Root Mean Square Velocity of a Gas

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.

rms

 

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:

R8314

 

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:

rms

 

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.

 

mm-he

 

Plugging our values into the root mean square velocity equation:

226

 

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:

mm-n2

 

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

mm-n2-kg

 

Plugging our values into the root mean square velocity equation:

493

 

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