The Air in Your Room Is Moving Faster Than Sound

A balloon stays round, a tyre stays firm, and a hot gas pushes harder than a cold one. None of these needs a mysterious force — they come from billions of molecules hitting the walls every instant.

Kinetic theory builds pressure and temperature out of the motion of molecules, using only Newton's laws and averages.

This part covers the assumptions of kinetic theory, the gas equation and work, the pressure formula, and the meaning of temperature. Take $R=8.314$ J/mol K and $k_B=1.38\times 10^{-23}$ J/K.

An ideal gas is pictured as a huge number of tiny molecules in random motion, whose own volume is negligible, which exert no forces on each other except during brief elastic collisions.

The assumptions:

- A gas contains a very large number of identical molecules
- The volume of the molecules is negligible compared with the container
- Molecules move randomly in all directions with a range of speeds
- Collisions are elastic and take negligible time
- Between collisions there are no intermolecular forces, so molecules move in straight lines

Worked example — how empty is a gas? At $0$ °C and $1.013\times 10^5$ Pa, the number of molecules per cubic metre is

$$n=\frac{P}{k_{B}T}=\frac{1.013\times 10^5}{1.38\times 10^{-23}\times 273}\approx 2.69\times 10^{25}{\text{ m}}^{-3}$$

Taking each molecule as a sphere of diameter $3\times 10^{-10}$ m, its volume is about $1.41\times 10^{-29}$ m${}^3$, so the molecules fill only

$$2.69\times 10^{25}\times 1.41\times 10^{-29}\approx 3.8\times 10^{-4}$$

of the space — **less than $0.04\%$.

An everyday example. The smell of an agarbatti spreads across a room because its molecules move and collide with air molecules in every direction.

The substance. Real gases depart from these assumptions at high pressure and low temperature**, where molecules are crowded and attract one another.

**A perfect gas obeys $PV=\mu RT=N{k}_{B}T$, and the work done on it during compression is $\int PdV$ — $P\Delta V$ at constant pressure and $\mu RT\ln \frac{V_1}{V_2}$ for slow compression at constant temperature.**

Here $k_B=\frac{R}{N_A}$ is Boltzmann's constant and $N$ the number of molecules.

Worked example 1 — counting molecules. $1$ L of gas at $1\times 10^5$ Pa and $300$ K:

$$N=\frac{PV}{k_{B}T}=\frac{1\times 10^5\times 10^{-3}}{1.38\times 10^{-23}\times 300}\approx 2.4\times 10^{22}$$

Worked example 2 — isothermal compression. $1$ mol at $300$ K is squeezed slowly from $20$ L to $10$ L:

$$W_{on}=\mu RT\ln \frac{V_1}{V_2}=8.314\times 300\times \ln 2\approx 1.73\times 10^3\text{ J}$$

Worked example 3 — constant pressure. The same change at a steady $1\times 10^5$ Pa needs

$$W_{on}=P\Delta V=1\times 10^5\times 0.010=1000\text{ J}$$

An everyday example. Pumping air into a football means doing work to squeeze more gas into a fixed space.

The substance. The work depends on the path — the same start and end volumes need different work at constant temperature and at constant pressure.

**Adding up the momentum given to a wall by molecular collisions gives $P=\frac{1}{3}nm\overline{v^2}=\frac{1}{3}\rho \overline{v^2}$, where $n$ is the number of molecules per unit volume and $\overline{v^2}$ the mean square speed.

Derivation outline.**

- A molecule with velocity component $v_x$ rebounds elastically from a wall, changing its momentum by $2m{v}_x$
- In time $\Delta t$, the molecules that reach area $A$ lie within $v_x\Delta t$; half of them move towards it, so $\frac{1}{2}nA{v}_x\Delta t$ hit
- Momentum delivered per second per area gives $P=nm{v}_x^2$
- Averaging over all molecules, with no preferred direction, $\overline{v_x^2}=\frac{1}{3}\overline{v^2}$

$$P=\frac{1}{3}nm\overline{v^2}$$

**Worked example — nitrogen at $0$ °C.** With $P=1.013\times 10^5$ Pa and density $\rho =1.25$ kg/m${}^3$:

$$\overline{v^2}=\frac{3P}{\rho }=\frac{3\times 1.013\times 10^5}{1.25}\approx 2.43\times 10^5{\text{ m}}^2/{\text{s}}^2{v}_{rms}\approx 493\text{ m/s}$$

It also follows that $P=\frac{2}{3}E$, where $E$ is the translational kinetic energy per unit volume.

An everyday example. A balloon stays blown up because the molecules inside strike its skin far more often than you could ever count.

The substance. Pressure depends on the mean of the squared speeds, not the square of the mean speed — the two are different numbers.

**Combining $P=\frac{1}{3}nm\overline{v^2}$ with $P=n{k}_{B}T$ shows that the average kinetic energy of a molecule is $\frac{3}{2}{k}_{B}T$, so temperature measures average molecular kinetic energy and $v_{rms}=\sqrt{\frac{3RT}{M}}$.

Worked example 1 — average energy at $300$ K.**

$$\overline{KE}=\frac{3}{2}\times 1.38\times 10^{-23}\times 300\approx 6.21\times 10^{-21}\text{ J}$$

**Worked example 2 — oxygen and hydrogen at $300$ K.** With $M$ in kg/mol:

$$v_{O_2}=\sqrt{\frac{3\times 8.314\times 300}{0.032}}\approx 484\text{ m/s}{v}_{H_2}=\sqrt{\frac{3\times 8.314\times 300}{0.002}}\approx 1.93\times 10^3\text{ m/s}$$

Hydrogen, $16$ times lighter, moves **$4$ times faster.

Worked example 3 — doubling the speed.** Since $v_{rms}\propto \sqrt{T}$, a gas at $300$ K must be heated to $4\times 300=1200$ K to double its rms speed.

An everyday example. Light hydrogen escapes Earth's atmosphere more readily than oxygen, because at the same temperature its molecules move much faster.

The substance. At the same temperature, all gases have the same average kinetic energy per molecule — but not the same speed.