The Partition Function: Connecting Molecules to Thermodynamics
One quantity that, once known, yields every thermodynamic property of the system. Understanding what it counts is more useful than memorising the formulas it generates.
The definition and what it means
The sum runs over energy levels, with gi the degeneracy of level i. Each term is the Boltzmann factor for that level, weighted by how many states share its energy.
Separating the contributions
Because the energy of a molecule is approximately the sum of independent contributions, the partition function factorises into a product:
This is enormously convenient: each mode is handled separately, and the approximation is good because the modes are largely independent at ordinary temperatures.
| Mode | Level spacing | Typical size of q at room temperature |
|---|---|---|
| Translational | Extremely small | Very large — enormous numbers of states accessible |
| Rotational | Small | Moderately large |
| Vibrational | Large | Close to 1 — mostly ground state only |
| Electronic | Very large | Equal to the ground-state degeneracy |
That table explains a great deal of ordinary chemistry. Vibrational levels are widely spaced, so at room temperature almost every molecule sits in its vibrational ground state — which is why the zero-point energy matters and why vibrational contributions to heat capacity only appear at higher temperature.
Getting thermodynamics out
Once q is known for a system of N particles, the standard results follow by differentiation:
The point worth grasping is the direction of the logic. Molecular data — energy levels obtained from spectroscopy — give q, and q gives bulk thermodynamic quantities. Statistical thermodynamics is therefore the bridge that lets a spectrum predict a heat capacity.
Distinguishable and indistinguishable particles
For localised particles, as in a crystal, each is distinguishable by its position and the total partition function is qN. For a gas, particles are indistinguishable and dividing by N! corrects for overcounting arrangements that are not physically different.
Omitting that factor gives an entropy that is not extensive — doubling the system does not double the entropy — which is the Gibbs paradox. Being asked why the N! appears is a standard conceptual question, and the extensivity argument is the answer.
The symmetry number
The rotational partition function is divided by a symmetry number, the number of indistinguishable orientations the molecule has. A homonuclear diatomic has symmetry number 2, since rotating it by 180° gives an identical configuration.
Omitting it double-counts states and gives entropy values that are systematically too high. It is a small factor with a clear physical meaning, and questions often include it precisely because it is easy to forget.
Frequently asked questions
Why is the partition function called that?
Because it describes how molecules are partitioned among the available energy levels. The name is descriptive of the Boltzmann distribution it encodes.
Why does the vibrational partition function stay near one?
Because vibrational levels are widely spaced compared with kT at room temperature, so excited levels have negligible population and only the ground state contributes appreciably.
What happens to q as temperature approaches zero?
It approaches the degeneracy of the ground state, since all other Boltzmann factors vanish. For a non-degenerate ground state, q approaches one.
How does this connect to the third law?
A non-degenerate ground state gives q of one at zero temperature, and the resulting entropy is zero. That is the statistical statement of the third law, and it also explains residual entropy where the ground state is degenerate.
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