# Boltzmann Factor Calculator

URL: https://speedycalculator.com/calculators/physics/boltzmann-factor/
Last updated: 2026-10-30

> The Boltzmann factor calculator finds the relative probability of a system being in a higher-energy state than a lower one at a given temperature.

## Inputs

- Lower energy E₁ (eV, keV, MeV, J, zJ)
- Higher energy E₂ (eV, keV, MeV, J, zJ)
- Temperature T (K, °C, °F)

## Results

- Boltzmann factor P₂/P₁
- Thermal energy k_B·T
- Energy gap in units of k_B·T

## Formula

```
P₂/P₁ = exp(−(E₂ − E₁)/(k_B·T))
```

## Explanation

At thermal equilibrium, higher-energy states are exponentially less likely to be occupied. The Boltzmann factor explains why reactions speed up with temperature, why the air thins with altitude and how lasers need population inversion.

### How to use the Boltzmann Factor Calculator

- **Lower energy E₁**
- **Higher energy E₂**
- **Temperature T**

Results update as you type, every field has its own unit menu, and everything is calculated in your browser.

### How it works

Enter the two energy levels and the temperature.

### Tips

- Use absolute temperature (K) in gas laws and efficiency formulas.
- Temperature differences are the same in K and °C.
- Heat = mass × specific heat × temperature change when no phase change occurs.
- Phase changes need latent heat at a constant temperature.

### Common mistakes to avoid

- Using °C in pV = nRT.
- Forgetting latent heat during melting or boiling.
- Confusing heat capacity (J/K) with specific heat (J/(kg·K)).

### Inputs explained

FieldWhat to enterLower energy E₁Units: eV, keV, MeV, J, zJHigher energy E₂Units: eV, keV, MeV, J, zJTemperature TUnits: K, °C, °F

### What each result means

- **Boltzmann factor P₂/P₁** = exp(-([Higher energy E₂] - [Lower energy E₁])÷(1.380649e-23×[Temperature T]))
- **Thermal energy k_B·T** = 1.380649e-23×[Temperature T]÷1.602176634e-19×1000
- **Energy gap in units of k_B·T** = ([Higher energy E₂] - [Lower energy E₁])÷(1.380649e-23×[Temperature T])

## Worked example

Inputs — Lower energy E₁: 0 eV; Higher energy E₂: 0.1 eV; Temperature T: 300 K.

Results — **Boltzmann factor P₂/P₁ = 0.020897**; Thermal energy k_B·T = 25.852 meV; Energy gap in units of k_B·T = 3.86817.

## Key terms

- **Boltzmann constant**: k_B = 1.380649 × 10⁻²³ J/K.

## FAQ

**What is k_B·T at room temperature?**
About 25.85 meV.

**Why does the factor grow with temperature?**
Thermal energy makes higher states more accessible.

**Is this the probability itself?**
It is the ratio of probabilities for non-degenerate states.

**Where is it used?**
Chemistry (Arrhenius), semiconductors, atmospheric physics and statistical mechanics.

**What formula does the Boltzmann Factor Calculator use?**
P₂/P₁ = exp(−(E₂ − E₁)/(k_B·T)).


## Sources

- [NIST – Fundamental physical constants](https://physics.nist.gov/cuu/Constants/)
- [OpenStax – University Physics Volume 2](https://openstax.org/details/books/university-physics-volume-2)
- [HyperPhysics (Georgia State University)](http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html)