About the Boltzmann Factor
The Boltzmann factor is a central expression in statistical mechanics: it describes how the population of an energy state falls off as that state's energy rises above a reference level, at a given absolute temperature. It underlies the Maxwell-Boltzmann speed distribution, chemical reaction rates through the Arrhenius equation, semiconductor carrier statistics, and the relative intensity of spectral lines.
The formula
For two states separated by an energy gap ΔE = E₂ − E₁, the Boltzmann factor is:
p = exp(−ΔE / kT)
- ΔE — energy of the upper state minus the reference state, entered here in electronvolts (eV).
- k — the Boltzmann constant, 8.617333262×10⁻⁵ eV/K (equivalently 1.380649×10⁻²³ J/K).
- T — absolute temperature in kelvin. Never substitute Celsius or Fahrenheit here.
When the upper state has degeneracy g₂ and the reference state has degeneracy g₁, the population ratio becomes N₂/N₁ = (g₂/g₁) × exp(−ΔE/kT). This calculator reports both the bare factor and the degeneracy-weighted ratio.
Reading the result
The Boltzmann factor is always between 0 and 1 when ΔE is positive — the higher a state sits above the reference, the smaller its relative population. The quantity kT sets the natural energy scale for comparison: when ΔE is much larger than kT, the upper state is essentially empty; when ΔE is comparable to or smaller than kT, thermal energy is enough to populate it significantly. At room temperature (T ≈ 300 K), kT ≈ 0.0259 eV, which is why energy gaps of a few tenths of an eV or more are rarely bridged by ordinary thermal motion.
Input tips
Always enter temperature in kelvin (K = °C + 273.15); an exponential formula is extremely sensitive to using the wrong temperature scale. Leave the degeneracy ratio at 1 for non-degenerate levels, or when you only want the bare exponential factor.