble and Molecular Partition Functions

Question 1

A diatomic gas has a vibrational frequency such that hν >> kT at room temperature. What happens to the vibrational contribution to the molar heat capacity as temperature is dramatically reduced?

• A. It increases, because lower temperature means slower vibration and greater sensitivity to small energy inputs
• B. It stays constant at R, because each vibrational mode always contributes exactly R to the heat capacity
• C. It approaches zero, because when kT << hν the Boltzmann factor for excited vibrational states is negligible and the mode is effectively frozen out
• D. It doubles, because the partition function increases as fewer states are thermally accessible

Answer: (C)

A vibrational mode contributes to heat capacity only when thermal energy kT is comparable to or larger than the energy spacing between vibrational levels (hν). When kT << hν, essentially all molecules remain in the ground vibrational state — the Boltzmann factor e^(−hν/kT) for the first excited state is nearly zero. The partition function approaches 1 (only the ground state counts), and its temperature derivative is negligible, so the heat capacity contribution approaches zero. This 'freezing out' of vibrational modes was a major triumph of quantum statistical mechanics over classical equipartition.

Question 2

From the canonical partition function Z, which thermodynamic quantities can be derived?

• A. Only average energy — other thermodynamic quantities require the microcanonical or grand canonical ensemble
• B. Average energy, Helmholtz free energy, entropy, pressure, and heat capacity can all be derived from Z and its temperature or volume derivatives
• C. Only entropy and Helmholtz free energy — average energy requires direct calculation from the energy spectrum
• D. Only properties of ideal gases — real interacting systems require fundamentally different partition functions

Answer: (B)

The partition function Z is a generating function for thermodynamics: the Helmholtz free energy is A = −kT ln Z, the average energy is ⟨E⟩ = −∂(ln Z)/∂β (where β = 1/kT), entropy follows from S = −(∂A/∂T)_V, pressure from P = −(∂A/∂V)_T, and heat capacity from C_V = (∂⟨E⟩/∂T)_V. All equilibrium thermodynamic properties flow from derivatives of Z — this is precisely why the partition function is the central object of statistical thermodynamics.

Question 3

The canonical partition function Z = Σ e^(−Eᵢ/kT) is simply a normalization constant that ensures microstate probabilities sum to one, with no deeper physical significance.

• T. True
• F. False

Answer: False

Z is far more than a normalization constant. It is a generating function from which all equilibrium thermodynamic properties can be derived via differentiation with respect to temperature or volume. The Helmholtz free energy equals −kT ln Z directly, and average energy, entropy, pressure, and heat capacity all follow from derivatives of ln Z. Calling Z 'merely' a normalization constant misses the central insight of the canonical ensemble: the logarithm of Z encodes the thermodynamic state of the system.

Question 4

For a molecule with independent translational, rotational, and vibrational modes, the total molecular partition function equals the product of the individual mode partition functions.

• T. True
• F. False

Answer: True

When different molecular degrees of freedom are independent (non-interacting), the total energy is the sum of contributions from each mode: E = E_trans + E_rot + E_vib + E_elec. Because the Boltzmann factor of a sum of energies equals the product of Boltzmann factors — e^(−E/kT) = e^(−E_trans/kT) · e^(−E_rot/kT) · e^(−E_vib/kT) · ... — the partition function factorizes into q = q_trans · q_rot · q_vib · q_elec. This factorization is what makes statistical mechanics tractable for real molecules.

Question 5

Why do the molar heat capacities of molecular gases depend on temperature, and what role does the partition function play in explaining this dependence?

Model answer: Heat capacity reflects how much thermal energy a system can absorb per degree of temperature rise, which depends on how many energy modes are thermally accessible. A mode contributes to heat capacity only when kT is comparable to or larger than its energy-level spacing. Translational and rotational levels are so closely spaced that they are fully excited at ordinary temperatures, contributing their classical values ((3/2)R and R or (3/2)R per mole). Vibrational levels are more widely spaced: when kT << hν, the Boltzmann factor for excited vibrational states is negligible and the mode is 'frozen out,' contributing nearly nothing to heat capacity. As temperature rises, kT eventually becomes comparable to hν and vibrational modes activate. The partition function captures this through the temperature dependence of the Boltzmann factors — modes only appear in the temperature derivative of ln Z when they have thermally accessible excited states.

This temperature dependence of heat capacity was inexplicable by classical statistical mechanics (which predicted constant heat capacities via equipartition). Quantum mechanics, encoded in the discrete energy levels of each mode, provides the correct answer: only modes with level spacings small compared to kT are fully excited. The partition function is the mathematical bridge between the quantum energy spectrum and the macroscopic heat capacity.