# Average Kinetic Energy of a Gas

**The Kinetic Energy, also known as the energy of motion, of gas particles is directly proportional to temperature.** The higher the temperature, the more kinetic energy the particles have. Another way of saying this is that, the higher the temperature, the faster the particles move.

This relationship between kinetic energy and temperature is described quantitatively through this equation:

This equation says that the Kinetic Energy (KE) per mole of particles is equal to three-halves times Rydberg’s constant times temperature.

Remember that temperature in all gas problems must be in the unit Kelvin.

Rydberg’s constant here is different than the one we have been using thus far, because the units are different. The Rydberg’s constant to use in this equation is:

This equation tells us that Kinetic Energy increases with temperature. But what about the relationship between KE and pressure? And between KE and volume?

Recall from Gay-Lussac’s Law that pressure is directly proportional to temperature:

As pressure increases, temperature increases, and vice versa.

Since pressure and temperature of a gas are directly proportional, and temperature and kinetic energy are directly proportional, we can surmise that pressure and kinetic energy are also directly proportional.

**Kinetic Energy and Pressure are directly proportional.**

What about Kinetic Energy and volume?

Recall from Charles’s Law that volume and temperature are directly proportional:

As volume increases, so does temperature, and vice versa.

Since volume and temperature of a gas are directly proportional, and temperature and kinetic energy are directly proportional, volume and kinetic energy must also be directly proportional.

**Kinetic Energy and Volume are directly proportional.**

Let’s practice using our new equation in a sample problem.

**Sample Problem**

A 5.00 L flask is filled with a sample of fluorine at a temp of 301.8 ºC. Calculate the average kinetic energy.

Our equation relating KE and temperature is:

Remember that, for all gas problems, we need to convert degrees Celsius into Kelvin.

Plugging in our value for temperature into the equation:

*The average kinetic energy is 7170. J/mol.*

**Sample Problem**

According to Kinetic Molecular Theory, if the temperature of a gas is raised from 100 °C to 200 °C, what will happen to the average kinetic energy?

Again, our equation relating kinetic energy and temperature is:

Remember, we need to convert temperature units to Kelvin.

To compare the kinetic energy of the two gases, we can look at the ratio of the kinetic energy of the second higher temperature of the gas to the lower first temperature:

Notice that both the fraction 3/2 and the constant *R* cancel out.

*The kinetic energy of the gas increases by a factor of 1.27.*