## Root-Mean-Square Speed

#### Learning Objective

• Recall the mathematical formulation of the root-mean-square velocity for a gas.

#### Key Points

• All gas particles move with random speed and direction.
• Solving for the average velocity of gas particles gives us the average velocity of zero, assuming that all particles are moving equally in different directions.
• You can acquire the average speed of gaseous particles by taking the root of the square of the average velocities.
• The root-mean-square speed takes into account both molecular weight and temperature, two factors that directly affect a material’s kinetic energy.

#### Term

• velocitya vector quantity that denotes the rate of change of position, with respect to time or a speed with a directional component

## Kinetic Molecular Theory and Root-Mean-Square Speed

According to Kinetic Molecular Theory, gaseous particles are in a state of constant random motion; individual particles move at different speeds, constantly colliding and changing directions. We use velocity to describe the movement of gas particles, thereby taking into account both speed and direction.

Although the velocity of gaseous particles is constantly changing, the distribution of velocities does not change. We cannot gauge the velocity of each individual particle, so we often reason in terms of the particles’ average behavior. Particles moving in opposite directions have velocities of opposite signs. Since a gas’ particles are in random motion, it is plausible that there will be about as many moving in one direction as in the opposite direction, meaning that the average velocity for a collection of gas particles equals zero; as this value is unhelpful, the average of velocities can be determined using an alternative method.

By squaring the velocities and taking the square root, we overcome the “directional” component of velocity and simultaneously acquire the particles’ average velocity. Since the value excludes the particles’ direction, we now refer to the value as the average speed. The root-mean-square speed is the measure of the speed of particles in a gas, defined as the square root of the average velocity-squared of the molecules in a gas.

It is represented by the equation: $v_{rms}=\sqrt{\frac{3RT}{M}}$, where vrms is the root-mean-square of the velocity, Mm is the molar mass of the gas in kilograms per mole, R is the molar gas constant, and T is the temperature in Kelvin.

The root-mean-square speed takes into account both molecular weight and temperature, two factors that directly affect the kinetic energy of a material.

## Example

• What is the root-mean-square speed for a sample of oxygen gas at 298 K?

$v_{rms}=\sqrt{\frac{3RT}{M_m}}=\sqrt{\frac{3(8.3145\frac{J}{K*mol})(298\;K)}{32\times10^{-3}\frac{kg}{mol}}}=482\;m/s$