Boltzmann Factor Calculator (e^(−ΔE/kT) and population ratio)

Understanding the Boltzmann factor calculator — concept and principle

The Boltzmann factor tells you how likely a higher-energy state is compared with a lower one at a given temperature. This calculator finds the factor, the population ratio including degeneracy and the share of particles in the upper level of a two-level system.

The gap is about 1.95 kT, so for every molecule below only 0.143 sit above: 12.5% of the pair.

How it is calculated

Boltzmann factor = e^(−ΔE ÷ kT).
Population ratio N₂ ÷ N₁ = (g₂ ÷ g₁) × e^(−ΔE ÷ kT).
Share in the upper state (two levels only) = r ÷ (1 + r).
Energy conversions: 1 eV = 1.602176634 × 10⁻¹⁹ J; per mole ÷ NA; a wavenumber ν̃ corresponds to E = hcν̃.
kB = 1.380649 × 10⁻²³ J/K; at 298.15 K, kT ≈ 25.7 meV (RT ≈ 2.48 kJ/mol).

The principle

In thermal equilibrium, energy is shared randomly among particles. Counting the ways energy can be distributed shows that the probability of a state falls exponentially with its energy, with kT as the natural energy scale. If a gap is small compared with kT, both states are almost equally populated; if it is many times kT, the upper state is nearly empty. This principle underlies reaction rates (the Arrhenius equation), spectroscopy, semiconductors and the atmosphere's density profile.

Worked example

Default inputs: ΔE = 0.05 eV, T = 298.15 K, g₂ ÷ g₁ = 1.

kT = 8.617333 × 10⁻⁵ eV/K × 298.15 K ≈ 25.693 meV; RT ≈ 2.479 kJ/mol.
ΔE ÷ kT = 0.05 ÷ 0.025693 ≈ 1.946.
Boltzmann factor = e^(−1.946) ≈ 0.1428; population ratio = 0.1428.
Share in the upper state = 0.1428 ÷ 1.1428 ≈ 12.5%.

Each point is the calculator re-run with only “Energy difference E₂ − E₁” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Energy difference E₂ − E₁	Boltzmann factor e^(−ΔE/kT)
0.0125	0.6148
0.025	0.3779
0.0375	0.2323
0.05	0.1428
0.0625	0.08781
0.075	0.05398
0.1	0.0204

How to read the result

A factor near 1 means the upper state is easily reached at this temperature; a factor near 0 means it is rarely occupied. Raising the temperature increases the factor. Degeneracy multiplies the ratio: a level with three times as many states holds three times as many particles at the same energy.

Common mistakes

Using kT in joules with ΔE in eV (or vice versa).
Using Celsius instead of kelvin.
Forgetting degeneracy factors in spectroscopy.
Applying the two-level share when more levels are populated.

Keep in mind: Assumes thermal equilibrium and classical Boltzmann statistics. Quantum gases at low temperature need Fermi–Dirac or Bose–Einstein statistics.

Frequently asked questions

What is the Boltzmann factor?

e^(−E ÷ kT), the relative probability of a state with energy E at temperature T.

What is kT at room temperature?

About 25.7 meV, or 2.48 kJ/mol, at 298 K.

How is it related to the Arrhenius equation?

The exponential term e^(−Ea ÷ RT) in reaction rates is a Boltzmann factor for the activation energy.

Sources: Atkins, P. & de Paula, J., Physical Chemistry (the Boltzmann distribution) · BIPM, SI Brochure 9th ed. (2019): exact values of kB, h, e, NA

Checked 9 Nov 2026. Q. Zhu, Physics 467/667 Thermal Physics, Lecture 21 “The Boltzmann Factor and Partition Function” (University of Nevada, Las Vegas, Spring 2019): P(s) ∝ e^−E(s)/kT, P(s₂)/P(s₁) = e^{−[E(s₂)−E(s₁)]/kT} (eq. 21.5). Degeneracy weighting g₂/g₁: standard formula. Constants: c = 299,792,458 m/s, e = 1.602176634 × 10⁻¹⁹ C, k_B = 1.380649 × 10⁻²³ J/K, h = 6.62607015 × 10⁻³⁴ J·s and N_A = 6.02214076 × 10²³ mol⁻¹ are exact in the 2019 SI.

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