Average Kinetic Energy & Temperature of a System

Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

As the temperature of a gas goes up, the molecules within it move faster and faster as they gain kinetic energy. In this lesson, we learn how to calculate the average speed of gas molecules given the temperature of the gas.

It's Getting Hot in Here!

On a cold winter day, Amy decides to build a fire in her living room fireplace to warm up. Soon, the fire is burning brightly and the room is much warmer. As Amy enjoys the warmth from the fire, she wonders what is happening to heat the air around her.

Temperature and Energy

According to the kinetic theory of gases, all gases are made of microscopic molecules that move in straight lines until they bump into another gas molecule or object. In Amy's living room, the energy from the fire is being transferred to these air molecules. This transfer of energy causes them to move around faster and bump into each other more.

Kinetic energy is proportional to the speed of the molecules. As the speed of the colliding molecules increases so does the total kinetic energy of all the gas molecules. It's pretty difficult to measure the speed of an individual gas molecule. Instead, temperature can be used as a measure of the average kinetic energy of all the molecules in the gas. As the gas molecules gain energy and move faster the temperature goes up. This is why Amy feels warmer!

To determine the average kinetic energy of gas molecules we need to know the temperature of the gas, the universal gas constant (R), and Avogadro's number (NA).

average kinetic energy of a gas

Let's assume that it is 75 degrees Fahrenheit in Amy's living room right now. What is the average kinetic energy of the air molecules in the room?

First, we need to convert the temperature of the room from degrees Fahrenheit to Kelvin. 75 degrees Fahrenheit is equal to 297 K. Remember that R and K are constants, so to calculate the average kinetic energy we just need to know the temperature! Use the formula given above to calculate the average kinetic energy as follows:

kinetic energy of a gas calculation

Therefore, the average kinetic energy of each molecule is 6.15x10-21 J.

Kinetic Energy and Average Speed

As temperature and average kinetic energy increases so does the average speed of the air molecules. However, all the molecules are NOT moving at exactly the same speed. One way to get an approximation of the average speed of the molecules in a gas is to calculate something called the root mean squared, or RMS speed. The RMS speed of the molecules is defined as the square root of the average of each individual velocity squared. For example, if three objects had speeds of 10 m/s, 5 m/s, and 7 m/s, respectively, then the RMS speed of the group would be 7.6 m/s. Of course, in Amy's living room, there are a lot more than 3 particles of gas!

rms velocity example

Since particles in motion have kinetic energy, and kinetic energy increases with speed, there is a relationship between the RMS speed of gas molecules and the average kinetic energy in the gas. This means that there is also a relationship between RMS speed and temperature! The average kinetic energy (K) is equal to one half of the mass (m) of each gas molecule times the RMS speed (vrms) squared.

definition of kinetic energy for a gas

So, what is the RMS speed of the gas molecules in Amy's living room? Remember that the average kinetic energy was 6.15x10-21 J. Although air in Earth's atmosphere contains lots of different gases almost 78% of the atmosphere is nitrogen. Therefore, we will use the mass of a molecule of nitrogen N2. Since a molecule of nitrogen gas contains two nitrogen atoms, we can find the mass of a molecule by first multiplying the atomic mass of a single nitrogen atom (0.014 g/mol) by two to get a molecular mass of 0.028 g/mol. Next, divide 28 g/mol by Avogradro's number (NA=6.022x1023 atoms/mol) to determine that the mass of a single molecule of nitrogen gas is 4.65x10-26 kg.

finding the mass of a nitrogen molecule

Once we have the mass of a nitrogen molecule (m = 4.65x10-26 kg) and the average kinetic energy (K =6.15x10-21 J) we can solve for the RMS speed of the molecules in the room.

rms velocity calculation

Therefore, we can calculate that the RMS speed of the molecules in Amy's living room is about 514 m/s. Pretty fast, right?!

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