Root Mean Square Velocity Calculator

Created by Joanna Śmietańska
Reviewed by Steven Wooding
Last updated: Jan 20, 2023

This online root mean square velocity calculator uses the kinetic theory of gases to find the root mean square velocity (or RMS velocity), median velocity, and average velocity of gas molecules. In the simplest model of gases, you can have a container filled with tiny molecules that move constantly and randomly and collide frequently. Their movements are so fast and disorderly that you can only use the concepts of average velocity.

Select the temperature and molar mass to calculate the root mean square velocity, median velocity, and average velocity of gas molecules. Would you like to know more about the theory of gases? Visit our ideal gas law calculator.

Kinetic theory of gases – basic concepts

Kinetic theory is used to explain the properties of gases by considering their molecular composition and motion.

The basic concepts of this theory are:

  • A gas is made up of small molecules separated by distances that are much larger than the size of the molecules.

  • The gas molecules are not bound by any attractive forces.

  • The molecules are in constant random motion and collide with each other or with the walls of the container.

  • Collisions are perfectly elastic and there is no energy loss.

  • The average kinetic energy of the gas molecules is directly proportional to the absolute temperature: Ek = 3/2RT, where gas constant R = 8.314 J/(K·mol).

Velocity of particles in a gas

Note that not all gas particles in a sample move at the same speed. There is a certain velocity distribution that is asymmetrical and dependent on the temperature and mass of the particles.

The higher the temperature, the faster the average velocity, and the wider the distribution. In other words, an increase in temperature causes the number of molecules with higher velocity to increase.

The less massive the particle, the faster the velocity, and the wider the distribution. This means that heavier gases will be slower on average than lighter ones.

Check this particles velocity calculator to find the average velocity of gas particles.

Formula for root mean square velocity of gases

In the kinetic theory of gases, the key observation is that the average kinetic energy for gas molecules is proportional to temperature. Note that, again, we are talking about "average" energy because the molecules in a sample have different energies, some lower than average and some higher than average.

🙋 Keep in mind that Ek is proportional to temperature. Reduce the temperature to zero kelvin and see that Ek = 0, which means that all motion stops. Double the temperature and you'll double Ek.

Compare formulas for average kinetic energy Ek and kinetic energy of motion of the gas molecules to find the root mean square velocity equation:

Ek=32RT\small E_{\rm k} = \frac{3}{2}RT

Ek=12mυ2\small E_{\rm k} = \frac{1}{2}m\upsilon^{2}
υrms=3RTM\small \upsilon_{\rm rms}=\sqrt{\frac{3RT}{M}}

where:

  • RR – Gas constant equal to 8.314 J/(K⋅mol);
  • TT – The temperature in Kelvins,
  • mm – Particle mass;
  • υ\upsilon – Average square velocity;
  • υrms\upsilon_{\rm rms} – RMS velocity; and
  • MM – Molar mass in kg/mol.

Check how to calculate root mean square for any data set, not only gas velocity distribution.

Average velocity of gas molecules

We found that at a selected temperature, a gas has a certain velocity distribution. From this distribution function, we can determine the average velocity (also called the mean square velocity of gas molecules) υave\upsilon_{\rm ave}. It is usually slightly lower than the RMS velocity, and we can write it as:

υave=8RTπM\small \upsilon_{\rm ave} = \sqrt{\frac{8RT}{\pi M}}

You can also derive the median velocity υm\upsilon_{\rm m} of gas molecules which is in the middle of the velocity distribution and describes the most probable velocity of particles in a gas:

υm=2RTM\small \upsilon_{\rm m} = \sqrt{\frac{2RT}{M}}

How to calculate root mean square velocity?

Imagine you want to determine the speed of oxygen molecules at 27 °C. To do this, proceed as follows:

  1. Input the temperature of 27 °C into the root mean square velocity calculator. It will convert to kelvin, or you can specify the temperature in kelvin straight away: K = °C + 273.15 = 27 + 273.15 = 300.15 K.

  2. Select the name "oxygen" from the list. The molar mass of the oxygen molecule is 0.032 kg/mol. You can choose from the molar masses of the common gases or give the molar mass of gas in kg/mol.

  3. The RMS velocity of oxygen at 27 °C is 483.68 m/s. Our calculator used the root mean square velocity equation: υrms = √(3RT)/M.

  4. Read the mean square velocity of gas molecules: υave = 445.63 m/s and median velocity: υm = 394.93 m/s.

FAQ

What is root mean square velocity?

The root mean square velocity (RMS velocity) of gas is a square root of the average square velocity. Like it, it has units of velocity. The higher the temperature of a given gas, the greater the RMS velocity of its molecules. The heavier the particle, the slower it moves, and the RMS velocity decreases.

How can I calculate root mean square velocity of CO₂ at 40 °C?

If you are wondering how to calculate the root mean square velocity of CO₂ (carbon dioxide) at 40 °C, follow these steps:

  1. Convert Celsius to kelvin: 40 °C + 273.15 = 313.15 K.

  2. Calculate the molar mass of CO₂:

    M = 1×M𝒸 + 2×Mₒ = 1×12.011 g/mol + 2×15.999 g/mol
    = 44.009 g/mol = 44.01 g/mol /1000 = 0.04401 kg/mol
    .

  3. Use the formula for root mean square velocity of a gas:

    υrms = √(3RT)/M = √(3×8.314×313.15)/0.04401
    = 421.275 m/s
    .

Does RMS velocity depend on volume?

No. RMS velocity is independent of pressure, volume, and the nature of the gas. In contrast, it depends only on the temperature and the mass of the gas. RMS velocity is always greater than the average velocity and median velocity.

What is the difference between RMS and average velocity?

RMS velocity and average velocity are two different quantities that describe how fast something is moving in one dimension. You can calculate RMS velocity by taking the square root of the sum of the squares of the velocities along that one axis. On the other hand, average velocity counts the arithmetic mean of the velocities of individual molecules in a given direction.

What is the RMS velocity ratio of molecules O₂ and H₂ at a given temperature?

The ratio is 1:4. RMS velocity of O₂ is √(3RT)/MO2, and for H2 is √(3RT)/MH2. Calculate the ratio of these gives:

υO2H2 = √(MH2/MO2) = √(2/32) = 1:4.

Joanna Śmietańska
Temperature (T)
K
Molar gas constant (R)
J/(mol·K)
Molar mass of common gases
Oxygen
Molar mass (M)
kg/mol
Velocities of gas molecules
Root mean square velocity (υᵣₘₛ)
m/s
Average velocity (υₐᵥₑ)
m/s
Median velocity (υₘ)
m/s
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