The molecular partition function

Result

The molecular partition function

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Statement

The bridge from energy levels to thermodynamics.

Why it matters

boltzmann-populations gives the fractional population of each individual energy level; the partition function \(q\) is the normalising sum sitting in the denominator of that distribution, and once it is known as a function of temperature (and volume), it alone determines every bulk thermodynamic quantity — internal energy, entropy, free energy, heat capacity — without needing to track individual level populations separately. This is the bridge from molecular energy levels to macroscopic thermodynamics that gives the unit its name, and it is exactly why the lectures immediately following this one build \(q\) up piece by piece into translational, rotational, vibrational, and electronic contributions.

Hypotheses

The system consists of independent particles, each with its own accessible set of quantum energy levels.Independence is what allows the total system partition function to be built simply from the single-particle partition function; genuinely interacting particles require more elaborate ensemble treatments beyond this picture. The relevant energy levels are known or can be modelled (e.g. particle-in-a-box for translation, a rigid rotor for rotation, a harmonic oscillator for vibration).This supplies the actual \(\varepsilon_i\) values the defining sum runs over; the later lectures in this unit evaluate \(q\) explicitly for each of these standard level structures. For a gas of \(N\) identical, indistinguishable particles, the naive \(q^N\) must be divided by \(N!\) to avoid overcounting.This correction, resolving the historical Gibbs paradox, is essential for extensive properties like entropy to come out properly additive; only with quantum mechanics's confirmation that identical particles are fundamentally indistinguishable did this correction receive a first-principles rather than ad hoc justification.

Proof

q = \sum_i g_i\, e^{-\varepsilon_i/kT}

The single-particle partition function is defined as a sum over every accessible energy level \(i\), each weighted by its degeneracy \(g_i\) and Boltzmann factor. A

T\to0: \ q\to g_0; \qquad T\to\infty: \ q \text{ grows, counting more levels}

At very low temperature only the ground state contributes appreciably; as temperature rises, progressively higher levels become thermally accessible and contribute to the sum — \(q\) is, in this sense, a running count of "effectively accessible states" at a given temperature. A

\frac{n_i}{N} = \frac{g_i e^{-\varepsilon_i/kT}}{q}

Comparing with boltzmann-populations' own result shows \(q\) is precisely the normalisation constant that makes the fractional populations across all levels sum to \(1\). A

Q = q^N \ (\text{distinguishable}); \qquad Q=\frac{q^N}{N!}\ (\text{indistinguishable})

For \(N\) independent particles, the total system partition function multiplies the single-particle \(q\) together \(N\) times; correct Boltzmann counting for indistinguishable particles (Hypotheses) requires dividing by \(N!\) to avoid counting the same physical state multiple times under different particle labellings. B

U-U(0) = kT^2\left(\frac{\partial \ln Q}{\partial T}\right)_V, \qquad A = -kT\ln Q

Standard statistical-thermodynamic relations generate internal energy, Helmholtz free energy, and (via further derivatives) entropy and heat capacity directly from \(\ln Q\), so once \(Q\) is known as a function of \(T\) and \(V\), every equilibrium thermodynamic property follows by differentiation alone. B

Result

q = \sum_i g_i\,e^{-\varepsilon_i/kT}, \qquad Q=\frac{q^N}{N!}, \qquad U-U(0)=kT^2\left(\frac{\partial\ln Q}{\partial T}\right)_V

Reading. The partition function is a temperature-dependent sum measuring how many energy states are thermally accessible; its logarithmic derivatives generate every macroscopic thermodynamic property directly, without ever tracking individual particles.

Scope. Valid for systems of non-interacting particles; generalises directly to translational, rotational, vibrational and electronic contributions multiplying together when a molecule's total energy separates into independent parts (Corollaries); genuinely interacting systems require ensemble treatments beyond this single-particle picture.

Corollaries & converses

When a molecule's total energy separates into translational, rotational, vibrational and electronic parts, \(q\) factorises as \(q_{\text{trans}}q_{\text{rot}}q_{\text{vib}}q_{\text{elec}}\), a factorisation exploited heavily by the lectures that follow in this unit.
heat-capacity-from-levels' Einstein and Debye models are a direct application: evaluating \(q\) for a lattice of vibrational oscillators and differentiating twice with respect to \(T\) yields the heat capacity.
statistical-entropy's \(S=k\ln W\) connects to \(q\) through \(S=U/T+k\ln Q\), linking the combinatorial and partition-function viewpoints of entropy.

Fails without

Omit the \(N!\) correction for a gas of indistinguishable particles (Hypotheses): the resulting entropy computed from \(Q=q^N\) fails to be properly extensive (the historical Gibbs paradox), giving physically inconsistent predictions for how entropy should scale with system size.
Treat genuinely interacting particles as independent (Hypotheses): the simple factorisation \(Q=q^N\) fails outright, since the total energy no longer partitions additively across particles, and a full interacting-system ensemble treatment is required instead.

