# Root Mean Square Velocity

You already know that the velocity (or speed) of gas particles increases with increasing temperature. Another factor affecting the velocity of gas particles is their size — specifically, their molar mass. The following equation relates the speed of gas particles to both temperature and molar mass.

where

$$\mu$$rms represents root mean square velocity, which is a measure of the average speed of gas particles.

R equals Rydberg’s constant, which for the units in the above equation, equals:

T = temperature in Kelvin

M = molar mass in kg/mol (to match the units of Rydberg’s constant, R)

Let’s do a sample problem.

Sample Problem

What is the root mean square velocity of helium at 1 atm and 8.20 K?

The equation for Root Mean Square Velocity is:

We are given a pressure of 1 atm and a temperature of 8.20 K. We can get the molar mass because we know the gas is helium. The molar mass of helium is 4.003 g/mol. We need to convert that to kg/mol.

Plugging our values into the root mean square velocity equation:

The root mean square velocity of helium at 8.20 K is 226 m/s.

Sample Problem

What is the root mean square speed of nitrogen gas at STP?

Recall that STP stands for Standard Temperature and Pressure which is 273.15 K and 1 atm.

Nitrogen is a diatomic molecule, so the molecular formula for nitrogen gas is N2. Its molar mass is:

We need to convert the molar mass into the unit kg/mol:

Plugging our values into the root mean square velocity equation:

The root mean square velocity of nitrogen gas at STP is 493 m/s.