Boltzmann Factor Calculator

The Boltzmann factor e^(-E/kT) describes the relative probability of a system being in a state with energy E at temperature T.

In a thermal equilibrium system, the ratio of particles in energy state E₂ versus state E₁ is:

n₂/n₁ = e^(-ΔE/kT) where ΔE = E₂ - E₁

Constants: k = 8.617 × 10⁻⁵ eV/K = 1.381 × 10⁻²³ J/K (Boltzmann constant)

At room temperature (T = 293 K), kT ≈ 0.0252 eV. If ΔE = 0.1 eV, the ratio is e^(-0.1/0.0252) ≈ 0.019, meaning the higher state is 52× less populated than the lower one.

Temperature effects

At very low temperatures (T → 0), nearly all particles occupy the ground state (lowest energy). The Boltzmann factor for any excited state approaches zero.

At very high temperatures, kT » ΔE, and the exponential approaches 1, so all states become equally populated.
This is why high-temperature systems appear “classical”: the quantisation is still there, it just stops making any difference to what you can measure.

Applications

Chemical reactions: the Arrhenius equation uses the Boltzmann factor with activation energy Ea. A higher temperature exponentially increases the fraction of molecules with enough energy to react.

Semiconductors: the number of conduction electrons is proportional to e^(-Eg/2kT) where Eg is the band gap. This explains why semiconductors conduct better at higher temperatures.

Lasers: population inversion (n₂ > n₁) is impossible in thermal equilibrium, because the Boltzmann factor guarantees the lower state is always more populated.
Lasers get around it by pumping energy in non-thermally, which is why a laser needs a power supply and a hot object does not lase however hot it gets.

NMR spectroscopy: the tiny population difference between spin-up and spin-down protons is what the signal comes from, and it really is tiny. On a 500 MHz instrument the gap is only 3.3 × 10⁻²⁵ J against a kT of 4.1 × 10⁻²¹ J at 300 K, giving a Boltzmann factor of 0.99992. Roughly 8 protons in every 100,000 are unpaired, and the whole of magnetic resonance imaging rests on that margin. It is also why NMR magnets keep getting stronger: the excess population grows with field strength.

Enter the energy gap and temperature. The result shows the Boltzmann factor and the population ratio of the higher state to the lower state.