## Kinetic Energy and Root Mean Square Velocity

The average kinetic energy of gas molecules is given by the equation:

Kinetic energy is thus directly related to the Kelvin temperature of a sample of gas. Temperature is thus a measure of the energy produced by the random motions of molecules and their collisions.

This relationship plays a role in calculating the average velocity of the gas. This velocity is known as the Root Mean Square Velocity.

Using the physical model for kinetic energy,

This relationship plays a role in calculating the average velocity of the gas. This velocity is known as the Root Mean Square Velocity.

Using the physical model for kinetic energy,

where u is the average velocity of the gas particles. Combination of both models for average kinetic energy results in:

The product nm is equivalent to the mass of n moles of gas weighing m kilograms.

The final equation thus becomes

The final equation thus becomes

where M is the molar mass in kilograms and R is 8.314 J / (K * mol)

## Example

Calculate the root mean square velocity for the atoms in a sample of oxygen gas at 32°C.

Solution:

Use the equation for room mean square velocity. M must be expressed in kilograms, so M for O2 equals 0.032 kg/mol

We must also convert 32°C to Kelvin.

32 + 273 = 305 K

Plugging in these values into the formula, we get

Solution:

Use the equation for room mean square velocity. M must be expressed in kilograms, so M for O2 equals 0.032 kg/mol

We must also convert 32°C to Kelvin.

32 + 273 = 305 K

Plugging in these values into the formula, we get

## Problems

1. Find the root mean square velocity of molecules of gaseous phosphoric acid at 158°C (its boiling point).

2. Find the root mean square velocity of radon at 452K

3. Find the root mean square velocity of uranium hexafluoride at 647 K.

4. Find the root mean square velocity of dichlorosilane (H2Cl2Si) at 238°F.

Solutions

2. Find the root mean square velocity of radon at 452K

3. Find the root mean square velocity of uranium hexafluoride at 647 K.

4. Find the root mean square velocity of dichlorosilane (H2Cl2Si) at 238°F.

Solutions