# Particles Velocity Calculator

Created by Miłosz Panfil, PhD
Reviewed by Bogna Szyk and Steven Wooding
Last updated: Sep 05, 2022

The particles velocity calculator helps you calculate the average velocity of gas particles. All you need to know is the temperature of the gas and the mass of its particles. Average speed is one of the pieces of information contained in the Maxwell-Boltzmann distribution. You can check out the calculator right now or keep on reading to learn more about the physics behind it.

## Maxwell-Boltzmann distribution

Particles in a gas move and collide according to Newton's equations of motion. However, the sheer number of particles, of the order of Avogadro's number (10²³), makes it unfeasible to trace the motion of each particle. Instead, we describe a gas of particles using emergent features such as temperature: check the ideal gas law calculator to learn more about the ideal gases and temperature.

The temperature is just the average kinetic energy of a particle in the gas. It does not make sense to talk about a temperature of a single particle, but it makes a lot of sense to talk about the temperature of the gas. If you want to get to know more about kinetic energy, check the kinetic energy calculator.

We describe systems containing many particles, like gases, in the language of statistics. Instead of asking what is the velocity of this particle, we ask: what is a speed of a random particle in a gas? The Maxwell-Boltzmann distribution answers this question. The probability that a particle has a velocity $v$ is:

$\small f(v) = \frac{4}{\sqrt{\pi}}\left(\frac{m}{2kT}\right)^{3/2}v^2e^{-mv^2/2kT}$

where:

• $m$ – Mass of a single gas particle;
• $T$ – Temperature of the gas;
• $k = 1.3806 \times 10^{−23}\ \text{J/K}$ – Boltzmann constant; and
• $v$ – Velocity of the particles.

## Average velocity of gas particles

From the Maxwell-Boltzmann equation, we infer that the average velocity $\bar{v}$ of particles is:

$\small \bar{v} = \int_0^\infin vf(v) = \sqrt{\frac{8}{\pi}\frac{kT}{m}}$

The formula means that if we draw a random particle from the gas, it would (on average) have velocity $\bar{v}$.

## Particles velocity calculator

You can quickly compute the average velocity with the particles velocity calculator. Just specify the mass of particles and the temperature. You will soon notice that the average velocity drops as the mass gets larger. The heavier the particles, the slower their motion at the same temperature.

To specify the mass of particles, it is convenient to use the atomic mass unit (u or Da). One atomic mass unit is approximately equal to the mass of a single nucleon, be it a proton or neutron. In this unit a carbon atom weighs 12 u, oxygen 16 u, and carbon dioxide weighs approximately 44 u.

Miłosz Panfil, PhD
Mass
u
Temperature
K
Velocity
m/s
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