The kinetic molecular theory of gases

From the laws to a picture

The gas laws came from experiments. They say that $PV$ is constant at a fixed temperature, and that the volume is proportional to the kelvin temperature. A good theory should explain these laws. The kinetic molecular theory of gases does it with a simple model of molecules that move.

The model

The model has five assumptions:

1. A gas is made of a very large number of tiny molecules. They are so far apart that their own volume is negligible compared with the volume of the container.
2. The molecules move all the time in random straight lines. They change direction only when they hit another molecule or the wall.
3. The molecules do not attract or repel each other, except in the moment of a collision.
4. Collisions are elastic: no kinetic energy is lost.
5. The average kinetic energy of the molecules is proportional to the temperature in kelvin, and it is the same for every gas at the same temperature.

An ideal gas is a gas that follows these assumptions exactly. Real gases follow them well when the pressure is low and the temperature is high. The next lessons show what goes wrong for a real gas.

Pressure as a result of collisions

Put $N$ molecules, each of mass $m$, in a cube of side $L$. The volume is $V = L^3$. Follow one molecule that moves with speed $u$. Split its velocity into three parts along the three edges of the cube. The part along the $x$ axis is $u_x$.

1. One hit. The molecule hits the wall at right angles to $x$ and bounces back. Its momentum goes from $+mu_x$ to $-mu_x$, so the wall gives it a momentum change of $2mu_x$. By Newton's third law, the molecule gives the wall the same momentum.
2. Hits per second. To hit the same wall again, the molecule travels to the opposite wall and back. This is a distance $2L$ at speed $u_x$, which takes the time $\dfrac{2L}{u_x}$. So it hits the wall $\dfrac{u_x}{2L}$ times each second.
3. Force of one molecule. Force is momentum change per second: $2mu_x \times \dfrac{u_x}{2L} = \dfrac{mu_x^2}{L}$.
4. Force of all molecules. Add them up: $F = \dfrac{Nm\,\overline{u_x^2}}{L}$. The bar means the average value over all molecules.
5. Pressure. The wall has an area $L^2$. $P = \dfrac{F}{L^2} = \dfrac{Nm\,\overline{u_x^2}}{L^3} = \dfrac{Nm\,\overline{u_x^2}}{V}$.

The molecules move equally in all directions, so $\overline{u_x^2} = \overline{u_y^2} = \overline{u_z^2}$. Since $u^2 = u_x^2 + u_y^2 + u_z^2$, we have $\overline{u_x^2} = \tfrac{1}{3}\overline{u^2}$. The square root of $\overline{u^2}$ is the root mean square speed, $u_{\text{rms}}$. It is the speed whose square is the average of all the squares. This gives the central result:

Some books write $c$ or $v$ for the speed, and $\overline{c^2}$ for the mean square speed. A-Level and IB use $\tfrac{1}{3}Nm\langle c^2\rangle$. These are all the same idea.

The speed and the temperature

For one mole, $N m$ is the molar mass $M$ in kg/mol. So $PV = \tfrac{1}{3}Mu_{\text{rms}}^2$. For one mole of an ideal gas, $PV = RT$. Put them together:

Use $M$ in kg/mol, and $R = 8.314\ \text{J K}^{-1}\ \text{mol}^{-1}$. A joule is $\text{kg m}^2\ \text{s}^{-2}$, so the answer comes out in m/s. Oxygen has $M = 32.00 \times 10^{-3}\ \text{kg/mol}$. At $300\ \text{K}$,

That is about $1740\ \text{km/h}$. Hydrogen, with $M = 2.016 \times 10^{-3}\ \text{kg/mol}$, has $u_{\text{rms}} = 1927\ \text{m/s}$ at the same temperature. Sound in air travels at about $340\ \text{m/s}$, which is slower than the molecules that carry it.

