Root Mean Square (RMS) Velocity
The Root Mean Square Velocity (vrms) is a measure of the speed of particles in a gas, which is the square root of the average of the squares of the velocity of individual gas molecules.
The Formula
Based on the Kinetic Molecular Theory, the RMS velocity of a gas is given by:
vrms = √(3RT / M)
vrms ∝ √T
If temperature is doubled, vrms increases by a factor of √2
Variables Defined:
- R: The Universal Gas Constant (8.314 J/mol·K)
- T: Absolute Temperature (in Kelvin)
- M: Molar Mass of the gas (in kg/mol)
Alternative Form
In terms of Boltzmann's constant (kB) and the mass of a single molecule (m):
vrms = √(3kBT / m)
Why use RMS Velocity?
In a gas, molecules are moving in random directions at different speeds. While the average velocity of a gas sample is zero (because velocity is a vector and directions cancel out), the RMS velocity provides a non-zero value that directly relates to the average kinetic energy of the particles.
NOTE: At a constant temperature, lighter gas molecules (lower M) will have a higher RMS velocity compared to heavier molecules.