How Countless Tiny Collisions Add Up to the Pressure of a Gas

Nobody can see air molecules, yet their constant, random motion explains why a tyre holds its pressure, why gases warm when compressed and why different gases need different amounts of heat. Kinetic theory builds all of this from a simple picture of tiny particles in motion.

This lesson covers the postulates of kinetic theory and gas pressure, the link between molecular kinetic energy and temperature, and equipartition of energy with the specific heats of gases.

**Kinetic theory pictures a gas as a huge number of tiny, identical molecules in random motion that collide elastically and exert no forces except during collisions, and the momentum they transfer to the walls produces a pressure $p=\frac{1}{3}\rho \overline{v^2}$.

Postulates:

- A gas consists of a very large number of identical molecules whose size is negligible compared with the spaces between them
- They move randomly in all directions with a range of speeds
- Collisions with each other and with the walls are perfectly elastic
- There are no intermolecular forces except during collisions, which take negligible time

Deriving pressure.** Take a cube of side L holding N molecules of mass m:

- A molecule with x-velocity $v_x$ changes momentum by $2m{v}_x$ at a wall and returns after time $\frac{2L}{v_x}$, so it exerts an average force $\frac{m{v}_x^2}{L}$
- Adding over all molecules gives a force $\frac{mN\overline{v_x^2}}{L}$, and random motion makes $\overline{v_x^2}=\frac{1}{3}\overline{v^2}$
- Dividing by the area $L^2$:

$$p=\frac{1}{3}\frac{mN}{V}\overline{v^2}=\frac{1}{3}\rho \overline{v^2}$$

Worked example. Nitrogen at STP has density 1.25 kg m${}^{-3}$ and pressure $1.01\times 10^5$ Pa, so its molecules have an rms speed of

$$v_{\text{rms}}=\sqrt{\frac{3p}{\rho }}=\sqrt{\frac{3\times 1.01\times 10^5}{1.25}}=492{\text{ m s}}^{-1}$$

An everyday example. Air in a bicycle tyre on a hot Chennai afternoon pushes harder on the tube, because faster molecules strike the walls harder and more often.

The substance. Pressure depends on the mean square speed, not the average speed — faster molecules count for much more, because both their momentum change and their collision rate grow with speed.

**The average translational kinetic energy of a gas molecule is directly proportional to absolute temperature, $\frac{1}{2}m\overline{v^2}=\frac{3}{2}{k}_{B}T$, so temperature measures how fast the molecules move.

Derivation.** Combining $pV=\frac{1}{3}Nm\overline{v^2}$ with the ideal gas equation $pV=N{k}_{B}T$:

$$\frac{1}{3}Nm\overline{v^2}=N{k}_{B}T\Rightarrow \frac{1}{2}m\overline{v^2}=\frac{3}{2}{k}_{B}T$$

where $k_B=1.38\times 10^{-23}$ J K${}^{-1}$ is Boltzmann's constant.

Consequences:

- At the same temperature, molecules of every gas have the same average kinetic energy
- One mole has total translational kinetic energy $\frac{3}{2}RT$
- $v_{\text{rms}}=\sqrt{\frac{3RT}{M}}$, so lighter molecules move faster at the same temperature

Worked example 1. At 400 K, the average kinetic energy of a molecule is

$$\frac{3}{2}\times 1.38\times 10^{-23}\times 400=8.28\times 10^{-21}\text{ J}$$

Worked example 2. To double the rms speed of a gas at 300 K, its absolute temperature must rise four times, to 1200 K.

An everyday example. The fragrance of incense spreads faster through a warm room than a cold one, because warmer air molecules move faster and mix more quickly.

The substance. Temperature is not the energy of any single molecule — molecules have a wide spread of speeds, and temperature reflects only their average kinetic energy.

**The law of equipartition states that energy is shared equally among all degrees of freedom, $\frac{1}{2}{k}_{B}T$ each per molecule, so a gas with f degrees of freedom has $C_v=\frac{f}{2}R$, $C_p=\left(\frac{f}{2}+1\right)R$ and $\gamma =1+\frac{2}{f}$.

Degrees of freedom — independent ways a molecule stores energy:

- Monatomic**, such as helium: 3 translational, so $f=3$
- Diatomic, such as oxygen at room temperature: 3 translational and 2 rotational, so $f=5$
- Non-linear triatomic, such as water vapour: 3 translational and 3 rotational, so $f=6$
- At high temperatures, vibrations add further degrees of freedom

Specific heats. One mole stores $U=\frac{f}{2}RT$, so

$$C_v=\frac{f}{2}R{C}_p=\left(\frac{f}{2}+1\right)R\gamma =1+\frac{2}{f}$$

Worked example 1, with $R=8.31$ J mol${}^{-1}$ K${}^{-1}$:

- Monatomic: $C_v=12.5$ J mol${}^{-1}$ K${}^{-1}$ and $\gamma =1.67$
- Diatomic: $C_v=20.8$ J mol${}^{-1}$ K${}^{-1}$ and $\gamma =1.40$
- Non-linear triatomic: $C_v=24.9$ J mol${}^{-1}$ K${}^{-1}$ and $\gamma =1.33$

Worked example 2 — a mixture. 1.0 mol of helium mixed with 1.0 mol of oxygen has

$$C_v=\frac{1.0\times \frac{3}{2}R+1.0\times \frac{5}{2}R}{2.0}=2R=16.6{\text{ J mol}}^{-1}{\text{ K}}^{-1}$$

An everyday example. LPG warms more slowly than the same number of moles of air when heated, because its larger molecules store energy in more ways.

The substance. Equipartition fails at low temperatures — rotations and vibrations switch on only above certain temperatures, a quantum effect classical theory cannot explain.