Boltzmann distribution
/ BOLT-smahn dis-tri-BYOO-shun /
Picture a hilly meadow with sheep wandering freely. Most sheep linger in the low, easy valleys; only a few ever climb the steep peaks, and the higher the peak, the rarer the sheep up there. The Boltzmann distribution is the exact rule for how the 'sheep' — molecules sharing a fixed budget of heat — spread themselves across energy levels: low energies are crowded, high energies are sparse.
More precisely, the Boltzmann distribution says the chance of finding a molecule in a state of energy E falls off in proportion to a factor that shrinks as E rises and as temperature drops, the so-called Boltzmann factor (it is e to the power of minus E divided by kT, where k is the Boltzmann constant and T the temperature). Double the energy gap and the upper state becomes far emptier; raise the temperature and molecules climb higher more easily.
Why it matters: this single law underlies reaction rates, the colours substances glow, the readings of spectrometers, and the very meaning of temperature. The honest caveat is that it assumes thermal equilibrium — a system left alone long enough to settle — and that energy levels of equal energy (degeneracies) must be counted, or your populations will come out wrong.
At room temperature kT is about 1/40 of an electron-volt. For two states one electron-volt apart, the upper one is roughly e to the power of minus 40 less populated — a vanishingly tiny fraction. That is why most molecules sit quietly in their lowest electronic state until you heat or light them up.
A one-electron-volt gap leaves the upper state almost empty at room temperature.
The closely related Maxwell-Boltzmann distribution is the Boltzmann distribution applied specifically to molecular speeds in a gas; the Boltzmann distribution itself is the general rule for any set of energy levels.