The Boltzmann distribution of populations

Result

The Boltzmann distribution of populations

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Statement

How molecules populate available energy levels.

Why it matters

Quantum mechanics gives the discrete, allowed energy levels available to a molecule (translational, rotational, vibrational, electronic); on its own, however, it says nothing about how a real sample of an enormous number of molecules actually distributes itself among those levels at a given temperature. The Boltzmann distribution answers exactly that question, and it is the single bridge connecting microscopic, quantum energy-level structure to macroscopic, measurable thermodynamic behaviour — the starting point molecular-partition-function immediately builds on to construct essentially every bulk thermodynamic quantity from molecular-level information.

Every temperature-dependent bulk property a chemist actually measures — heat capacity (heat-capacity-from-levels), the position of a chemical equilibrium, a reaction's rate (via the Arrhenius equation's own reliance on essentially this same distribution) — ultimately traces back to how populations redistribute among energy levels as temperature changes, making this the conceptual foundation the rest of statistical thermodynamics is built on.

Hypotheses

The system is at thermal equilibrium, with a large, fixed total number of particles \(N\) and a fixed total energy \(E\).The derivation below finds the single most probable distribution of particles among energy levels consistent with these two constraints; away from equilibrium (immediately after a fast perturbation, for instance), the population distribution can differ substantially from the Boltzmann form until the system relaxes. Particles are treated as distinguishable enough, and \(N\) large enough, that the most probable distribution overwhelmingly dominates all other distributions consistent with the same \(N\) and \(E\).For the macroscopic particle numbers relevant to real chemical samples (of order Avogadro's number), the most probable distribution is so overwhelmingly more likely than any other consistent distribution that its properties are, for all practical purposes, exactly the system's observed, average behaviour — a key simplification that would not hold for a system of only a handful of particles.

Proof

W = \frac{N!}{\prod_i n_i!}, \qquad \sum_i n_i = N, \quad \sum_i n_i\varepsilon_i = E

The number of distinct ways \(W\) of distributing \(N\) distinguishable particles among energy levels \(\varepsilon_i\) with populations \(n_i\) is a standard combinatorial (multinomial) count, subject to the two physical constraints of fixed total particle number and fixed total energy. B

\ln W \approx N\ln N - N - \sum_i (n_i\ln n_i - n_i) \quad \text{(Stirling's approximation, valid for large } n_i\text{)}

Because \(N\) and each significantly populated \(n_i\) are astronomically large for a macroscopic sample, Stirling's approximation for the factorial (cited, a standard result) converts the unwieldy factorial expression into a smooth function that can be maximised using ordinary calculus. B

\frac{\partial}{\partial n_i}\Big[\ln W - \alpha\big(\textstyle\sum n_i - N\big) - \beta\big(\textstyle\sum n_i\varepsilon_i - E\big)\Big] = 0

The most probable distribution is the one that maximises \(\ln W\) (equivalently \(W\) itself, since \(\ln\) is monotonic) subject to the two constraints; introducing Lagrange multipliers \(\alpha\) and \(\beta\) (the standard method for constrained optimisation, cited here) for particle number and energy conservation respectively allows the maximisation to be carried out level by level. B

n_i \propto e^{-\beta\varepsilon_i}, \qquad \beta \equiv \frac{1}{kT}

Carrying out the differentiation of Step 3 for each level and solving gives populations falling off exponentially with energy; identifying \(\beta=1/kT\) (established by comparing this statistical result against the known thermodynamic behaviour of, for example, an ideal gas) fixes the multiplier's physical meaning as inverse temperature. B

\frac{n_i}{n_j} = \frac{g_i}{g_j}\,e^{-(\varepsilon_i-\varepsilon_j)/kT}

Including a degeneracy factor \(g_i\) (the number of distinct states sharing energy \(\varepsilon_i\)) generalises the result to levels that are not individually non-degenerate; the ratio of populations in any two levels depends only on their energy gap and the temperature, not on the absolute energy scale chosen. A

Result

\frac{n_i}{n_j} = \frac{g_i}{g_j}\,e^{-(\varepsilon_i-\varepsilon_j)/kT}

Reading. At thermal equilibrium, the ratio of populations in any two energy levels is set entirely by their energy gap relative to the thermal energy \(kT\) (and by their relative degeneracies): higher-energy levels are always less populated than lower ones, and the population gap narrows as temperature rises.

Scope. Valid at thermal equilibrium for a large number of particles (Hypotheses); molecular-partition-function extends this single-level-pair ratio into a normalised distribution across all available levels simultaneously.

Corollaries & converses

As \(T\to0\), essentially all population collapses into the lowest available energy level (or levels, if the ground state is degenerate); as \(T\to\infty\), the exponential factor for every level approaches \(1\) and population spreads out, weighted only by degeneracy.
heat-capacity-from-levels follows directly: as temperature rises, an increasing population fraction gains access to higher, closely spaced vibrational or rotational levels, and it is precisely this redistribution that gives rise to a temperature-dependent heat capacity in the Einstein and Debye models of solids.
Converse: measuring the relative intensities of two spectroscopic transitions originating from two populated levels lets the temperature be inferred directly from the Result, run in reverse — a standard diagnostic technique in rotational and vibrational spectroscopy.

