Calculator
Formula used
The Boltzmann factor for an energy gap ΔE at temperature T is: f = exp( −ΔE / (kBT) ).
When degeneracy differs between states, the population ratio becomes: n₂/n₁ = (g₂/g₁) · exp( −ΔE / (kBT) ).
Constants used: kB = 1.380649×10−23 J/K, NA = 6.02214076×1023 mol−1.
How to use
- Select a mode to compute factor, ΔE, or temperature.
- Enter ΔE and/or the factor based on your mode.
- Enter the temperature and choose its unit.
- Optionally add g₂/g₁ for degeneracy effects.
- Press Calculate to see results above the form.
Example data table
| ΔE (eV) | T (K) | g₂/g₁ | Boltzmann factor f | Population ratio n₂/n₁ |
|---|---|---|---|---|
| 0.025 | 300 | 1 | ≈ 0.38 | ≈ 0.38 |
| 0.10 | 300 | 1 | ≈ 0.02 | ≈ 0.02 |
| 0.050 | 600 | 2 | ≈ 0.38 | ≈ 0.76 |
| -0.020 | 300 | 1 | ≈ 2.16 | ≈ 2.16 |
Notes for interpretation
- Higher T reduces sensitivity to ΔE, increasing excited-state weighting.
- Larger ΔE makes f smaller, strongly suppressing the higher-energy state.
- Degeneracy can offset an energy penalty by increasing available microstates.
- Units matter: kJ/mol and kcal/mol are converted per particle using NA.
Professional article
1) What the Boltzmann factor represents
In thermal physics, the Boltzmann factor quantifies how strongly a system favors lower energy states at a given temperature. For two states separated by an energy gap ΔE, the weighting is exp(−ΔE/kBT). A small factor means the higher-energy state is rarely occupied, while a larger factor means thermal agitation can populate it more often.
2) Key constants and typical magnitudes
This calculator uses the exact Boltzmann constant kB = 1.380649×10⁻²³ J/K. At T = 300 K, the thermal energy scale is kBT ≈ 4.14×10⁻²¹ J, which is about 0.0259 eV. This is why energy gaps near 0.01–0.1 eV often show strong temperature sensitivity.
3) From energy gaps to probabilities
Beyond the factor itself, the tool reports a population ratio and state fractions. With equal degeneracy (g₂/g₁ = 1), the ratio is n₂/n₁ = f. The fractions follow p₂ = (n₂/n₁)/(1 + n₂/n₁) and p₁ = 1 − p₂, giving an immediate sense of expected occupancy under equilibrium assumptions.
4) Degeneracy as an “entropy lever”
Many physical systems have multiple microstates at the same energy. If state 2 has more microstates than state 1, the degeneracy ratio g₂/g₁ increases the population ratio to (g₂/g₁)·exp(−ΔE/kBT). For example, g₂/g₁ = 3 can triple the expected occupancy even when ΔE is unchanged.
5) Unit handling with practical science workflows
The calculator supports Joules and electronvolts for per-particle energies, plus kJ/mol and kcal/mol for chemistry. Molar energies are converted to per-particle energies using Avogadro’s number NA = 6.02214076×10²³ mol⁻¹. This lets you compare spectroscopy-scale gaps with thermochemical gaps consistently.
6) Solving for ΔE or temperature
Inverse modes help with design and inference. If you know a desired factor at a temperature, the tool solves ΔE = −kBT ln(f). If you know ΔE and a factor, it solves T = −ΔE/(kB ln(f)). These forms are common in Arrhenius-type comparisons and equilibrium population estimates.
7) Interpreting edge cases and signs
A negative ΔE means state 2 is lower in energy than state 1, so the factor can exceed 1 and the “excited” label may be misleading. If f ≈ 1, the two states are nearly equally weighted, and solving for temperature becomes ill-conditioned because ln(f) approaches zero.
8) Where the factor appears in real systems
Boltzmann weighting underpins partition functions, molecular conformer populations, carrier distributions in semiconductors, and rotational or vibrational level occupancy. As temperature rises, factors increase for fixed ΔE, shifting measurable averages such as magnetization, reaction rates, and spectral line intensities in predictable ways.
FAQs
1) What is a “typical” ΔE at room temperature?
At 300 K, kBT is about 0.026 eV. Energy gaps of that size give factors around e⁻¹ ≈ 0.37, so occupancy changes noticeably with small temperature shifts.
2) Can the Boltzmann factor be larger than 1?
Yes. If ΔE is negative (meaning state 2 is lower than state 1), then exp(−ΔE/kBT) exceeds 1, indicating state 2 is favored relative to state 1.
3) What does the degeneracy ratio change?
Degeneracy multiplies the ratio: n₂/n₁ = (g₂/g₁)·f. Larger g₂/g₁ increases the fraction in state 2 even if the energy gap stays the same.
4) Why does the tool convert kJ/mol to per-particle energy?
The Boltzmann factor compares a single particle’s energy to kBT. Converting molar values using NA ensures the exponent is dimensionless and physically consistent.
5) When does solving for temperature fail?
If f is 1 or extremely close to 1, then ln(f) is near zero and the computed temperature becomes unstable. Use a factor meaningfully different from 1 for reliable inversion.
6) Is this valid for non-equilibrium systems?
The factor is an equilibrium weighting. In driven or rapidly changing systems, populations may not follow Boltzmann statistics. Use these results as a baseline, then compare with kinetics or transport constraints.
7) How should I choose ΔE’s sign?
Use ΔE = E₂ − E₁. If state 2 is higher, ΔE is positive and the factor is below 1. If state 2 is lower, ΔE is negative and the factor is above 1.