At 298 K, the vibrational partition function q_vib for a diatomic molecule like N₂ is very close to 1, while q_trans is on the order of 10³⁰. What explains this enormous difference?
ATranslational motion is quantized with smaller energy level spacings than vibrational motion, so far more translational states are thermally accessible
BVibrational modes are not allowed in diatomic molecules at room temperature
CThe translational partition function includes a volume factor that artificially inflates it
DRotational modes borrow energy from vibrational modes, reducing q_vib
The key comparison is energy level spacing vs. kT. Translational energy level spacings in a macroscopic container are incredibly tiny (≈10⁻³⁸ J), so kT at 298 K (≈4×10⁻²¹ J) is enormously larger — millions of translational states are thermally accessible, giving a huge q_trans. Vibrational energy spacings hν are typically on the order of 10⁻²⁰ J, comparable to or larger than kT, so nearly all molecules sit in the ground vibrational state, giving q_vib ≈ 1. The factorization q = q_trans · q_rot · q_vib · q_elec makes this mode-by-mode comparison clean.
Question 2 True / False
For an ideal gas of N identical molecules, the N-molecule partition function Q equals q^N, where q is the single-molecule partition function.
TTrue
FFalse
Answer: False
Q = q^N/N! for identical indistinguishable molecules. The N! corrects for overcounting: naively treating each permutation of N identical molecules as a distinct microstate would violate the quantum-mechanical principle that swapping identical particles does not create a new state. This correction is essential — without it, the calculated entropy is too large (the Gibbs paradox), and the entropy of mixing of identical gases would be nonzero, which is unphysical.
Question 3 Short Answer
What does it mean physically for a partition function q to be large, and which molecular partition function is typically largest at room temperature?
Think about your answer, then reveal below.
Model answer: A large partition function means many quantum states are thermally accessible at temperature T — the Boltzmann factors for many excited states are non-negligible. The translational partition function q_trans is by far the largest at room temperature (often 10²⁵–10³⁰ for 1 mole of gas in a liter), because translational energy levels are so closely spaced that an astronomical number are populated. A large q also implies high entropy: S = k(ln q + T d(ln q)/dT), so modes with large q contribute most to the total entropy.
Recognizing which partition functions are large versus small at a given temperature gives immediate insight into what dominates thermodynamic properties. At room temperature: q_trans >> q_rot >> q_vib ≈ 1 (for most molecules), and q_elec = ground-state degeneracy (usually 1). This hierarchy shifts at high temperatures where vibrational modes become activated.