# Boltzmann Factor Calculator

Created by Miłosz Panfil, PhD
Reviewed by Dominik Czernia, PhD and Adena Benn
Last updated: Jun 05, 2023

You can use this Boltzmann factor calculator to compute the relative probability of two states of a system at thermal equilibrium. If you want to learn more about the thermal equilibrium and the Boltzmann distribution (also called Gibbs distribution).

## Boltzmann distribution (Gibbs distribution)

The Boltzmann distribution specifies a probability with which an individual state of a system appears at thermal equilibrium characterized by the temperature $T$. The equation is:

$P = \frac{1}{Z}e^{-E/k_BT}$

where

• $Z$ – Normalization constant;
• $E$ – Energy of the state (in joules);
• $k_B = 1.38065 \times 10^{-23}\rm J/K$ is the Boltzmann constant;
• $T$ – Temperature (in kelvins); and
• $P$ – Probability that this state occurs.

An essential feature of the Boltzmann distribution is that the probability P depends only on the energy E of the state. Boltzmann distribution is central to our understanding of condensed matter. You can check our particles velocity calculator to learn about its close relative, the Boltzmann-Maxwell distribution.

## Boltzmann factor

The Boltzmann factor tells us the relative probability with which two states of energies, $E_1$ and $E_2$ occur. Dividing the Boltzmann distribution for these two states, we find

$\frac{P_1}{P_2} = e^{(E_1 - E_2)/k_BT}$

Here it is important to note that the relative probability depends only on the difference in energies. For example, two states of the same energy are equally probable. The other factor that plays a role in the Gibbs factor is temperature. The lower the temperature, the more probable the state of the lower energy.

## Boltzmann factor calculator

With our Boltzmann factor calculator, you can test the above predictions. You can also use it to get a quantitative answer. The suitable energy scale is electronvolt ($\rm 1 eV = 1.60217 \times 10^{−19} J$) because the Boltzmann constant is a very small number. For example, we can ask how much more probable a state with energy $E_1 = 0.1\rm\ eV$ compared to a state with energy $E_2 = 0.2\rm\ eV$ at temperature $T = 0\rm \ \degree C$. It turns out that it is around $70$ times more probable.

In the Boltzmann factor equation, we have to use the temperature scale in kelvins. However, in our Boltzmann factor calculator, you can use any temperature scale you want, and we will transform it to Kelvin for you. Also, check out our energy conversion calculator if you are interested in learning more about how we convert energy.

The Boltzmann factor calculator brings us to the micro-world and tiny energy scales. You can check the hydrogen energy levels calculator to get more information on the energy scales of the world.

Miłosz Panfil, PhD
E₁
eV
E₂
eV
Temperature
°C
Boltzmann Factor
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