Common errors

Omitting the \(N!\) indistinguishability correction for a gas of identical particles, which leads to a non-extensive, Gibbs-paradox entropy.
Confusing the single-particle partition function \(q\) with the full system partition function \(Q=q^N\) (or \(q^N/N!\)); thermodynamic formulas require the correct one for the quantity being computed.
Assuming \(q\) is always a small, simple integer count of states, when in fact it is a continuous, temperature-dependent real number that can be enormous — e.g. of order \(10^{30}\) for translational motion of a gas molecule in a macroscopic container.
Forgetting the reference-energy convention: \(U-U(0)\) formulas require care about exactly where the zero of energy \(\varepsilon_i\) is defined.

Discussion

The partition function was developed by Ludwig Boltzmann and given its modern, rigorous ensemble formulation largely by Josiah Willard Gibbs in the late nineteenth century; the name itself (from the German Zustandssumme, "sum over states") describes exactly its definition.

Resolving the Gibbs paradox — why naive \(q^N\) overcounts states for indistinguishable particles, producing an entropy that fails to be properly extensive — was historically significant well before a full quantum-mechanical justification existed; only quantum mechanics's confirmation that identical particles are fundamentally, not merely practically, indistinguishable gave the \(N!\) correction a first-principles rather than ad hoc basis.

Common misconception: that a large partition function value means many particles occupy excited states. Rather, \(q\) reflects the number of states that are thermally accessible, weighted by their Boltzmann factor; this can be large even when only a modest fraction of particles actually populate excited levels, simply because many nearly degenerate levels are available — translational motion is the standard example.

Worked examples

\text{Two-level system: } \varepsilon_0=0,\ \varepsilon_1=\varepsilon,\ g_0=g_1=1 \ \Rightarrow\ q=1+e^{-\varepsilon/kT}

As \(T\to0\), \(q\to1\) (only the ground state contributes); as \(T\to\infty\), \(q\to2\) (both levels equally accessible) — the partition function interpolates smoothly between these two limits as temperature rises. A

\frac{n_1}{N} = \frac{e^{-\varepsilon/kT}}{1+e^{-\varepsilon/kT}}

Using Step 3 of the Proof with this two-level \(q\), the excited-state fraction approaches \(1/2\) at high \(T\) and vanishes at low \(T\), directly matching the two limiting values of \(q\) found above. A

q(T\to0)=1; \qquad q(T\to\infty)=2

Reading. Even the simplest possible level structure shows \(q\) directly tracking the number of thermally accessible states, from just the ground state at low \(T\) to both levels equally weighted at high \(T\).

Scope. The same limiting-case reasoning (low-\(T\) and high-\(T\) behaviour of \(q\)) applies to any discrete level structure, and underlies the qualitative behaviour of vibrational and electronic partition functions evaluated later in the unit.

Problems

A three-level system has \(\varepsilon_0=0\), \(\varepsilon_1=\varepsilon\), \(\varepsilon_2=2\varepsilon\), all non-degenerate (\(g_i=1\)). Write \(q\) at temperature \(T\).
Solution
\(q=1+e^{-\varepsilon/kT}+e^{-2\varepsilon/kT}\), directly summing the Boltzmann factor for each of the three levels as defined in Step 1.
Show that \(q\to g_0\) as \(T\to0\) for any level structure, and interpret this physically.
Solution
As \(T\to0\), \(e^{-\varepsilon_i/kT}\to0\) for every level with \(\varepsilon_i>\varepsilon_0\) (since the exponent diverges to \(-\infty\)), while the ground-state term \(g_0e^{-\varepsilon_0/kT}\) remains finite (taking \(\varepsilon_0=0\) as reference, it equals \(g_0\)). Physically, at absolute zero only the ground state (with its full degeneracy \(g_0\)) is populated at all.
Explain, without detailed calculation, why translational motion typically dominates a gas's heat capacity at room temperature while vibrational modes remain largely "frozen out," in terms of the relative sizes of \(q_{\text{trans}}\) and \(q_{\text{vib}}\).
Solution
Translational energy levels are extremely closely spaced (a macroscopic container gives a near-continuum of accessible states), so \(q_{\text{trans}}\) is enormous and changes smoothly and substantially with temperature near room temperature, contributing fully to the heat capacity. Vibrational level spacings are, by contrast, typically large compared with \(kT\) at room temperature, so \(q_{\text{vib}}\) stays close to its low-temperature limiting value and changes very little with modest temperature increases — contributing little to the heat capacity until much higher temperature (heat-capacity-from-levels develops this quantitatively).