Fails without

Apply the distribution to a system far from thermal equilibrium: immediately after a fast perturbation (e.g. a laser pulse), the population distribution can differ substantially from the equilibrium Boltzmann form until the system has time to relax back toward it (Hypotheses, first point).
Apply the most-probable-distribution reasoning to a system of only a few particles: for small \(N\), the most probable distribution is not overwhelmingly more likely than nearby distributions (Hypotheses, second point), so the sharp, single-distribution result of Step 4 no longer describes the system's actual, fluctuating behaviour well.

Common errors

Forgetting the degeneracy factor \(g_i/g_j\) in Step 5 when comparing populations of levels that are not equally degenerate, particularly for rotational levels, whose degeneracy itself grows with quantum number.
Assuming the Boltzmann distribution implies every level is at least somewhat populated; levels with \(\varepsilon_i-\varepsilon_j\gg kT\) have a population ratio so close to zero as to be, for practical purposes, entirely unpopulated at that temperature.
Confusing the Boltzmann distribution (populations across quantum states of one system, at equilibrium) with the completely separate Maxwell-Boltzmann speed distribution (over molecular speeds in a gas); the two share underlying statistical logic but describe different physical quantities.
Treating \(\beta=1/kT\) as an arbitrary fitting parameter rather than as literally defining the thermodynamic temperature within this statistical framework (Step 4).

Discussion

Ludwig Boltzmann developed the statistical framework underlying this distribution in the 1870s, providing a microscopic, probabilistic foundation for macroscopic thermodynamic quantities that had until then been understood only phenomenologically; his work (alongside James Clerk Maxwell's closely related contributions on molecular speed distributions) founded the field of statistical mechanics.

The derivation given here (maximising \(W\) subject to constraints) finds only the single most probable distribution, not literally the only possible one; statistical-entropy's result, \(S=k\ln W\), evaluated for this specific most-probable \(W\), is what connects this microscopic population-counting exercise directly back to the macroscopic thermodynamic entropy.

Common misconception: that at any given instant, every molecule in a sample sits in the single lowest-energy level, with only "some" molecules exceptionally promoted to higher levels. In reality the Boltzmann distribution describes a genuine, continuously repopulating equilibrium spread across all accessible levels simultaneously, with the relative population of each level fixed (on average, for a large sample) by the Result, not a static assignment of most molecules to one level alone.

Worked examples

\Delta\varepsilon = 8.0\times10^{-21}\,\text{J}, \qquad T=298\,\text{K}, \qquad g_i=g_j=1

Two non-degenerate vibrational-like levels separated by a small energy gap are compared at room temperature; the population ratio follows directly from the Result. A

\frac{n_i}{n_j} = e^{-\Delta\varepsilon/kT} = \exp\!\left(\frac{-8.0\times10^{-21}}{1.381\times10^{-23}\times298}\right) = e^{-1.94} \approx 0.14

Using Boltzmann's constant \(k=1.381\times10^{-23}\,\text{J/K}\), the excited level is populated at roughly \(14\%\) of the lower level's population at room temperature — a small but non-negligible fraction, typical of low-lying vibrational levels of modest energy gap. A

n_i/n_j \approx 0.14 \text{ at } 298\,\text{K}

Reading. Even a comparatively small energy gap, of order a few \(kT\), leaves a substantial minority population in the higher level at room temperature rather than leaving it essentially empty.

Scope. Repeating this calculation at a much higher or lower temperature shows the same exponential sensitivity that drives essentially all temperature-dependent spectroscopic and thermodynamic behaviour discussed throughout this unit.

Problems

Two energy levels are separated by \(\Delta\varepsilon = 4\times10^{-20}\,\text{J}\) and are equally degenerate. Estimate the population ratio \(n_i/n_j\) at \(500\,\text{K}\).
Solution
\(n_i/n_j = e^{-\Delta\varepsilon/kT} = \exp(-4\times10^{-20}/(1.381\times10^{-23}\times500)) = e^{-5.79}\approx3.1\times10^{-3}\), a strongly suppressed excited-level population even at this elevated temperature.
A higher energy level has degeneracy \(g_i=3\) while the lower level has \(g_j=1\), separated by \(2kT\). Find the population ratio and comment on the effect of the degeneracy factor.
Solution
\(n_i/n_j = (g_i/g_j)e^{-2} = 3\times0.135\approx0.41\). Without the degeneracy factor the ratio would be only \(0.135\); the threefold degeneracy of the upper level substantially increases its total population relative to a naive energy-only estimate, since three distinct states are each independently populated according to the same Boltzmann exponential.
Explain, using the Result, why raising the temperature of a two-level system always increases the population of the higher-energy level relative to the lower one, but the ratio can never exceed \(g_i/g_j\).
Solution
As \(T\) increases, \(-(\varepsilon_i-\varepsilon_j)/kT\) becomes less negative, so \(e^{-(\varepsilon_i-\varepsilon_j)/kT}\) increases monotonically toward \(1\) as \(T\to\infty\); the population ratio therefore rises toward, but never exceeds, \(g_i/g_j\), since the exponential factor itself is always strictly less than or equal to \(1\) for \(\varepsilon_i>\varepsilon_j\) at any finite positive temperature.