The kinetic energy of one mole is $\tfrac{1}{2}Mu_{\text{rms}}^2 = \tfrac{3}{2}RT$. For one molecule, $\tfrac{1}{2}mu_{\text{rms}}^2 = \tfrac{3}{2}kT$, where $k = \dfrac{R}{N_A} = 1.381 \times 10^{-23}\ \text{J K}^{-1}$ is the Boltzmann constant. At $300\ \text{K}$, the energy of a mole is $3.74\ \text{kJ/mol}$ and that of one molecule is $6.21 \times 10^{-21}\ \text{J}$. This is the same for hydrogen and for oxygen. A hydrogen molecule has a smaller mass and so it must move faster to carry the same energy.

So the temperature of a gas measures the average kinetic energy of its molecules. At absolute zero this motion would stop.

Set the molar mass to $28\ \text{g/mol}$ and the temperature to $300\ \text{K}$. Double the temperature. The speed grows by the factor $\sqrt{2} = 1.41$ and not by $2$. Quadruple the molar mass and the speed halves.

How the model explains the gas laws

• Boyle's law. At a fixed temperature the molecules have the same speed. A smaller volume means that a molecule hits the walls more often, so the pressure rises.
• Charles' law. A higher temperature gives faster molecules. They hit harder and more often. For the pressure to stay the same, the volume must grow.
• Avogadro's law. The pressure depends on the number of molecules per volume, and on their mass and speed through $\tfrac{1}{3}mu^2$, which is $\tfrac{2}{3}$ of the kinetic energy. At the same $T$ the energy is the same for every gas. So equal numbers of molecules in equal volumes give equal pressures, whatever the gas.
• Dalton's law. The molecules do not act on each other, so each kind hits the wall on its own, and the pressures add up.
• Graham's law. At the same $T$, $\tfrac{1}{2}m_1u_1^2 = \tfrac{1}{2}m_2u_2^2$, so $\dfrac{u_1}{u_2} = \sqrt{\dfrac{m_2}{m_1}}$.

The speed of the lightest molecules explains why Earth has almost no free hydrogen or helium in the air. A few molecules in the fast tail of the distribution reach the escape speed of $11.2\ \text{km/s}$ and leave. Over millions of years all the light gas goes. The next lesson shows how to count the molecules in that tail.

Practise with new numbers

Try it in Python

The program finds the rms speed of five gases at $298.15\ \text{K}$, the kinetic energy of a mole, and the speed of hydrogen compared with carbon dioxide.

Common mistakes

• Molar mass in g/mol. In $u = \sqrt{3RT/M}$ with $R$ in joules, $M$ is in kg/mol. The mistake makes every speed about $31.6$ times too small.
• Temperature in Celsius. Use kelvin.
• Thinking that every molecule moves at the rms speed. It is a kind of average. The molecules have a spread of speeds.
• Thinking that a heavier gas has more kinetic energy. At the same temperature the average kinetic energy is the same for all gases.
• Forgetting the one third. The factor $\tfrac{1}{3}$ comes from the three directions of motion.

Remember these

• $PV = \tfrac{1}{3}Nmu_{\text{rms}}^2$, and for a mole $u_{\text{rms}} = \sqrt{3RT/M}$ with $M$ in kg/mol.
• Average kinetic energy of one molecule: $\tfrac{3}{2}kT$. Of a mole: $\tfrac{3}{2}RT$. It depends on $T$ only.
• $k = R/N_A = 1.381 \times 10^{-23}\ \text{J K}^{-1}$.
• At the same $T$, the lighter gas is faster, with $u \propto 1/\sqrt{M}$.
• The theory explains Boyle, Charles, Avogadro, Dalton and Graham.

Check yourself

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

1. Which is larger at $300\ \text{K}$, the average kinetic energy of a hydrogen molecule or that of a nitrogen molecule?

2. By what factor does the rms speed change when the temperature is raised from $300\ \text{K}$ to $1200\ \text{K}$?

3. Calculate the rms speed of methane, $M = 16.04\ \text{g/mol}$, at $300\ \text{K}$.

4. Why does a gas exert pressure in a closed vessel even though no molecule pushes steadily?

5. In the derivation, why is $\overline{u_x^2}$ equal to $\tfrac{1}{3}\overline{u^2}$?

References

• OpenStax, Chemistry 2e, Chapter 9, section 9.5 (the kinetic-molecular theory).
• The Boltzmann constant and the gas constant are the values fixed in the 2019 SI.