Kinetic energy and temperature

What does a thermometer measure?

Heat a sealed box of gas. Its pressure rises. In the last lesson we saw that pressure comes from molecules hitting the walls. So heating must make the molecules move faster. Can we say exactly how temperature and molecular motion are linked?

Putting two results together

We now have two equations for the same gas.

• From the kinetic theory: $pV = \tfrac23 E$, where $E$ is the total translational kinetic energy of the molecules.
• From experiments, the ideal gas equation: $pV = nRT$, for $n$ moles at the kelvin temperature $T$.

The left sides are the same. So the right sides are equal too:

The total translational kinetic energy of an ideal gas depends only on the amount of gas and its temperature.

One molecule at a time

One mole holds $N_A = 6.02 \times 10^{23}$ molecules. So $n$ moles hold $N = nN_A$ molecules. Divide $E$ by $N$ to get the average kinetic energy of one molecule:

Here $k_B = R/N_A = 1.38 \times 10^{-23}$ J/K is the Boltzmann constant. It is the gas constant for one molecule.

The average translational kinetic energy of a molecule is $\tfrac32 k_BT$. It depends only on the temperature.

This is the meaning of temperature in the kinetic theory. Temperature is a measure of the average kinetic energy of the random motion of the molecules.

At $300\ \text{K}$, $\bar{K} = \tfrac32 \times 1.38 \times 10^{-23}\ \text{J/K} \times 300\ \text{K} = 6.21 \times 10^{-21}$ J. That is a tiny energy. But one mole of gas has a huge number of molecules. Its total is $\tfrac32 \times 8.314\ \text{J/(mol K)} \times 300\ \text{K} = 3741$ J.

Double the temperature. What happens to the average kinetic energy? Now change the amount of gas. Does the energy of one molecule change, or only the total?

Every gas, the same energy

Notice what is missing from $\tfrac32 k_BT$: the mass of the molecule. At the same temperature, a light hydrogen molecule and a heavy xenon atom have the same average kinetic energy.

The heavy molecule must then move more slowly. Its $\tfrac12 mv^2$ is the same, with a larger $m$. We use this in the next lesson.

A sealed box, heated or cooled

A sealed box has a fixed volume and a fixed amount of gas. Heat it from $300\ \text{K}$ to $600\ \text{K}$:

• The average kinetic energy of a molecule doubles, because it is proportional to $T$.
• The pressure doubles too, because $p = \tfrac23 E/V$ and $V$ is fixed.

Cool the box instead, and the molecules slow down. In this model, at $T = 0$ K the molecules would have no kinetic energy at all. A real gas turns into a liquid long before that.

Always use the kelvin temperature in these formulas. At $27\ ^\circ\text{C}$ the gas is at $300$ K, not at $27$ K.

Practise with new numbers

Try it in Python

The program makes a table of the average kinetic energy of a molecule and the total for one mole. Then it works backwards, from an energy to a temperature.

Common mistakes

• Using °C. $\tfrac32 k_BT$ and $\tfrac32 nRT$ need the kelvin temperature.
• Thinking heavy molecules have more energy. At one temperature, every gas has the same average kinetic energy per molecule.
• Mixing $k_B$ and $R$. $k_B$ is for one molecule. $R$ is for one mole.
• Counting the energy of spin. $\tfrac32 k_BT$ is the energy of moving from place to place. Molecules with two or more atoms can also spin. That is for the next course.

Remember these

• $E = \tfrac32 nRT$ for the translational kinetic energy of $n$ moles.
• $\bar{K} = \tfrac32 k_BT$ for one molecule, with $k_B = 1.38 \times 10^{-23}$ J/K.
• Temperature measures the average kinetic energy of the molecules.
• At one temperature, all gases have the same average kinetic energy per molecule.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. What is the average translational kinetic energy of a gas molecule at $0\ ^\circ\text{C}$?

2. A box holds helium and xenon at $300$ K. Which atoms have the larger average kinetic energy?

3. At what temperature, in °C, is the average kinetic energy of a molecule twice its value at $27\ ^\circ\text{C}$?

4. A sealed box of gas is cooled so that its kelvin temperature halves. What happens to its pressure?