Kinetic Theory of Gases

After thermodynamics established that heat is energy, the idea returned. In 1857 Rudolf Clausius calculated from measured pressures that air molecules must travel at hundreds of metres per second. The Dutch meteorologist Buys Ballot raised an obvious objection: then a smell released across a room should arrive instantly, yet it takes minutes. Clausius answered with the mean free path. Molecules are so crowded that each collides billions of times a second, and its path is a random zigzag that makes little progress.

In 1860 James Clerk Maxwell took a decisive step. The molecules do not all move at the same speed. Their speeds are spread out in a pattern fixed by the temperature, the first law of physics expressed as a probability distribution. From it he predicted that a gas's viscosity should be the same at any pressure, which seemed absurd. In 1866 he and Katherine Maxwell, whose role in the experiments he acknowledged, measured it in their attic. The prediction held.

Ludwig Boltzmann wanted more: to derive the second law of thermodynamics from mechanics. In 1872 he wrote an equation for how collisions change the distribution of speeds and proved that a quantity, $H$, always decreases until Maxwell's distribution is reached. Josef Loschmidt objected. Newton's laws run equally well backwards, so reversing every molecule's velocity would make $H$ increase. Boltzmann's answer changed physics. The second law is not absolutely certain but overwhelmingly probable. Entropy-decreasing states are possible, just unimaginably rare. That idea became statistical mechanics.

Kinetic theory gives the typical speed of a molecule from its mass $m$ and the temperature $T$:

where $k_B = 1.38 \times 10^{-23}$ J/K is Boltzmann's constant. A nitrogen molecule, the main component of air, has mass $28 \times 1.66 \times 10^{-27} \approx 4.65 \times 10^{-26}$ kg. At room temperature, 300 K:

That is about twice the cruising speed of an airliner, and faster than the speed of sound, about 350 m/s in air. The two are related: sound is a disturbance passed from molecule to molecule, so it cannot outrun the molecules carrying it.

So why does a smell take minutes to cross a room? Each cubic metre of air holds about $2.4 \times 10^{25}$ molecules. With so many, a nitrogen molecule travels only about 70 nanometres, around a thousandth of the width of a human hair, before hitting another. At 500 m/s that means about 7 billion collisions every second, each sending it off in a new random direction. A random walk of $N$ steps gets only about $\sqrt N$ steps from its start. So diffusion alone moves a scent molecule only a few millimetres in a second, and in a real room it is mostly carried by air currents.

The same calculation, run backwards, measured the size of molecules. In 1865 Loschmidt combined the mean free path, inferred from viscosity, with the density of liquids to estimate how many molecules a gas contains: the first estimate of what is now Avogadro's number.

Even so, many physicists and chemists around 1900, led by Ernst Mach and Wilhelm Ostwald, regarded atoms as a convenient fiction. Boltzmann, who felt he was fighting alone, took his own life in 1906. Proof came within two years. Albert Einstein had predicted in 1905 how far a pollen-sized particle in water should wander under random molecular impacts. In 1908 Jean Perrin measured it and obtained the number of molecules in a mole. Several other independent methods gave the same value. Ostwald conceded, and atoms became as real as anything in physics. Deriving fluid equations rigorously from the motion of molecules, Hilbert's sixth problem, took much longer. A proof for an idealised gas of hard spheres came only in 2025.