Physical chemistry
Distribution functions, partition functions
How molecules are shared among speeds and energy levels: the Boltzmann weight, the Maxwell–Boltzmann speed law, and the partition function that packages a molecule's accessible states into one number.
IntuitionIntuition: temperature is a rule for sharing
A gas at 300 K contains roughly 10²³ molecules, yet nobody needs to know where each one is. What survives from that microscopic chaos is a distribution: the fraction of molecules in each speed range or energy level. Heating the sample does not simply "add energy"; it redistributes population — pushing a few more molecules into high-energy states while thinning the crowd at the bottom. The sharing rule turns out to be universal: each state is weighted by e^(−ε/k_BT).
For speeds in an ideal gas this rule becomes the Maxwell–Boltzmann law, a bell-shaped curve that flattens and shifts to higher v as T rises. Crucially, the curve never truly dies at high speed: it is exactly the exponentially thin "tail" of fast molecules that supplies the reactants for activated chemistry.
SchoolSchool: which molecules are fast, which are excited?
Definition: The Boltzmann distribution
At thermal equilibrium at temperature T, the probability that a molecule occupies a state of energy ε_i is proportional to e^(−ε_i/k_BT), with k_B = 1.381×10⁻²³ J K⁻¹. States higher in energy are occupied, but exponentially more sparsely; the ratio of two populations depends only on the energy gap: N_i/N_j = e^(−(ε_i−ε_j)/k_BT).
At room temperature k_BT ≈ 4.1×10⁻²¹ J ≈ 25.7 meV (2.5 kJ mol⁻¹). Any level lying several times k_BT above the ground state is essentially empty — which is why most molecules sit in their electronic ground state and lowest vibrational level at 300 K. Temperature is what relaxes this censorship.
Example: An excited spin population
A spectroscopic measurement places an excited level Δε = 500 cm⁻¹ above the ground state (both non-degenerate). What fraction of molecules occupies it at 300 K?
Solution
Convert to temperature units first: Δε/k_B = hc(500 cm⁻¹)/k_B ≈ 720 K. Then P_exc/P_0 = e^(−720/300) = e^(−2.4) ≈ 0.09 — about 9%. The lesson: a few hundred cm⁻¹ is enough to keep ~90% of the population in the ground state, yet 9% of an excited state is easily detected by spectroscopy.
UndergraduateUniversity: the partition function as a generator
The Boltzmann factor gives ratios; to get absolute populations one must normalise. The sum of Boltzmann weights over all states — the partition function q — is the normalising constant, but it is much more: nearly every thermodynamic property of an ideal gas can be generated from q and its temperature derivatives.
q counts the thermally accessible states, weighted by availability. At low T, q → g₀ (only the ground state counts); at high T, q grows — for rotation q_rot ≈ T/θ_rot grows linearly, for translation q_trans ∝ T^(3/2). Each degree of freedom contributes its own factor, so a molecule's total partition function factorises.
| Molecule | θ_rot (K) | θ_vib (K) |
|---|---|---|
| H₂ | 85.4 | 6215 |
| N₂ | 2.88 | 3390 |
| CO₂ (bend) | 0.56 | 960 |
AdvancedAdvanced: degeneracies, limits, and where the law comes from
Two refinements complete the picture. First, degeneracy: rotational level J is (2J+1)-fold degenerate, so populations P(J) first rise before the Boltzmann tail cuts them — the maximum sits at J_max ≈ √(T/2θ_rot) − 1/2. Second, the classical limit: when k_BT exceeds typical level spacings, the quantum sum becomes an integral and q acquires simple power-law forms — this is why heat capacities approach classical values at high T.
References
- Physical Chemistry: Quanta, Matter, and Change · P. Atkins, J. de Paula, R. Friedman, 2014
- Statistical Mechanics · D. A. McQuarrie, 2000